Zearn Math–Teacher Edition: Mission 2, G6 : simplebooklet.com

Table of Contents MISSION OVERVIEW viii ASSESSMENTS xv TOPIC A WHAT ARE RATIOS LESSON 1 Introducing Ratios and Ratio Language LESSON 2 Representing Ratios with Diagrams 1 12 TOPIC B EQUIVALENT RATIOS LESSON 3 Recipes 27 LESSON 4 Color Mixtures 39 LESSON 5 Defining Equivalent Ratios 48 TOPIC C REPRESENTING EQUIVALENT RATIOS LESSON 6 Introducing Double Number Line Diagrams 61 LESSON 7 Creating Double Number Line Diagrams 73 LESSON 8 How Much for One 84 LESSON 9 Constant Speed 96 LESSON 10 Comparing Situations by Examining Ratios 108 TOPIC D SOLVING RATIO AND RATE PROBLEMS LESSON 11 Representing Ratios with Tables 120 LESSON 12 Navigating a Table of Equivalent Ratios 132 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license iii

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ZEARN MATH MISSION OVERVIEW G6M2 CURRICULUM MAP 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 K M3 M2 M1 2D 3D Shapes Numbers to 10 Numbers to 5 Digital Activities 50 M1 G1 M1 G3 M2 Add Subtract Friendly Numbers Explore Length M1 Add Subtract Round G5 Place Value with Decimal Fractions G7 G8 Key Base Ten Operations M1 M2 Area and Surface Area Introducing Ratios M2 M1 Scale Drawings Introducing Proportional Relationships M1 Rigid Transformations and Congruence Whole Numbers and Operations M3 M2 M3 Rates and Percentages M4 M3 Add Subtract Fractions M4 Dividing Fractions Proportional Measuring Relationships Circles and Percentages M2 Dilations Similarity and Introducing Slope M3 Linear Relationships Expanding Whole Numbers and Operations to Fractions and Decimals M5 Rational Number Arithmetic M4 Linear Equations and Linear Systems M6 Equal Groups Fractions as Numbers M4 M5 M5 M7 Functions and Volume Algebraic Thinking and Reasoning Leading to Functions M6 Associations in Data Geometry M6 M9 M8 Rational Numbers Angles Triangles and Prisms Multiply Measure The Coordinate Plane M7 Expressions and Equations M7 Decimal Fractions Volume Area Shapes M6 M6 Shapes Measurement Display Data M6 Multiply and Divide Fractions Decimals Expressions Equations and Inequalities M7 M6 M5 Arithmetic in Base Ten M8 Shapes Time Fractions Length Money Data Equivalent Fractions M5 M6 Add Subtract to 100 M7 M5 M4 Construct Lines Angles Shapes Multiply Divide Big Numbers M5 M4 Find the Area Numbers to 20 Digital Activities 35 Work with Shapes Add Subtract Big Numbers M3 M3 M1 M4 M5 Multiply Divide Tricky Numbers Measure It Numbers to 15 Digital Activities 35 Add Subtract Big Numbers Add Subtract Solve M2 M2 M1 M3 Measure Length M4 Counting Place Value Multiply Divide Friendly Numbers G4 G6 M3 Measure Solve G2 M2 Meet Place Value M6 Analyzing Comparing Composing Shapes Numbers 10 20 Count to 100 by Ones and Tens Number Pairs Addition Subtraction to 10 Numbers to 10 Digital Activities 50 Add Subtract Small Numbers M5 M4 Comparison of Length Weight Capacity Numbers to 10 Putting It ALL Together 1 Data Sets and Distributions M8 Probability and Sampling M7 Exponents and Scientific Notation M9 Putting It ALL Together M8 Pythagorean Theorem and Irrational Numbers M9 Putting It ALL Together WEEK Measurement Statistics and Probability 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license v

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ZEARN MATH MISSION OVERVIEW G6M2 Overview of Topics and Lesson Objectives Each mission is broken down into topics A topic is a group of lessons that teach the same concept There is a balance of Independent Digital Lessons and Concept Explorations in each topic of a mission to ensure every student learns with a mix of modalities feedback and support while engaging in grade level content Throughout each mission students work on grade level content with embedded remediation to address unfinished learnings Objective Topic A What Are Ratios Lesson 1 Use ratio language and notation to describe an association between two or more quantities Lesson 2 Draw a diagram that represents a ratio and explain what the diagram means Topic B Equivalent Ratios Lesson 3 Use recipes to understand equivalent ratios and represent multiple batches of a recipe with a diagram Lesson 4 Use color mixtures to explain equivalent ratios Lesson 5 Write equivalent ratios and explain why two ratios are equivalent or not equivalent Topic C Representing Equivalent Ratios Lesson 6 Use double number line diagrams to represent equivalent ratios Lesson 7 Use double number lines to find equivalent ratios Lesson 8 Use equivalent ratios to find unit prices Lesson 9 Use ratios and diagrams to understand how fast things move Lesson 10 Use equivalent ratios to see if situations happen at the same rate Mid Mission Assessment Topics A C vi 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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ZEARN MATH MISSION OVERVIEW G6M2 Objective Topic D Solving Ratio and Rate Problems Lesson 11 Use tables to find equivalent ratios Lesson 12 Solve equivalent ratio problems by finding the rate per 1 in a table Lesson 13 Compare double number lines and tables that represent the same situation Lesson 14 Solve word problems involving equivalent ratios Topic E Part Part Whole Ratios Lesson 15 Create tape diagrams to solve problems involving a ratio and a total amount Lesson 16 Apply number lines tables and tape diagrams to solve problems about ratios End of Mission Assessment Topics D E Topic F Let s Put It to Work Lesson 17 Use ratio reasoning and representations to tackle difficult problems Note on Pacing for Differentiation If you are using the Zearn Math recommended weekly schedule that consists of four Core Days when students learn grade level content and one Flex Day that can be tailored to meet students needs we recommend omitting the optional lessons in this mission during the Core Days Students who demonstrate a need for further support can explore these concepts with you and peers as part of a flex day as needed This schedule ensures students have sufficient time each week to work through grade level content and includes built in weekly time you can use to differentiate instruction to meet student needs 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license vii

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ZEARN MATH MISSION OVERVIEW G6M2 Mission Overview Work with ratios in grade 6 draws on earlier work with numbers and operations In elementary school students worked to understand represent and solve arithmetic problems involving quantities with the same units In grade 4 students began to use two column tables e g to record conversions between measurements in inches and yards In grade 5 they began to plot points on the coordinate plane building on their work with length and area These early experiences were a brief introduction to two key representations used to study relationships between quantities a major focus of work that begins in grade 6 with the study of ratios Starting in grade 3 students worked with relationships that can be expressed in terms of ratios and rates e g conversions between measurements in inches and in yards however they did not use these terms In grade 4 students studied multiplicative comparison In grade 5 they began to interpret multiplication as scaling preparing them to think about simultaneously scaling two quantities by the same factor They learned what it means to divide one whole number by another so they are well equipped to consider the quotients a b and b a associated with a ratio a b for non zero whole numbers a and b In this mission students learn that a ratio is an association between two quantities e g 1 teaspoon of drink mix to 2 cups of water Students analyze contexts that are often expressed in terms of ratios such as recipes mixtures of different paint colors constant speed an association of time measurements with distance measurements and uniform pricing an association of item amounts with prices One of the principles that guided the development of these materials is that students should encounter examples of a mathematical concept in various contexts before the concept is named and studied as an object in its own right The development of ratios equivalent ratios and unit rates in this mission and the next mission is in accordance with that principle In this mission equivalent ratios are first encountered in terms of multiple batches of a recipe and equivalent is first used to describe a perceivable sameness of two ratios for example two mixtures of drink mix and water taste the same or two mixtures of red and blue paint are the same shade of purple Building on these experiences students analyze situations involving both discrete and continuous quantities and involving ratios of quantities with units that are the same and that are different Several lessons later equivalent acquires a more precise meaning All ratios that are equivalent to a b can be made by multiplying both a and b by the same non zero number note that students are not yet considering negative numbers This mission introduces discrete diagrams and double number line diagrams representations that students use to support thinking about equivalent ratios before their work with tables of equivalent ratios 0 4 8 12 0 3 6 9 Initially discrete diagrams are used because they are similar to the kinds of diagrams students might have used to represent multiplication in earlier grades Next come double number line diagrams These can be drawn more quickly than discrete diagrams but are more similar to tables while allowing reasoning based on the lengths of intervals on the number lines After some work with double number line diagrams students use tables to represent equivalent ratios Because equivalent pairs of ratios can be written in any order in a table and there is no need to attend to the distance between values tables are the most flexible and concise of the three representations for equivalent ratios but they are viii 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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ZEARN MATH MISSION OVERVIEW G6M2 also the most abstract Use of tables to represent equivalent ratios is an important stepping stone toward use of tables to represent linear and other functional relationships in grade 8 and beyond Because of this students should learn to use tables to solve all kinds of ratio problems but they should always have the option of using discrete diagrams and double number line diagrams to support their thinking When a ratio involves two quantities with the same units we can ask and answer questions about ratios of each quantity and the total of the two Such ratios are sometimes called part part whole ratios and are often used to introduce ratio work However students often struggle with them so in this mission the study of part part whole ratios occurs at the end Note that tape diagrams are reserved for ratios in which all quantities have the same units The major use of partpart whole ratios occurs with certain kinds of percentage problems which come in the next mission On using the terms ratio rate and proportion In these materials a quantity is a measurement that is or can be specified by a number and a unit e g 4 oranges 4 centimeters my height in feet or my height with the understanding that a unit of measurement will need to be chosen The term ratio is used to mean an association between two or more quantities and the fractions a b and b a are never called ratios Ratios of the form 1 a b or 1 b a which are equivalent to a b are highlighted as useful but a b and b a are not identified as unit rates for the ratio a b until the next mission However the meanings of these fractions in contexts are very carefully developed The word per is used with students in interpreting a unit rate in context as in 3 per ounce and at the same rate is used to signify a situation characterized by equivalent ratios In the next unit students learn the term unit rate and that if two ratios a b and c d are equivalent then the unit rates a b and c d are equal The terms proportion and proportional relationship are not used anywhere in the grade 6 materials A proportional relationship is a collection of equivalent ratios and such collections are objects of study in grade 7 In high school after their study of ratios rates and proportional relationships students discard the term unit rate referring to a to b a b and a b as ratios Progression of Disciplinary Language In this mission teachers can anticipate students using language for mathematical purposes such as interpreting explaining and comparing Throughout the unit students will benefit from routines designed to grow robust disciplinary language both for their own sense making and for building shared understanding with peers Teachers can formatively assess how students are using language in these ways particularly when students are using language to Interpret ratio notation Lesson 1 different representations of ratios Lesson 6 situations involving equivalent ratios Lesson 8 situations with different rates Lesson 9 tables of equivalent ratios Lessons 11 and 12 questions about situations involving ratios Lesson 17 Explain features of ratio diagrams Lesson 2 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license ix

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ZEARN MATH MISSION OVERVIEW G6M2 reasoning about equivalence Lesson 4 reasoning about equivalent rates Lesson 10 reasoning with reference to tables Lesson 14 reasoning with reference to tape diagrams Lesson 15 Compare situations with and without equivalent ratios Lesson 3 representations of ratios Lessons 6 and 13 situations with different rates Lessons 9 and 12 situations with same rates and different rates Lesson 10 representations of ratio and rate situations Lesson 16 In addition students are expected to describe and represent ratio associations represent doubling and tripling of quantities in a ratio represent equivalent ratios justify whether ratios are or aren t equivalent and why information is needed to solve a ratio problem generalize about equivalent ratios and about the usefulness of ratio representations and critique representations of ratios The table shows lessons where new terminology is first introduced including when students are expected to understand the word or phrase receptively and when students are expected to produce the word or phrase in their own speaking or writing Terms defined in Student and Teacher Lesson Materials are bolded Teachers should continue to support students use of a new term in the lessons that follow where it was first introduced New Terminology x Lesson Receptive Productive 1 ratio __ to __ __ for every __ 2 diagram 3 recipe batch same taste ratio __ to __ __ for every __ 4 mixture same color check an answer batch 5 equivalent ratios 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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ZEARN MATH MISSION OVERVIEW G6M2 New Terminology Lesson Receptive Productive 6 double number line diagram tick marks representation diagram 7 per 8 unit price how much for 1 at this rate 9 meters per second constant speed 10 same rate 11 table row column 14 calculation 15 tape diagram parts suppose 16 double number line equivalent ratios per table tape diagram Digital Lessons Students also learn the concepts from this mission in their Independent Digital Lessons There are 16 Digital Lessons for Mission 2 It s important to connect teacher instruction and digital instruction at the mission level So when you start teaching Mission 2 set students to the first digital lesson of Mission 2 The intentional balance of learning with teachers and peers and learning independently in digital lessons ensures every student has multiple opportunities to represent engage with and express their math reasoning In the digital lessons students explore the concepts through interactive problem solving with embedded support that launches at the moment of misconception As students complete digital lessons they will automatically progress to the next lesson Go online to Zearn org to explore more of the digital lessons for this mission 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license xi

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ZEARN MATH MISSION OVERVIEW G6M2 Terminology Double number line diagram A double number line diagram uses a pair of parallel number lines to represent equivalent ratios The locations of the tick marks match on both number lines The tick marks labeled 0 line up but the other numbers are usually different 0 3 6 9 12 0 5 10 15 20 Red paint teaspoons Yellow paint teaspoons Equivalent ratios Two ratios are equivalent if you can multiply each of the numbers in the first ratio by the same factor to get the numbers in the second ratio For example 6 8 is equivalent to 3 4 because 8 1 2 4 and 6 1 2 3 A recipe for lemonade says to use 8 cups of water and 6 lemons If we use 4 cups of water and 3 lemons it will make half as much lemonade Both recipes taste the same because 8 6 and 4 3 are equivalent ratios Cups of water Number of lemons 8 6 4 3 Per A recipe for cinnamon rolls uses 2 tablespoons of sugar per teaspoon of cinnamon for the filling Complete the double number line diagram to show the amount of cinnamon and sugar in 3 4 and 5 batches Ratio A ratio is an association between two or more quantities For example the ratio 3 2 could describe a recipe that uses 3 cups of flour for every 2 eggs or a boat that moves 3 meters every 2 seconds One way to represent the ratio 3 2 is with a diagram that has 3 blue squares for every 2 green squares Same rate We use the words same rate to describe two situations that have equivalent ratios For example a sink is filling with water at a rate of 2 gallons per minute If a tub is also filling with water at a rate of 2 gallons per minute then the sink and the tub are filling at the same rate Table A table organizes information into horizontal rows and vertical columns The first row or column usually tells what the numbers represent For example here is a table showing the tail lengths of three different pets This table has four rows and two columns xii Pet Tail length inches Dog 225 Cat 150 Mouse 475 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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ZEARN MATH MISSION OVERVIEW G6M2 Tape diagram A tape diagram is a group of rectangles put together to represent a relationship between quantities For example this tape diagram shows a ratio of 30 gallons of yellow paint to 50 gallons of blue paint If each rectangle were labeled 5 instead of 10 then the same picture could represent the equivalent ratio of 15 gallons of yellow paint to 25 gallons of blue paint 10 10 10 10 10 10 10 10 Unit price The unit price is the cost for one item or for one unit of measure For example if 10 feet of chain link fencing cost 150 then the unit price is 150 10 or 15 per foot Required Materials Beakers Colored pencils Copies of template Drink mix Empty containers Food coloring Graduated cylinders Graph paper Markers Masking tape Meter sticks Paper cups Pre printed slips cut from copies of the template Rulers Snap cubes Stopwatches String Students collections of objects Teacher s collection of objects 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license xiii

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PAGE 1 Name Date GRADE 6 MISSION 2 End of Mission Assessment 1 Giovanni is making meatballs and his recipe calls for 10 eggs to every 4 pounds of meat Giovanni uses a double number line diagram to figure out the recipe for large batches 0 10 20 30 0 4 8 12 Eggs Meat pounds Eggs Meat pounds a Use the information presented in the double number line diagram to complete the first three rows of the table b Use the table to determine the number of eggs needed for 24 pounds of meat eggs c Use the table to determine the pounds of meat needed if 5 eggs are used lbs of meat d How many eggs are needed if 30 pounds of meat are used Show your work and or explain your reasoning eggs 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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GRADE 6 MISSION 2 LESSON 1 Introducing Ratios and Ratio Language LEARNING GOALS LEARNING GOALS STUDENT FACING LEARNING TARGETS STUDENT FACING G6M2 LESSON 1 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Comprehend the word ratio in written and spoken language and the notation a b in written language to refer to an association between quantities Describe orally and in writing associations between quantities using the language For every a of these there are b of those and The ratio of these to those is a b or a to b Let s describe two quantities at the same time I can write or say a sentence that describes a ratio I know how to say words and numbers in the correct order to accurately describe the ratio Teacher s collection of objects Students collections of objects REQUIRED MATERIALS REQUIRED PREPARATION Tools for creating a visual display any way for students to create work that can be easily displayed to the class Examples chart paper and markers whiteboard space and markers shared online drawing tool access to a document camera A few days before this lesson ask students to bring a personal collection of 10 50 small objects Examples include rocks seashells trading cards or coins Bring in your personal collection and display it ahead of time Think of possible ways to sort your collection See the Launch section of the first activity for details Prepare a few extra collections for students who don t bring one 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 1

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G6M2 LESSON 1 ZEARN MATH TEACHER LESSON MATERIALS In this lesson students use collections of objects to make sense of and use ratio language Students see that there are several different ways to describe a situation using ratio language For example if we have 12 squares and 4 circles we can say the ratio of squares to circles is 12 4 and the ratio of circles to squares is 4 to 12 We may also see a structure that prompts us to regroup them and say that there are 6 squares for every 2 circles or 3 squares for every one circle YOUR NOTES Expressing associations of quantities in a context as students will be doing in this lesson requires students to use ratio language with care Making groups of physical objects that correspond with for every language is a concrete way for students to make sense of the problem It is important that in this first lesson students have physical objects they can move around Later they will draw diagrams that reflect the same structures and learn to reason with and interpret abstract representations like double number line diagrams and tables Working with objects that can be physically rearranged in the beginning of the mission can help students make sense of increasingly abstract representations they will encounter as the mission progresses Students will continue to develop ratio language throughout the mission and will learn about equivalent ratios in a future lesson Warm Up WHAT KIND AND HOW MANY Instructional Routine Think Pair Share In this warm up students compare figures and sort them into categories The first two questions are very straightforward to allow all students access to the activity and prime them to think about other ways the figures can be sorted Students create their own categories in the third question and explain their reasoning LAUNCH Display the image for all to see Give students quiet think time followed by partner discussion 2 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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G6M2 LESSON 1 WARM UP TASK 1 ZEARN MATH TEACHER LESSON MATERIALS Use this diagram to answer the questions below YOUR NOTES 1 If you sorted this set by color how many groups would you have 2 If you sorted this set by area how many groups would you have 3 Think of a third way you could sort these figures What categories would you use How many groups would you have STUDENT RESPONSE 1 Four red blue green yellow 2 Four 2 3 4 5 square units 3 Answers vary Sample responses Two rectangles and non rectangles Three rectangles two different squares glued together and L shapes Four squares rectangles two different squares glued together and L shapes Seven small medium and large rectangles 2 by 2 squares small L big L and a small and a big square glued together DISCUSSION GUIDANCE After briefly inviting students to share responses to the first two questions record all the ways students answered the third question for all to see Ask a student to explain how they sorted 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 3

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G6M2 LESSON 1 ZEARN MATH TEACHER LESSON MATERIALS the figures Ask if anyone saw it a different way until all the different ways of seeing the shapes have been shared Emphasize that the important thing is to describe the way they sorted them clearly enough that everyone agrees that it is a reasonable way to sort them Tell students we will be looking at different ways of seeing the same set of objects in the next activity YOUR NOTES Concept Exploration Activity 1 THE TEACHER S COLLECTION Instructional Routines Anticipate Monitor Select Sequence Connect MLR7 Compare and Connect This activity introduces students to ratio language and notation through examples based on a collection of everyday objects Students learn that a ratio is an association between quantities and that this association can be expressed in multiple ways After discussing examples of ratio language and notation for one way of categorizing the objects in the collection students write ratios to describe the quantities for another way of categorizing objects in the collection As students work circulate and identify those who Create different categories from the given collection Create categories whose quantities can be rearranged into smaller groups e g 6 A s and 4 B s can be expressed as for every 3 A s there are 2 B s Express the same ratio in opposite order or by using different words e g the ratio of A to B is 7 to 3 and for every 7 A s there are 3 B s Have a collection of objects ready to display for the launch Make sure there are different ways the collection can be sorted For example the dinosaurs below can be categorized by color black green orange and purple by the number of legs they stand on standing on 4 legs or on 2 legs or by the features along their backs crest white stripe or nothing 4 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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G6M2 LESSON 1 Familiar classroom objects such as binder clips or pattern blocks can also be used to form collections This picture shows a collection of binder clips that could be categorized by size small medium and large or by color black green and blue ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES LAUNCH Display a collection of objects for all to see Give students quiet think time to come up with as many different categories for sorting the collection as they can think of Record students categories for all to see Sort the collection into one of the student suggested categories and count the number of items in each Record the number of objects in each category and display for all to see For example Amount in each category Category A black Category B green Category C orange Category D purple 1 2 2 4 Explain that we can talk about the quantities in the different categories using something called ratios Tell students A ratio is an association between two or more quantities We use a colon or the word to between two values we are associating Share the following examples adapt them to suit your collection and display them for all to see Keep the examples visible for the duration of the lesson The ratio of purple to orange dinosaurs is 4 to 2 The ratio of purple to orange dinosaurs is 4 2 The ratio of orange to purple dinosaurs is 2 to 4 The ratio of orange to purple dinosaurs is 2 4 Explain that we can also associate two quantities using the phrase for every a of these there are b of those Add the following examples to the display For every 2 green dinosaurs there are 4 purple dinosaurs 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 5

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G6M2 LESSON 1 There are 4 purple dinosaurs for every 2 orange dinosaurs ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Finally find two categories whose items can be rearranged into smaller groups e g 4 purple dinosaurs to 2 orange dinosaurs Point out that in some cases we can associate the same categories using different numbers Share the following example and add it to the display For every 2 purple dinosaurs there is 1 orange dinosaur Have students write two or three sentences to describe ratios between the categories they suggested ACTIVITY 1 TASK 1 2 Think of a way to sort your teacher s collection into two or three categories 1 Record your categories in the top row of the table and the amounts in the second row Category Name Category Amount 2 Write at least two sentences that describe ratios in the collection Remember there are many ways to write a ratio The ratio of one category to another category is to The ratio of one category to another category is There are of one category for every of another category STUDENT RESPONSE Answers vary depending on the particulars of the teacher s collection and the choices made by students 6 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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G6M2 LESSON 1 DISCUSSION GUIDANCE Invite several students to share their categories and sentences Display them for all to see attending to correct ratio language Be sure to include students who express the same categories in reverse order in different words or with a different set of numbers which students will later call an equivalent ratio Leave several sentences displayed for students to see and use as a reference while working on the next task ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES ANTICIPATED MISCONCEPTIONS Students may write ratios with no descriptive words 8 2 is a good start but part of writing a ratio is stating what those numbers mean Draw students attention to the sentence stems in the task statement encourage them to use those words SUPPORT FOR STUDENTS WITH DISABILITIES Executive Functioning Visual Aids Create an anchor chart i e definition and visual for ratio as described in the lesson publicly displaying important definitions rules formulas or concepts for future reference SUPPORT FOR ENGLISH LANGUAGE LEARNERS Speaking MLR 7 Compare and Connect Use this routine when students present their sentences Ask students to consider what is the same and what is different between sentences Draw students attention to the association between quantities in each sentence These exchanges strengthen students mathematical language use and reasoning based on ratios Design Principle s Maximize meta awareness 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 7

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G6M2 LESSON 1 ZEARN MATH TEACHER LESSON MATERIALS Concept Exploration Activity 2 THE STUDENT S COLLECTION Instructional Routines Group Presentations MLR8 Discussion Supports MLR3 Clarify Critique Correct YOUR NOTES In this activity students write ratios to describe objects in their own collection They create a display of objects and circulate to look at their classmates work Students see that there are several ways to write ratios to describe the same situation LAUNCH Invite students to share what types of items are in their personal collections If students did not bring in a collection pair them with another student or provide them with an extra collection that you have brought in for that purpose Provide access to tools for creating a visual display Tell students they will pause their work before creating a visual display to get their sentences approved ACTIVITY 2 TASK 1 3 Sort your collection into three categories You can experiment with different ways of arranging these categories 1 Count the items in each category and record the information in the table Category Name Category Amount 2 Write at least two sentences that describe ratios in the collection Remember there are many ways to write a ratio 8 The ratio of one category to another category is to The ratio of one category to another category is There are of one category for every of another category 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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3 Make a visual display of your items that clearly shows one of your statements Be prepared to share your display with the class G6M2 LESSON 1 ZEARN MATH TEACHER LESSON MATERIALS STUDENT RESPONSE Answers vary YOUR NOTES DISCUSSION GUIDANCE Once students have had enough time to create their displays circulate through each display and listen to how students describe their ratios As students present their displays point out the various ways that students chose to showcase their work including different ways to say the same ratio Ask students who used two sets of numbers to describe the same categories e g 8 to 2 and 4 for every 1 to demonstrate the two ways of grouping the objects ANTICIPATED MISCONCEPTIONS Watch for students simply writing a numerical ratio such as 3 7 without any descriptive words Draw their attention to the sentence stems in the task statement SUPPORT FOR ENGLISH LANGUAGE LEARNERS Conversing MLR 8 Discussion Supports To help students identify and have language for the types of categories that can be quantified ask students to count out loud how many there are of a particular type e g blue items Publicly record a list of quantities that surfaced e g number of blue items Design Principle s Cultivate conversation Writing MLR 3 Clarify Critique Correct Present an incorrect ratio statement that reflects a possible misunderstanding For example in a collection of hair clips that includes 2 blue hair clips and 6 yellow hair clips an incorrect statement is For every yellow hair clip there are 3 blue hair clips Prompt students to identify the error and then write a correct version This helps students evaluate and improve upon the written mathematical arguments of others Design Principle s Maximize meta awareness 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 9

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G6M2 LESSON 1 ZEARN MATH TEACHER LESSON MATERIALS Digital Lesson In your own words explain a ratio You may choose to support your thinking with a picture A ratio is YOUR NOTES STUDENT RESPONSE Answers vary Sample response A ratio is a way to associate two or more quantities Students might also draw a picture and then give example ratios that correspond to their picture Wrap Up LESSON SYNTHESIS This lesson is all about how to use ratio language and notation to describe an association between two or more quantities Wrap up the lesson by drawing a diagram for all to see of say 6 squares and 3 circles Say One way to write this ratio is there are 6 squares for every 3 circles What are some other ways to write this ratio Some correct options might be The ratio of squares to circles is 6 3 The ratio of circles to squares is 3 to 6 There are 2 squares for every 1 circle Display this diagram and the associated sentences the class comes up with somewhere in the classroom so students can refer back to the correct ratio and rate language during subsequent lessons 10 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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Consider posing some more general questions such as Explain what a ratio is in your own words G6M2 LESSON 1 ZEARN MATH TEACHER LESSON MATERIALS What things must you pay attention to when writing a ratio What are some words and phrases that are used to write a ratio YOUR NOTES TERMINOLOGY Ratio A ratio is an association between two or more quantities For example the ratio 3 2 could describe a recipe that uses 3 cups of flour for every 2 eggs or a boat that moves 3 meters every 2 seconds One way to represent the ratio 3 2 is with a diagram that has 3 blue squares for every 2 green squares EXIT TICKET Here is a collection of dogs mice and cats Write two sentences that describe a ratio of types of animals in this collection STUDENT RESPONSE Answers vary Sample responses The ratio of dogs to cats is 6 4 There are 3 dogs for every 2 cats There is 1 mouse for every 2 cats The ratio of cats to mice is 4 2 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 11

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G6M2 LESSON 2 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES GRADE 6 MISSION 2 LESSON 2 Representing Ratios with Diagrams Coordinate discrete diagrams and multiple written sentences describing the same ratios LEARNING GOALS Draw and label discrete diagrams to represent situations involving ratios Practice reading and writing sentences describing ratios e g The ratio of these to those is a b The ratio of these to those is a to b For every a of these there are b of those LEARNING GOALS STUDENT FACING LEARNING TARGETS STUDENT FACING Let s use diagrams to represent ratios I include labels when I draw a diagram representing a ratio so that the meaning of the diagram is clear I can draw a diagram that represents a ratio and explain what the diagram means Copies of Template for Concept Exploration Activity 2 Colored pencils REQUIRED MATERIALS Tools for creating a visual display any way for students to create work that can be easily displayed to the class Examples chart paper and markers whiteboard space and markers shared online drawing tool access to a document camera Pre printed slips cut from copies of the template REQUIRED PREPARATION 12 For the Card Sort Spaghetti Sauce activity make 1 copy of the template for each group of 2 students plus a few extras The template shows the correct matches Keep the extra copies whole to serve as answer keys Cut up the rest of the slips for students to use and throw away the cut slips that say The above diagram also matches this sentence It may be helpful to copy each group s slips on a different color of paper so that misplaced slips can quickly be put back 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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Students used physical objects to learn about ratios in the previous lesson Here they use diagrams to represent situations involving ratios and continue to develop ratio language The use of diagrams to represent ratios involves some care so that students can make strategic choices about the tools they use to solve problems Both the visual and verbal descriptions of ratios demand careful interpretation and use of language Students should see diagrams as a useful and efficient ways to represent ratios There is not really a right or wrong way to draw a diagram what is important is that it represents the mathematics and makes sense to the student and the student can explain how the diagram is being used However a goal of this lesson is to help students draw useful diagrams efficiently G6M2 LESSON 2 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES For example here is a diagram to show 6 cups of juice and 3 cups of soda water in a recipe juice cups water cups When students are asked to draw diagrams they often include unnecessary details such as making each cup look like an actual cup which makes the diagrams inefficient to use for solving problems Examples of very simple diagrams help guide students toward more abstract representations while still relying on visual or spatial cues to support reasoning Diagrams can also help students see associations between quantities in different ways For example we can see there are 2 cups of juice for 1 cup of soda water by grouping the items as shown below While students may say for every 2 cups of juice there is 1 cup of soda note that for now we will not suggest writing the association as 2 1 Equivalent ratios will be carefully developed in upcoming lessons Diagrams like the one above are referred to as discrete diagrams in these materials but students do not need to know this term In student facing materials they are simply called diagrams juice cups water cups The discrete diagrams in this lesson are meant to reflect the parallel structure of double number lines that students will learn later in the mission But for now students do not need to draw them this way as long as they can explain their diagrams and interpret discrete diagrams like the ones shown in the lesson 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 13

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G6M2 LESSON 2 ZEARN MATH TEACHER LESSON MATERIALS Warm Up NUMBER TALK DIVIDING BY 4 AND MULTIPLYING BY 1 4 Instructional Routines Number Talk MLR8 Discussion Supports YOUR NOTES This number talk helps students recall that dividing by a number is the same as multiplying by its reciprocal Four problems are given however they do not all require the same amount of time In grade 4 students multiplied a fraction by a whole number using their understanding of multiplication as groups of a number as the basis for their reasoning In grade 5 students multiply fractions by whole numbers reasoning in terms of taking a part of a part either by using division or partitioning a whole In both grade levels the context of the problem played a significant role in how students reasoned and notated the problem and solution Two important ideas that follow from this work and that will be relevant to future work should be emphasized during discussions Dividing by a number is the same as multiplying by its reciprocal We can multiply numbers in any order if it makes it easier to find the answer LAUNCH Display one problem at a time Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy Allow students to share their answers with a partner and note any discrepancies Pause after the third question and ask What do you notice about the first three questions Do you notice the same thing if we divide 5 by 4 Why WARM UP TASK 1 Find the value of each expression mentally 24 4 1 4 24 24 1 4 5 4 14 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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G6M2 LESSON 2 STUDENT RESPONSE ZEARN MATH TEACHER LESSON MATERIALS 1 24 4 6 Possible strategies Divide 24 into 4 equal groups or know that 4 6 24 2 1 4 24 6 Possible strategies Divide 24 into 4 equal groups or know that 4 6 24 1 4 YOUR NOTES 3 24 6 Possible strategies Divide 24 into 4 equal groups or know that 4 6 24 or Commutative Property from the second question 4 5 4 54 or equivalent Possible strategies Distributive Property 4 1 4 4 4 1 4 or know that 5 14 54 DISCUSSION GUIDANCE Ask students to share what they noticed about the first three problems Record student explanations that connect dividing by a number with multiplying by its reciprocal Revisit the meaning of reciprocal when the term comes up or bring it up if it s not mentioned by students Help students recall that the product of a number and its reciprocal is 1 Discuss how students could use their observations on the first three questions to divide 5 by 4 and then any two whole numbers SUPPORT FOR STUDENTS WITH DISABILITIES Memory Processing Time Provide sticky notes or mini whiteboards to aid students with working memory challenges SUPPORT FOR ENGLISH LANGUAGE LEARNERS Speaking MLR 8 Discussion Supports Provide sentence frames to support students with explaining their strategies For example I noticed that or First I because When students share their answers with a partner prompt them to rehearse what they will say when they share with the full group Rehearsing provides opportunities to clarify their thinking Design Principle s Optimize output for explanation 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 15

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G6M2 LESSON 2 ZEARN MATH TEACHER LESSON MATERIALS Concept Exploration Activity 1 BLUE PAINT AND ART PASTE Instructional Routines MLR2 Collect and Display Anticipate Monitor Select Sequence Connect Group Presentations MLR8 Discussion Supports YOUR NOTES In this activity students continue to draw connections between a diagram and the ratios it represents Students work in pairs to discuss different ways to use ratio language to describe discrete diagrams They first identify statements that would correctly describe a given diagram Then they create both a diagram and corresponding statements to represent a new situation involving ratio As students work monitor for different ways in which students draw and discuss diagrams of the paste recipe Identify a few pairs who draw different diagrams and use ratio language differently to share later A few things to anticipate Some students may draw very literal drawings of cups and pints Encourage them to use simpler representations Students may choose to draw letters X s or other symbols or marks instead of squares and rectangles Students may use equivalent ratios to describe a situation even though these have not been explicitly taught e g they may say the ratio of cups of flour to pints of water is 4 1 instead of 8 2 Though this is correct be careful here We have previously regrouped objects and might say for example that with a ratio 8 2 for every 4 cups of flour there is 1 cup of water but we have not asserted that this ratio can be written as 4 1 yet The idea of equivalent ratios is sophisticated and will be developed over the next several lessons Correct descriptions may include fractions e g for every tablespoon of blue paint there is 13 cup of white paint Although students are not expected to work with fractions in this lesson responses involving fractions are fine LAUNCH Arrange students in groups of 2 Provide them with the tools needed for creating a large visual display for the second part of the task Ensure students understand they are supposed to select more than one statement for the first question Consider having students take turns reading each statement and deciding whether they think it describes the situation or not 16 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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G6M2 LESSON 2 ACTIVITY 1 TASK 1 2 ZEARN MATH TEACHER LESSON MATERIALS Elena mixed 2 cups of white paint with 6 tablespoons of blue paint YOUR NOTES Here is a diagram that represents this situation white paint cups blue paint tablespoons Discuss the statements that follow and circle all those that correctly describe this situation Make sure that both you and your partner agree with each circled answer a The ratio of cups of white paint to tablespoons of blue paint is 2 6 b For every cup of white paint there are 2 tablespoons of blue paint c There is 1 cup of white paint for every 3 tablespoons of blue paint d There are 3 tablespoons of blue paint for every cup of white paint e For each tablespoon of blue paint there are 3 cups of white paint f For every 6 tablespoons of blue paint there are 2 cups of white paint g The ratio of tablespoons of blue paint to cups of white paint is 6 to 2 STUDENT RESPONSE The following statements describe the paint mixture a The ratio of cups of white paint to tablespoons of blue paint is 2 6 c There is 1 cup of white paint for every 3 tablespoons of blue paint d There are 3 tablespoons of blue paint for every cup of white paint f For every 6 tablespoons of blue paint there are 2 cups of white paint g The ratio of tablespoons of blue paint to cups of white paint is 6 to 2 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 17

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G6M2 LESSON 2 The following statements do not describe the paint mixture ZEARN MATH TEACHER LESSON MATERIALS b For every cup of white paint there are 2 tablespoons of blue paint e For each tablespoon of blue paint there are 3 cups of white paint YOUR NOTES ACTIVITY 1 TASK 2 3 Jada mixed 8 cups of flour with 2 pints of water to make paste for an art project a Draw a diagram that represents the situation b Write at least two sentences describing the ratio of flour and water STUDENT RESPONSE Answers vary Sample responses The ratio of cups of flour to pints of water is 8 2 The ratio of pints of water to cups of flour is 2 to 8 For each pint of water there are 4 cups of flour For every 8 cups of flour there are 2 pints of water For every 4 cups of flour there is 1 pint of water There are 2 pints of water for every 8 cups of flour DISCUSSION GUIDANCE Select students to share their paste diagrams and sentences with the class Sequence the diagrams from most elaborate to most simple Connect the many ways in which the paste can be represented and described Compare more detailed pictures with a discrete diagram point out how the discrete diagram is a more efficient way of showing the paste recipe 18 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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G6M2 LESSON 2 ANTICIPATED MISCONCEPTIONS ZEARN MATH TEACHER LESSON MATERIALS Some students may think all of the statements about the paint mixture are accurate descriptions If so suggest that there are two false statements Have students discuss the statements again in determining which two are false YOUR NOTES SUPPORT FOR STUDENTS WITH DISABILITIES Social Emotional Functioning Peer Tutors Pair students with their previously identified peer tutors SUPPORT FOR ENGLISH LANGUAGE LEARNERS Speaking Writing MLR 2 Collect and Display Circulate and listen to student talk during partner or group work and display publicly common or important words and phrases e g for every the ratio of for each students are using Refer back to this list and ask students to clarify their meaning explain how they are useful and to reflect on which words and phrases help to communicate ideas more precisely This will provide access to important language for students to use as they are needed Design Principle s Support sense making Concept Exploration Activity 2 CARD SORT SPAGHETTI SAUCE MATERIALS TEMPLATE FOR CONCEPT EXPLORATION ACTIVITY 2 Instructional Routines Take Turns MLR8 Discussion Supports MLR7 Compare and Connect Writing and using ratio language requires attention to detail This task further develops students ability to describe ratio situations precisely by attending carefully to the quantities their units and their order in the ratio Students work in pairs to match ratios of sauce ingredients to discrete diagrams and to explain reasoning 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 19

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G6M2 LESSON 2 ZEARN MATH TEACHER LESSON MATERIALS LAUNCH Arrange students in groups of 2 Place two copies of uncut templates in envelopes to serve as answer keys YOUR NOTES Demonstrate how to set up and play the matching game Choose a student to be your partner Discuss what all the symbols mean Mix up the cards and place them face up Point out that the cards contain either diagrams or sentences Select one of each style of card and then explain to your partner why you think the cards do or do not match Demonstrate productive ways to agree or disagree e g by explaining your mathematical thinking asking clarifying questions etc Give each group cut up cards for matching Tell students to check their matches after they complete the activity using the answer keys ACTIVITY 2 TASK 1 43 Your teacher will give you cards describing different recipes for spaghetti sauce In the diagrams a circle represents a cup of tomato sauce a square represents a tablespoon of oil a triangle represents a teaspoon of oregano 1 Take turns with your partner to match a sentence with a diagram a For each match that you find explain to your partner how you know it s a match b For each match that your partner finds listen carefully to their explanation If you disagree discuss your thinking and work to reach an agreement 2 After you and your partner have agreed on all of the matches check your answers with the answer key If there are any errors discuss why and revise your matches 20 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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3 There were two diagrams that each matched with two different sentences Which were they Diagram matched with both sentences and Diagram matched with both sentences and G6M2 LESSON 2 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES 4 Select one of the other diagrams and invent another sentence that could describe the ratio shown in the diagram STUDENT RESPONSE 1 Diagram A matches with sentence 4 Diagram B matches with sentences 2 and 8 Diagram C matches with sentence 1 Diagram D matches with sentence 5 Diagram E matches with sentences 3 and 7 Diagram F matches with sentence 6 2 No answer necessary 3 a Diagram B matches with sentences 2 and 8 b Diagram E matches with sentences 3 and 7 4 Answers vary Sample responses For diagram A the ratio of cups of tomato sauce to tablespoons of oil is 3 1 For diagram D the ratio of tablespoons of oil to cups of tomato sauce is 2 to 5 DISCUSSION GUIDANCE Once all groups have completed the matching discuss the following Which matches were tricky Explain why Did any pairs need to make adjustments in their matches What might have caused an error What adjustments were made 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 21

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G6M2 LESSON 2 What if you were making this tasty sauce and got the ratios wrong What would happen ZEARN MATH TEACHER LESSON MATERIALS ANTICIPATED MISCONCEPTIONS YOUR NOTES If students disagree about a match encourage them to figure out the correct answer through discussion and use of the answer key Make sure that when students use the answer key they discuss any errors rather than just make changes Students may think the shapes in the diagram need to be drawn in the same order the ingredients appear in the description This is not the case You could turn a diagram card upside down and it would still represent the same situation The diagram just shows ingredients that get mixed together in a pot It is the case however that within the description the order of the words in the sentence must correspond with the terms within the ratio SUPPORT FOR STUDENTS WITH DISABILITIES Expressive Language Eliminate Barriers Provide sentence frames for students to explain their reasoning i e are a match because and SUPPORT FOR ENGLISH LANGUAGE LEARNERS Representing MLR 7 Compare and Connect To help students connect the words of the sentence with the visual representation in the diagram suggest that students with the sentence cards circle the two quantities e g tablespoons of oil and cups of tomato sauce Then ask the student with the diagram card to show where those quantities are in the diagram Design Principle s Maximize meta awareness Digital Lesson These two diagrams show the same ratio of circles to triangles Diagram A Diagram B What is the ratio of circles to triangles Which diagram was more helpful to you in describing the ratio Why 22 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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The ratio of circles to triangles is Diagram A Diagram B was more helpful to me in describing the ratio because G6M2 LESSON 2 ZEARN MATH TEACHER LESSON MATERIALS STUDENT RESPONSE YOUR NOTES The ratio of circles to triangles is 2 to 6 Diagram A was more helpful in describing the ratio because I can easily see the number of triangles and circles It s organized into groups by shape Wrap Up LESSON SYNTHESIS This lesson used diagrams to represent ratios These diagrams omit details that are not necessary for understanding and solving the problem at hand Discuss What are some good things to remember when you draw a diagram of a ratio You only need necessary information You could include shapes color coded boxes or initials to represent each object within the set It is helpful to organize the types of items in rows and to arrange smaller groups so they are easier to see How can a diagram help you make sense of a situation involving a ratio It is easier to write correct statements about them Also you can see how the objects can be grouped EXIT TICKET There are 3 cats in a room and no other creatures Each cat has 2 ears 4 paws and 1 tail 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 23

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G6M2 LESSON 2 ZEARN MATH TEACHER LESSON MATERIALS 1 Draw a diagram that shows an association between numbers of ears paws and tails in the room 2 Complete each statement a The ratio of YOUR NOTES to to b There are paws for every tail c There are paws for every ear is STUDENT RESPONSE 1 Answers vary Sample response 2 a The ratio of ears to paws to tails is 6 12 3 b There are 4 paws for every tail c There are 4 paws for every 2 ears This means that there are 2 paws for every ear ANTICIPATED MISCONCEPTIONS In the second question students may not realize that the order of the words in the sentence must correspond with the terms within the ratio Ears paws tails must correspond with 6 12 3 In the fourth question students may not write the sentence for every one ear If this is the case prompt them to draw a circle around each set of two paws and one ear to help them see this relationship 24 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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Template for Lesson 2 Concept Exploration Activity 2 page 1 of 2 Blackline Master for Classroom Activity 6 2 2 4 Card Sort Spaghetti Sauce Card Sort Spaghetti Sauce Card Sort Spaghetti Sauce Diagram A Sentence 4 There are 3 cups of tomato sauce for every tablespoon of oil Card Sort Spaghetti Sauce Card Sort Spaghetti Sauce Diagram B Sentence 2 There are 3 tablespoons of oil for every cup of tomato sauce Card Sort Spaghetti Sauce Sentence 8 The above diagram also matches this sentence The ratio of cups of tomato sauce to tablespoons of oil is 1 3 Card Sort Spaghetti Sauce Card Sort Spaghetti Sauce Diagram C Sentence 1 The ratio of tablespoons of oil to cups of tomato sauce is 5 to 2 Card Sort Spaghetti Sauce Card Sort Spaghetti Sauce Diagram D Sentence 5 The ratio of cups of tomato sauce to tablespoons of oil is 5 to 2 Download for free at openupresources org 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 25

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Template for Lesson 2 Concept Exploration Activity 2 page 2 of 2 Blackline Master for Classroom Activity 6 2 2 4 Card Sort Spaghetti Sauce Card Sort Spaghetti Sauce Card Sort Spaghetti Sauce Diagram E Sentence 3 For every tablespoon of oil there are 2 cups of tomato sauce and 5 teaspoons of oregano Card Sort Spaghetti Sauce Sentence 7 The above diagram also matches this sentence Tablespoons of oil teaspoons of oregano and cups of tomato sauce are in the ratio 1 5 2 Card Sort Spaghetti Sauce Card Sort Spaghetti Sauce Diagram F Sentence 6 Cups of tomato sauce tablespoons of oil and teaspoons of oregano are in the ratio 1 5 2 Download for free at openupresources org 26 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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GRADE 6 MISSION 2 LESSON 3 Recipes G6M2 LESSON 3 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Draw and label a discrete diagram with circled groups to represent multiple batches of a recipe LEARNING GOALS Explain equivalent ratios orally and in writing in terms of different sized batches of the same recipe having the same taste Understand that doubling or tripling a recipe involves multiplying the amount of each ingredient by the same number yielding something that tastes the same LEARNING GOALS STUDENT FACING LEARNING TARGETS STUDENT FACING Let s explore how ratios affect the way a recipe tastes I can use a diagram to represent a recipe a double batch and a triple batch of a recipe I know what it means to double or triple a recipe I can explain the meaning of equivalent ratios using a recipe as an example Markers Teaspoon Water REQUIRED MATERIALS Paper cups Empty containers Drink mix A powder that is mixed with water to create a fruit flavored or chocolate flavored drink Using a sugar free drink mix is recommended but not a mix that calls for adding a separate sweetener when mixing up the drink Create two separate drink mixtures Container A has one cup of water and one teaspoon of powdered drink mix Container B has one cup of water and four teaspoons of powdered drink mix You might have to stir the mixtures vigorously for a minute or more to ensure all the powder dissolves REQUIRED PREPARATION Get 6 small paper cups Do not mark the cups Put a small amount of mixture A in three of the cups and a small amount of mixture B in the other three cups Keep track of which is which as you will give each of three volunteers one of each cup Discard the rest of the mixtures for now You will do a dramatic performance creating each mixture during class During class you will need three empty mixing containers with at least a 2 cup capacity each One marked A one marked B and one marked C You will also need a supply of water a supply of drink mix a measuring cup and a teaspoon 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 27

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G6M2 LESSON 3 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES This is the first of two lessons that develop the idea of equivalent ratios through physical experiences A key understanding is that if we scale a recipe up or down to make multiple batches or a fraction of a batch the result will still be the same in some meaningful way Students see this idea in two contexts taste and color In this lesson a mixture containing two batches of a recipe tastes the same as a mixture containing one batch For example 2 cups of water mixed thoroughly with 8 teaspoons of powdered drink mix tastes the same as 1 cup of water mixed with 4 teaspoons of powdered drink mix In the next lesson a mixture containing two batches of a recipe for colored water will produce the same shade of the color as a mixture containing one batch For example 10 ml of blue mixed with 30 ml of yellow produces the same shade of green as 5 ml of blue mixed with 15 ml of yellow The fact that two equivalent ratios yield the same taste or produce the same color is a physical manifestation of the equivalence of the ratios In this lesson students start to use the term equivalent ratios Students see that scaling a recipe up or down requires multiplying the amount of each ingredient by the same factor e g doubling a recipe means doubling the amount of each ingredient They also gain more experience using a discrete diagram as a tool to represent a situation Warm Up FLOWER PATTERN Instructional Routine Think Pair Share The purpose of this warm up is to quickly remind students of different ways to write ratios They also have an opportunity to multiply the number of each type of shape by 2 to make two copies of the flower which previews the process introduced in this lesson for making a double batch of a recipe LAUNCH Arrange students in groups of 2 Ensure students understand there are 6 hexagons 2 trapezoids and 9 triangles in the picture and that their job is to write ratios about the numbers of shapes Give quiet work time and then invite students to share their sentences with their partner followed by discussion 28 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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G6M2 LESSON 3 WARM UP TASK 1 ZEARN MATH TEACHER LESSON MATERIALS This flower is made up of hexagons trapezoids and triangles YOUR NOTES 1 Write sentences to describe the ratios of the shapes that make up this pattern 2 How many of each shape would be in two copies of this flower pattern STUDENT RESPONSE 1 Answers vary Sample responses For every 2 hexagons there are 3 triangles There are 3 hexagons for every trapezoid The ratio of trapezoids to triangles is 2 to 9 The ratio of hexagons to trapezoids to triangles is 6 2 9 2 There would be 12 hexagons 4 trapezoids and 18 triangles DISCUSSION GUIDANCE Invite a student to share a sentence that describes the ratios of shapes in the picture Ask if any students described the same relationship a different way For example three ways to describe the same ratio are The ratio of hexagons to trapezoids is 6 2 The ratio of trapezoids to hexagons is 2 to 6 There are 3 hexagons for every trapezoid Ask a student to describe why two copies of the picture would have 12 hexagons 4 trapezoids and 18 triangles If no student brings it up be sure to point out that each number in one copy of the picture can be multiplied by 2 to find the number of each shape in two copies 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 29

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G6M2 LESSON 3 ZEARN MATH TEACHER LESSON MATERIALS ANTICIPATED MISCONCEPTIONS Students might get off track by attending to the area each shape covers Clarify that this task is only concerned with the number of each shape and not the area covered YOUR NOTES SUPPORT FOR STUDENTS WITH DISABILITIES Visual Spatial Processing Visual Aids Provide handouts of the representations for students to draw on or highlight Concept Exploration Activity 1 POWDERED DRINK MIX Instructional Routine MLR8 Discussion Supports In this activity three student volunteers participate in a taste test of two drink mixtures Mixture A is made with 1 cup of water and 1 teaspoon of drink mix Mixture B is made with 1 cup of water and 4 teaspoons of drink mix The taste testers match diagrams with each mixture and explain their reasoning After the taste test in front of students recreate Mixture A 1 cup water with 1 teaspoon of drink mix and Mixture B 1 cup water with 4 teaspoons of drink mix Ask students to describe how the diagrams correspond with these mixtures Then conduct a demonstration in which 3 teaspoons of drink mix are added to Mixture A and a new diagram is drawn Once Mixture A and Mixture B both contain one cup of water and 4 teaspoons of drink mix both mixtures are combined in a third container labeled Mixture C As part of their work on the task students reason that this combined mixture tastes the same as each individual batch Students then conclude that a mixture containing two batches tastes the same as the mixture that contains just one batch because mixing two things together that taste the same will produce a mixture that tastes the same They should also note that each ingredient was doubled in the mixture ACTIVITY 1 LAUNCH 1 Display the diagram for all to see 30 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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A B G6M2 LESSON 3 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Taste test Recruit three volunteers for a taste test Give each volunteer two unmarked cups one each of a small amount of Mixture A and Mixture B Explain that their job is to take a tiny sip of each sample match the diagrams to the samples and explain their matches Demonstration Conduct a dramatic demonstration of mixing powdered drink mix and water Start with two empty containers labeled A and B To Container A add 1 cup of water and 1 teaspoon of drink mix To Container B add 1 cup of water and 4 teaspoons of drink mix Mix them both thoroughly The first diagram should still be displayed Discuss Which mixture has a stronger flavor B has more drink mix in the same quantity of water How can we make Mixture A taste like Mixture B Put 3 more teaspoons of drink mix into Container A ACTIVITY 1 LAUNCH 2 Add 3 more teaspoons of drink mix to Container A Display a new diagram to represent the situation A B Discuss Describe the ratio of ingredients that is now in Container A The ratio of cups of water to teaspoons of drink mix is 1 4 Describe the ratio of ingredients that is in Container B The ratio of cups of water to teaspoons of drink mix is also 1 to 4 How do you think they compare in taste They taste the same If desired you can have volunteers verify that they taste the same but this might not be necessary 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 31

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G6M2 LESSON 3 Pour the contents of both A and B into a larger container labeled C and mix them thoroughly ZEARN MATH TEACHER LESSON MATERIALS Discuss How would the taste of Mixture C compare to the taste of Mixture A and Mixture B The new mixture would taste the same as each component mixture YOUR NOTES Following this demonstration students individually interpret the drink mixture diagrams The work in the task will reiterate what happened in the demonstration ACTIVITY 1 TASK 1 2 Here are diagrams representing three mixtures of powdered drink mix and water A B Key 1 teaspoon drink mix C 1 cup water 1 How would the taste of Mixture A compare to the taste of Mixture B 2 Use the diagrams to complete each statement a Mixture B uses cups of water and teaspoons of drink mix The ratio of cups water to teaspoons of drink mix in Mixture B is b Mixture C uses cups of water and teaspoons of drink mix The ratio of cups of water to teaspoons of drink mix in Mixture C is 3 How would the taste of Mixture B compare to the taste of Mixture C STUDENT RESPONSE 1 Mixtures A and B will taste the same because the have the same amount of water and drink mix 2 a Mixture B uses 1 cup of water and 4 teaspoons of drink mix The ratio of cups of water to teaspoons of drink mix in Mixture B is 1 4 32 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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b Mixture C uses 2 cups of water and 8 teaspoons of drink mix The ratio of cups of water to teaspoons of drink mix in mixture C is 2 8 G6M2 LESSON 3 ZEARN MATH TEACHER LESSON MATERIALS 3 Mixtures B and C will taste the same This is because Mixture C was made by doubling Mixture B or by mixing A and B together which taste the same so the mixture would still taste the same YOUR NOTES DISCUSSION GUIDANCE Mixing 1 cup of water with 4 teaspoons of powdered drink mix makes a mixture that tastes exactly the same as mixing 2 cups of water with 8 teaspoons of powdered drink mix We say that 1 4 and 2 8 are equivalent ratios Ask students to discuss what they think equivalent means Some ways they might respond are Mixtures that taste the same use equivalent ratios A double batch of a recipe doubling each ingredient is an equivalent ratio to a single batch ANTICIPATED MISCONCEPTIONS Students may not initially realize that Mixtures C and B taste the same You could ask them to imagine ordering a smoothie from a takeout window Would a small size smoothie taste the same as a size that is double that amount If you double the amount of each ingredient the mixture tastes the same SUPPORT FOR STUDENTS WITH DISABILITIES Strengths based Approach This activity leverages many natural strengths of students with ADHD LD and other concrete learners in terms of its multi modal and hands on nature This may be an opportunity for the teacher to highlight this strength in class and allow an individual with disability to lead peer interactions discussions increasing buy in SUPPORT FOR ENGLISH LANGUAGE LEARNERS Conversing MLR 8 Discussion Supports To help students justify their reasoning provide a sentence frame such as Mixtures B and C will taste because Using the same frame ask What if I add a half cup of water to C or What if I add a teaspoon of drink mix to B Monitor these conversations for ideas and as a way to start the discussion 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 33

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G6M2 LESSON 3 Design Principle s Support sense making ZEARN MATH TEACHER LESSON MATERIALS Speaking MLR 8 Discussion Supports Ask partners to share what they think equivalent means by finding other ratios that are equivalent to 1 4 Provide a sentence frame for discussion Equivalent means For example when you have two mixtures that YOUR NOTES Design Principle s Optimize output for explanation Concept Exploration Activity 2 BATCH OF COOKIES Instructional Routine MLR8 Discussion Supports Students continue to use diagrams to represent the ratio of ingredients in a recipe as well as mixtures that contain multiple batches They come to understand that a change in the number of batches changes the quantities of the ingredients but the end product tastes the same They then use this observation to come up with a working definition for equivalent ratio LAUNCH Launch the task with a scenario and a question Let s say you are planning to make cookies using your favorite recipe and you re going to double the recipe What does it mean to double a recipe There are a few things you want to draw out in this conversation If we double a recipe we need to double the amount of every ingredient If the recipe calls for 3 eggs doubling it means using 6 eggs If the recipe calls for 13 teaspoon of baking soda we use 23 teaspoon of baking soda etc We expect to end up with twice as many cookies when we double the recipe as we would when making a single batch However we expect the cookies from 2 batches of a recipe to taste exactly the same as those from a single batch Tell students they will now think about making different numbers of batches of a cookie recipe 34 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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G6M2 LESSON 3 ACTIVITY 2 TASK 1 3 ZEARN MATH TEACHER LESSON MATERIALS A recipe for one batch of cookies calls for 5 cups of flour and 2 teaspoons of vanilla YOUR NOTES 1 Draw a diagram that shows the amount of flour and vanilla needed for two batches of cookies 2 How many batches can you make with 15 cups of flour and 6 teaspoons of vanilla Indicate the additional batches by adding more ingredients to your diagram 3 How much flour and vanilla would you need for 5 batches of cookies 4 Whether the ratio of cups of flour to teaspoons of vanilla is 5 2 10 4 or 15 6 the recipes would make cookies that taste the same We call these equivalent ratios a Find another ratio of cups of flour to teaspoons of vanilla that is equivalent to these ratios b How many batches can you make using this new ratio of ingredients STUDENT RESPONSE 1 Diagrams may look different but should clearly show two groups of 5 and 2 Here are two ways that students might circle the batches If they see each batch individually they might draw something like this If they think of a double batch as a single thing they might circle it like this 2 You can make 3 batches Sample diagram 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 35

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G6M2 LESSON 3 ZEARN MATH TEACHER LESSON MATERIALS 3 You would need 25 cups of flour and 10 teaspoons of vanilla The diagram may be expanded to reflect this 4 Answers vary Sample responses a 20 8 YOUR NOTES b 4 batches DISCUSSION GUIDANCE Invite a few students to share their responses and diagrams with the group A key point to emphasize during discussion is that when we double or triple a recipe we also have to double or triple each ingredient Record a working but not final definition for equivalent ratio that can be displayed for at least the next several lessons Here is an example Cups of flour and teaspoons of vanilla in the ratio 5 2 10 4 or 15 6 are equivalent ratios because they describe different numbers of batches of the same recipe Include a diagram in this display ANTICIPATED MISCONCEPTIONS For the fourth question students may not multiply both the amount of flour and the amount of vanilla by the same number If this happens refer students to the previous questions in noting that the amount of each ingredient was changed in the same way SUPPORT FOR STUDENTS WITH DISABILITIES Receptive Expressive Language Peer Tutors Pair students with their previously identified peer tutors to aid in comprehension and expression of understanding SUPPORT FOR ENGLISH LANGUAGE LEARNERS Conversing Representing MLR 8 Discussion Supports Display sentence frames for students to use during discussion For example In my diagram represents and To find an equivalent ratio first I because This will help students produce mathematical language as they make sense of equivalent ratios using diagrams Design Principle s Cultivate conversation Support sense making 36 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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G6M2 LESSON 3 Digital Lesson ZEARN MATH TEACHER LESSON MATERIALS Cheryl makes a chocolate milk recipe that uses 2 cups of milk and 7 tablespoons of chocolate powder Draw a diagram to represent double the recipe YOUR NOTES Write a ratio to describe the quantities The ratio of milk to chocolate powder is STUDENT RESPONSE Diagram should show 4 total items representing cups of milk and 14 total items representing tablespoons of chocolate powder Students may break their diagram into 2 groups showing 2 cups of milk for every 7 teaspoons of chocolate powder or leave as 1 large group Students may choose to write a variety of ratios but it should be a ratio equivalent to 4 14 Wrap Up LESSON SYNTHESIS The four main ideas you want to draw out to conclude the lesson are To double a recipe you need to double the amount of each ingredient To scale a recipe generally you need to multiply each ingredient by the same number Scaling a recipe results in a substance that tastes the same as the original recipe We say that a ratio that represents a recipe is equivalent to a ratio that represents multiple batches of the same recipe Discuss When doubling a recipe how does the amount of each individual ingredient change Each ingredient is doubled We call the new ratio of ingredients an equivalent ratio When tripling a recipe how does the amount of each individual ingredient change Each ingredient is tripled We call the new ratio of ingredients an equivalent ratio 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 37

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G6M2 LESSON 3 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES How do different numbers of batches of the same recipe taste They taste exactly the same EXIT TICKET LAUNCH If necessary explain that some people like to observe birds These people might put bird food in a bird feeder outside their homes to attract birds so they can watch them through a window TASK Usually when Elena makes bird food she mixes 9 cups of seeds with 6 tablespoons of maple syrup However today she is short on ingredients Think of a recipe that would yield a smaller batch of bird food but still taste the same Explain or show your reasoning STUDENT RESPONSE Two likely valid answers are 3 cups of seeds and 2 tablespoons of syrup 6 cups of seeds and 4 tablespoons of syrup Explanations and diagrams may vary Here are some possibilities 3 2 represents the cups of seeds to the tablespoons of syrup 3 2 is equivalent to 9 6 seeds cups syrup tablespoons 38 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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GRADE 6 MISSION 2 LESSON 4 Color Mixtures G6M2 LESSON 4 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Comprehend and respond orally and in writing to questions asking whether two ratios are equivalent in the context of color mixtures LEARNING GOALS Draw and label a discrete diagram with circled groups to represent multiple batches of a color mixture Explain equivalent ratios orally and in writing in terms of the amounts of each color in a mixture being multiplied by the same number to create another mixture that is the same shade LEARNING GOALS STUDENT FACING Let s see what color mixing has to do with ratios I know what it means to double or triple a color mixture LEARNING TARGETS STUDENT FACING I can use a diagram to represent a single batch a double batch and a triple batch of a color mixture I can explain the meaning of equivalent ratios using a color mixture as an example Graduated cylinders Markers REQUIRED MATERIALS Food coloring Beakers Paper cups REQUIRED PREPARATION Mix blue water and yellow water each group of 2 students will need 1 cup of each To make colored water add 1 teaspoon of food coloring to 1 cup of water It would be best to give each mixture to students in a beaker or another container with a pour spout If possible conduct this lesson in a room with a sink 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 39

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G6M2 LESSON 4 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES This is the second of two lessons that help students make sense of equivalent ratios through physical experiences In this lesson students mix different numbers of batches of a recipe for green water by combining blue and yellow water created ahead of time with food coloring to see if they produce the same shade of green They also change the ratio of blue and yellow water to see if it changes the result The activities here reinforce the idea that scaling a recipe up or down requires scaling the amount of each ingredient by the same factor Students continue to use discrete diagrams as a tool to represent a situation For students who do not see color the lesson can be adapted by having students make batches of dough with flour and water 1 cup of flour to 5 tablespoons of water makes a very stiff dough and 1 cup of flour to 6 tablespoons of water makes a soft but not sticky dough In this case doubling a recipe yields dough with the same tactile properties just as doubling a coloredwater recipe yields a mixture with the same color The invariant property is stiffness rather than color The principle that equivalent ratios yield products that are identical in some important way applies to both types of experiments Warm Up NUMBER TALK ADJUSTING A FACTOR Instructional Routine Number Talk This number talk encourages students to use the structure of base ten numbers and the properties of operations to find the product of two whole numbers While many strategies may emerge the focus of this string of problems is for students to see how adjusting a factor impacts the product and how this insight can be used to reason about other problems Four problems are given however it may not be possible to share every possible strategy Consider gathering only two or three different strategies per problem Each problem was chosen to elicit a slightly different reasoning so as students explain their strategies ask how the factors impacted how they approached the problem LAUNCH Display one problem at a time Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy Follow with a whole class discussion 40 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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G6M2 LESSON 4 WARM UP TASK 1 ZEARN MATH TEACHER LESSON MATERIALS Find the value of each product mentally YOUR NOTES 1 6 15 2 12 15 3 6 45 4 13 45 STUDENT RESPONSE 1 90 Possible strategy 6 10 6 5 90 2 180 Possible strategy Since the 6 from the first question doubled to 12 and the 15 stayed the same the product doubles to 180 This is because there are twice as many groups of 15 than in the first question 3 270 Possible strategy Since the 6 is the same as in the first question and the 15 tripled to 45 the product triples to 270 This is because the number of groups stayed the same but the amount in each group got three times as large 4 585 Possible strategy Since the 45 is the same as the previous question we can double the 6 and the product to get 540 We need one more group of 45 and 540 45 585 DISCUSSION GUIDANCE Ask students to share their strategies for each problem Record and display their explanations for all to see Ask students if or how the factors in the problem impacted the strategy choice To involve more students in the conversation consider asking Who can restate s reasoning in a different way Did anyone solve the problem the same way but would explain it differently Did anyone solve the problem in a different way Does anyone want to add on to Do you agree or disagree Why s strategy 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 41

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G6M2 LESSON 4 SUPPORT FOR STUDENTS WITH DISABILITIES ZEARN MATH TEACHER LESSON MATERIALS Memory Processing Time Provide sticky notes or mini whiteboards to aid students with working memory challenges YOUR NOTES Concept Exploration Activity 1 TURNING GREEN Instructional Routine MLR8 Discussion Supports In this activity students mix different numbers of batches of a color recipe to obtain a certain shade of green They observe how multiple batches of the same recipe produce the same shade of green as a single batch which suggests that the ratios of blue to yellow for the two situations are equivalent They also come up with a ratio that is not equivalent to produce a mixture that is a different shade of green As students make the mixtures ensure that they measure accurately so they will get accurate outcomes As students work note the different diagrams students use to represent their recipes Select a few examples that could be highlighted in discussion later LAUNCH Arrange students in groups of 2 4 Smaller groups are better but group size might depend on available equipment Each group needs a beaker of blue water and one of yellow water one graduated cylinder a permanent marker a craft stick and 3 opaque white cups either styrofoam white paper or with a white plastic interior Show students the blue and yellow water Demonstrate how to pour from the beakers to the graduated cylinder to measure and mix 5 ml of blue water with 15 ml of yellow water Demonstrate how to get an accurate reading on the graduated cylinder by working on a level surface and by reading the measurement at eye level Tell students they will experiment with different mixtures of green water and observe the resulting shades 42 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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G6M2 LESSON 4 ACTIVITY 1 TASK 1 2 Your teacher mixed milliliters of blue water and milliliters of yellow water in the ratio 5 15 Use this mixture to help with the recipes below ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES 1 Doubling the original recipe a Draw a diagram to represent the amount of each color that you will combine to double your teacher s recipe b Use a marker to label an empty cup with the ratio of blue water to yellow water in this double batch c Predict whether these amounts of blue and yellow will make the same shade of green as your teacher s mixture Next check your prediction by measuring those amounts and mixing them in the cup d Is the ratio in your mixture equivalent to the ratio in your teacher s mixture Explain your reasoning 2 Tripling the original recipe a Draw a diagram to represent triple your teacher s recipe b Label an empty cup with the ratio of blue water to yellow water c Predict whether these amounts will make the same shade of green Next check your prediction by mixing those amounts d Is the ratio in your new mixture equivalent to the ratio in your teacher s mixture Explain your reasoning 3 Next invent your own recipe for a bluer shade of green water a Draw a diagram to represent the amount of each color you will combine b Label the final empty cup with the ratio of blue water to yellow water in this recipe c Test your recipe by mixing a batch in the cup Does the mixture yield a bluer shade of green d Is the ratio you used in this recipe equivalent to the ratio in your teacher s mixture Explain your reasoning 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 43

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G6M2 LESSON 4 ZEARN MATH TEACHER LESSON MATERIALS STUDENT RESPONSE 1 Doubling the recipe YOUR NOTES a Here is one example of a diagram Students may arrange the groups differently or use different symbols to represent 1 ml of water b A cup is labeled 10 30 or 10 to 30 c If the recipe is correct the shade of green is identical to the teacher s d 10 30 is equivalent to 5 15 because it is 2 batches of the same recipe It creates an identical shade of green 2 Tripling the recipe a Like the previous diagram except showing 3 batches b A cup is labeled 15 45 or 15 to 45 c If the recipe is correct the shade of green is identical to the teacher s d 15 45 is equivalent to 5 15 because it is 3 batches of the same recipe It creates an identical shade of green 3 A bluer shade of green a Answers vary You might use more blue for the same amount of yellow or less yellow for the same amount of blue Sample response b Answers vary Sample responses 10 15 more blue for the same amount of yellow or 5 10 less yellow for the same amount of blue 44 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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c If a correct ratio is used the mixture should be a bluer shade of green than the other mixtures G6M2 LESSON 4 ZEARN MATH TEACHER LESSON MATERIALS d No it was not the same shade of green The first and second parts were not respectively obtained by multiplying 5 and 15 by the same number YOUR NOTES DISCUSSION GUIDANCE After each group has completed the task have the students rotate through each group s workspace to observe the mixtures and diagrams As they circulate pose some guiding questions Are each group s results for the first two mixtures the same shade of green Are the ratios representing the double batch the triple batch and your new mixture all equivalent to each other How do you know What are some different ways groups drew diagrams to represent the ratios Highlight the idea that a ratio is equivalent to another if the two ratios describe different numbers of batches of the same recipe ANTICIPATED MISCONCEPTIONS If any students come up with an incorrect recipe consider letting this play out A different shade of green shows that the ratio of blue to yellow in their mixture is not equivalent to the ratio of blue to yellow in the other mixtures Rescuing the incorrect mixture to display during discussion may lead to meaningful conversations about what equivalent ratios mean SUPPORT FOR STUDENTS WITH DISABILITIES Strengths based Approach This activity leverages many natural strengths of students with ADHD LD and other concrete learners in terms of its multi sensory and hands on nature This may be an opportunity for the teacher to highlight this strength in class and allow an individual with disability to lead peer interactions discussions increasing buy in 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 45

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G6M2 LESSON 4 SUPPORT FOR ENGLISH LANGUAGE LEARNERS ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Conversing MLR 8 Discussion Supports Assign one member from each group to stay behind to answer questions from students visiting from other groups Provide visitors with question prompts such as These look like the same shade how can we be sure the ratios are equivalent If you want a smaller amount with the same shade what can you do or What is the ratio for your new mixture And if I want a triple batch what do I need to do Design Principle s Cultivate Conversation Digital Lesson To make 1 batch of pink paint Joe mixes 2 liters of red paint with 1 gallon of white paint Make a diagram of an equivalent ratio Consider doubling or tripling this mixture to find an equivalent ratio Diagram Equivalent ratio The ratio of liters of red paint to gallons of white paint is STUDENT RESPONSE Responses vary but should show an equivalent ratio to 2 1 Sample response Diagram shows 2 groups of 2 squares and 1 circle The ratio of liters of white paint to gallons of red paint is 4 2 Wrap Up The important take aways from this lesson are To create more batches of a color recipe that will come out to be the same shade of the color multiply each ingredient by the same number We can think of equivalent ratios as representing different numbers of batches of the same recipe 46 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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G6M2 LESSON 4 LESSON SYNTHESIS ZEARN MATH TEACHER LESSON MATERIALS Remind students of the work done and observations made in this lesson Some questions to guide the discussion might include How did you decide that 10 ml blue and 30 ml yellow would make 2 batches of 5 ml blue and 15 ml yellow Multiply each part by 2 YOUR NOTES How did you decide that 15 ml blue and 45 ml yellow would make 3 batches Multiply each part by 3 How did we know that 5 15 10 30 and 15 45 were equivalent They created the same shade of green Also 10 30 has both parts of the original recipe multiplied by 2 and 15 45 has both parts of the original recipe multiplied by 3 EXIT TICKET A recipe for orange water says Mix 3 teaspoons yellow water with 1 teaspoon red water For this recipe we might say The ratio of teaspoons of yellow water to teaspoons of red water is 3 1 1 Write a ratio for 2 batches of this recipe 2 Write a ratio for 4 batches of this recipe 3 Explain why we can say that any two of these three ratios are equivalent STUDENT RESPONSE 1 The ratio of teaspoons of yellow to teaspoons of red is 6 2 or any sentence that accurately states this ratio Note a statement like The ratio of yellow to red is 6 2 describes the association between the colors but does not indicate the amount of stuff in the mixture 2 The ratio of teaspoons of yellow to teaspoons of red is 12 4 or any sentence that accurately states this ratio 3 These are equivalent ratios because they describe different numbers of batches of the same recipe To make 2 batches multiply the amount of each color by 2 To make 4 batches multiply the amount of each color by 4 As long as you multiply the amounts for both colors by the same number you will get a ratio that is equivalent to the ratio in the recipe 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 47

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G6M2 LESSON 5 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES GRADE 6 MISSION 2 LESSON 5 Defining Equivalent Ratios Generate equivalent ratios and justify that they are equivalent LEARNING GOALS LEARNING GOALS STUDENT FACING LEARNING TARGETS STUDENT FACING REQUIRED MATERIALS 48 Present in words and through other representations a definition of equivalent ratios including examples and non examples Let s investigate equivalent ratios some more If I have a ratio I can create a new ratio that is equivalent to it If I have two ratios I can decide whether they are equivalent to each other Tools for creating a visual display Any way for students to create work that can be easily displayed to the class Examples chart paper and markers whiteboard space and markers shared online drawing tool access to a document camera 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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Previously students understood equivalent ratios through physical perception of different batches of recipes In this lesson they work with equivalent ratios more abstractly both in the context of recipes and in the context of abstract ratios of numbers They understand and articulate that all ratios that are equivalent to a b can be generated by multiplying both a and b by the same number By connecting concrete quantitative experiences to abstract representations that are independent of a context students develop their skills in reasoning abstractly and quantitatively They continue to use diagrams words or a combination of both for their explanations The goal in subsequent lessons is to develop a general definition of equivalent ratios G6M2 LESSON 5 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Warm Up DOTS AND HALF DOTS In this warm up students are asked to determine the number of dots in an image and explain how they arrived at that answer The goal is to prompt students to visualize and articulate different ways in which they can decompose the dots using what they know about arrays symmetry and multiplication to arrive at the total number of dots To encourage students to refer to the image in their explanation but not count every dot this image is flashed for a few seconds and then hidden It is flashed once more for students to check their thinking Ask students how they saw the dots instead of how they found the number of dots so they focus on the structure of the dots in the image As students share how they saw the dots ask how the expressions they used to describe the arrangements and grouping of the dots in the two problems are similar This prompts students to make connections between the properties of multiplication LAUNCH Tell students you will show them an image made up of dots for 3 seconds Their job is to find how many dots are in the image and explain how they saw them Display the image for all to see for 3 seconds and then hide it Do this twice Give students quiet think time between each flash of the image Encourage students who have one way of seeing the dots to think of another way while they wait 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 49

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G6M2 LESSON 5 ZEARN MATH TEACHER LESSON MATERIALS STUDENT RESPONSE 21 dots Answers vary Possible strategies 6 groups with 3 and a half in each group 6 3 1 2 21 YOUR NOTES 3 groups with 7 in each group 3 7 21 DISCUSSION GUIDANCE Ask students to share how many dots they saw and how they saw them Record and display student explanations for all to see Consider re displaying the image for reference while students are explaining what they saw To involve more students in the conversation consider asking Who can restate the way saw the dots in different words Did anyone see the dots the same way but would explain it differently Does anyone want to add an observation to the way saw the dots Who saw the dots differently Do you agree or disagree Why SUPPORT FOR STUDENTS WITH DISABILITIES Memory Processing Time This instructional routine can be very taxing to a student s working memory For students with challenges in this area show the image for a longer period of time or repeat the image flash as needed Students also benefit from being explicitly told not to count the dots but instead to look for helpful structure within the image 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 51

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G6M2 LESSON 5 ZEARN MATH TEACHER LESSON MATERIALS Concept Exploration Activity 1 TUNA CASSEROLE Instructional Routine MLR8 Discussion Supports YOUR NOTES Students use a realistic food recipe to find equivalent ratios that represent different numbers of batches Students use the original recipe to form ratios of ingredients that represent double half five times and one fifth of the recipe Then they examine given ratios of ingredients and determine how many batches they represent LAUNCH Ask students if they have ever cooked something by following a recipe If so ask them what they made and what some of the ingredients were Ask How might we use ratios to describe the ingredients in your recipe The ratios could associate the quantities of each ingredient being used Explain to students that in this task they will think about the ratios of ingredients for a tuna casserole and how to adjust them for making different numbers of batches ACTIVITY 1 TASK 1 3 Use the ingredients from a tuna casserole recipe to answer the questions in your notes Ingredients 3 cups cooked elbow shaped pasta 6 ounce can tuna drained 10 ounce can cream of chicken soup 1 cup shredded cheddar cheese 1 52 1 2 cups French fried onions 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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G6M2 LESSON 5 Instructions Combine the pasta tuna soup and half of the cheese Transfer into a 9 inch by 13 inch baking dish Put the remaining cheese on top Bake 30 minutes at 350 degrees During the last 5 minutes add the French fried onions Let sit for 10 minutes before serving 1 What is the ratio of the ounces of soup to the cups of shredded cheese to the cups of pasta in one batch of casserole ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES 2 How much of each of these 3 ingredients would be needed to make a twice the amount of casserole b half the amount of casserole c five times the amount of casserole d one fifth the amount of casserole 3 What is the ratio of cups of pasta to ounces of tuna in one batch of casserole 4 How many batches of casserole would you make if you used the following amounts of ingredients a 9 cups of pasta and 18 ounces of tuna b 36 ounces of tuna and 18 cups of pasta c 1 cup of pasta and 2 ounces of tuna STUDENT RESPONSE 1 The ratio of the ounces of soup to cups of shredded cheese to cups of pasta is 10 1 3 2 The ratio of these ingredients for different numbers of batches are a 20 ounces 2 cups 6 cups b 5 ounces 1 2 cup 1 1 2 cups c 50 ounces 5 cups 15 cups d 2 ounces 1 5 cup 3 5 cup 3 The ratio of cups of pasta to ounces of tuna is 3 6 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 53

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G6M2 LESSON 5 4 ZEARN MATH TEACHER LESSON MATERIALS a 3 batches b 6 batches c YOUR NOTES 1 3 batch DISCUSSION GUIDANCE Display the recipe for all to see Ask students to share and explain their responses List their responses along with the specified number of batches for all to see Ask students to analyze the list and describe how the ratio of quantities relate to the number of batches in each case Draw out the idea that each quantity within the recipe was multiplied by a number to obtain each batch size and that each ingredient amount is multiplied by the same value In finding one half and one fifth of a batch students may speak in terms of dividing by 2 and dividing by 5 Point out that dividing by 2 has the same outcome as multiplying by one half and dividing by 5 has the same outcome as multiplying by one fifth Students multiplied a whole number by a fraction in grade 5 Later we will want to state our general definition of equivalent ratios as simply as possible as multiplying both a and b in the ratio a b by the same number not multiplying or dividing ANTICIPATED MISCONCEPTIONS Students who are not yet fluent in fraction multiplication from grade 5 may have difficulty understanding how to find half or one fifth of the recipe ingredient amounts Likewise they may have difficulty identifying one third of a batch Suggest that they draw a picture of 12 of 10 remind them that finding 12 of a number is the same as dividing it by 2 or remind them that 1 of a number means 1 times that number 2 2 SUPPORT FOR ENGLISH LANGUAGE LEARNERS Listening MLR 8 Discussion Supports When asking the question How might we use ratios to describe the ingredients in your recipe act out or use images that demonstrate the meaning of the terms ratio recipe and ingredients in the context of cooking Demonstrate combining specific ingredients in their stated ratios This will help students connect the language found in a recipe with the ratio reasoning needed for different batches of that recipe Design Principle s Support sense making 54 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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G6M2 LESSON 5 SUPPORT FOR STUDENTS WITH DISABILITIES Conceptual Processing Processing Time Check in with individual students as needed to assess for comprehension during each step of the activity ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Executive Functioning Eliminate Barriers Provide a task checklist which makes all the required components of the activity explicit Concept Exploration Activity 2 WHAT ARE EQUIVALENT RATIOS Instructional Routines Group Presentations MLR1 Stronger and Clearer Each Time In this activity students identify what equivalent ratios have in common a ratio equivalent to a b can be generated by multiplying both a and b by the same number and generate equivalent ratios It is at this point in the mission where students will explicitly define the term equivalent ratios LAUNCH Arrange students in groups of 3 4 Provide each group with tools for creating a visual display Summarize what we know so far about equivalent ratios When we double or triple a color recipe the ratios of the amount of ingredients in the mixtures are equivalent to those in the original recipe For example 24 9 and 8 3 are equivalent ratios because we can think of 24 9 as a mixture that contains three batches of purple water where a single batch is 8 3 When we make multiple batches of a food recipe we say the ratios of the amounts of the ingredients are equivalent to the ratios in a single batch For example 3 6 1 2 and 9 18 are equivalent ratios because they correspond to the amount of the ingredients in different numbers of batches of tuna noodle casserole and they all taste the same In this activity we ll write a definition for equivalent ratios When students pause after question 5 have a discussion about the first five questions Then assign each group a different ratio to use as their example for their visual display Some possibilities 4 5 3 2 5 6 3 4 2 5 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 55

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G6M2 LESSON 5 ACTIVITY 2 TASK 1 ZEARN MATH TEACHER LESSON MATERIALS 43 The ratios 5 3 and 10 6 are equivalent ratios Use these ratios to answer the questions in your notes YOUR NOTES 1 Is the ratio 15 12 equivalent to these Explain your reasoning 2 Is the ratio 30 18 equivalent to these Explain your reasoning 3 Give two more examples of ratios that are equivalent to 5 3 4 How do you know when ratios are equivalent and when they are not equivalent 5 Write a definition of equivalent ratios Pause here so your teacher can review your work and assign you a ratio to use for your visual display STUDENT RESPONSE 1 15 12 is not equivalent to 5 3 because 15 is 5 3 but 12 is 3 4 2 30 18 is equivalent to 5 3 because 30 is 5 6 and 18 is 3 6 3 Answers vary and might include 15 9 20 12 and 50 30 4 Answers vary and should include some version of multiply both parts by the same number 5 Answers vary Sample response A ratio is equivalent to a b when both a and b are multiplied by the same number ACTIVITY 2 TASK 2 53 Create a visual display that includes 56 the title Equivalent Ratios 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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your best definition of equivalent ratios the ratio your teacher assigned to you at least two examples of ratios that are equivalent to your assigned ratio an explanation of how you know these examples are equivalent at least one example of a ratio that is not equivalent to your assigned ratio an explanation of how you know this example is not equivalent G6M2 LESSON 5 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Be prepared to share your display with the class STUDENT RESPONSE Answers vary DISCUSSION GUIDANCE Each group will share their visual display as they explain their definitions Highlight phrases or explanations that are similar in each display Make one class display that incorporates all valid definitions This display should be kept posted in the classroom for the remaining lessons in this mission It should look something like Equivalent Ratio A ratio is equivalent to a b when both a and b are multiplied by the same number ANTICIPATED MISCONCEPTIONS Students may incorporate recipes specific examples or batch thinking into their definitions These are important ways of thinking about equivalent ratios but challenge them to come up with a definition that only talks about the numbers involved and not what the numbers represent If groups struggle to get started thinking generally about a definition give them a head start with A ratio is equivalent to a b when If students include or divide in their definition remind them that for example dividing by 5 gives the same result as multiplying by one fifth Therefore we can just use multiply in our definition 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 57

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G6M2 LESSON 5 SUPPORT FOR STUDENTS WITH DISABILITIES ZEARN MATH TEACHER LESSON MATERIALS Executive Functioning Eliminate Barriers Provide a task checklist which makes all the required components of the activity explicit YOUR NOTES SUPPORT FOR ENGLISH LANGUAGE LEARNERS Writing and speaking Math Language Routine 1 Stronger and Clearer Each Time This is the first time Math Language Routine 1 is suggested as a support in this course In this routine students are given a thought provoking question or prompt and asked to create a first draft response Students meet together in 2 3 partners to share and refine their response through conversation While meeting listeners ask questions such as What did you mean by and Can you say that another way Finally students write a second draft of their response that reflects ideas from their partners and improvement on their writing The purpose of this routine is to provide a structured and interactive opportunity for students to revise and refine their ideas through verbal and written means Design Principle s Optimize output for generalization How It Happens 1 Use this routine to provide students a structured opportunity to revise and refine their response to How do you know when ratios are equivalent and when they are not equivalent Allow students 2 3 minutes to individually create first draft responses 2 Invite students to meet with 2 3 other partners for feedback Instruct the speaker to begin by sharing their ideas without looking at their written draft if possible Provide the listener with these prompts for feedback that will help teams strengthen their ideas and clarify their language Can you explain how You should expand on Can you give an example of equivalent ratios and Could you justify that differently Be sure to have the partners switch roles Allow 1 2 minutes to discuss 3 Signal for students to move on to their next partner and repeat this structured meeting 4 Close the partner conversations and invite students to revise and refine their writing in a second draft Students can borrow ideas and language from each partner to strengthen the final product Provide these sentence frames to help students organize their thoughts in a clear precise way I know ratios are equivalent not equivalent when and An example of this is because Here is an example of a second draft I know that ratios are equivalent when I multiply both parts of one ratio by the same number and I get the other ratio For example I know that 5 3 is equivalent to 30 18 because when I multiply 5 by 6 I get 30 and when I multiply 3 by 6 I get 18 so 6 is the same number used to multiply both parts But 5 3 is not equivalent to 15 12 because when I multiply 5 by 3 I get 15 and when I multiply 3 by 4 I get 12 So since a different number is used to multiply to get the second ratio they re not equivalent 58 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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5 If time allows instruct students to compare their first and second drafts If not the students can continue on with the lesson by returning to their first partner and creating the visual G6M2 LESSON 5 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Digital Lesson What are equivalent ratios Explain how you could find a ratio that is equivalent to 3 5 STUDENT RESPONSE Equivalent ratios are ratios in which every number in the first ratio can be multiplied by the same value to equal the numbers in the second ratio To find a ratio equivalent to 3 5 multiply both 3 and 5 by the same value For example if you multiply 3 and 5 by 2 you get the equivalent ratio 6 10 Wrap Up In this lesson you came to an understanding of what equivalent ratios are LESSON SYNTHESIS Discuss How do you make different amounts of a colored water mixture that have the same color The amount of each color being mixed must be multiplied by the same value If you want to make a different amount of a food recipe how can you ensure that the resulting food will taste the same Each ingredient in the recipe must be multiplied by the same value 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 59

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G6M2 LESSON 5 ZEARN MATH TEACHER LESSON MATERIALS What are equivalent ratios and how are they generated Each number is multiplied by the same value TERMINOLOGY YOUR NOTES Equivalent ratios Two ratios are equivalent if you can multiply each of the numbers in the first ratio by the same factor to get the numbers in the second ratio For example 8 6 is equivalent to 4 3 because 8 12 4 and 6 12 3 A recipe for lemonade says to use 8 cups of water and 6 lemons If we use 4 cups of water and 3 lemons it will make half as much lemonade Both recipes taste the same because 8 6 and 4 3 are equivalent ratios Cups of water Number of lemons 8 6 4 3 EXIT TICKET 1 Write another ratio that is equivalent to the ratio 4 6 2 How do you know that your new ratio is equivalent to 4 6 Explain or show your reasoning STUDENT RESPONSE 1 Answers vary Sample responses 2 3 16 24 400 600 2 Answers vary 2 3 is equivalent to 4 6 because both 4 and 6 are multiplied by 12 16 24 is equivalent because both 4 and 6 are multiplied by 4 400 600 is equivalent because both 4 and 6 are multiplied by 100 ANTICIPATED MISCONCEPTIONS If students are not clear about the meaning of equivalent ratios refer them to the visual displays created in the previous activity 60 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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GRADE 6 MISSION 2 LESSON 6 Introducing Double Number Line Diagrams G6M2 LESSON 6 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Compare and contrast orally and in writing discrete diagrams and double number line diagrams representing the same situation LEARNING GOALS Explain orally how to use a double number line diagram to find equivalent ratios Label and interpret a double number line diagram that represents a familiar context LEARNING GOALS STUDENT FACING LEARNING TARGETS STUDENT FACING REQUIRED MATERIALS Let s use number lines to represent equivalent ratios When I have a double number line that represents a situation I can explain what it means I can label a double number line diagram to represent batches of a recipe or color mixture Rulers 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 61

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G6M2 LESSON 6 ZEARN MATH TEACHER LESSON MATERIALS This lesson introduces the double number line diagram a useful efficient and sophisticated tool for reasoning about equivalent ratios The lines in a double number line diagram are similar to the number lines students have seen in earlier grades in that YOUR NOTES Numbers correspond to distances on the line so that the distance between say 0 and 12 is three times the distance between 0 and 4 We can choose what scale to use i e whether each interval represents 1 unit 2 units 5 units etc The lines can be extended as needed In a double number line diagram we use two parallel number lines one line for each quantity in the ratio and choose a scale on each line so equivalent ratios line up vertically For example if the ratio of number of eggs to cups of milk in a recipe is 4 to 1 we can draw a number line for the number of eggs and one for the cups of milk On the number lines the quantity of 4 for the number of eggs and the 1 for cups of milk would line up vertically as would 8 eggs and 2 cups of milk and so on 0 4 8 12 16 20 0 1 2 3 4 5 Number of eggs Cups of milk Because they represent quantities with length on a number line rather than with counts of objects double number lines are both more abstract and more general than discrete diagrams Later in this mission students will learn an even more abstract representation of equivalent ratios the table of values Connecting the concrete to the abstract helps students connect quantitative reasoning to abstract reasoning Though some activities are designed to hone students facility with particular representations students should continue to have autonomy in choosing representations to solve problems as long as they can explain their meaning 62 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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G6M2 LESSON 6 Warm Up NUMBER TALK ADJUSTING ANOTHER FACTOR ZEARN MATH TEACHER LESSON MATERIALS Instructional Routines Number Talk MLR8 Discussion Supports This Number Talk encourages students to think about the numbers in computation problems and rely on what they know about structure patterns whole number multiplication and properties of operations to mentally solve a problem YOUR NOTES While many strategies may emerge the focus of this string of problems is for students to see how adjusting a factor impacts the product and how this insight can be used to reason about other problems Four problems are given however given limited time it may not be possible to share every strategy Consider gathering only two or three different strategies per problem Each problem was chosen to elicit slightly different reasoning so as students explain their strategies ask how the factors impacted their product LAUNCH Display one problem at a time Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy Follow with a whole class discussion WARM UP TASK 1 Find the value of each product mentally Then write your answer in your notes 1 4 5 4 2 4 5 8 3 1 10 65 4 2 10 65 STUDENT RESPONSE 1 4 5 4 18 Possible strategies 4 4 0 5 4 double and halve 9 2 2 4 5 8 36 Possible strategies double the product from the first question because a factor doubled 8 4 0 5 8 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 63

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G6M2 LESSON 6 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES 3 1 10 4 2 10 65 6 5 Possible strategies 65 10 65 0 1 65 13 Possible strategies double the product from the previous question because a factor doubled 65 10 2 DISCUSSION GUIDANCE Ask students to share their strategies for each problem Record and display their explanations for all to see Ask students if or how the factors in the problem impacted the strategy choice To involve more students in the conversation consider asking Who can restate s reasoning in a different way Did anyone solve the problem the same way but would explain it differently Did anyone solve the problem in a different way Does anyone want to add on to s strategy Do you agree or disagree Why SUPPORT FOR STUDENTS WITH DISABILITIES Memory Processing Time Provide sticky notes or mini whiteboards to aid students with working memory challenges SUPPORT FOR ENGLISH LANGUAGE LEARNERS Speaking MLR 8 Discussion Supports Display sentence frames to support students when they explain their strategy For example I noticed that or First I because When students share their answers with a partner prompt them to rehearse what they will say when they share with the whole class Rehearsing provides students with additional opportunities to clarify their thinking and to consider how they will communicate their reasoning Design Principle s Optimize output for explanation 64 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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G6M2 LESSON 6 Concept Exploration Activity 1 DRINK MIX ON A DOUBLE NUMBER LINE ZEARN MATH TEACHER LESSON MATERIALS Instructional Routines Think Pair Share MLR8 Discussion Supports In this activity a double number line a new representation is presented and interpreted alongside the more familiar discrete diagrams and in the familiar context of recipes YOUR NOTES Students learn that just like discrete diagrams double number lines represent equivalent ratios They see that alignment between the numbers of the two lines matters that it is through the alignment that the association of two quantities are shown Students notice pairs of numbers that line up vertically are equivalent ratios Because double number lines are quicker to draw and can be extended easily to show many more equivalent ratios they are more efficient than discrete diagrams especially for dealing with larger quantities As students work monitor for those who contrast the two representations in terms of using graphic symbols versus numbers and those who think about equivalent ratios in terms of the alignment of numbers in the double number line diagram LAUNCH Ask students to recall the mixture of powdered drink mix and water from a previous lesson Ask How much drink mix and water was in one batch 4 teaspoons of drink mix and 1 cup of water What would you need to mix a double batch 8 teaspoons of drink mix and 2 cups of water Explain that they are going to show batches of a mixture using a double number line diagram Give students quiet think time to make sense of the new representation and answer the questions followed by time to share their response with a partner and a discussion afterwards 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 65

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G6M2 LESSON 6 ACTIVITY 1 TASK 1 ZEARN MATH TEACHER LESSON MATERIALS 2 YOUR NOTES The other day we made drink mixtures by mixing 4 teaspoons of powdered drink mix for every cup of water Below are two ways to represent multiple batches of this recipe Drink mix teaspoons Water cups 0 4 8 12 16 0 1 2 3 4 Drink mix teaspoons Water cups 1 How can we tell that 4 1 and 12 3 are equivalent ratios 2 How are these representations the same How are these representations different 3 How many teaspoons of drink mix should be used with 3 cups of water 4 How many cups of water should be used with 16 teaspoons of drink mix 5 What numbers should go in the empty boxes on the double number line diagram What do these numbers mean STUDENT RESPONSE 1 12 and 3 are 3 times 4 and 1 respectively On the number line diagram you can see that 4 and 1 line up vertically as do 12 and 3 2 Same Each representation shows the amount of drink mix and water for one batch and two batches They each show teaspoons of drink mix along the top and cups of water along the bottom Different The first diagram uses squares to represent each teaspoon of drink mix and cup of water but the number line diagram has these amounts written with numbers The first diagram shows only two batches and the number line diagram shows 0 1 2 3 and 4 batches with space for 5 batches 3 12 teaspoons 4 4 cups 66 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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5 The numbers 20 and 5 should go in the missing places These numbers mean that the result of mixing 20 teaspoons of drink mix with 5 cups of water would taste the same as the other mixtures or that these amounts would make 5 batches of the recipe G6M2 LESSON 6 ZEARN MATH TEACHER LESSON MATERIALS DISCUSSION GUIDANCE YOUR NOTES Select students to share their observations about how the two representations are alike and how they differ As students discuss solutions to the questions circle pairs of associated quantities on the double number line Help students connect information as it is represented in the different diagrams On the last question ask students how they knew that 20 was the next number on the line representing teaspoons of drink mix Skip counting by 4 multiply the next number of cups of water by 4 Ask students to think more generally for a minute about the representations at hand What is a double number line diagram What do they do What do the numbers on the tick marks represent and how should they be scaled What might be some benefits of using double number lines instead of diagrams We can use them to show many more batches they are quicker to draw ANTICIPATED MISCONCEPTIONS While the double number line diagram is given here some students may not feel comfortable with seeing the same numbers the 4 s in different positions Remind students that each number line represents a different quantity and that the two 4 s have different meanings SUPPORT FOR STUDENTS WITH DISABILITIES Expressive Language Eliminate Barriers Provide sentence frames for students to explain their reasoning i e the representations are the same because the representations are different because Conceptual Processing Manipulatives Use a kinesthetic representation of the number line on a clothesline Students can place and adjust numbers on folded paper cardstock on the clothesline 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 67

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G6M2 LESSON 6 SUPPORT FOR ENGLISH LANGUAGE LEARNERS ZEARN MATH TEACHER LESSON MATERIALS Reading Writing MLR 8 Discussion Supports If necessary remind students of the meaning of these terms recipe batch mixture and diagram This will support student understanding of the context so that they can make sense of the double number line YOUR NOTES Design Principle s Support sense making Optimize output for explanation Concept Exploration Activity 2 BLUE PAINT ON A DOUBLE NUMBER LINE Instructional Routine MLR1 Stronger and Clearer Each Time Anticipate Monitor Select Sequence Connect The purpose of this activity is for students to practice labeling the tick marks on a double number line diagram with equivalent ratios This activity revisits a familiar context from a previous lesson so students can apply reasoning about different sized batches of a recipe to help them understand the more abstract representation of a double number line diagram Some students may interpret the diagram as showing 3 tablespoons of blue paint for every 1 cup of white paint and may choose to label the top line of the double number line diagram counting by 1s instead of 2s and the bottom line counting by 3s instead of 6s This is also an acceptable correct answer There are two reasons to monitor for students using this alternate representation First when students are comparing their diagrams with their partner if one partner counted by 2s and the other partner counted by 1s they may need guidance in determining that these are both correct answers Second during the whole class discussion consider selecting a student with the less common representation to share their solution at the end LAUNCH Give students quiet work time followed by time to share their response with a partner and then a discussion afterwards 68 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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G6M2 LESSON 6 ACTIVITY 2 TASK 1 3 ZEARN MATH TEACHER LESSON MATERIALS Here is a diagram showing Elena s recipe for light blue paint YOUR NOTES White paint cups Blue paint tablespoons 1 Complete the double number line diagram to show the amounts of white paint and blue paint in different sized batches of light blue paint 0 0 2 Compare your double number line diagram with your partner Discuss your thinking If needed revise your diagram 3 How many cups of white paint should Elena mix with 12 tablespoons of blue paint How many batches would this make 4 How many tablespoons of blue paint should Elena mix with 6 cups of white paint How many batches would this make 5 Use your double number line diagram to find another amount of white paint and blue paint that would make the same shade of light blue paint 6 How do you know that these mixtures would make the same shade of light blue paint STUDENT RESPONSE 1 0 2 4 6 8 10 12 0 6 12 18 24 30 36 White paint cups Blue paint tablespoons 2 Answers vary 3 4 cups of white paint because that is the number on the top line that lines up with the 12 on the bottom line This would make 2 batches of paint 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 69

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G6M2 LESSON 6 ZEARN MATH TEACHER LESSON MATERIALS 4 18 tablespoons of blue paint because that is the number on the bottom line that lines up with the 6 on the top line This would make 3 batches of paint 5 Answers vary Sample responses 8 cups of white paint and 24 tablespoons of blue paint 10 cups of white paint and 30 tablespoons of blue paint 3 cups of white paint and 9 tablespoons of blue paint 1 cup of white paint and 3 tablespoons of blue paint YOUR NOTES 6 I know these mixtures would make the same shade of light blue paint because the ratios of the amounts of each paint color are equivalent to the ratio in the original recipe DISCUSSION GUIDANCE Select students to present their solutions Help them connect the different ways in which the information is represented in the different diagrams Emphasize the importance of labeling everything clearly so the interpretation is easy to make SUPPORT FOR STUDENTS WITH DISABILITIES Receptive Expressive Language Peer Tutors Pair students with their previously identified peer tutors to aid in comprehension and expression of understanding SUPPORT FOR ENGLISH LANGUAGE LEARNERS Representing Writing Listening MLR 1 Stronger and Clearer Each Time Use this routine to provide students with a structured opportunity to revise and refine their ideas Give students 2 3 minutes to draft an initial response to the final question Ask each student to meet with 2 3 other partners in a row for feedback Students should take turns playing the role of speaker and listener Provide listeners with prompts for feedback that will help their partners strengthen their ideas and clarify their language e g Can you explain how you used your double number line diagram or You should expand on etc Students can borrow ideas and language from each partner to strengthen the final product Design Principle s Optimize output for justification 70 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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G6M2 LESSON 6 Digital Lesson ZEARN MATH TEACHER LESSON MATERIALS Water cups Lemonade mix tsp 0 1 YOUR NOTES Water cups Lemonade mix tsp 0 1 Complete the double number line diagram to show the amount of each ingredient in 1 2 3 4 and 5 batches of lemonade 2 How are these two representations the same How are they different STUDENT RESPONSE 1 0 1 2 3 4 5 0 3 6 9 12 15 Water cups Lemonade mix tsp 2 They re the same because they represent the same recipe They re different because the diagram represents the recipe in pictures and the double number line represents the recipe in numbers Wrap Up LESSON SYNTHESIS The main ideas to draw out of this lesson are the reasons for using a double number line diagram Double number lines easily display equivalent ratios with the numbers in each equivalent ratio lining up vertically A double number line diagram can be used when a discrete diagram would be cumbersome or even impossible 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 71

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G6M2 LESSON 6 ZEARN MATH TEACHER LESSON MATERIALS The other major goal of this lesson is building up students fluency in creating double number lines Students will have further opportunities in upcoming lessons but watch for common errors such as inconsistent labeling and failing to align the equivalent ratios TERMINOLOGY YOUR NOTES Double Number Line Diagram 0 Red paint teaspoons A double number line diagram uses a pair of parallel Yellow paint teaspoons number lines to represent equivalent ratios The 0 locations of the tick marks match on both number lines The tick marks labeled 0 line up but the other numbers are usually different 3 6 9 12 5 10 15 20 EXIT TICKET A recipe for one batch of cookies uses 5 cups of flour and 2 teaspoons of vanilla 1 Complete the double number line diagram to show the amount of flour and vanilla needed for 1 2 3 4 and 5 batches of cookies 0 0 2 If you use 20 cups of flour how many teaspoons of vanilla should you use 3 If you use 6 teaspoons of vanilla how many cups of flour should you use STUDENT RESPONSE 1 0 5 10 15 20 25 0 2 4 6 8 10 Flour cups Vanilla teaspoons 2 You should use 8 teaspoons of vanilla 3 You should use 15 cups of flour 72 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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GRADE 6 MISSION 2 LESSON 7 Creating Double Number Line Diagrams G6M2 LESSON 7 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Comprehend and use the word per orally and in writing to mean for each LEARNING GOALS Draw and label a double number line diagram from scratch with parallel lines and equally spaced tick marks Use double number line diagrams to find a wider range of equivalent ratios LEARNING GOALS STUDENT FACING LEARNING TARGETS STUDENT FACING REQUIRED MATERIALS REQUIRED PREPARATION Let s draw double number line diagrams like a pro I can create a double number line diagram and correctly place and label tick marks to represent equivalent ratios I can explain what the word per means Rulers It may be helpful but not required to bring back the blue and yellow water mixtures 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 73

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G6M2 LESSON 7 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES In this lesson students create double number line diagrams from scratch They see that it is important to use parallel lines equally spaced tick marks and descriptive labels They are also introduced to using the word per to refer to how much of one quantity there is for every one unit of the other quantity Double number lines are included in the first few activity statements to help students find an equivalent ratio involving one item or one unit In later activities and lessons students make their own strategic choice of an appropriate representation to support their reasoning Regardless of method students indicate the units that go with the numbers in a ratio in both verbal statements and diagrams Note that students are not expected to use or understand the term unit rate in this lesson Warm Up ORDERING ON A NUMBER LINE Instructional routine Think Pair Share MLR8 Discussion Supports In this warm up students partition a number line and locate fraction and decimal equivalents in preparation for working with double number lines in this mission Students are purposely not asked to locate 1 on the number line to see how they reason about locating the 12 and 14 It is important for students to be able to identify the fractions or decimals and label tick marks correctly interpreting the distance between tick marks rather than the number of tick marks as the fractional size As students discuss with their partner select students to share their answers to the first question during the whole class discussion LAUNCH Arrange students in groups of 2 Display the number line for all to see Give students quiet think time and ask them to give a signal when they have an answer and a strategy Ask students to compare their number line with a partner and share the fractions or decimals they chose to place on the number line for the second question WARM UP TASK 1 74 Answer the questions about the number line 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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G6M2 LESSON 7 1 Locate and label the following numbers on the number line 1 2 1 4 1 3 4 1 5 0 1 75 2 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES 2 Based on where you placed the numbers locate and label four more fractions or decimals on the number line STUDENT RESPONSE 1 Here is the number line 0 1 4 1 2 1 5 1 75 1 2 3 4 2 Answers vary DISCUSSION GUIDANCE Select students to explain how they reasoned about the location of each number on the number line After each number ask the class whether they agree or disagree and if anyone else had a different way of thinking about that number If time permits ask students to share the fractions or decimals they located for the second question Discuss why they chose those numbers and how they decided on their location ANTICIPATED MISCONCEPTIONS Students may place 12 in the center of the number line reasoning that it is half of the number line Explain to the students they are placing the number 12 which has a specific value and location on the number line SUPPORT FOR STUDENTS WITH DISABILITIES Conceptual Processing Manipulatives Use a kinesthetic representation of the number line on a clothesline Students can place and adjust numbers on folded paper cardstock on the clothesline 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 75

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G6M2 LESSON 7 SUPPORT FOR ENGLISH LANGUAGE LEARNERS ZEARN MATH TEACHER LESSON MATERIALS Reading Writing MLR 8 Discussion Supports Briefly review the meaning of the terms label and tick marks as you or a student points to these features in the student task statement Review the meaning of the term locate by acting out and thinking aloud YOUR NOTES Design Principle s Support sense making Concept Exploration Activity 1 JUST A LITTLE GREEN Instructional Routines Think Pair Share MLR1 Stronger and Clearer Each Time MLR2 Collect and Display Students continue to use double number lines to reason about equivalent ratios Here students attention is directed to the 1 3 blue to yellow ratio in the green water recipe which can then be used to determine any equivalent ratio The task is also the beginning of students exploration of finding and using ratios containing a 1 One key idea to convey here is that finding a ratio associated with 1 unit of a quantity can be very helpful Another is that the intervals on double number lines can be subdivided to help us find such ratios As students work identify those who use division to determine the 1 3 ratio and then use multiplication to determine the ratios for 8 ml and 13 ml of blue water This is a key insight for a type of reasoning that is broadly useful and will be developed further LAUNCH Ask students to recall what double number lines are and how they can be used to represent problems involving equivalent ratios Explain that they are going to investigate the structure of double number lines in more detail Give students quiet think time and then time to discuss their responses with a partner 76 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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G6M2 LESSON 7 ACTIVITY 1 TASK 1 2 ZEARN MATH TEACHER LESSON MATERIALS We made green water by mixing 5 ml of blue water with 15 ml of yellow water Use the double number line to answer the questions below 0 5 10 0 15 30 YOUR NOTES Blue water ml Yellow water ml 1 On the number line for blue water label the four tick marks shown 2 On the number line for yellow water draw and label tick marks to show the amount of yellow water needed for each amount of blue water 3 How much yellow water should be used for 1 ml of blue water Circle where you can see this on the double number line 4 How much yellow water should be used for 11 ml of blue water 5 How much yellow water should be used for 8 ml of blue water 6 Why is it useful to know how much yellow water should be used with 1 ml of blue water STUDENT RESPONSE 1 Write 1 2 3 4 at the tick marks because the fifth tick mark is 5 2 Write 3 6 9 12 at the tick marks because the fifth tick mark is 15 3 3 ml of yellow water is needed 4 33 ml of yellow water is needed because 33 3 11 5 24 ml of yellow water is needed because 24 3 8 6 Using this you can multiply to figure out any amount of yellow water needed for a given amount of blue water 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 77

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G6M2 LESSON 7 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES DISCUSSION GUIDANCE Debrief as a group after students have a chance to share their work with a partner Focus discussions on how students determine the amount of yellow water for 1 ml of blue and how they determine the amounts of yellow for 8 ml and 11 ml of blue Select students who used division to find the former and multiplication to find the latter to share If students do not do so frame the relationship of blue to yellow using phrases such as for every 1 ml of or per milliliter of There are 3 milliliters of yellow water for every 1 milliliter of blue water There are 3 milliliters of yellow water per milliliter of blue water The word per means for every Ask students to think of any other situation in which they may use the word per as it is used here e g price per bottle of water cost per ticket etc and discuss why knowing the value of one item would be helpful ANTICIPATED MISCONCEPTIONS Students may have trouble figuring out that the length of a segment between consecutive tick marks is 15 of the interval from 0 to 5 especially since there are four tick marks not five When focusing on blue students first guess about the tick marks is generally correct For yellow remind them that the numbers on the tick marks are made by skip counting they are then likely to try 3 s and 5 s since both can make 15 Students who label the spaces between tick marks rather than the tick marks themselves may need additional work with important measurement conventions SUPPORT FOR ENGLISH LANGUAGE LEARNERS Writing Speaking Listening MLR 1 Stronger and Clearer Each Time To help speakers optimize for generalization related to the importance of for every 1 ml listeners should press for details and clarity as appropriate based on what each speaker produces This will help students begin to develop generalizations about multiplying and dividing using a common ratio to produce batches of water of the same color For students who need extra support in creating their first or second drafts provide these sentence frames at the relevant point in the routine If we know then we can find Design Principle s Optimize output for generalization Maximize metaawareness 78 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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Representing MLR 2 Collect and Display Capture the language students use to describe per and display this language during the synthesis G6M2 LESSON 7 ZEARN MATH TEACHER LESSON MATERIALS Design Principle s Support sense making and Maximize meta awareness YOUR NOTES SUPPORT FOR STUDENTS WITH DISABILITIES Conceptual Processing Manipulatives Use a kinesthetic representation of the number line on a clothesline Students can place and adjust numbers on folded paper cardstock on the clothesline Concept Exploration Activity 2 ART PASTE ON A DOUBLE NUMBER LINE Instructional Routine MLR8 Discussion Supports In the previous lesson students were given blank double number line diagrams and were only responsible for labeling them to match the situation In this activity students draw their own double number line diagram from scratch and identify which elements are important to create a useful double number line diagram LAUNCH You just used a double number line to solve some problems Now you ll create a double number line from scratch Once you know how to make double number lines you can use them for any situation with equivalent ratios Arrange students in groups of 2 Ensure each student has access to a ruler Have students check with a partner and come to an agreement about how to draw the diagrams before moving on to question 3 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 79

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G6M2 LESSON 7 ACTIVITY 2 TASK 1 ZEARN MATH TEACHER LESSON MATERIALS 3 A recipe for art paste says For every 2 pints of water mix in 8 cups of flour Answer the questions about this recipe below YOUR NOTES 1 Follow the instructions to draw a double number line diagram representing the recipe for art paste a Use a ruler to draw two parallel lines b Label the first line pints of water Label the second line cups of flour c Draw at least 6 equally spaced tick marks that line up on both lines d Along the water line label the tick marks with the amount of water in 0 1 2 3 4 and 5 batches of art paste e Along the flour line label the tick marks with the amount of flour in 0 1 2 3 4 and 5 batches of art paste 2 Compare your double number line diagram with your partner s Discuss your thinking If needed revise your diagram 3 Next use your double number line to answer these questions a How much flour should be used with 10 pints of water b How much water should be used with 24 cups of flour c How much flour per pint of water does this recipe use STUDENT RESPONSE 1 The correctly drawn and labeled number line should look like the one below The places to find answers to the following questions are circled 0 2 4 6 8 10 12 0 8 16 24 32 40 48 Pints of water Cups of flour 2 No answer necessary 80 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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G6M2 LESSON 7 3 ZEARN MATH TEACHER LESSON MATERIALS a Use 40 cups of flour for 10 pints of water b Use 6 pints of water for 24 cups of flour c 4 cups of flour per pint of water YOUR NOTES SUPPORT FOR STUDENTS WITH DISABILITIES Receptive Expressive Language Peer Tutors Pair students with their previously identified peer tutors to aid in comprehension and expression of understanding SUPPORT FOR ENGLISH LANGUAGE LEARNERS Representing MLR 8 Discussion Supports Briefly review the meaning of the following terms parallel equal increments and line up Use visuals to show what these terms are in the context of the problem Design Principle s Support sense making Digital Lesson The ratio of apples to oranges needed to make a smoothie is 12 apples to 6 oranges Explain how you can determine how many apples are needed per orange to make a smoothie that tastes the same Consider using a double number line to support your thinking STUDENT RESPONSE Answers vary Example response I can use a double number line and divide both number lines into equal segments to find equivalent ratios Per means for every so if I can show how many apples are needed for every 1 orange I can find how many apples are needed per orange 0 2 4 6 8 10 12 0 1 2 3 4 5 6 Apples Oranges 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 81

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G6M2 LESSON 7 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Wrap Up LESSON SYNTHESIS Create a double number line with the help of the class Start by asking What are some important things to pay attention to when you create a double number line Then What situation should we represent It is fine to choose a situation that students have already encountered in this lesson or an earlier lesson As you are creating the double number line together write down anything mentioned that it is important to pay attention to For example The two lines you draw should be parallel to each other One practice is to use both edges of a ruler to create two parallel lines But double number lines are tools for reasoning so they don t have to be perfect Each line should be labeled with what it represents Include units of measure Tick marks should be evenly spaced and the two sets of tick marks should be lined up vertically in pairs One strategy might be to intentionally do something wrong and ask students how you should fix it For example draw tick marks that are very obviously not evenly spaced or neglect to include units of measure in your labels TERMINOLOGY Per The word per means for each For example if the price is 5 per ticket that means you will pay 5 for each ticket Buying 4 tickets would cost 20 because 4 5 20 EXIT TICKET Each of these cats has 2 ears 4 paws and 1 tail 1 Draw a double number line diagram that represents a ratio in the situation 82 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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G6M2 LESSON 8 ZEARN MATH TEACHER LESSON MATERIALS GRADE 6 MISSION 2 LESSON 8 How Much for One YOUR NOTES Calculate equivalent ratios between prices and quantities and present the solution method using words and other representations LEARNING GOALS Calculate unit price and express it using the word per orally and in writing Understand the phrase at this rate indicates that equivalent ratios are involved LEARNING GOALS STUDENT FACING Let s use ratios to describe how much things cost I can choose and create diagrams to help me reason about prices LEARNING TARGETS STUDENT FACING I can explain what the phrase at this rate means using prices as an example If I know the price of multiple things I can find the price per thing Rulers REQUIRED MATERIALS 84 Tools for creating a visual display Any way for students to create work that can be easily displayed to the class Examples chart paper and markers whiteboard space and markers shared online drawing tool access to a document camera 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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This lesson introduces students to the idea of unit price Students use the word per to refer to the cost of one apple one pound one bottle one ounce etc as in 6 per pound or 1 50 per avocado The phrase at this rate is used to indicate that the ratios of price to quantity are equivalent For example Pizza costs 1 25 per slice At this rate how much for 6 slices They find unit prices in different situations and notice that unit prices are useful in computing prices for other amounts G6M2 LESSON 8 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Students choose whether to draw double number lines or other representations to support their reasoning They continue to use precision in stating the units that go with the numbers in a ratio in both verbal statements and diagrams Note that students are not expected to use or understand the term unit rate in this lesson Warm Up NUMBER TALK REMAINDERS IN DIVISION Instructional Routine Number Talk MLR8 Discussion Supports This number talk encourages students to think about the numbers in a computation problem and rely on what they know about structure patterns and division with remainders to mentally solve a problem Only one problem is presented to allow students to share a variety of strategies for division Notice how students handle a remainder in a problem which may depend on their prior experiences with division Students may write it as r_ and struggle with either fraction or decimal notation In the next lesson when students begin finding unit price they will need to be able to interpret the remainder in either decimal or fraction form LAUNCH Display the problem for all to see Give students quiet think time and ask them to give a signal when they have an answer and a strategy Follow with a whole class discussion 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 85

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G6M2 LESSON 8 WARM UP TASK ZEARN MATH TEACHER LESSON MATERIALS 1 Find the quotient mentally YOUR NOTES 246 12 STUDENT RESPONSE 246 12 20 5 or 246 12 20 1 2 DISCUSSION GUIDANCE Invite students to share their strategies Record and display student explanations for all to see Ask students to explain if or how the dividend or divisor impacted their choice of strategy and how they decided to write their remainder To involve more students in the conversation consider asking Who can restate s reasoning in a different way Did anyone solve the problem the same way but would explain it differently Did anyone solve the problem in a different way Does anyone want to add on to s strategy Do you agree or disagree Why At the end of discussion if time permits ask a few students to share a story problem or context that 246 12 20 5 would represent ANTICIPATED MISCONCEPTIONS Students may struggle to interpret the remainder as a decimal or fraction and may instead write r 6 86 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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SUPPORT FOR STUDENTS WITH DISABILITIES Memory Processing Time Provide sticky notes or mini whiteboards to aid students with working memory challenges Support for English Language Learners G6M2 LESSON 8 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Speaking MLR 8 Discussion Supports Display sentence frames to support students when they explain their strategy For example I noticed that or First I because When students share their answers with a partner prompt them to rehearse what they will say when they share with the whole class Rehearsing provides students with additional opportunities to clarify their thinking and to consider how they will communicate their reasoning Design Principle s Optimize output for explanation Concept Exploration Activity 1 GROCERY SHOPPING Instructional Routines Anticipate Monitor Select Sequence Connect Think Pair Share MLR5 Co Craft Questions and Problems This activity continues the work on ratios involving one unit of something Students determine the prices of grocery items and learn to use the term unit price to describe cost per unit To determine unit prices students may Divide the cost by the number of items Use discrete diagrams Use a double number line As students work monitor for students using different methods If students choose to draw a double number line diagram remind them to label each number line and to circle the ratio where they find the answer 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 87

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G6M2 LESSON 8 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES LAUNCH Frame the task in shopping terms Say that when most of us go shopping we often see prices for multiple items or units e g 2 bottles for 3 or 1 99 for three pounds etc But sometimes we want to know how much it costs to buy just one item or one unit of something Tell students they will explore the use of per in the context of shopping Ask students to solve the problems involving price for one using any method and to be ready to explain their reasoning Provide access to rulers in case students choose to draw double number lines Give students quiet think time Pause after the first question and if any students say the answer is 2 point out that is dollars per avocado rather than avocados per dollar Allow students work time and time to discuss with a partner ACTIVITY 1 TASK 1 2 Answer each question and explain or show your reasoning If you get stuck consider drawing a double number line diagram 1 Eight avocados cost 4 a How much do 16 avocados cost b How much do 20 avocados cost c How much do 9 avocados cost 2 Twelve large bottles of water cost 9 a How many bottles can you buy for 3 b What is the cost per bottle of water c How much would 7 bottles of water cost 3 A 10 pound sack of flour costs 8 a How much does 40 pounds of flour cost b What is the cost per pound of flour 88 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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G6M2 LESSON 8 ZEARN MATH TEACHER LESSON MATERIALS STUDENT RESPONSE 1 0 4 8 9 10 12 16 20 Number of avocados YOUR NOTES Cost in dollars 0 2 4 5 6 8 10 a 16 avocados cost 8 because 4 2 8 and 8 2 16 b 20 avocados cost 10 c 9 avocados cost 4 50 2 0 1 2 4 1 5 3 7 8 12 6 9 Number of bottles Cost in dollars 0 a You can buy 4 bottles for 3 because 9 1 3 3 and 12 1 3 4 b The cost per bottle is 0 75 because 9 12 0 75 c The cost for 7 bottles is 5 25 because 0 75 7 5 25 3 a 40 pounds costs 32 because 10 4 40 and 8 4 32 b The cost per pound is 0 80 because 8 10 0 8 DISCUSSION GUIDANCE Select students who used unique methods to share their reasoning as listed in the narrative If no one used double number lines represent one of the statements with a double number line diagram and display it for all to see Although double number lines are not required in the task their use in the context of problem situations helps students see their merits and illustrates how they might be used in other problems especially as students transition from unit prices to constant speed and other contexts Draw connections between the double number line strategy and the dividing by the numbers of items strategy 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 89

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G6M2 LESSON 8 ZEARN MATH TEACHER LESSON MATERIALS Tell students that each cost per one unit being sold avocado pound or bottle is an example of a unit price Ask them to name as many kinds of unit prices they can and to think of a situation where they might be used starting with the list from the task Cost per avocado YOUR NOTES Cost per pound Cost per bottle Other possibilities include cost per liter cost per ounce cost per jelly bean and so on ANTICIPATED MISCONCEPTIONS Some students may have difficulty with the answers not being integers Either fractions or decimals are acceptable Fractions provide the most direct route but decimals are common for working with dollars and cents Also students may use the larger numbers as the dividend simply because they are larger Encourage students to check the reasonableness of their answers SUPPORT FOR ENGLISH LANGUAGE LEARNERS Writing and Conversing Math Language Routine 5 Co Craft Questions This is the first time Math Language Routine 5 is suggested as a support in this course In this routine students are given a context or situation often in the form of a problem stem with or without numerical values Students develop mathematical questions that can be asked about the situation A typical prompt is What mathematical questions could you ask about this situation The purpose of this routine is to allow students to make sense of a context before feeling pressure to produce answers and to develop students awareness of the language used in mathematics problems Design Principle s Maximize meta awareness How It Happens 1 After students complete the first question bring the group together and present only the stem Twelve large bottles of water cost 9 Do not allow students to see the follow up questions for this situation Ask students What mathematical questions could you ask about this situation 2 Give students 1 minute of individual time to jot some notes and then 3 minutes to share ideas with a partner As pairs discuss support students in using conversation skills to generate and refine their questions collaboratively by seeking clarity referring to students written notes and revoicing oral responses as necessary Listen for students use of per as they talk 90 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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3 Ask each pair of students to contribute one written question to a poster the whiteboard or digital projection Call on 2 3 pairs of students to present their question to the small group and invite the group to make comparisons among the questions shared and their own questions Listen for questions intended to ask about the unit price for a single water bottle and listen for their use of per Revoice student ideas with an emphasis on the use of per wherever it serves to clarify a question 4 Reveal the follow up questions for this situation and give students a couple of minutes to compare these three questions to their own and those of their classmates Identify similarities and differences How many bottles can you buy for 3 What is the cost per bottle of water How much would 7 bottles of water cost 5 Invite students to choose one question to answer from the small group or from the curriculum and then have students move on to the following problems G6M2 LESSON 8 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES SUPPORT FOR STUDENTS WITH DISABILITIES Fine Motor Skills Peer Tutors Pair students with their previously identified peer tutors and allow students who struggle with fine motor skills to dictate how to draw double number line diagrams as needed Concept Exploration Activity 2 MORE SHOPPING Instructional Routines Group Presentations MLR2 Collect and Display In this task students practice finding unit prices using different reasoning strategies and articulating their reasoning They also learn about the term at this rate As students work observe their work and then assign one problem for each group to own and present to the group The problems can each be assigned to more than one group Have them work together to create a visual display of their problem and its solution LAUNCH Arrange students in groups of 3 4 Provide tools for creating a visual display and access to rulers Explain that they will work together to solve some shopping problems run their work by 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 91

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G6M2 LESSON 8 ZEARN MATH TEACHER LESSON MATERIALS you and prepare to present an assigned problem to the group Tell students that they can use double number lines if they wish Display the problem and read it aloud Pizza costs 1 25 per slice At this rate how much will 6 slices cost YOUR NOTES Ask students what they think at this rate means in the question Ensure they understand that at this rate means we know that equivalent ratios are involved The ratio of cost to number of slices is 1 25 to 1 That is pizza costs 1 25 per slice The ratio of cost to number of slices is something to 6 That is pizza costs something for 6 slices The something is the thing we are trying to figure out and at this rate tells us that the two ratios in this situation are equivalent Another way to understand at this rate in this context is at this price per unit and that the price per unit is the same no matter how many items or units are purchased Discuss any expectations for the group presentation For example each group member might be assigned a specific role for the presentation ACTIVITY 2 TASK 1 3 Answer the following questions about unit rates 1 Four bags of chips cost 6 a What is the cost per bag b At this rate how much will 7 bags of chips cost 2 At a used book sale 5 books cost 15 a What is the cost per book b At this rate how many books can you buy for 21 3 Neon bracelets cost 1 for 4 a What is the cost per bracelet 92 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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b At this rate how much will 11 neon bracelets cost Pause here so you teacher can review your work G6M2 LESSON 8 ZEARN MATH TEACHER LESSON MATERIALS 4 Your teacher will assign you one of the problems Create a visual display that shows your solution to the problem Be prepared to share your solution with the group YOUR NOTES STUDENT RESPONSE 1 a The cost per bag is 1 50 b Seven bags cost 10 50 2 a The cost per book is 3 b You can buy 7 books for 21 3 a The cost per bracelet is 25 cents b Eleven bracelets cost 2 75 DISCUSSION GUIDANCE Invite each group to present its assigned problem After each group presents highlight the group s strategy accurate uses of the terms at this rate and per and the ways in which a double number line might have been used when working with unit price ANTICIPATED MISCONCEPTIONS The first and third questions involve using decimals to represent cents If the decimal point is forgotten remind students that the cost of the bracelet is less than one dollar and the cost of the chips is in between one and two dollars Watch for students working in cents instead of dollars for the bracelets They may come up with an answer of 275 cents For these students writing 25 cents as 0 25 should help or consider reminding them of the avocados from a previous activity which had a unit price of 0 50 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 93

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G6M2 LESSON 8 SUPPORT FOR ENGLISH LANGUAGE LEARNERS ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Representing MLR 2 Collect and Display Capture the language students use to describe at this rate and per and display this language during the synthesis Design Principle s Support sense making Digital Lesson The cost of 9 boxes of cereal is 36 What is the unit price Create a visual representation that shows your solution STUDENT RESPONSE The unit price is 4 per box of cereal Here is a double number line that shows this solution double number lines may vary 0 1 2 3 4 5 6 7 8 9 0 4 8 12 16 20 24 28 32 36 Boxes of cereal Price dollars Wrap Up LESSON SYNTHESIS The main ideas to develop in this lesson are techniques for finding a unit price and the things that can be done once a unit price is known Discuss with students the methods they use to find a unit price The likely answers are Division if 2 bags of rice cost 3 then 1 bag costs 3 2 1 50 94 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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Double number line adding tick marks to a double number line signifying 1 bag can determine the cost per bag Briefly discuss with students the meaning of the word per for each Discuss with students the things they can do once they know a unit price Specifically they can directly compute any cost when the number of items is known by multiplying the unit price by the number of items You may want to point out to students that by multiplying they are finding part of an equivalent ratio For example the ratio 30 for 20 bags is equivalent to the ratio 3 for 2 bags G6M2 LESSON 8 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES TERMINOLOGY Unit price The unit price is the cost for one item or for one unit of measure For example if 10 feet of chain link fencing cost 150 then the unit price is 150 10 or 15 per foot EXIT TICKET Here is a double number line showing that it costs 3 to buy 2 bags of rice 0 3 0 2 Cost dollars Rice number of bags 1 At this rate how many bags of rice can you buy with 12 2 Find the cost per bag 3 How much do 20 bags of rice cost STUDENT RESPONSE 1 8 bags cost 12 0 1 5 3 6 9 12 15 0 1 2 4 6 8 10 Cost dollars Rice number of bags 2 The cost per bag is 1 50 3 20 bags cost 30 Multiply by the price for one bag or use an equivalent ratio 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 95

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G6M2 LESSON 9 ZEARN MATH TEACHER LESSON MATERIALS GRADE 6 MISSION 2 LESSON 9 Constant Speed YOUR NOTES Calculate the distance an object travels in 1 unit of time and express it using a phrase like meters per second orally and in writing LEARNING GOALS For an object moving at a constant speed use a double number line diagram to represent equivalent ratios between the distance traveled and elapsed time Justify orally and in writing which of two objects is moving faster by identifying that it travels more distance in the same amount of time or that it travels the same distance in less time LEARNING GOALS STUDENT FACING LEARNING TARGETS STUDENT FACING Let s use ratios to work with how fast things move I can create double number line diagrams to help me reason about constant speed I can decide which of two objects is moving faster based on information about the distances they travel and the amounts of time Stopwatches REQUIRED MATERIALS String Meter sticks Masking tape REQUIRED PREPARATION 96 Before class set up 4 paths with a 1 meter warm up zone and a 10 meter measuring zone See image in Activity 1 Launch 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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In the previous lesson students used the context of shopping to explore how equivalent ratios and ratios involving one can be used to find unknown amounts In this lesson they revisit these ideas in a new context constant speed and through concrete experiences Students measure the time it takes them to travel a predetermined distance first by moving slowly then quickly and use it to calculate and compare the speed they traveled in meters per second Here double number lines are used to represent the association between distance and time and to convey the idea of constant speed as a set of equivalent ratios e g 10 meters traveled in 20 seconds at a constant speed means that 0 5 meters is traveled in 1 second and 5 meters is traveled in 10 seconds Students come to understand that like price speed can be described using the terms per and at this rate G6M2 LESSON 9 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES The idea of a constant speed relating the quantities of distance and time is foundational for the later more abstract idea of a constant rate and is important in the development of students ability to reason abstractly about quantities Warm Up NUMBER TALK DIVIDING BY POWERS OF 10 Instructional Routines Number Talk MLR8 Discussion Supports This number talk encourages students to use the structure of base ten numbers to find the quotient of a base ten number and 10 The goal is to get students to see how understanding each quotient helps them find the next quotient Reasoning about this computation will be important in both this lesson and future lessons where students are working with the metric system and percentages LAUNCH Display one problem at a time Give students quiet think time and ask them to give a signal when they have an answer and a strategy Follow with a whole class discussion WARM UP TASK 1 Find the quotient mentally 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 97

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G6M2 LESSON 9 30 10 ZEARN MATH TEACHER LESSON MATERIALS 34 10 3 4 10 YOUR NOTES 34 100 STUDENT RESPONSE 30 10 3 Possible strategy 3 10 30 or students may say 10 goes into 30 three times 34 10 3 4 Possible strategy Students may say 10 goes into 34 three times with four tenths written as a fraction or decimal left over 3 4 10 0 34 Possible strategy Using the previous problem since 34 was divided by 10 students may divide their previous answer by 10 3 4 10 0 34 34 100 0 34 Possible strategy Students may notice both the dividend and divisor were multiplied by 10 to get this problem so the quotient is the same DISCUSSION GUIDANCE Ask students to share their strategies for each problem Record and display their explanations for all to see Emphasize student strategies based in place value to explain methods students may have learned about moving the decimal point left or right or crossing out zeros To involve more students in the conversation consider asking Who can restate s reasoning in a different way Did anyone solve the problem the same way but would explain it differently Did anyone solve the problem in a different way Does anyone want to add on to s strategy Do you agree or disagree Why SUPPORT FOR STUDENTS WITH DISABILITIES Memory Processing Time Provide sticky notes or mini whiteboards to aid students with working memory challenges 98 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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G6M2 LESSON 9 SUPPORT FOR ENGLISH LANGUAGE LEARNERS Speaking MLR 8 Discussion Supports Provide sentence frames to support students with explaining their strategies For example I noticed that or First I because When students share their answers with a partner prompt them to rehearse what they will say when they share with the full group Rehearsing provides opportunities to clarify their thinking ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Design Principle s Optimize output for explanation Concept Exploration Activity 1 MOVING 10 METERS MATERIALS STOPWATCHES STRING METER STICKS MASKING TAPE Instructional Routine MLR7 Compare and Connect This activity gives students first hand experience in relating ratios of time and distance to speed Students time one another as they move 10 meters at a constant speed first slowly and then quickly and then reason about the distance traveled in 1 second Double number lines play a key role in helping students see how time and distance relate to constant speed allowing us to compare how quickly two objects are moving in two ways We can look at how long it takes to move 10 meters a shorter time needed to move 10 meters means faster movement or at how far one travels in 1 second a longer distance in one second means faster movement Along the way students see that the language of per and at this rate which was previously used to talk about unit price is also relevant in the context of constant speed They begin to use meters per second to express measurements of speed As students work notice the different ways they use double number lines or other means to reason about distance traveled in one second LAUNCH Before class set up 4 paths with a 1 meter warm up zone and a 10 meter measuring zone Start 1m Finish 10 m Warm up Mark 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 99

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G6M2 LESSON 9 ZEARN MATH TEACHER LESSON MATERIALS Arrange students into 4 groups with one for each path Provide a stopwatch Explain that they will gather some data on the time it takes to move 10 meters Select a student to be your partner and demonstrate the activity for the class Share that the experiment involves timing how long it takes to move the distance from the start line to the finish line YOUR NOTES Explain that each person in the pair will play two roles the mover and the timer Each mover will go twice once slowly and once quickly starting at the warm up mark each time The initial 1 meter long stretch is there so the mover can accelerate to a constant speed before the timing begins Demonstrate the timing protocol as shown in the task statement Stress the importance of the mover moving at a constant speed while being timed The warmup segment is intended to help them reach a steady speed To encourage students to move slowly consider asking them to move as if they are balancing something on their head or carrying a full cup of water trying not to spill it Alternatively set up one path and ask for two student volunteers to demonstrate while the rest of the class watches ACTIVITY 1 TASK 1 2 Your teacher will set up a straight path with a 1 meter warm up zone and a 10 meter measuring zone Follow the following instructions to collect the data Start 1m Finish 10 m Warm up Mark 1 a The person with the stopwatch the timer stands at the finish line The person being timed the mover stands at the warm up line b On the first round the mover starts moving at a slow steady speed along the path When the mover reaches the start line they say Start and the timer starts the stopwatch 100 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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c The mover keeps moving steadily along the path When they reach the finish line the timer stops the stopwatch and records the time rounded to the nearest second in the table G6M2 LESSON 9 ZEARN MATH TEACHER LESSON MATERIALS d On the second round the mover follows the same instructions but this time moving at a quick steady speed The timer records the time the same way YOUR NOTES e Repeat these steps until each person in the group has gone twice once at a slow steady speed and once at a quick steady speed Your slow moving time seconds Your fast moving time seconds 2 After you finish collecting the data use the double number line diagrams to answer the questions Use the times your partner collected while you were moving Moving slowly 0 10 Distance traveled meters Elapsed time seconds 0 Moving quickly 0 10 Distance traveled meters Elapsed time seconds 0 a Estimate the distance in meters you traveled in 1 second when moving slowly b Estimate the distance in meters you traveled in 1 second when moving quickly c Trade diagrams with someone who is not your partner How is the diagram representing someone moving slowly different from the diagram representing someone moving quickly STUDENT RESPONSE 1 Data will vary 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 101

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G6M2 LESSON 9 2 ZEARN MATH TEACHER LESSON MATERIALS a Estimates will vary but should align to student s data b Estimates will vary but should align to student s data YOUR NOTES c Answers vary Sample response The diagram that represents moving quickly will have a higher number of meters per second or a lower number of seconds at 10 meters DISCUSSION GUIDANCE Select students who used different methods to reason about distance traveled in 1 second first question to share During the discussion demonstrate the use of the phrase meters per second or emphasize it if it comes up naturally in students explanations Discuss how we can use double number lines to distinguish faster movement from slower movement second question If it hasn t already surfaced in discussion help students see we can compare the time it takes to travel the same distance in this case 10 meters and the distance traveled in the same amount of time say 1 second Explain to students that when we represent time and distance on a double number line we are saying the object is traveling at a constant speed or a constant rate This means that the ratios of meters traveled to seconds elapsed or miles traveled to hours elapsed are equivalent the entire time the object is traveling The object does not move faster or slower at any time The equal intervals on the double number line show this steady rate ANTICIPATED MISCONCEPTIONS Students may have difficulty estimating the distance traveled in 1 second Encourage them to mark the double number line to help For example marking 5 meters halfway between 0 and 10 and determining the elapsed time as half the recorded total may cue them to use division SUPPORT FOR STUDENTS WITH DISABILITIES Strengths based Approach This activity leverages many natural strengths of students with ADHD LD and other concrete learners in terms of its multi sensory and hands on nature This may be an opportunity for the teacher to highlight this strength in class and allow an individual with disability to lead peer interactions discussions increasing buy in 102 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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SUPPORT FOR ENGLISH LANGUAGE LEARNERS Representing MLR 7 Compare and Connect As students share different approaches for reasoning about distance traveled in 1 second ask students to identify what is the same and what is different about the approaches Help students connect approaches by asking Where do you see the measurement of speed meters per second in each approach This helps students connect the concept of rate and a visual representation of that rate G6M2 LESSON 9 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Design Principle s Maximize meta awareness Concept Exploration Activity 2 MOVING FOR 10 SECONDS Instructional Routines Anticipate Monitor Select Sequence Connect Think Pair Share MLR5 Co Craft Questions and Problems In the previous activity students traveled the same distance in differing amounts of time In this activity students analyze a situation where two people travel for the same amount of time each at a constant speed but go different distances The use of double number lines is suggested but not required Monitor students work and notice different ways students compare speeds For the first question students may use meters per second or compare the distance traveled in the same number of seconds For the last question they may draw a double number line for each of the scenarios being compared or calculate each speed in meters per second LAUNCH Keep students with the same partners Give students quiet think time and then time to share their responses with their partners Tell students that in the last activity everyone traveled the same distance but in different times Now they will analyze a situation in which two people travel for the same amount of time but cover different distances 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 103

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G6M2 LESSON 9 ACTIVITY 2 TASK 1 ZEARN MATH TEACHER LESSON MATERIALS 3 Lin and Diego both ran for 10 seconds each at their own constant speed Lin ran 40 meters and Diego ran 55 meters YOUR NOTES 1 Who was moving faster Explain your reasoning 2 How far did each person move in 1 second If you get stuck consider drawing double number line diagrams to represent the situations 3 Use your data from the previous activity to find how far you could travel in 10 seconds at your quicker speed 4 Han ran 100 meters in 20 seconds at a constant speed Is this speed faster slower or the same as Lin s Diego s Yours STUDENT RESPONSE 1 Diego ran faster covering a greater distance in the same amount of time 2 Lin ran 4 meters per second and Diego ran 5 5 meters per second 3 Answers vary but 10 times the distance traveled in 1 second 4 Han ran 5 meters per second which is faster than Lin s speed but slower than Diego s DISCUSSION GUIDANCE Select students who reasoned differently to share Some students will know that Diego ran faster simply because he ran further but this reasoning is not always correct Han runs further but is slower than Diego because he had more time Be sure to attend to both distance and time when making the comparison Help students draw connections between the different ways they represented and reasoned about the problem During the discussion keep as much emphasis as possible on the concept of meters per second ANTICIPATED MISCONCEPTIONS Instead of dividing 40 by 10 some students may instead calculate 10 40 Ask them to articulate what the resulting number means 0 25 seconds to travel 1 meter and contrast that meaning with what the problem is asking how many meters in one second Another approach 104 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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would be to encourage them to draw a double number line and think about how they can figure out what value for distance corresponds to 1 second on the line for elapsed time 0 G6M2 LESSON 9 ZEARN MATH TEACHER LESSON MATERIALS 40 Distance traveled meters YOUR NOTES Elapsed time seconds 0 1 2 3 4 5 6 7 8 9 10 SUPPORT FOR ENGLISH LANGUAGE LEARNERS Lighter Support MLR 5 Co Craft Questions and Problems Present students with the situation Lin and Diego both ran for 10 seconds each at their own constant speed Lin ran 40 meters and Diego ran 55 meters without revealing the questions that follow and ask students to write possible mathematical questions about the situation Then invite students to share their questions with a partner before sharing with the whole class This helps students produce the language of mathematical questions and talk about the relationships between the two speeds in this task Ask students to use the phrase at a constant speed so that they must reason about its mathematical meaning Design Principle s Maximize meta awareness Digital Lesson Evan ran 20 meters in 5 8 seconds Yelena ran 20 meters in 5 4 seconds Who ran faster Explain how you know STUDENT RESPONSE Yelena ran faster because she went the same distance in less time 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 105

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G6M2 LESSON 9 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Wrap Up LESSON SYNTHESIS The work in this lesson parallels the work in the previous lesson Knowing a speed in meters per second gives the same kind of information as knowing a unit price in dollars per item The overall objective is for students to see consistencies in the underlying mathematical structure of these contexts Time permitting discuss the ways in which this work was similar to the work on unit prices asking students to state in their own words which actions and methods felt consistent TERMINOLOGY Meters per second Meters per second is a unit for measuring speed It tells how many meters an object goes in one second For example a person walking 3 meters per second is going faster than another person walking 2 meters per second EXIT TICKET Two trains are traveling at constant speeds on different tracks Train A 0 12 5 0 1 100 Distance traveled meters Elapsed time seconds Train B 0 100 0 4 Distance traveled meters Elapsed time seconds Which train is traveling faster Explain your reasoning 106 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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G6M2 LESSON 10 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES GRADE 6 MISSION 2 LESSON 10 Comparing Situations by Examining Ratios Choose and create diagrams to help compare two situations and explain whether they happen at the same rate LEARNING GOALS Justify that two situations do not happen at the same rate by finding a ratio to describe each situation where the two ratios share one value but not the other i e a b and a c or x z and y z Recognize that a question asking whether two situations happen at the same rate is asking whether the ratios are equivalent LEARNING GOALS STUDENT FACING Let s use ratios to compare situations I can choose and create diagrams to help me compare two situations LEARNING TARGETS STUDENT FACING 108 I can explain what it means when two situations happen at the same rate I know some examples of situations where things can happen at the same rate 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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In previous lessons students learned that if two situations involve equivalent ratios we can say that the situations are described by the same rate In this lesson students compare ratios to see if two situations in familiar contexts involve the same rate The contexts and questions are G6M2 LESSON 10 ZEARN MATH TEACHER LESSON MATERIALS Two people run different distances in the same amount of time Do they run at the same speed YOUR NOTES Two people pay different amounts for different numbers of concert tickets Do they pay the same cost per ticket Two recipes for a drink are given Do they taste the same In each case the numbers are purposely chosen so that reasoning directly with equivalent ratios is a more appealing method than calculating how many per one and then scaling The reason for this is to reinforce the concept that equivalent ratios describe the same rate before formally introducing the notion of unit rate and methods for calculating it However students can use any method Regardless of their chosen approach students need to be able to explain their reasoning in the context of the problem Warm Up TREADMILLS Instructional Routines Anticipate Monitor Select Sequence Connect Think Pair Share In this activity students encounter two distance time ratios in which one quantity distance has the same value and the other quantity time has different values Students interpret what the ratios mean in context i e in terms of the speeds of two runners There are several ways to reason about this with or without double number lines Students may argue that since both runners ran for the same distance but Mai ran a shorter amount of time she ran at a greater speed Students may also say that if Mai could run 3 miles in 24 minutes at that speed she would run more than 3 miles in 30 minutes Since Jada only ran 3 miles in 30 minutes Mai ran faster The double number line that corresponds to these arguments may be as shown below 0 3 0 24 distance run miles elapsed time minutes 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 109

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G6M2 LESSON 10 more than 3 ZEARN MATH TEACHER LESSON MATERIALS 0 3 0 24 distance run miles elapsed time minutes YOUR NOTES 30 As students work monitor for the different ways they reason about the situation and identify a few students to share different approaches later WARM UP LAUNCH Explain to students that a treadmill is an exercise machine for walking or running Explain that while the runner does not actually go anywhere on a treadmill a computer inside the treadmill keeps track of the distance traveled as if she were running outside Show the video of Mr Sawicki running on a treadmill at a constant speed for a few seconds Students work on all parts of the activity silently and individually then share their explanation with a partner WARM UP TASK 1 Mai and Jada each ran on a treadmill The treadmill display shows the distance in miles each person ran and the amount of time it took them in minutes and seconds Here is Mai s treadmill display Here is Jada s treadmill display 1 What is the same about their workouts What is different about their workouts 2 If each person ran at a constant speed the entire time who was running faster Explain your reasoning 110 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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G6M2 LESSON 10 STUDENT RESPONSE ZEARN MATH TEACHER LESSON MATERIALS 1 They both ran the same distance 3 miles Also the incline level and pulse are the same The amount of time it took them is different 24 minutes versus 30 minutes Also the pace and calories differ YOUR NOTES 2 Mai ran faster She ran 3 miles in less time than it took Jada DISCUSSION GUIDANCE Select a few students to share their reasoning about the speeds of the runners If no students use a double number line to make an argument illustrate one of their explanations using a double number line Remind students that even though a double number line is not always necessary it can be a helpful tool to support arguments about ratios in different contexts ANTICIPATED MISCONCEPTIONS Because a person running on a treadmill does not actually go anywhere it may be challenging to think about a distance covered If this comes up suggest that students think about running the given distances outside on a straight flat road at a constant speed Concept Exploration Activity 1 CONCERT TICKETS Instructional Routines MLR8 Discussion Supports MLR6 Three Reads Think Pair Share Anticipate Monitor Select Sequence Connect Previously students worked with ratios in which one quantity distance run had the same value and the other time elapsed did not In the context of running they concluded that the runners did not run at the same rate Here students work with two ratios in which neither quantity number of tickets bought and money paid has the same value and decide if the two people in the situation bought tickets at the same rate Students may approach the task in several ways They may use a double number line to generate ratios that are equivalent to 47 3 representing the prices Diego would pay for different number of tickets Once the price for 9 tickets is determined by scaling up the first ratio they can compare it to the amount that Andre paid for the same number of tickets They may remember without drawing number lines that multiplying two values of a ratio by the same number produces a ratio that is equivalent They may notice that multiplying 3 tickets 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 111

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G6M2 LESSON 10 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES by 3 results in 9 tickets making the values for one quantity in the ratios match They can then compare the two prices for 9 tickets Alternatively they may divide Andre s 9 tickets and 141 payment by 3 to get his price for 3 tickets Another approach is to calculate the price of 1 ticket as students did in a previous lesson To make this approach less attractive here the numbers have been deliberately chosen so the price of a single ticket 15 23 is not a whole number As students work monitor for those who reason correctly using the three approaches described above LAUNCH Ask students what comes to mind when they hear the term at the same rate Ask if they can think of any examples of situations that happen at the same rate An example is two things traveling at the same speed Any distance traveled will have the same associated time for things traveling at the same rate Remind students that when situations happen at the same rate they can be described by ratios that are equivalent Give students quiet think time to complete the activity and then time to share their explanation with a partner ACTIVITY 1 TASK 1 2 Diego paid 47 for 3 tickets to a concert Andre paid 141 for 9 tickets to a concert Did they pay at the same rate Explain your reasoning STUDENT RESPONSE Yes Andre paid at the same rate Sample explanation Since 9 is 3 3 multiply 47 by 3 47 3 141 Diego would have paid 141 for 9 tickets if he paid at the same rate he did for 3 tickets Since this is what Andre paid for 9 tickets they paid at the same rate 3 see if amount paid matches the other ratio 0 47 94 141 188 0 3 6 9 12 amount paid in dollars number of tickets 1 start here 112 2 go to 9 tickets 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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G6M2 LESSON 10 DISCUSSION GUIDANCE ZEARN MATH TEACHER LESSON MATERIALS The main strategy to highlight here is one that could tell us what Diego would pay for 9 tickets if he paid at the same rate as he did for 3 tickets For 9 tickets he would have paid 141 which is what Andre paid for 9 tickets and which tells us that they paid at the same rate YOUR NOTES Select 2 3 students to present their work to the group in the order of their methods Correct use of a double number line to show that the given ratios are equivalent Correct use of multiplication or division without using a double number line Optional Correct use of unit price i e by finding out 141 9 and 47 3 Though it s a less efficient approach here the outcome also shows that the two people paid at the same rate Recap that using equivalent ratios to make one of the corresponding quantities the same can help us compare the other quantity and tell whether the situations involve the same rate SUPPORT FOR STUDENTS WITH DISABILITIES Conceptual Processing Eliminate Barriers Allow students to use calculators to ensure inclusive participation in the activity SUPPORT FOR ENGLISH LANGUAGE LEARNERS Reading MLR 6 Three Reads This is the first time Math Language Routine 6 is suggested as a support in this course In this routine students are supported in reading a mathematical text situation or word problem three times each with a particular focus During the first read students focus on comprehending the situation during the second read students identify quantities during the third read students brainstorm possible strategies to answer the question The question to be answered does not become a focus until the third read so that students can make sense of the whole context before rushing to a solution The purpose of this routine is to support students reading comprehension as they make sense of mathematical situations and information through conversation 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 113

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G6M2 LESSON 10 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES 114 Design Principle s Support sense making How It Happens 1 In the first read students read the problem with the goal of comprehending the situation Invite a student to read the problem aloud while everyone else reads with them and then ask What is this situation about Allow time to discuss with a partner and then share with the group A clear response would be Diego and Andre both bought tickets to a concert 2 In the second read students analyze the mathematical structure of the story by naming quantities Invite students to read the problem aloud with their partner or select a different student to read to the group and then prompt students by asking What can be counted or measured in this situation For now we don t need to focus on how many or how much of anything but what can we count in this situation Give students quiet think time followed by time to share with their partner Quantities may include the number of tickets Diego bought the number of tickets Andre bought the amount Diego paid the amount Andre paid the rate of dollars per ticket that Diego paid the rate of dollars per ticket that Andre paid Call attention to the fact that whether we are talking about Andre or Diego the two important quantities in this situation are number of tickets and amount paid in dollars 3 In the third read students brainstorm possible strategies to answer the question Did they pay at the same rate Invite students to read the problem aloud with their partner or select a different student to read to the group Instruct students to think of ways to approach the question without actually solving the problem Consider using these questions to prompt students How would you approach this question What strategy or method would you try first and Can you think of a different way to solve it Give students quiet think time followed by time to discuss with their partner Provide these sentence frames as partners discuss To compare the rates I would use a double number line by One way to approach the question would be to Sample responses include I would figure out how much each person paid for one ticket I know that if I multiply 3 tickets by 3 I get 9 so I would see what happens when I multiply 47 by 3 I would draw a diagram to figure out how much 3 groups of three tickets would cost at Diego s rate and I would use a double number line to scale up the rate for Diego s tickets or scale down the rate for Andre s tickets to see if they are the same rate This will help students concentrate on making sense of the situation before rushing to a solution or method 4 As partners are discussing their strategies select 1 2 students to share their ideas with the group As students are presenting ideas to the group create a display that summarizes ideas about the question Listen for quantities that were mentioned during the second read and take note of strategies for relating number of tickets to cost amount paid 5 Post the summary where all students can use it as a reference and suggest that students consider using the Three Reads routine for the next activity as well 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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Speaking Writing MLR 8 Discussion Supports Provide a sentence frame to support students language when justifying their reasoning For example Diego and Andre did did not pay at the same rate because _____ The helps students place extra attention on language used to engage in mathematical communication without over scaffolding their reasoning It also emphasizes the importance of justification G6M2 LESSON 10 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Design Principle Maximize meta awareness Concept Exploration Activity 2 FRUIT PUNCH Instructional Routine MLR1 Stronger and Clearer Each Time Think Pair Share Here students compare the tastes of two fruit punch mixtures which involves reasoning about whether the two situations involve equivalent ratios The problem is more challenging because no values of the quantities match or are multiples of one another Instead of finding an equivalent ratio for one recipe so that it matches the other students need to do so for both recipes To answer the question students can either make the values of the strawberry juice match and then compare the orange juice amounts or make the orange juice amounts match and compare the values for strawberry juice Again they may use a double number line multiplication and possibly finding how much of one quantity per 1 unit of the other quantity LAUNCH Introduce the task by saying that some people make fruit punch by mixing orange juice and strawberry juice Ask students to predict how the drink would taste if we mixed a huge amount of strawberry juice with just a little bit of orange juice it would not have a very orange y flavor or the other way around Explain that they will now compare the tastes of two fruit punch recipes Remind them that we previously learned that making larger or smaller batches of the same recipe does not change its taste Give students quiet think time to complete the activity and then time to share their explanation with a partner 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 115

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G6M2 LESSON 10 ACTIVITY 2 TASK 1 ZEARN MATH TEACHER LESSON MATERIALS 3 YOUR NOTES Lin and Noah each have their own recipe for making fruit punch Lin mixes 3 liters of orange juice with 4 liters of strawberry juice Noah mixes 4 liters of orange juice with 5 liters of strawberry juice How do the two mixtures compare in taste Explain your reasoning STUDENT RESPONSE Lin s mixture tastes a little more like strawberry juice and Noah s mixture tastes a little more like orange juice Double number line for Lin s 3 4 recipe 0 3 6 9 12 15 0 4 8 12 16 20 0 4 8 12 16 20 0 5 10 15 20 25 Orange juice L Strawberry juice L Double number line for Noah s 4 5 recipe Orange juice L Strawberry juice L With these double number line diagrams we can see that the mixtures do not taste the same For 12 liters of orange juice the first recipe has 16 liters of strawberry juice and the second recipe has 15 liters of strawberry juice so Noah s mixture has a stronger orange flavor We can also see that for 20 liters of strawberry juice the first recipe has less orange juice than the second recipe so Lin s mixture has a weaker orange flavor ACTIVITY 2 RECAP Display two double number line diagrams for all to see one that represents batches of the 3 4 recipe and another that represents batches of the 4 5 recipe Scale them up to show enough batches of each recipe to be able to make comparisons 116 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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Ask students to explain how they can tell that the 4 5 recipe tastes more orange y Elicit both explanations comparing the amount of strawberry juice for the same amount of orange juice and the other way around Ensure students can articulate why each way of comparing means that the second recipe is more orange y If any students calculated a unit rate for each recipe you might consider inviting them to share but help them be careful with their choice of words It is important to say for example In the first recipe there is 34 or 0 75 liter of orange juice per liter of strawberry juice but in the second recipe there is 45 or 0 8 liter of orange juice per liter of strawberry juice A student with this response would be comparing the number of cups of orange juice for every 1 liter of strawberry juice in each mixture Note if this approach comes up consider taking this opportunity to discuss fraction comparison methods that students should know from earlier grades In particular 34 1 14 is less than 45 1 15 because 15 is smaller than 14 G6M2 LESSON 10 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES ANTICIPATED MISCONCEPTIONS Some students may say that these two recipes would taste the same because they each use 1 more liter of strawberry juice than orange juice an additive comparison instead of a multiplicative comparison Remind them of when we made batches of drink mix and that mixtures have the same taste when mixed in equivalent ratios SUPPORT FOR STUDENTS WITH DISABILITIES Receptive Expressive Language Peer Tutors Pair students with their previously identified peer tutors to aid in comprehension and expression of understanding SUPPORT FOR ENGLISH LANGUAGE LEARNERS Writing Conversing MLR 1 Stronger and Clearer Each Time Use this routine to help students improve their writing by providing them with multiple opportunities to clarify their explanations through conversation Give students time to meet with 2 3 partners to share their response to the question How do the two mixtures compare in taste Students should first check to see if they agree with each other about how Lin and Noah s mixtures compare Provide listeners with prompts for feedback that will help their partner add detail to strengthen and clarify their ideas For example students can ask their partner How did you use double number lines to solve this problem or Can you say more about what each ratio means Next provide students with a few minutes to revise their initial draft based on feedback from their partner s This will help strengthen students understanding of how they determine whether the two situations involve equivalent ratios Design Principle s Support sense making Optimize output for explanation 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 117

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G6M2 LESSON 10 ZEARN MATH TEACHER LESSON MATERIALS Digital Lesson Tracy rides her bike 5 meters in 5 seconds Peter rides his scooter 10 meters in 10 seconds Both travel at a constant rate Peter says I traveled faster because I went a greater distance Do you agree with Peter Explain or show your reasoning Consider using a double number line to help YOUR NOTES I agree disagree with Peter because STUDENT RESPONSE Explanations vary Students may show a double number line to show Tracy s and Peter s rate per 1 second or Tracy s and Peter s rate for the same distance Sample response I disagree with Peter because he traveled at a rate of 1 meter in 1 second Tracy also traveled 1 meter in 1 second So they traveled at the same rate per 1 second Wrap Up This lesson is all about figuring out whether two situations happen at the same rate by comparing one quantity when the other quantity is the same In order to do that it s helpful to generate equivalent ratios LESSON SYNTHESIS Briefly review the strategies used in the three activities in this lesson How did we know that the people on the treadmill were not going the same speed They went the same distance in different amounts of time How did we know the people paid the same rate for the concert tickets We figured out how much one person would have paid for 9 tickets at the same rate he paid for 3 We compared that to what the other person paid for 9 tickets How did we know that the fruit punch recipes did not taste the same We made equivalent ratios so we could compare orange juice for the same amount of strawberry juice or compare strawberry juice for the same amount of orange juice How were all these problems alike We used equivalent ratios to make one part of the ratio the same and compared the other part 118 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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G6M2 LESSON 10 TERMINOLOGY ZEARN MATH TEACHER LESSON MATERIALS Same rate We use the words same rate to describe two situations that have equivalent ratios For example a sink is filling with water at a rate of 2 gallons per minute If a tub is also filling with water at a rate of 2 gallons per minute then the sink and the tub are filling at the same rate YOUR NOTES EXIT TICKET Andre ran 2 kilometers in 15 minutes and Jada ran 3 kilometers in 20 minutes Both ran at a constant speed Did they run at the same speed Explain your reasoning STUDENT RESPONSE They did not run at the same speed There are many ways to justify this response Here are some examples Andre would have run 6 kilometers in 45 minutes and Jada would have run 6 kilometers in 40 minutes Jada completes the 6 kilometers in less time so she runs at a faster speed than Andre Andre would have run 8 kilometers in 60 minutes and Jada would have run 9 kilometers in 60 minutes Jada travels further in the same amount of time so she runs at a faster speed than Andre These examples while they explain why Jada runs faster also explain why the two runners did not run at the same speed 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 119

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G6M2 LESSON 11 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES GRADE 6 MISSION 2 LESSON 11 Representing Ratios with Tables Comprehend the words row and column in written and spoken language as they are used to describe a table of equivalent ratios LEARNING GOALS Explain orally and in writing how to find a missing value in a table of equivalent ratios Interpret a table of equivalent ratios that represents different sized batches of a recipe LEARNING GOALS Let s use tables to represent equivalent ratios If I am looking at a table of values I know where the rows are and where the columns are LEARNING TARGETS STUDENT FACING When I see a table representing a set of equivalent ratios I can come up with numbers to make a new row When I see a table representing a set of equivalent ratios I can explain what the numbers mean 120 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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Over the course of this mission students learn to work with ratios using different representations They began by using discrete diagrams to represent ratios and to identify equivalent ratios Later they reasoned more efficiently about ratios using double number lines Here they encounter situations in which using a double number line poses challenges and for which a different representation would be helpful Students learn to organize a set of equivalent ratios in a table which is a more abstract but also a more flexible tool for solving problems G6M2 LESSON 11 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Although different representations are encouraged at different points in the mission allowing students to use any representation that accurately represents a situation and encouraging them to compare the efficiency of different methods will develop their ability to make strategic choices about representations Whatever choices they make they should be encouraged to explain how their method works in solving a problem Warm Up HOW IS IT GROWING Instructional Routine Think Pair Share This warm up encourages students to look for regularity in how the tiles in the image are growing Students may use each color to reason about the total while others may reason about the way the total tiles increase each time Emphasize both insights as students share their strategies LAUNCH Arrange students in groups of 2 Display the image for all to see and tell students that the collection of images of light gray and dark gray tiles is growing Ask how many total tiles will be in the 4th 5th and 10th image if it keeps growing in the same way Tell students to give a signal when they have an answer and strategy Give students quiet think time and then time to discuss their responses and reasoning with their partner WARM UP TASK 1 Look for a pattern in the figures 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 121

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G6M2 LESSON 11 1 How many total tiles will be in ZEARN MATH TEACHER LESSON MATERIALS a the 4th figure b the 5th figure YOUR NOTES c the 10th figure 2 How do you see it growing STUDENT RESPONSE 1 Answers vary Sample response 4th image 28 tiles 5th image 35 tiles 10th image 70 tiles 2 Answers vary Sample response I see light grey increasing by 3 each time and dark grey increasing by 4 DISCUSSION GUIDANCE Invite students to share their responses and reasoning Record and display the different ways of thinking for all to see If possible record the relevant reasoning on or near the images themselves After each explanation ask the class if they agree or disagree and to explain alternative ways of thinking referring back to what is happening in the images each time SUPPORT FOR STUDENTS WITH DISABILITIES Conceptual Processing Manipulatives Provide manipulatives i e snap cubes to aid students who benefit from hands on activities Concept Exploration Activity 1 A HUGE AMOUNT OF FRUIT PUNCH Instructional Routines MLR7 Compare and Connect Anticipate Monitor Select Sequence Connect Here students are asked to find missing values for significantly scaled up ratios The activity serves several purposes 122 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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To uncover a limitation of a double number line i e that it is not always practical to extend it to find significantly scaled up equivalent ratios G6M2 LESSON 11 ZEARN MATH TEACHER LESSON MATERIALS To reinforce the multiplicative reasoning needed to find equivalent ratios especially in cases when drawing diagrams or skip counting is inefficient and YOUR NOTES To introduce a table as a way to represent equivalent ratios To find equivalent ratios involving large values some students may simply try to squeeze numbers on the extreme right side of the paper ignoring the previously equal intervals Others may use multiplication or division and write expressions or equations to capture the given scenarios Notice students reasoning processes especially any struggles with the double number line e g the lines not being long enough requiring much marking and writing the numbers being too large etc as these can motivate a need for a more efficient strategy LAUNCH Give students time to work on the first two questions and then ask them to pause As a group discuss the two approaches students are likely to take counting multiples of 4 and 5 up to 36 and 45 and multiplicative reasoning asking What number times 4 equals 36 Also discuss how a double number line like the one below might be used to support reasoning 0 4 8 12 16 36 0 5 10 15 20 45 Orange juice liters Strawberry juice liters 9 Reiterate the multiplicative relationship between equivalent ratios before students move on ACTIVITY 1 TASK 1 2 Noah s recipe for one batch of fruit punch uses 4 liters of orange juice and 5 liters of strawberry juice 1 Use the double number line to show how many liters of each ingredient to use for different sized batches of fruit punch Orange juice liters Strawberry juice liters 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 123

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G6M2 LESSON 11 ZEARN MATH TEACHER LESSON MATERIALS 2 If someone mixes 36 liters of orange juice and 45 liters of strawberry juice how many batches would they make 3 If someone uses 400 liters of orange juice how much strawberry juice would they need YOUR NOTES 4 If someone uses 455 liters of strawberry juice how much orange juice would they need 5 Explain the trouble with using a double number line diagram to answer the last two questions STUDENT RESPONSE 0 4 8 12 16 36 0 5 10 15 20 45 Orange juice liters Strawberry juice liters 9 1 Amounts for the first four batch sizes are shown 2 9 batches 4 9 36 and 5 9 45 3 500 liters of strawberry juice 4 100 400 and 5 100 500 4 364 liters of orange juice 455 5 91 and 4 91 364 5 The numbers I needed to find were too big to fit on the number lines ACTIVITY 1 RECAP After students have a chance to share with a partner select a few to share their reasoning with the group for the last few questions Start with students who tried to extend the double number line if anyone did so Discuss any challenges of using the double number line and merits of alternative methods students might have come up with Explain that there is a more appropriate tool a table that can be used to represent equivalent ratios Display for all to see the double number line from the activity above and a table of equivalent ratios Explain that even though the table is oriented vertically and the double number line is oriented horizontally the two representations represent the same ratios Explain what we mean by row and column and demonstrate the use of these words Fill in 124 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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the table using the values from the orange strawberry ratios and along the way compare and contrast how the two representations work A few other key insights to convey G6M2 LESSON 11 ZEARN MATH TEACHER LESSON MATERIALS Just as it was important to label the double number line it is important to label the columns of the table to indicate what the values represent Each row of a table shows a pair of values from a collection of equivalent ratios Unlike a number line distances between values do not matter YOUR NOTES On each line of a double number line numbers are shown in order In each column of a table order is not important i e pairs of values can be placed in any order that is convenient When complete the display should look something like this Orange juice liters Strawberry juice liters 4 5 8 10 12 15 16 20 36 45 400 500 364 455 100 100 100 Orange juice liters Strawberry juice liters 0 4 8 12 16 0 5 10 15 20 36 400 500 45 9 100 ANTICIPATED MISCONCEPTIONS Students may become frustrated when they run out of number line but remind them of what they know about how to find ratios equivalent to 4 5 they need to multiply both 4 and 5 by the same number Consider directing their attention to a definition of equivalent ratios displayed in your room or in a previous lesson or suggesting they reexamine some of the simpler cases e g the relationship between 4 5 and 36 45 Be on the lookout for students trying to tape on more paper to extend their number lines 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 125

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G6M2 LESSON 11 SUPPORT FOR STUDENTS WITH DISABILITIES ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Conceptual Processing Eliminate Barriers Allow students to use calculators to ensure inclusive participation in the activity Receptive Expressive Language Peer Tutors Pair students with their previously identified peer tutors to aid in comprehension and expression of understanding Executive Functioning Graphic Organizers Consider having students create a Venn diagram with appropriate uses of the double number line and table of equivalent ratios for future reference It can then be pasted into the student s notebook SUPPORT FOR ENGLISH LANGUAGE LEARNERS Representing MLR 7 Compare and Connect Use this routine to help students make connections between specific features of tables and double number lines Ask students to describe to a partner how multiplication appears in each representation and then invite listeners to restate or revoice what they heard back to their partner using mathematical language e g product row column table equivalent ratio etc After students have a chance to share with a partner select a few to share their reasoning with the class Design Principle s Maximize meta awareness Concept Exploration Activity 2 BATCHES OF TRAIL MIX Instructional Routines MLR2 Collect and Display MLR8 Discussion Supports This task gets students to interact with a table in a way that discourages skip counting Numbers within each column are deliberately out of order This is intended to encourage students to multiply the pairs of values from a given ratio by the same number and to emphasize that the order in which pairs of values appear is not a necessary part of the structure of a table Order within rows however is necessary The last question reinforces the definition of equivalent ratios Students may use the given values 7 and 5 as the basis for every calculation e g for every row they think 7 times what or 5 times what They may also reason with values from another row e g they may see 250 as 10 25 rather than as 5 50 As students work notice different approaches 126 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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G6M2 LESSON 11 ZEARN MATH TEACHER LESSON MATERIALS LAUNCH Explain that a table is just a list of equivalent ratios In this case one column contains amounts of almonds and the other column contains corresponding amounts of raisins Each row shows the amount of each ingredients in a particular batch YOUR NOTES Reiterate that multiplying both parts of a ratio by the same non zero number always creates a ratio that is equivalent to the original ratio ACTIVITY 2 TASK 1 3 A recipe for trail mix says Mix 7 ounces of almonds with 5 ounces of raisins Here is a table that has been started to show how many ounces of almonds and raisins would be in different sized batches of this trail mix 1 Complete the table so that ratios represented by each row are equivalent 2 What methods did you use to fill in the table 3 How do you know that each row shows a ratio that is equivalent to 7 5 Explain your reasoning Almonds oz Raisins oz 7 5 28 10 3 5 250 56 STUDENT RESPONSE 1 Here is the table 2 Answers vary Sample response To find the first equivalent ratio I multiplied 5 by 4 since 7 times 4 equals 28 I used the same strategy multiplying 7 and 5 by the same factor to fill in the rest of the table 3 To find each row multiply 7 and 5 by the same thing This means that each row has values of a ratio equivalent to 7 5 Almonds oz Raisins oz 7 5 28 20 14 10 3 5 2 5 350 250 56 40 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 127

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G6M2 LESSON 11 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES DISCUSSION GUIDANCE Invite one or more students who used multiplicative approaches to share their reasoning with the group Consider displaying the table and using it to facilitate gesturing and arrow drawing while students explain Highlight the strategy of multiplying the 7 and 5 values by the same number ANTICIPATED MISCONCEPTIONS Students may make patterns that do not yield equivalent ratios For example they may think 7 minus 2 is 5 so for the next row 28 minus 2 is 26 Or they may think 7 plus 21 is 28 so then 5 plus 21 is 26 If so consider Appealing to what students know about batches of recipes The second row represents how many batches of trail mix 4 because 28 is 7 4 Okay so to make 4 batches of trail mix how will we figure out how many raisins Also multiply the 5 by 4 Refreshing what students learned about equivalent ratios We need a ratio that is equivalent to the ratio represented in row 1 So what do we need to do to the 7 and the 5 Multiply them by the same number Students may be unsure about how to find the missing value in the row with 3 5 Encourage them to reason about it the same way they reasoned about the other rows We need a ratio that is equivalent to the ratio represented in row 1 So what do we need to do to the 7 and the 5 They may have to get there by way of division 7 divided by 2 is 3 5 so 7 times 12 is 3 5 this means multiplying 5 by 12 as well SUPPORT FOR ENGLISH LANGUAGE LEARNERS Conversing MLR 2 Collect and Display Circulate and listen to student talk about their methods for completing the table during pair work or group work and jot notes about their reasoning To support students mathematical language use scribe students words on a visual display to refer back to during whole class discussions throughout the lesson Design Principle s Support sense making 128 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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G6M2 LESSON 11 SUPPORT FOR STUDENTS WITH DISABILITIES Conceptual Processing Eliminate Barriers Assist students in seeing the connections between new problems and prior work Students may benefit from a review of different representations to activate prior knowledge ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Representing MLR 8 Discussion Supports Display the table and use it to facilitate gesturing and arrow drawing while students explain Highlight the strategy of multiplying the 7 and 5 values by the same number Design Principle s Support sense making Digital Lesson Fill in the table and explain the strategies you used to fill it in Choc chips oz Pretzels oz 2 6 24 14 40 120 20 STUDENT RESPONSES Responses vary Sample response I multiplied 2 and 6 by the same number to find equivalent ratios of chocolate chips to pretzels For example 6 pretzels times 4 equals 24 pretzels so I need to multiply 2 chocolate chips by 4 to get 8 chocolate chips I used the same strategy for the next row 2 chocolate chips times 7 equals 14 chocolate chips so I multiplied 6 pretzels times 7 to get 42 pretzels To fill out the last row I saw that 20 chocolate chips is half of 40 chocolate chips so I took half of 120 pretzels too That s 60 pretzels Choc chips oz Pretzels oz 2 6 8 24 14 42 40 120 20 60 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 129

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G6M2 LESSON 11 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Wrap Up LESSON SYNTHESIS Sometimes it is easier to use a table rather than a double number line to represent equivalent ratios Each row contains a ratio that is equivalent to all the other ratios so if we know one row we can multiply both of its values by the same number to find another row s values TERMINOLOGY Table A table organizes information into horizontal rows and vertical columns The first row or column usually tells what the numbers represent For example here is a table showing the tail lengths of three different pets This table has four rows and two columns Pet Tail length inches Dog 22 Cat 12 Gerbil 2 EXIT TICKET In previous lessons we worked with a diagram and a double number line that represent this cookie recipe Here is a table that represents the same situation Flour cups Vanilla tablespoons 5 2 15 6 1 2 1 2 1 Write a sentence that describes a ratio shown in the table 2 What does the second row of numbers represent 130 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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3 Complete the last row for a different batch size that hasn t been used so far in the table Explain or show your reasoning G6M2 LESSON 11 ZEARN MATH TEACHER LESSON MATERIALS STUDENT RESPONSE 1 Answers vary Sample responses The ratio of cups of flour to teaspoons of vanilla is 5 2 This recipe uses 5 cups of flour for every 2 teaspoons of vanilla This recipe uses 2 1 2 YOUR NOTES cups of flour per teaspoon of vanilla 2 For 15 cups of flour you need 6 teaspoons of vanilla 3 Answers vary Sample response 10 cups of flour and 4 teaspoons of vanilla 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 131

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G6M2 LESSON 12 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES GRADE 6 MISSION 2 LESSON 12 Navigating a Table of Equivalent Ratios Choose multipliers strategically while solving multi step problems involving equivalent ratios LEARNING GOALS Describe orally and in writing how a table of equivalent ratios was used to solve a problem about prices and quantities Remember that dividing by a whole number is the same as multiplying by an associated unit fraction LEARNING GOALS STUDENT FACING LEARNING TARGETS STUDENT FACING 132 Let s use a table of equivalent ratios like a pro I can solve problems about situations happening at the same rate by using a table and finding a 1 row I can use a table of equivalent ratios to solve problems about unit price 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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The purpose of this lesson is to develop students ability to work with a table of equivalent ratios It also provides opportunities to compare and contrast different ways of solving equivalent ratio problems Students see that a table accommodates different ways of reasoning about equivalent ratios with some being more direct than others They notice that to find an unknown quantity they can G6M2 LESSON 12 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Find the multiplier that relates two corresponding values in different rows e g What times 5 equals 8 and use that multiplier to find unknown values This follows the multiplicative thinking developed in previous lessons Find an equivalent ratio with one quantity having a value of 1 and use that ratio to find missing values 15 8 Amount earned Time worked hours 90 5 18 1 144 8 15 8 All tasks in the lesson aim to strengthen students understanding of the multiplicative relationships between equivalent ratios that given a ratio a b an equivalent ratio may be found by multiplying both a and b by the same factor They also aim to build students awareness of how a table can facilitate this reasoning to varying degrees of efficiency depending on one s approach Ultimately the goal of this mission is to prepare students to make sense of situations involving equivalent ratios and solve problems flexibly and strategically rather than to rely on a procedure such as set up a proportion and cross multiply without an understanding of the underlying mathematics To reason using ratios in which one of the quantities is 1 students are likely to use division In the example above they are likely to divide the 90 by 5 to obtain the amount earned per hour Remind students that dividing by a whole number is the same as multiplying by its reciprocal a unit fraction and encourage the use of multiplication as shown in the activity about hourly wages whenever possible Doing so will better prepare students to 1 scale down i e to find equivalent ratios involving values that are smaller than the given ones 2 relate fractions to percentages later in the course and 3 understand division of fractions including the invert and multiply rule in a later mission 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 133

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G6M2 LESSON 12 ZEARN MATH TEACHER LESSON MATERIALS Warm Up NUMBER TALK MULTIPLYING BY A UNIT FRACTION Instructional Routines Number Talk MLR8 Discussion Supports YOUR NOTES The purpose of this number talk is to encourage students to use the meaning of fractions and the properties of operations to find the product of fractions and decimals In grade 4 students multiplied a fraction by a whole number reasoning about these problems based on their understandings of multiplication as groups of a number In grade 5 students multiply fractions by whole numbers reasoning in terms of taking a part of a part whether that be by using division or partitioning a whole In both grade levels the context of the problem played a significant role in how students reasoned and notated the problem and solution Based on these understandings two ideas will be relevant to future work in the mission and are important to emphasize during discussions Dividing by a number is the same as multiplying by its reciprocal The commutative property of multiplication can help us solve a problem regardless of the context LAUNCH Display one problem at a time Tell students to give a signal when they have an answer and a strategy After each problem give students quiet think time and follow with a whole class discussion WARM UP TASK Find the product mentally 1 1 3 21 1 6 21 5 6 1 4 134 1 8 5 6 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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G6M2 LESSON 12 ZEARN MATH TEACHER LESSON MATERIALS STUDENT RESPONSE 1 1 3 2 1 6 21 7 Possible strategies 21 3 or 3 7 21 3 5 Possible strategies 21 6 or divide the product from the first question by 2 because 16 is half of 13 3 5 6 4 1 8 YOUR NOTES 0 7 Possible strategies 5 6 8 or 8 0 7 1 4 5 6 1 4 Possible strategies 5 6 4 or multiply the product from the third question by 2 because 14 is twice as much as 18 DISCUSSION GUIDANCE Ask students to share their strategies for each problem Record and display their explanations for all to see Ask students if or how the factors in the problem impacted their strategy choice To involve more students in the conversation consider asking Who can restate s reasoning in a different way Did anyone solve the problem the same way but would explain it differently Did anyone solve the problem in a different way Does anyone want to add on to s strategy Do you agree or disagree Why SUPPORT FOR STUDENTS WITH DISABILITIES Memory Processing Time Provide sticky notes or mini whiteboards to aid students with working memory challenges SUPPORT FOR ENGLISH LANGUAGE LEARNERS Speaking MLR 8 Discussion Supports Provide sentence frames to support students with explaining their strategies For example I noticed that or First I because When students share their answers with a partner prompt them to rehearse what they will say when they share with the full group Rehearsing provides opportunities to clarify their thinking 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 135

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G6M2 LESSON 12 Design Principle s Optimize output for explanation ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Concept Exploration Activity 1 COMPARING TACO PRICES Instructional Routine MLR5 Co Craft Questions and Problems The purpose of this activity is to encourage students to use a table to find the price for one taco for two different situations Students are likely to divide the cost of the tacos by the number of tacos to find the cost for one taco which is appropriate Use the opportunity to remind students that dividing by a whole number is the same as multiplying by its reciprocal a unit fraction This insight will come in handy in future activities and lessons LAUNCH Tell students that we usually use tables to show equivalent ratios but since we do not know in advance whether the ratios of number of tacos to price will be the same we might want to keep track of them in separate tables Arrange students in groups of 2 Give students quiet think time and then time to discuss their responses and reasoning with their partner ACTIVITY 1 TASK 1 2 Use the table to help you solve these problems Explain or show your reasoning Number of tacos 136 Price in dollars 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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1 Noah bought 4 tacos and paid 6 At this rate how many tacos could he buy for 15 2 Jada s family bought 50 tacos for a party and paid 72 Were Jada s tacos the same price as Noah s tacos STUDENT RESPONSE G6M2 LESSON 12 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES 1 10 tacos 2 No Noah s tacos cost 1 50 each Jada s cost 1 44 each DISCUSSION GUIDANCE The focus should be on how students found the cost of a single taco in each situation Be sure to remind students that dividing by a whole number is the same as multiplying by its reciprocal a unit fraction SUPPORT FOR ENGLISH LANGUAGE LEARNERS Conversing MLR 5 Co Craft Questions Display only the first line of this task Noah bought 4 tacos and paid 6 without revealing the questions that follow Invite pairs of students to discuss possible mathematical questions they could ask about this situation Listen for questions that ask about the price for one taco or the price of multiple tacos and select these students to share their questions with the class This will help draw students attention to the relationships between the two quantities in this task e g tacos and money prior to being asked to calculate any values Design Principle s Cultivate conversation Support sense making Concept Exploration Activity 2 HOURLY WAGES Instructional Routines MLR8 Discussion Supports Think Pair Share This task introduces students to the strategy of using an equivalent ratio with one quantity having a value of 1 to find other equivalent ratios Students look at a worked out example of the strategy make sense of how it works and later apply it to solve other problems 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 137

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G6M2 LESSON 12 There are a couple of key insights to uncover here ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES The ratios we deal with do not always have corresponding quantities that are multiples of each other e g in the task 5 is not a multiple of 8 or vice versa In those situations finding an equivalent ratio where one of the quantities is 1 can be a helpful intermediate step Also highlighted and reinforced here is an idea students learned in Grade 5 that dividing by a whole number is equivalent to multiplying by its reciprocal e g dividing by 5 is the same as multiplying by 15 Expect some students to initially overlook the benefit of using a ratio involving a 1 to rely on methods from previous work and to potentially get stuck especially when dealing with a decimal value in the last row For example since the table shows an arrow and a multiplication from the first to the second row and from the second to third students may try to do the same to find the missing value in the fourth row While finding a factor that can be multiplied to 8 to obtain 3 is valid encourage students to consider an alternative given what they already know about the situation i e how much the person earned in 1 hour If needed scaffold their thinking by asking how much Lin would earn in 2 hours and then in 3 hours Identify a student or two who can articulate why 15 is used as a multiplier Also notice those who can correctly reason why using a ratio with one of the values being 1 helps to find other equivalent ratios and students who reason differently Invite these students to share later LAUNCH This may be some students first time reasoning about money earned by the hour Take a short time to ensure everyone understands the concept Ask if anyone has earned money based on the number of hours doing a job Some students may have experience being paid by the hour for helping with house cleaning a family business babysitting dog walking or doing other jobs Give students quiet think time to complete the activity and a minute to share their responses especially to the last two questions with a partner before discussing as a group ACTIVITY 2 TASK 1 3 138 Lin is paid 90 for 5 hours of work She used the following table to calculate how much she would be paid at this rate for 8 hours of work 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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Amount earned Time worked hours 90 5 18 1 144 8 15 8 G6M2 LESSON 12 15 ZEARN MATH TEACHER LESSON MATERIALS 8 YOUR NOTES 1 What is the meaning of the 18 that appears in the table 2 Why was the number 1 5 used as a multiplier 3 Explain how Lin used this table to solve the problem 4 At this rate how much would Lin be paid for 3 hours of work For 2 1 hours of work STUDENT RESPONSE 1 Lin earned 18 for 1 hour of work or for every hour of work 2 We wanted to turn the 5 into a 1 so the 1 could be multiplied by 8 because 5 15 1 1 5 was chosen 3 First they found how much Lin made in 1 hour by multiplying both the 90 and the 5 by 15 or dividing them both by 5 Then they multiplied both the 18 and the 1 by 8 to find that she earned 144 in 8 hours 4 Lin would be paid 54 for 3 hours of work and 37 80 for 2 1 hours of work DISCUSSION GUIDANCE Select a few students to share about the use of 15 as a multiplier and to explain the reasoning process shown in the table If different approaches are used take the opportunity to compare and contrast the efficacy of each If students had trouble reasoning to find the pay for 2 1 hours of work help them articulate what they have done in each preceding case and urge them to think about the 2 1 the same way If they are unsure whether multiplying 18 by 2 1 would work encourage them to check whether the answer makes sense For two hours of work Lin would earn 36 so it stands to reason that she would earn a bit more than 36 for 2 1 hours In doing so students practice decontextualizing and contextualizing their reasoning and solutions 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 139

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G6M2 LESSON 12 SUPPORT FOR STUDENTS WITH DISABILITIES ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Strengths based Approach This activity leverages many natural strengths of students with ADHD LD and other concrete learners in terms of the use of real world contexts and personal student interest This may be an opportunity for the teacher to highlight this strength in class and allow an individual with disability to lead peer interactions discussions increasing buy in Receptive Expressive Language Peer Tutors Pair students with their previously identified peer tutors to aid in comprehension and expression of understanding SUPPORT FOR ENGLISH LANGUAGE LEARNERS Speaking MLR 8 Discussion Supports Provide a sentence frame to scaffold students language use when discussing the table representation For example Lin multiplied both 90 and 5 by 1 5 because This helps students have a structure for language use while focusing on the mathematical reasoning for the each step in the table Design Principle s Support sense making Digital Lesson A printer prints 8 sheets of paper in 16 seconds Explain how you could find each unknown in the table Paper sheets Seconds 8 16 1 13 STUDENT RESPONSE Answers vary Sample response I know that 8 sheets of paper divided by 8 equals 1 sheet of paper so I can also divide 16 seconds by 8 to keep the ratio the same That s 2 seconds for every 1 sheet 140 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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of paper Then I can use that rate to find how many seconds it takes to print 13 sheets of paper 1 sheet of paper times 13 equals 13 sheets of paper so I multiplied 2 seconds by 13 too That s 26 seconds G6M2 LESSON 12 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Wrap Up LESSON SYNTHESIS This lesson is about using a table of equivalent ratios in an efficient way To wrap up highlight a few important points In problems with equivalent ratios finding an equivalent ratio containing a 1 is often a good strategy To create a new row in a table of equivalent ratios take an existing row and multiply both values by the same number Remember that we can multiply whole numbers by unit fractions to get smaller numbers EXIT TICKET A shop sells bagels for 5 per dozen You can use the table to answer the questions Explain your reasoning Number of bagels Price in dollars 12 5 1 At this rate how much would 6 bagels cost 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 141

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GRADE 6 MISSION 2 LESSON 13 Tables and Double Number Line Diagrams LEARNING GOALS LEARNING GOALS STUDENT FACING G6M2 LESSON 13 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Compare and contrast orally double number line diagrams and tables representing the same situation Draw and label a table of equivalent ratios from scratch to solve problems about constant speed Let s contrast double number lines and tables I can create a table that represents a set of equivalent ratios LEARNING TARGETS STUDENT FACING I can explain why sometimes a table is easier to use than a double number line to solve problems involving equivalent ratios I include column labels when I create a table so that the meaning of the numbers is clear REQUIRED MATERIALS REQUIRED PREPARATION Pre printed slips cut from copies of the template Template for Concept Exploration Activity 2 Make 1 copy of the Template for Concept Exploration Activity 2 for every 4 students and cut them up ahead of time 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 143

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G6M2 LESSON 13 ZEARN MATH TEACHER LESSON MATERIALS In this lesson students explicitly connect and contrast double number lines and tables They also encounter a problem involving relatively small fractions so the flexibility of a table makes it preferable to a double number line Students have used tables in earlier grades to identify arithmetic patterns and record measurement equivalents In grade 6 a new feature of working with tables is considering the relationship between values in different rows Two features of tables make them more flexible than double number lines YOUR NOTES On a double number line differences between numbers are represented by lengths on each number line While this feature can help support reasoning about relative sizes it can be a limitation when large or small numbers are involved which may consequently hinder problem solving A table removes this limitation because differences between numbers are no longer represented by the geometry of a number line A double number line dictates the ordering of the values on the line but in a table pairs of values can be written in any order 5 pounds of coffee cost 40 How much does 8 5 pounds cost You can see in the table below how being able to skip around makes for more nimble problem solving Weight of coffee pounds Cost dollars 5 40 1 8 8 5 68 At this point in the mission students should have a strong sense of what it means for two ratios to be equivalent so they can fill in a table of equivalent ratios with understanding instead of just by following a procedure Students can also always fall back to other representations if needed Warm Up NUMBER TALK CONSTANT DIVIDEND Instructional Routines Number Talk MLR8 Discussion Supports This number talk helps students think about what happens to a quotient when the divisor is doubled In this lesson and in upcoming work on ratios and unit rates students will be asked to find a fraction of a number and identify fractions on a number line 144 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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G6M2 LESSON 13 ZEARN MATH TEACHER LESSON MATERIALS LAUNCH Display one problem at a time Tell students to give a signal when they have an answer and a strategy After each problem give students quiet think time followed by a whole class discussion Pause after discussing the third question and give students quiet think time to place the quotients on the number line YOUR NOTES WARM UP TASK 1 Find the quotients mentally then locate and label the quotients on the number line 150 2 150 4 150 8 0 150 STUDENT RESPONSE 150 2 75 Possible strategies 2 75 150 100 2 50 2 75 150 4 37 5 Possible strategies 75 2 37 5 from the previous question 148 4 2 4 37 5 150 8 18 75 Possible strategies 37 5 2 18 75 from the previous question 144 8 6 8 18 75 0 150 8 150 4 150 2 18 75 37 5 75 150 DISCUSSION GUIDANCE Ask students to share their strategies for each problem Record and display their explanations for all to see To involve more students in the conversation consider asking 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 145

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G6M2 LESSON 13 Who can restate ZEARN MATH TEACHER LESSON MATERIALS s reasoning in a different way Did anyone solve the problem the same way but would explain it differently Did anyone solve the problem in a different way YOUR NOTES Does anyone want to add on to s strategy Do you agree or disagree Why For the fourth question display the number line for all to see and invite a few students to share their reasoning about the location of each quotient on the number line Discuss students observations from when they placed the numbers on the number line SUPPORT FOR STUDENTS WITH DISABILITIES Memory Processing Time Provide sticky notes or mini whiteboards to aid students with working memory challenges SUPPORT FOR ENGLISH LANGUAGE LEARNERS Speaking MLR 8 Discussion Supports Provide sentence frames to support students with explaining their strategies For example I noticed that or First I because When students share their answers with a partner prompt them to rehearse what they will say when they share with the full group Rehearsing provides opportunities to clarify their thinking Design Principle s Optimize output for explanation Concept Exploration Activity 1 MOVING 3 000 METERS Instructional Routines Anticipate Monitor Select Sequence Connect MLR3 Clarify Critique Correct In this activity students use tables of equivalent ratios to solve three problems with decreasing scaffolding throughout the activity For the first problem students start by examining a table of 146 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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equivalent ratios noticing that descriptive column headers are important in helping you use a table of equivalent ratios to solve a problem For the second problem there is an empty table students can fill in The third problem does not provide any scaffolding allowing students to choose their own method of solving the problem G6M2 LESSON 13 ZEARN MATH TEACHER LESSON MATERIALS Monitor for students solving the last problem in different ways YOUR NOTES LAUNCH Arrange students in groups of 2 Give quiet work time and then have students work with their partner ACTIVITY 1 TASK 1 2 The other day we saw that Han can run 100 meters in 20 seconds Han wonders how long it would take him to run 3 000 meters at this rate He made a table of equivalent ratios 1 Do you agree that this table represents the situation Explain your reasoning 2 Complete the last row with the missing number 3 What question about the situation does this number answer 20 100 10 50 1 5 3 000 4 What could Han do to improve his table 5 Priya can bike 150 meters in 20 seconds At this rate how long would it take her to bike 3 000 meters 6 Priya s neighbor has a dirt bike that can go 360 meters in 15 seconds At this rate how long would it take them to ride 3 000 meters STUDENT RESPONSE 1 Answers vary Sample response I agree with the first three rows but the last row would be for 3 000 seconds instead of 3 000 meters so it wouldn t help Han answer the question 2 The empty cell should contain 15 000 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 147

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G6M2 LESSON 13 3 How far he would go in 3 000 seconds ZEARN MATH TEACHER LESSON MATERIALS 4 He should label what quantities are in each column 5 400 seconds YOUR NOTES 6 125 seconds DISCUSSION GUIDANCE Select students with different methods for the last question to explain their solutions to the class Highlight the connections between different strategies especially between tables and double number line diagrams SUPPORT FOR ENGLISH LANGUAGE LEARNERS Conversing This activity uses MLR 3 Clarify Critique Correct to give students an opportunity to use mathematical language to critique the reasoning of others This routine fortifies output and engages students in meta awareness Design Principle s Cultivate Conversation Maximize meta awareness Concept Exploration Activity 2 THE INTERNATIONAL SPACE STATION MATERIALS TEMPLATE FOR CONCEPT EXPLORATION ACTIVITY 2 Instructional Routines MLR7 Compare and Connect MLR2 Collect and Display This activity prompts students to compare and contrast two representations of equivalent ratios Students work collaboratively to observe similarities and differences of using a double number line and using a table to express the same situation Below are some key distinctions 148 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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Double number line Table Distances between numbers and lengths of lines matter Distances and lengths do not matter because there are no lines The numbers on each line must be in order Rows of ratios can be out of order within a column numbers can go in any order that is convenient Each value of a ratio is shown on a line Each value of a ratio is shown in a column Pairs of values of a ratio are aligned vertically Pairs of values of a ratio appear in the same row G6M2 LESSON 13 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES You will need The International Space Station template for this activity ACTIVITY 2 LAUNCH To help students build some intuition about kilometers begin by connecting it with contexts that are familiar to them Tell students that kilometer is a unit used in the problem Then ask a few guiding questions Can you name two things in our town or city that are about 1 kilometer apart Consider finding some examples of 1 kilometer distances near your school ahead of time How long do you think it would take you to walk 1 kilometer Typical human walking speed is about 5 kilometers per hour so it takes a person about 12 minutes to walk 1 kilometer What might be a typical speed limit on a highway in kilometers per hour 100 kilometers per hour is a typical highway speed limit Students might be more familiar with a speed limit such as 65 miles per hour Since there are about 1 6 kilometers in every mile the same speed will be expressed as a higher number in kilometers per hour than in miles per hour Arrange students in groups of 2 Give one person a slip with the table and the other a slip with a double number line shown below Ask students to first do what they can independently and then to obtain information from their partners to fill in all the blanks Explain that when the blanks are filled the two representations will show the same information 0 Distance traveled km Elapsed time sec 0 5 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 149

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G6M2 LESSON 13 ZEARN MATH TEACHER LESSON MATERIALS Distance traveled km Elapsed time sec 0 0 80 10 1 YOUR NOTES ACTIVITY 2 TASK 1 3 The International Space Station orbits around the Earth at a constant speed You and your partner will each be given a double number line or a table that represents this situation 1 Complete the parts of your representation that you can figure out for sure 2 Share information with your partner and use the information that your partner shares to complete your representation 3 What is the speed of the International Space Station 4 Place the two completed representations side by side Discuss with your partner some ways in which they are the same and some ways in which they are different 5 Record at least one way that they are the same and one way they are different STUDENT RESPONSE 150 Distance traveled km Elapsed time sec 0 0 80 10 8 1 40 5 56 7 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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G6M2 LESSON 13 1 See table and double number line ZEARN MATH TEACHER LESSON MATERIALS 2 See table and double number line 3 The ISS is traveling in its orbit at a speed of 8 kilometers per second You can see this because both representations show the distance traveled in 1 second YOUR NOTES 4 Answers vary see Activity Recap 5 See Activity Recap ACTIVITY 2 RECAP Display completed versions of both representations for all to see Invite students to share the ways the representations are alike and different Consider writing some of these on the board or this could just be a verbal discussion Highlight the distinctions in terms of distances between numbers order of numbers and the vertical or horizontal orientations of the representations Although it is not a structural distinction students might describe the direction in which multiplying happens as a difference between the two representations They might say that we multiply up or down to find equivalent ratios in a table and we multiply across to do the same on a double number line You could draw arrows to illustrate this fact 7 0 8 40 56 80 0 1 5 7 10 Distance traveled km Elapsed time sec 7 7 Distance traveled km Elapsed time sec 0 0 80 10 8 1 40 5 56 7 7 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 151

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G6M2 LESSON 13 ZEARN MATH TEACHER LESSON MATERIALS The vertical orientation of tables and the horizontal orientation of double number lines are conventions we decided to consistently use in these materials Mathematically there is nothing wrong with orienting each representation the other way Students may encounter tables oriented horizontally in a later course Later in this course they will encounter number lines oriented vertically YOUR NOTES ANTICIPATED MISCONCEPTIONS Students with the double number line representation may decide to label every tick mark instead of just the ones indicated with dotted rectangles This is fine Make sure they understand that the tick marks with dotted rectangles are the ones they are supposed to record in the table SUPPORT FOR STUDENTS WITH DISABILITIES Receptive Expressive Language Peer Tutors Pair students with their previously identified peer tutors to aid in comprehension and expression of understanding SUPPORT FOR ENGLISH LANGUAGE LEARNERS Reading MLR 2 Collect and Display While pairs are working circulate and listen to student talk about the similarities and differences between the table and double number line representations and write down common or important phrases you hear students say about the representations e g order or numbers direction of their multiplying Write the students words onto to a visual display of either or both the table or double number This will help students read and use mathematical language during their paired and whole group discussions Design Principle s Support sense making Digital Lesson The double number line shows how much sour cream and how much seasoning to mix to make different amounts of onion dip 152 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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0 0 5 1 1 5 Sour cream cups G6M2 LESSON 13 ZEARN MATH TEACHER LESSON MATERIALS Seasoning tablespoons 0 2 4 6 YOUR NOTES 1 Make a table to represent the same situation 2 What does the number 4 represent STUDENT RESPONSE 1 Sour cream cups Seasoning tablespoons 0 0 0 5 2 1 4 1 5 6 2 The number 4 represents the number of tablespoons of seasoning you need when you use 1 cup of sour cream to make the recipe taste the same as the other batches Wrap Up LESSON SYNTHESIS Briefly revisit the two tasks displaying the representations for all to see and pointing out ways in which tables and double number lines are the same and different Emphasize that tables are sometimes easier to work with In one task we looked at the distance the ISS travels in its orbit and the time it takes to orbit the Earth How are the table and the double number line similar to each other How are they different 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 153

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G6M2 LESSON 14 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES GRADE 6 MISSION 2 LESSON 14 Solving Equivalent Ratio Problems LEARNING GOALS Determine what information is needed to solve a problem involving equivalent ratios Ask questions to elicit that information Understand the structure of a what why info gap activity LEARNING GOALS STUDENT FACING LEARNING TARGETS STUDENT FACING REQUIRED MATERIALS REQUIRED PREPARATION 156 Let s practice getting information from our partner I can decide what information I need to know to be able to solve problems about situations happening at the same rate I can explain my reasoning using diagrams that I choose Template for Concept Exploration Activity 1 Pre printed slips cut from copies of the template You will need the Template for Concept Exploration Activity 1 for this lesson Make 1 copy for every 4 students and cut them up ahead of time 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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The purpose of this lesson is to gives students further practice in solving equivalent ratio problems and introduce them to the info gap activity structure The info gap structure requires students to make sense of problems by determining what information is necessary and then to ask for information they need to solve it This may take several rounds of discussion if their first requests do not yield the information they need It also allows them to refine the language they use and ask increasingly more precise questions until they get the information they need G6M2 LESSON 14 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Warm Up WHAT DO YOU WANT TO KNOW The warm up prepares students for the next info gap activity by first asking them to brainstorm what information they would need to know to solve an equivalent ratio problem Next the teacher demonstrates asking students to share what they want to know and why they want to know it before giving them the information LAUNCH Give students quiet think time WARM UP TASK 1 What information would you need to solve the problem below A red car and a blue car enter the highway at the same time and travel at a constant speed How far apart are they after 4 hours STUDENT RESPONSE Answers vary Sample responses How fast is each car traveling Are the cars going the same direction Did the cars enter the highway at the same location 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 157

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G6M2 LESSON 14 What is the difference between the speeds of the two cars ZEARN MATH TEACHER LESSON MATERIALS DISCUSSION GUIDANCE YOUR NOTES Demonstrate asking students the questions they will use in the Info Gap in the next activity Ask them What specific information do you need As students pose questions write them down and ask Why do you need that information When students explain why they need the information provide it to them After sharing each piece of information ask the class whether they have enough information to solve the problem When they think they do give them quiet think time to solve the problem and then have them share their strategies The red car is traveling faster than the blue car One car is traveling 5 miles per hour faster than the other car The slower car is traveling at 60 miles per hour The blue car is traveling at 60 miles per hour The faster car is traveling at 65 miles per hour The red car is traveling as 65 miles per hour Both cars entered the highway at the same location Both cars are traveling in the same direction Concept Exploration Activity 1 INFO GAP HOT CHOCOLATE AND POTATOES MATERIALS TEMPLATE FOR CONCEPT EXPLORATION ACTIVITY 1 Instructional Routine MLR4 Information Gap In this info gap activity students solve problems involving equivalent ratios If students use a table it may take different forms Some students may produce a table that has many rows that require repeated multiplication Others may create a more abbreviated table and use more efficient multipliers Though some approaches may be more direct or efficient than others it is important for students to choose their own method for solving them and to explain their method so that their partner can understand 158 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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The info gap structure requires students to make sense of problems by determining what information is necessary and then to ask for information they need to solve it This may take several rounds of discussion if their first requests do not yield the information they need It also allows them to refine the language they use and ask increasingly more precise questions until they get the information they need Here is the text of the cards for reference and planning G6M2 LESSON 14 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Problem Card 1 Jada mixes milk and cocoa powder to make hot chocolate She wants to use all the cocoa powder she has left How much milk should Jada use Data Card 1 One batch of Jada s recipe calls for 3 cups of milk One batch of Jada s recipe calls for 2 tablespoons of cocoa powder Jada has 2 gallons of milk left Jada has 9 tablespoons of cocoa powder left There are 16 cups in 1 gallon Problem Card 2 Noah needs to peel a lot of potatoes before a dinner party He has already peeled some potatoes If he keeps peeling at the same rate will he finish all the potatoes in time Data Card 2 Noah has already been peeling potatoes for 10 minutes Noah has already peeled 8 potatoes Noah needs to peel 60 more potatoes Noah needs to be finished peeling potatoes in 1 hour and 10 minutes There are 60 minutes in 1 hour LAUNCH Arrange students in groups of 2 In each group distribute a problem card to one student and a data card to the other student 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 159

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G6M2 LESSON 14 ACTIVITY 1 TASK 1 ZEARN MATH TEACHER LESSON MATERIALS 2 YOUR NOTES You will be given either a problem card or a data card Do not show or read your card to your partner Follow the directions in your notes to work with your partner If your teacher gives you the problem card 1 Read your card silently and think about what you need to know to be able to answer the questions 2 Ask your partner for the specific information that you need 3 Explain how you are using the information to solve the problem 4 Solve the problem and show your reasoning to your partner If your teacher gives you the data card 1 Read your card silently 2 Ask your partner What specific information do you need and wait for them to ask for information If your partner asks for information that is not on the card do not do the calculations for them Tell them you don t have that information 3 Have them explain Why do you need that information before telling them the information 4 After your partner solves the problem ask them to explain their reasoning even if you understand what they have done Both you and your partner should record a solution to each problem STUDENT RESPONSE Jada should use 13 5 cups of milk Possible strategies Finding a multiplier that relates 2 to 9 tablespoons of cocoa They may ask 2 times what is 9 and use 4 5 as the factor to multiply by 3 Multiplying by 12 or dividing by 2 to find the number of cups of milk that correspond to 1 tablespoon of cocoa and then multiplying that number by 9 for 9 tablespoons of cocoa 160 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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Multiplying the number of cups of milk by 9 to correspond to 18 tablespoons of cocoa and then dividing it by 2 for 9 tablespoons of cocoa as shown in the table below Milk cups Cocoa tablespoons 3 2 or 1 5 1 or 13 5 9 1 2 3 2 9 13 1 2 1 2 G6M2 LESSON 14 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES 9 No Noah does not have enough time It will take him 75 minutes to finish peeling all the potatoes Possible strategies Elapsed time minutes Potatoes peeled 10 2 or 1 25 1 75 60 10 5 8 4 1 8 4 10 5 or 0 8 25 20 70 56 1 It will take 75 minutes to peel 60 potatoes 1 25 minutes per potato 2 He could peel 56 potatoes in 70 minutes 0 8 potatoes per minute DISCUSSION GUIDANCE Select one student to explain each distinct approach Highlight how multiplicative reasoning and using the table are similar or different in each case When all approaches have been discussed ask students When might it be helpful to first find the amount that corresponds to 1 unit of one quantity and scale that amount up to any value we want Encourage students to refer to all examples seen in this lesson so far 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 161

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G6M2 LESSON 14 ZEARN MATH TEACHER LESSON MATERIALS ANTICIPATED MISCONCEPTIONS Students may misinterpret the meaning of the numbers or associate quantities incorrectly and multiply 8 by 6 because 10 6 is 60 Encourage them to organize the given information in a table or a double number line YOUR NOTES SUPPORT FOR STUDENTS WITH DISABILITIES Social Emotional Functioning Peer Tutors Pair students with their previously identified peer tutors SUPPORT FOR ENGLISH LANGUAGE LEARNERS Conversing This activity uses MLR 4 Information Gap to give students a purpose for discussing information necessary to solve problems involving equivalent ratios Consider offering questions or question starters for students who need a starting point on their first few questions in the Problem Data Card or Info Gap Games Design Principle s Cultivate Conversation Digital Lesson Jayson reads the first 48 pages of a 480 page book in 3 days Marcus reads the first 50 pages of a 400 page book in 4 days Nina reads the first 120 pages of a 600 page book in 5 days 1 If they continue to read at these rates who will finish first 2 How many days will each student take to finish his or her book STUDENT RESPONSE 1 162 Nina will finish the book first 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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2 It will take Jayson 30 days to finish his book It will take Marcus 32 days to finish his book It will take Nina 25 days to finish her book Students may show their thinking with a representation such as the tables below Marcus Jayson G6M2 LESSON 14 ZEARN MATH TEACHER LESSON MATERIALS Nina Pages Days Pages Days Pages Days 48 3 48 3 48 3 480 30 480 30 480 30 YOUR NOTES Wrap Up LESSON SYNTHESIS When solving problems involving equivalent ratios we often have three pieces of information and need to find a fourth For example If you eat 12 strawberries in 3 minutes how long will it take to eat 8 strawberries at that rate We can use a table to solve this problem very quickly For example Number of strawberries Number of minutes 12 3 1 1 4 8 2 If you jump 8 times in 10 seconds how many jumps can you make in 45 seconds at that rate Where would you put the one in this table What is the answer to the question 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 163

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Template for Lesson 14 Concept Exploration Activity 1 page 1 of 1 Info Gap Hot Chocolate and Potatoes Info Gap Hot Chocolate and Potatoes Problem Card 1 DataCard 1 Jada mixes milk and cocoa powder to make hot chocolate She wants to use all of the cocoa powder she has left How much milk should Jada use Jada s recipe calls for 3 cups of milk Jada s recipe calls for 2 tablespoons of cocoa powder Jada has 2 gallons of milk Jada has 9 tablespoons of cocoa powder There are 16 cups in 1 gallon Info Gap Hot Chocolate and Potatoes Info Gap Hot Chocolate and Potatoes Problem Card 2 Data Card 2 Noah needs to peel a lot of potatoes before a large dinner He has already peeled some potatoes If Noah keeps peeling at the same rate will he finish all the potatoes in time Noah has already peeled 8 potatoes Noah has been peeling for 10 minutes Noah needs to peel 60 more potatoes Noah needs to be finished peeling in 1 hour and 10 minutes There are 60 minutes in 1 hour Info Gap Hot Chocolate and Potatoes Info Gap Hot Chocolate and Potatoes Problem Card 1 DataCard 1 Jada mixes milk and cocoa powder to make hot chocolate She wants to use all of the cocoa powder she has left How much milk should Jada use Jada s recipe calls for 3 cups of milk Jada s recipe calls for 2 tablespoons of cocoa powder Jada has 2 gallons of milk Jada has 9 tablespoons of cocoa powder There are 16 cups in 1 gallon Info Gap Hot Chocolate and Potatoes Info Gap Hot Chocolate and Potatoes Problem Card 2 Data Card 2 Noah needs to peel a lot of potatoes before a large dinner He has already peeled some potatoes If Noah keeps peeling at the same rate will he finish all the potatoes in time Noah has already peeled 8 potatoes Noah has been peeling for 10 minutes Noah needs to peel 60 more potatoes Noah needs to be finished peeling in 1 hour and 10 minutes There are 60 minutes in 1 hour 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 165

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G6M2 LESSON 15 ZEARN MATH TEACHER LESSON MATERIALS GRADE 6 MISSION 2 LESSON 15 Part Part Whole Ratios YOUR NOTES LEARNING GOALS LEARNING GOALS STUDENT FACING LEARNING TARGETS STUDENT FACING Comprehend the word parts as an unspecified unit in sentences written and spoken describing ratios Draw and label a tape diagram to solve problems involving ratios and the total amount Explain orally the solution method Let s look at situations where you can add the quantities in a ratio together I can create tape diagrams to help me reason about problems involving a ratio and a total amount I can solve problems when I know a ratio and a total amount Snap cubes Graph paper REQUIRED MATERIALS REQUIRED PREPARATION 166 Tools for creating a visual display Any way for students to create work that can be easily displayed to the class Examples chart paper and markers whiteboard space and markers shared online drawing tool access to a document camera Prepare a set of 50 red snap cubes and 30 blue snap cubes for each group of students 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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Up to this point students have worked with ratios of quantities where the units are the same e g cups to cups and ratios of quantities where the units are different e g miles to hours Sometimes in the first case the sum of the quantities makes sense in the context and we can ask questions about the total amount as well as the component parts For example when mixing 3 cups of yellow paint with 2 cups of blue paint we get a total of 5 cups of green paint Notice that this does not always work 3 cups of water mixed with 2 cups of dry oatmeal will not make 5 cups of soggy oatmeal In the paint scenario the ratio of yellow paint to blue paint to green paint is 3 2 5 Furthermore if we double the amount of both yellow and blue paint we will double the amount of green paint In general if the ratio of yellow to blue paint is equivalent the ratio of yellow to blue to green paint will also be the equivalent We can see this is always true because of the distributive property G6M2 LESSON 15 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES a b a b is equivalent to 2a 2b 2a 2b because 2a 2b 2 a b These ratios are sometimes called part part whole ratios In this lesson students learn about tape diagrams as a handy tool to represent ratios with the same units and as a way to reason about individual quantities the parts and the total quantity the whole Here students also see ratios expressed not in terms of specific units milliliters cups square feet etc but in terms of parts e g the recipe calls for 2 parts of glue to 1 part of water Warm Up TRUE OR FALSE MULTIPLYING BY A UNIT FRACTION Instructional Routine True or False This warm up encourages students to use the meaning of fractions and properties of operations to reason about equations While students may evaluate each side of the equation to determine if it is true or false encourage students to think about the following ideas in each The first question Dividing is the same as multiplying by the reciprocal of the divisor The second question Adjusting the factors adjusts the products If both factors increase the resulting product will be greater than the original The third question The commutative property of multiplication The fourth question Decomposing a dividend into two numbers and dividing each by the divisor is a way to find the quotient of the original dividend 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 167

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G6M2 LESSON 15 ZEARN MATH TEACHER LESSON MATERIALS LAUNCH Display one problem at a time Tell students to give a signal when they have an answer and a strategy After each problem give students quiet think time and follow with a whole class discussion YOUR NOTES WARM UP TASK 1 Determine if each problem is true or false 1 5 45 45 5 1 5 20 1 4 24 1 6 1 6 42 42 486 1 12 480 12 6 12 STUDENT RESPONSE True Division is the same as multiplying by the reciprocal False Both factors increased True Commutative Property of Multiplication True Partial quotients DISCUSSION GUIDANCE Ask students to share their strategies for each problem Record and display their explanations for all to see Ask students if or how the factors in the problem impacted the strategy choice To involve more students in the conversation consider asking Who can restate s reasoning in a different way Does anyone want to add on to s strategy Do you agree or disagree Why 168 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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After each true equation ask students if they could rely on the reasoning used on the given problem to think about or solve other problems that are similar in type After each false equation ask students how we could make the equation true SUPPORT FOR STUDENTS WITH DISABILITIES G6M2 LESSON 15 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Memory Processing Time Provide sticky notes or mini whiteboards to aid students with working memory challenges Concept Exploration Activity 1 CUBES OF PAINT Instructional Routine MLR8 Discussion Supports Up until now students have worked with ratios of quantities given in terms of specific units such as milliliters cups teaspoons etc This task introduces students to the use of the more generic parts as a unit in ratios and the use of tape diagrams to represent such ratios In addition to thinking about the ratio between two quantities students also begin to think about the ratio between the two quantities and their total Two important ideas to make explicit through the task and discussion A ratio can associate quantities given in terms of a specific unit as in 4 teaspoons of this to 3 teaspoons of that A ratio can also associate quantities of the same kind without specifying particular units in terms of parts as in 4 parts of this to 3 parts of that Any appropriate unit can be used in place of parts without changing the 4 to 3 ratio A ratio can tell us about how two or more quantities relate to one another but it can also tell us about the combined quantity when that makes sense and allow us to solve problems As students work notice in particular how they approach the last two questions Identify students who add snap cubes to represent the larger amount of paint and those who use the original number of snap cubes but adjust their reasoning about what each cube represents Be sure to leave enough time to debrief as a class and introduce tape diagrams afterwards LAUNCH Explain to students that they will explore paint mixtures and use snap cubes to represent them Say To make a particular green paint we need to mix 1 ml of blue paint to 3 ml of yellow 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 169

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G6M2 LESSON 15 ZEARN MATH TEACHER LESSON MATERIALS Represent this recipe with 1 blue snap cube and 3 yellow ones and display each set horizontally to mimic the appearance of a tape diagram YOUR NOTES Ask How much green paint will this recipe yield 4 ml of green paint If each cube represents 2 ml instead of 1 ml how much of blue and yellow do the snap cubes represent How many ml of green paint will we have 2 ml of blue 6 ml of yellow and 8 ml of green Is there another way to represent 2 ml of blue and 6 ml of yellow using snap cubes We could use 2 blue snap cubes and 6 yellow ones How do we refer to 2 ml of blue and 6 ml of yellow in terms of batches 2 batches Highlight the fact that they could either represent 2 ml of blue and 6 ml of yellow with 2 blue snap cubes and 6 yellow ones show this representation if possible or with 1 blue snap cube and 3 yellow ones show representation with the understanding that each cube stands for 2 ml of paint instead of 1 ml Explain to students that in the past they had thought about different amounts of ingredients in a recipe in terms of batches but in this task they will look at another way to mix the right amounts specified by a ratio Arrange students in groups of 3 5 Provide 50 red snap cubes and 30 blue snap cubes to each group Give groups time to complete the activity and then debrief as a small group ACTIVITY 1 TASK 1 2 A recipe for maroon paint says to mix 5 ml of red paint with 3 ml of blue paint Answer the following questions about this recipe 1 Use snap cubes to represent the amounts of red and blue paint in the recipe Then draw a sketch of your snap cube representation of the maroon paint 170 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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G6M2 LESSON 15 a What amount does each cube represent ZEARN MATH TEACHER LESSON MATERIALS b How many milliliters of maroon paint will there be 2 a Suppose each cube represents 2 ml How much of each color paint is there YOUR NOTES Red ml Blue ml Maroon ml b Suppose each cube represents 5 ml How much of each color paint is there Red ml Blue ml Maroon ml 3 a Suppose you need 80 ml of maroon paint How much red and blue paint would you mix Be prepared to explain your reasoning Red ml Blue ml Maroon 80 ml b If the original recipe is for one batch of maroon paint how many batches are in 80 ml of maroon paint STUDENT RESPONSE 1 Show 5 red snap cubes and 3 blue ones a Each snap cube represents 1 ml b 1 1 1 1 1 5 so there is 5 ml of red paint 1 1 1 3 so there is 3 ml of blue paint 5 3 8 so there is 8 ml of maroon paint 2 a 2 2 2 2 2 10 so there is 10 ml of red paint 2 2 2 6 so there is 6 ml of blue paint 10 6 16 so there is 16 ml of maroon paint b There is 25 ml of red since 5 5 25 and 15 ml of blue since 5 3 15 25 15 40 so there is 40 ml of maroon paint 3 a 80 8 10 and 10 5 50 so there is 50 ml red 10 3 30 so there is 30 ml blue 50 30 80 so there is 80 ml maroon b There are 10 batches of paint because each part changed from a value 1 ml to a value of 10 ml 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 171

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G6M2 LESSON 15 ZEARN MATH TEACHER LESSON MATERIALS ACTIVITY 1 RECAP Discussion should center around how students used snap cubes to answer the questions and their approach to the last two questions Invite some students to share their group s approach Ask YOUR NOTES How did the snap cubes help you solve the first few problems In one of the problems you were only given the total amount of maroon paint How did you find out the amounts of blue and yellow paint needed to produce 80 ml of maroon How did you approach the last question Add more cubes or use the same representation of 5 red cubes and 3 blue ones Discuss how the same 5 red cubes and 3 blue ones can be used to represent a total of 80 ml of blue paint Explain that this situation can be represented with a tape diagram A tape diagram is a horizontal strip that is partitioned into parts Each part like each snap cube represents a value It can be any value as long as the same value is used throughout Show a tape diagram representing a 5 3 ratio of red paint to blue paint yielding 80 ml of maroon paint Ask students where they see the 5 the 3 and the 80 being represented in the diagram Discuss how many batches of paint are represented 10 10 10 10 10 10 10 10 Show the tape diagram for green paint mixture discussed earlier Students should be able to say that the ratio of blue to yellow paint is 1 3 Ask What value each part of the diagram would have to take to show a 20 ml mixture of green paint How do you know 5 5 5 5 Guide students to see that if each of the 4 total parts must be equal in value and amount to 20 ml we could divide 20 by 4 to find out what each part represents 20 4 5 so each part represents 5 ml of paint 172 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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G6M2 LESSON 15 ANTICIPATED MISCONCEPTIONS Students may need help interpreting Suppose each cube represents 2 ml If necessary suggest they keep using one cube to represent 1 ml of paint So for example the second question would be represented by 5 stacks of 2 red cubes and 3 stacks of 2 blue cubes If they use that strategy each part of the tape diagram would represent one stack ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES SUPPORT FOR STUDENTS WITH DISABILITIES Strengths based Approach This activity leverages many natural strengths of students with ADHD LD and other concrete learners in terms of the use of hands on manipulatives This may be an opportunity for the teacher to highlight this strength in class and allow an individual with a disability to lead peer interactions discussions increasing buy in SUPPORT FOR ENGLISH LANGUAGE LEARNERS Representing MLR 8 Discussion Supports To help students connect ratio language and ratio reasoning invite a student to represent their reasoning using the snap cubes or with a tape diagram Press for details by requesting that students challenge an idea elaborate on an idea or give an example This will help students communicate with precise language Design Principle s Support sense making Concept Exploration Activity 2 SNEAKERS CHICKEN AND FRUIT JUICE Instructional Routine Anticipate Monitor Select Sequence Connect MLR5 Co Craft Questions and Problems This activity allows students to practice reasoning about situations involving ratios of two quantities and their sum It also introduces students to using parts in recipes e g 3 parts oil with 2 parts soy sauce and 1 part orange juice instead of more familiar units such as cups teaspoons milliliters etc Students may use tape diagrams to support their reasoning or they may use other representations learned so far discrete diagrams number lines tables 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 173

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G6M2 LESSON 15 ZEARN MATH TEACHER LESSON MATERIALS or equations All approaches are welcome as long as students use them to represent the situations appropriately to support their reasoning As students work monitor for different ways students reason about the problems with or without using tape diagrams YOUR NOTES LAUNCH Keep students in the same groups Provide graph paper and snap cubes any three colors Explain that they will now practice solving problems involving ratios and their combined quantities similar to the green and maroon paint in the previous task Draw students to a ratio that uses parts as its unit Ask students what they think one part means or amounts to and how situations expressed in terms of parts could be diagrammed Before students begin working make sure they understand that parts do not represent specific amounts that the value of one part can vary but the size of all parts is equal and that a tape diagram can be used to show these parts ACTIVITY 2 TASK 1 3 Solve each of the following problems and show your thinking If you get stuck consider drawing a tape diagram to represent the situation 1 The ratio of students wearing sneakers to those wearing boots is 5 to 6 If there are 33 students in the class and all of them are wearing either sneakers or boots how many of them are wearing sneakers 2 A recipe for chicken marinade says Mix 3 parts oil with 2 parts soy sauce and 1 part orange juice If you need 42 cups of marinade in all how much of each ingredient should you use 3 Elena makes fruit punch by mixing 4 parts cranberry juice to 3 parts apple juice to 2 parts grape juice If one batch of fruit punch includes 30 cups of apple juice how large is this batch of fruit punch 174 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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G6M2 LESSON 15 ZEARN MATH TEACHER LESSON MATERIALS STUDENT RESPONSE 1 15 students are wearing sneakers 33 11 3 The value of each unit is 3 3 5 15 Sneakers 3 3 3 3 3 Boots 3 3 3 3 3 3 YOUR NOTES 2 14 cups of soy sauce 42 6 7 The value of each unit is 7 7 3 21 There are 21 cups of oil 7 2 14 There are 14 cups of soy sauce 7 1 7 There are 7 cups of orange juice Oil 7 7 Soy Sauce 7 7 Orange Juice 7 7 3 90 cups of punch 30 3 10 The value of each unit is 10 10 3 30 There are 30 cups of apple juice 10 4 40 There are 40 cups cranberry juice 10 2 20 There are 20 cups grape juice 40 30 20 90 Cranberry Apple Grape DISCUSSION GUIDANCE Select students to share their reasoning Help students make connections between different representations especially any tape diagrams ANTICIPATED MISCONCEPTIONS Students may think of each segment of a tape diagram as representing each cube rather than as a flexible representation of an increment of a quantity Help them set up the tapes with the correct number of sections and then discuss how many parts there are in all 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 175

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G6M2 LESSON 15 SUPPORT FOR STUDENTS WITH DISABILITIES ZEARN MATH TEACHER LESSON MATERIALS Executive Functioning Eliminate Barriers Chunk this task into more manageable parts e g presenting one question at a time which will aid students who benefit from support with organizational skills in problem solving YOUR NOTES SUPPORT FOR ENGLISH LANGUAGE LEARNERS Speaking Reading MLR 5 Co Craft Questions Use this routine to help students interpret the language of ratios and to increase awareness of language used to talk about ratios Display only the first sentence of this problem The ratio of students wearing sneakers to those wearing boots is 5 to 6 and ask students to write down possible mathematical questions that could be asked about the situation Invite students to compare their questions before revealing the remainder of the question Listen for and amplify any questions involving ratios and their combined quantities Design Principle s Maximize meta awareness Support sense making Digital Lesson The ratio of students wearing sandals to those wearing sneakers is 5 1 If there are 30 students in the class and all of them are wearing either sandals or sneakers how many of them are wearing sandals Draw a tape diagram to support your thinking STUDENT RESPONSE If there are 30 students in the class then 25 students are wearing sandals 176 Sandals 5 Sneakers 5 5 5 5 5 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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G6M2 LESSON 15 Wrap Up ZEARN MATH TEACHER LESSON MATERIALS LESSON SYNTHESIS YOUR NOTES Today s ratio problems were different from the ones we ve worked on so far because they include an additional piece of information Can anyone identify what made these problems different They include the combined or total amount of the quantities in the ratio This is possible because in each problem there was only one unit of measure and the total of the quantities made sense in the context How can a tape diagram represent these types of situations Each part of the tape represents a particular value and the sum of those values represents the total amount How does changing the value of each part of the tape affect the total amount If the value is different the combined sum will be different Review the use of a tape diagram for representing and solving a problem involving the total amount TERMINOLOGY Tape diagram A tape diagram is a group of rectangles put together to represent a relationship between quantities For example this tape diagram shows a ratio of 30 gallons of yellow paint to 50 gallons of blue paint 10 10 10 If each rectangle were labeled 5 instead of 10 then the same picture could represent the equivalent ratio of 15 gallons of yellow paint to 25 gallons of blue paint 10 10 10 10 10 EXIT TICKET LAUNCH Provide access to graph paper 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 177

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G6M2 LESSON 15 ZEARN MATH TEACHER LESSON MATERIALS TASK The first floor of a house consists of a kitchen playroom and dining room The areas of the kitchen playroom and dining room are in the ratio 4 3 2 The combined area of these three rooms is 189 square feet What is the area of each room YOUR NOTES STUDENT RESPONSE Kitchen 21 21 21 Play room 21 21 21 Dining room 21 21 21 All three rooms amount to 9 units All three rooms make 189 square feet 189 9 21 so each part of the tape diagram represents 21 square feet The area of the kitchen is 84 square feet the area of the playroom is 63 square feet and the area of the dining room is 42 square feet 178 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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GRADE 6 MISSION 2 LESSON 16 Solving More Ratio Problems LEARNING GOALS LEARNING GOALS STUDENT FACING G6M2 LESSON 16 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Choose and create diagrams to help solve problems involving ratios and the total amount Compare and contrast orally different representations of and solution methods for the same problem Let s compare all our strategies for solving ratio problems I can choose and create diagrams to help think through my solution LEARNING TARGETS STUDENT FACING I can solve all kinds of problems about equivalent ratios I can use diagrams to help someone else understand why my solution makes sense Graph paper Rulers REQUIRED MATERIALS Tools for creating a visual display Any way for students to create work that can be easily displayed to the class Examples chart paper and markers whiteboard space and markers shared online drawing tool access to a document camera 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 179

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G6M2 LESSON 16 ZEARN MATH TEACHER LESSON MATERIALS In this lesson students use all representations they have learned in this mission double number lines tables and tape diagrams to solve ratio problems that involve the sum of the quantities in the ratio They consider when each tool might be useful and preferable in a given situation and why In so doing they make sense of situations and representations and are strategic in their choice of solution method YOUR NOTES Warm Up YOU TELL THE STORY Instructional Routines Think Pair Share MLR5 Co Craft Questions and Problems This warm up reminds students of previous work with tape diagrams and encourages a different way to reason with them Students are given only a tape diagram and are asked to generate a concrete context to go with the representation This warm up is a modified language routine MLR 5 and supports language development by using conversation skills to generate mathematical situations Students stories should have the following components the same unit for both quantities in the ratio a ratio of 7 3 scaling by 3 or 3 units per part They may also have a quantity of 30 that represents the sum of the two quantities in the ratio As students work identify a few different students whose stories are clearly described and are consistent with the diagram so that they can share later LAUNCH Ask students to share a few things they remember about tape diagrams from the previous lesson Students may recall that We draw one tape for every quantity in the ratio Each tape has parts that are the same size We draw as many parts as the numbers in the ratio show e g a 2 3 ratio we draw 2 parts in a tape and 3 parts in another tape 180 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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G6M2 LESSON 16 Each part represents the same value Tape diagrams can be used to think about a ratio of parts and the total amount ZEARN MATH TEACHER LESSON MATERIALS Tell students their job is to come up with a valid situation to match a given tape diagram Give students some quiet thinking time and then time to share their response with a partner YOUR NOTES WARM UP TASK 1 Describe a situation with two quantities that this tape diagram could represent 3 3 3 3 3 3 3 3 3 3 STUDENT RESPONSE 1 Answers vary Examples might be a One batch of purple paint is made by mixing 7 cups of blue with 3 cups of red In three batches there are 30 cups of purple paint which is made of 21 cups of blue paint and 9 cups of red paint b There are 30 fish in an aquarium The ratio of blue fish to red fish is 7 3 There are 21 blue fish and 9 red fish DISCUSSION GUIDANCE Invite a few students to share their stories with the class As they share consider recording key details about each story for all students to see Then ask students to notice similarities in the different scenarios Guide students to see that they all involve the same units for both quantities of the ratio a ratio of 7 to 3 and either 3 units per part or scaling by 3 They may also involve an amount of 30 units representing the sum of the two quantities 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 181

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G6M2 LESSON 16 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES ANTICIPATED MISCONCEPTIONS Students may misunderstand the meaning of the phrase with two quantities and simply come up with a situation involving ten identical groups of three Point out that the phrase means that the row of seven groups of three should represent something different than the row of three groups of three Students may also come up with a situation involving different units for example quantity purchased and cost or distance traveled and time elapsed Remind them that the parts of tape are meant to represent the same value so we need a situation that uses the same units for both parts of the ratio SUPPORT FOR STUDENTS WITH DISABILITIES Conceptual Processing Eliminate Barriers Assist students to see the connections between new problems and prior work Students may benefit from a review of different representations to activate prior knowledge Concept Exploration Activity 1 A TRIP TO THE AQUARIUM Instructional Routines Poll the Class MLR6 Three Reads MLR8 Discussion Supports This task prompts students to solve a single problem using a triple number line a table or a tape diagram Either assign each student a representation or allow students to choose the representation they prefer During the following discussion they will compare and contrast the three representations and identify the relative merits of the different representations for the different problems Students will have varying opinions about which representation they prefer and why Their views may stem from observations such as Number lines and tables involve scaling up individual quantities to find the total Tape diagrams involve starting with the total and breaking it down into equal groups It is hard to take shortcuts with number line diagrams We can use the number line diagram or the table efficiently by thinking 17 times what is 85 This value tells us how many batches of tickets the teacher had to order 182 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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When using a tape diagram it was easier to count 17 groups and compute 85 17 to find how many tickets each group needs G6M2 LESSON 16 ZEARN MATH TEACHER LESSON MATERIALS Students may struggle to represent the problem with a tape diagram As they work notice any trends that may need to be addressed with the group YOUR NOTES LAUNCH Tell students that they will now solve a ratio problem in one of three different ways Either assign each student one of the representations or instruct them to choose one representation to use Give students quiet think time to complete the activity and then optionally time to share their responses with a small group or in pairs ACTIVITY 1 TASK 1 2 A teacher is planning a class trip to the aquarium The aquarium requires 2 chaperones for every 15 students The teacher plans accordingly and orders a total of 85 tickets How many tickets are for chaperones and how many are for students 1 Solve this problem in one of three ways a Use a triple number line Kids 0 15 0 2 0 17 Chaperones Total b Use a table Fill rows as needed 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 183

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G6M2 LESSON 16 c Kids 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 85 Chaperones 5 ZEARN MATH TEACHER LESSON MATERIALS 5 There are a total of 17 boxes 85 17 5 Each part of tape is worth 5 tickets so we can see that the teacher ordered 75 tickets for kids and 10 tickets for chaperones YOUR NOTES 2 Answers vary DISCUSSION GUIDANCE Before debriefing as a small group consider arranging students in groups of 3 where a group includes one student who used each representation Give them time to see if they got the same answer and compare and contrast the representations Solicit students reactions to the three strategies encouraging them to identify what is similar and different about the approaches Consider polling the small group for their preferred strategy Ask 1 2 students favoring each method to explain why This can also be a discussion about what worked well in this or that approach and what might make this or that approach more complete or easy to understand While there is no right or wrong answer with regards to their preferred strategy look out for unsupported reasoning or misunderstandings Some students may prefer the tape diagram because the solution path seems more direct but caution them against trying to use a tape diagram any time they see a ratio problem Because tape diagrams involve equal sized parts they can only be used to represent quantities with the same unit If different units are involved the parts of one tape and those of the other will not represent an identical quantity ANTICIPATED MISCONCEPTIONS The number line and table representations are organized similarly For example one could make progress with both of them simply by skip counting and keeping an eye out for a total of 85 people The tape diagram though is organized in a much different way Equivalent ratios are not listed out but rather equivalent ratios arise from thinking about how the diagram could represent any number of batches Students may thus mistakenly treat the tape diagram like a double number line diagram they may start writing 15 30 45 etc in the kids tape for example Once this plays out students may self regulate once they notice there are only 2 boxes in the chaperones row But they may decide to just draw more boxes Reorient these students by asking how many parts of tape there are 17 and reminding them each part of tape represents equal numbers of people and that there are 85 total people The presentation of correct work during the discussion could be used as an opportunity to remediate as well For example 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 185

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G6M2 LESSON 16 ZEARN MATH TEACHER LESSON MATERIALS consider asking a student to explain what they understand about another student s correct work SUPPORT FOR ENGLISH LANGUAGE LEARNERS YOUR NOTES Reading MLR 6 Three Reads This is one of the first times Math Language Routine 6 is suggested as a support in this course In this routine students are supported in reading a mathematical text situation or word problem three times each with a particular focus During the first read students focus on comprehending the situation during the second read students identify quantities during the third read students brainstorm possible strategies to answer the question The question to be answered does not become a focus until the third read so that students can make sense of the whole context before rushing to a solution The purpose of this routine is to support students reading comprehension as they make sense of mathematical situations and information through conversation with a partner Design Principle s Support sense making How It Happens 1 In the first read students read the problem with the goal of comprehending the situation Do not reveal the questions with the three options for solving at this point Invite a student to read the description of the situation aloud while everyone else reads with them and then ask What is this situation about Allow one minute to discuss with a partner and then share with the whole small group A clear response would be buying tickets for a class trip to an aquarium 2 In the second read students analyze the mathematical structure of the story by naming quantities Invite students to read the problem aloud with their partner or select a different student to read to the small group and then prompt students by asking What can be counted or measured in this situation For now we don t need to focus on how many or how much of anything but what can we count in this situation Give students quiet think time followed by time to share with their partner Quantities may include total number of tickets bought number of students number of chaperones ratio of chaperones to students 3 In the third read students brainstorm possible mathematical strategies to answer the question How many tickets are for chaperones and how many are for students Still do not reveal the three given options for solving Invite students to read the problem aloud with their partner or select a different student to read to the small group Instruct students to think of ways to approach the question without actually solving the problem Consider using these questions to prompt students What strategy or method would you try first How could a diagram help you approach this question and Can you think of a different way to solve it Give students quiet think time followed by time to discuss with their partner Provide this sentence frame as partners discuss One way to approach the question would be to 186 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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Sample responses include I would figure out possibilities for the number of students and number of chaperones and see which one adds up to 85 I know that if I divide 85 by 2 it won t work because there are more students than chaperones so the groups are not equal so I know I have to split it up differently I would draw a tape diagram to figure out different lengths that fit the ratio of students to chaperones and I would use a double number line table to scale up the ratio of students to chaperones This will help students concentrate on making sense of the situation before rushing to a solution or method 4 As partners are discussing their strategies select 1 2 students to share their ideas with the whole small group Listen for quantities that were mentioned during the second read and take note of strategies that make explicit the relationships between number of tickets and number of students number of tickets and number of chaperones or number of students and number of chaperones As students are presenting their strategies to the whole small group create a display that summarizes ideas about the question 5 Post the display where all students can use it as a reference and finally reveal the actual problems and ensure that a variety of strategies are chosen by students G6M2 LESSON 16 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES Speaking MLR 8 Discussion Supports Provide sentence frames for students to state their reasoning e g I liked this method of solving the problem because This way worked best because The strategy is the same as different from the strategy because The helps students place extra attention on the language used to engage in mathematical communication and reasoning Design Principle s Maximize meta awareness Optimize output for generalization SUPPORT FOR STUDENTS WITH DISABILITIES Conceptual Processing Processing Time Check in with individual students as needed to assess for comprehension during each step of the activity 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 187

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G6M2 LESSON 16 ZEARN MATH TEACHER LESSON MATERIALS Digital Lesson Write a word problem that could describe this situation Red paint ml YOUR NOTES 0 5 10 15 20 25 0 3 6 9 12 15 0 8 16 24 32 40 Blue paint ml Purple paint ml Red paint ml 5 5 5 Blue paint ml 5 5 5 5 5 Red paint ml Blue paint ml Purple paint ml 5 3 8 25 15 40 STUDENT RESPONSE Answers vary Example response It takes 5 ml of red paint and 3 ml of blue paint to make 1 batch of purple paint I bought a total of 40 ml of blue and red paint so I can make 5 batches of purple paint Wrap Up LESSON SYNTHESIS This lesson was all about understanding that there are different valid representations to use for problems involving equivalent ratios For some problems one representation is easier to 188 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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use than others There are no hard and fast rules As long as the diagram correctly shows the mathematics and the problem solver can explain it it s okay Some guidelines to draw out are G6M2 LESSON 16 ZEARN MATH TEACHER LESSON MATERIALS Tape diagrams are most likely to be helpful when the parts of the ratio have the same kind of units and the sum of the quantities is meaningful in the context For example cups to cups miles to miles boxes moved to boxes moved YOUR NOTES Number lines are a good choice when it helps to visualize how far apart numbers are from each other They are harder to use with very big or very small numbers Tables work well in almost all situations Ask students to articulate good habits when solving equivalent ratio problems with the different representations Some ideas might include Label each part of the diagram with what it represents Use brackets to indicate total amounts Make sure you read what the question is asking and answer it Make sure you make the answer easy to find Include units in your answer For example instead of just writing 4 write 4 cups EXIT TICKET You are having a pizza making party You will need 6 ounces of dough and 4 ounces of sauce for each person at the party including yourself the host Once you have a total count of guests you buy exactly the needed amount of all the ingredients The dough and sauce that you buy weigh 130 ounces all together 1 How many ounces of dough did you buy 2 How many ounces of sauce did you buy 3 How many guests are coming to the party STUDENT RESPONSE 1 78 ounces of dough 2 52 ounces of sauce 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 189

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GRADE 6 MISSION 2 LESSON 17 A Fermi Problem G6M2 LESSON 17 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES TEACHER INSTRUCTION ONLY ZEARN MATH TIP There is no Independent Digital Lesson for this Lesson We recommend teaching this lesson whole group with your students Apply reasoning developed throughout this mission to an unfamiliar problem LEARNING GOALS Decide what information is needed to solve a real world problem Make simplifying assumptions about a real world situation LEARNING GOALS STUDENT FACING LEARNING TARGETS STUDENT FACING REQUIRED MATERIALS Let s solve a Fermi problem I can decide what information I need to know to be able to solve a realworld problem about ratios and rates I can apply what I have learned about ratios and rates to solve a more complicated problem Tools for creating a visual display Any way for students to create work that can be easily displayed to the class Examples chart paper and markers whiteboard space and markers shared online drawing tool access to a document camera 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 191

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G6M2 LESSON 17 ZEARN MATH TEACHER LESSON MATERIALS This mission concludes with an opportunity for students to apply the reasoning developed so far to solve an unfamiliar Fermi type problem Students must take a problem that is not well posed and make assumptions and approximations to simplify the problem so that it can be solved which requires sense making and perseverance To understand what the problem entails students break down larger questions into more manageable sub questions They need to make assumptions plan an approach and reason with the mathematics they know YOUR NOTES Warm Up FIX IT This activity encourages students to reason informally about equivalent ratios in a context similar to what they will see in the next lesson During the discussion emphasize the use of ratios and ratio reasoning in explaining how we know two batches of a recipe will taste the same LAUNCH Arrange students in groups of 2 Display the image for all to see Optionally instead of the abstract image you could bring in a clear glass milk and cocoa powder Pour 1 cup of milk into the glass add 5 tablespoons of cocoa powder and introduce the task that way Tell students to give a signal when they have an answer and a strategy Give students some quiet think time WARM UP TASK 1 192 Andre likes a hot cocoa recipe with 1 cup of milk and 3 tablespoons of cocoa He poured 1 cup of milk but accidentally added 5 tablespoons of cocoa 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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1 How can you fix Andre s mistake and make his hot cocoa taste like the recipe 2 Explain how you know your adjustment will make Andre s hot cocoa taste the same as the one in the recipe STUDENT RESPONSE G6M2 LESSON 17 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES 1 Answers vary Possible strategies Add 1 more tablespoon of cocoa and 1 cup of milk or add 23 cup of milk 2 The ratios for the recipe and for the fixed mixture are equivalent DISCUSSION GUIDANCE Invite students to share their strategies with the class and record them for all to see After each explanation ask the class if they agree or disagree and how they know two hot cocoas will taste the same SUPPORT FOR STUDENTS WITH DISABILITIES Conceptual Processing Eliminate Barriers Begin with a physical demonstration of the actions that occur in a situation Concept Exploration Activity 1 WHO WAS FERMI Instructional Routine MLR3 Clarify Critique Correct In this first activity students are introduced to the type of thinking useful for Fermi problems The purpose of this activity is not to come up with an answer but rather to see different ways to break a Fermi problem down into smaller questions that can be measured estimated or calculated Much of the appeal of Fermi problems is in making estimates for things that in modern times we could easily look up To make this lesson more fun and interesting challenge students to work without performing any internet searches 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 193

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G6M2 LESSON 17 ZEARN MATH TEACHER LESSON MATERIALS As students work notice the range of their estimates and the sub questions they formulate to help them answer the large questions Some examples of productive sub questions might be What information do we know What information can be measured YOUR NOTES What information cannot be measured but can be calculated What assumptions should we make LAUNCH Open the activity with one or two questions that your students may find thought provoking Some ideas How many times does your heart beat in a year How many hours of television do you watch in a year Some research has shown that it takes 10 000 hours of practice for a person to achieve the highest level of performance in any field sports music art chess programming etc If you aspire to be a top performer in a field you love as Michael Jordan in basketball Tiger Woods in golf Maya Angelou in literature etc how many years would it take you to meet that 10 000 hour benchmark if you start now How old would you be Give students a moment to ponder a question and make a rough estimate Then share that the questions above are called Fermi problems named after Enrico Fermi an Italian physicist who loved to think up and discuss problems that are impossible to measure directly but can be roughly estimated using known facts and calculations Here are some other examples of Fermi problems How long would it take to paddle across the Pacific Ocean How much would it cost to replace all the windows on all the buildings in the United States Share the questions above or select a few other Fermi type questions that are likely to intrigue your students Have some resources on hand to support the investigation on your chosen questions e g have a globe handy if the question about paddling across the Pacific is on your short list As a class decide on one question to pursue For this activity consider giving students the option to either work independently or in groups of two 194 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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G6M2 LESSON 17 ACTIVITY 1 TASK 1 2 ZEARN MATH TEACHER LESSON MATERIALS Answer the problems about the Fermi question your class chose in your notes YOUR NOTES 1 Record the Fermi question that your class will explore together 2 Make an estimate of the answer If making an estimate is too hard consider writing down a number that would definitely be too low and another number that would definitely be too high 3 What are some smaller sub questions we would need to figure out to reasonably answer our bigger question 4 Think about how the smaller questions above should be organized to answer the big question Label each smaller question with a number to show the order in which they should be answered If you notice a gap in the set of sub questions i e there is an unlisted question that would need to be answered before the next one could be tackled write another question to fill the gap STUDENT RESPONSE Answers vary depending on the question explored For How long would it take to paddle across the Pacific Ocean some sub questions might be What is the distance across the Pacific Ocean At what speed can you paddle a boat Do we assume that someone paddles continuously or that they take breaks to sleep DISCUSSION GUIDANCE First ask students to share their estimates Note the lowest and highest estimates and point out that it is perfectly acceptable for an estimate to be expressed as a range of values rather than a single value Ask students to share some of their smaller questions Then discuss how you might come up with answers to these smaller questions which likely revolve around what information is known can be measured or can be computed Also discuss how our assumptions about the situation affect how we solve the problem 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 195

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G6M2 LESSON 17 SUPPORT FOR STUDENTS WITH DISABILITIES ZEARN MATH TEACHER LESSON MATERIALS Executive Functioning Eliminate Barriers Provide a project checklist that chunks the various steps of the project into a set of manageable tasks YOUR NOTES SUPPORT FOR ENGLISH LANGUAGE LEARNERS Conversing MLR 3 Clarify Critique Correct Tell students you are going to show them examples of questions that are not Fermi problems then display How many students are in our classroom right now and How tall is a stack of 20 pennies Ask pairs of students to select and critique one of the questions and then collaborate to write a new version that represents a Fermitype question Invite students to share their new Fermi questions and ask the class to identify the changes that made them Fermi questions This provides students with an opportunity to produce language that communicates their understanding of the characteristics of a Fermi problem Design Principle s Cultivate conversation Concept Exploration Activity 2 RESEARCHING YOUR OWN FERMI PROBLEM Instructional Routine MLR7 Compare and Connect Group Presentations This activity asks students to choose or pose a Fermi problem and solve it with the aim of promoting the reasoning and tools developed in this mission Students brainstorm potential problems choose one and after your review use a graphic organizer to help them formulate the sub questions that could support their problem solving They go on to solve their chosen Fermi problem To encourage ratio reasoning and the use of tools such as double number lines and tables look for problems that involve two quantities Questions that involve one quantity can be solved with multi step multiplication and without ratio reasoning e g How many pens are there at the school involves only one quantity the number of pens But a problem such as How much would it cost to replace all the windows on all buildings in the U S or How long would it take to paddle across the Pacific Ocean involves accounting for two quantities at the same time cost and number of windows or time and distance across the Pacific and is more likely to elicit ratio reasoning Keep this in mind as you help students sift through their ideas 196 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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G6M2 LESSON 17 LAUNCH ZEARN MATH TEACHER LESSON MATERIALS Explain to students that they will now brainstorm some Fermi problems they are interested in answering and select one to solve Consider sharing a few more examples of Fermi problems to jumpstart their thinking YOUR NOTES How much would it cost to charge all the students cell phones in the school for a month How much does it cost to operate a car for a year How long would it take to make a sandwich for everyone living in our town How long would it take to read the dictionary out loud How long would it take to give every dog in America a bath Tell students that once they have a few good ideas they should pause and get your attention so that you could help to decide on the one problem to pursue Arrange students in groups of 2 if desired Provide tools for creating a visual display ACTIVITY 2 TASK 1 3 Answer the problems about your own Fermi questions in your notes 1 Brainstorm at least five Fermi problems that you want to research and solve If you get stuck consider starting with How much would it cost to or How long would it take to 2 Pause here so your teacher can review your questions and approve one of them 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 197

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G6M2 LESSON 17 3 Use the graphic organizer to break your problem down into sub questions ZEARN MATH TEACHER LESSON MATERIALS Subquestion Subquestion Answer Answer YOUR NOTES Fermi problem Subquestion Subquestion Answer Answer 4 Find the information you need to get closer to answering your question Measure make estimates and perform any necessary calculations If you get stuck consider using tables or double number line diagrams 5 Create a visual display that includes your Fermi problem and your solution Organize your thinking so it can be followed by others STUDENT RESPONSE Answers vary DISCUSSION GUIDANCE Display students posters or visual presentations throughout the classroom Consider asking some students or all if time permits to present their problems and solutions to the class Notice and highlight instances of ratio and rate reasoning particularly productive use of double number lines or tables ANTICIPATED MISCONCEPTIONS Students may think of problems that do not lend themselves to ratio reasoning because they only involve one quantity If they have trouble coming up with any good options offer them some examples It may also be helpful to have a list of sample problems that students could refer to in creating their own problem 198 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license

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SUPPORT FOR STUDENTS WITH DISABILITIES Executive Functioning Eliminate Barriers Provide a project checklist that chunks the various steps of the project into a set of manageable tasks Receptive Expressive Language Peer Tutors Pair students with their previously identified peer tutors to aid in comprehension and expression of understanding G6M2 LESSON 17 ZEARN MATH TEACHER LESSON MATERIALS YOUR NOTES SUPPORT FOR ENGLISH LANGUAGE LEARNERS Representing Conversing MLR 7 Compare and Connect Use this routine to prepare students for the whole class discussion Invite students to quietly circulate and read at least 2 of the posters or visual presentations in the room Give students quiet think time to consider what is the same and what is different about the questions and displays Next ask students to find a partner to discuss what they noticed Listen for and amplify observations that include mathematical language and reasoning about double number lines or tables Design Principle s Cultivate conversation Wrap Up LESSON SYNTHESIS The debrief and presentation of student work provides opportunities to summarize takeaways from this lesson Aside from opportunities to point out how ratio reasoning and the use of representations can help us tackle difficult problems this lesson makes explicit some aspects of mathematical modeling Highlight instances where students had to make an estimate in order to proceed figured out what additional information they would need to make progress or made simplifying assumptions 2023 Zearn Licensed to you pursuant to Zearn s Terms of Use This work is a derivative of Open Up Resources 6 8 Math curriculum which is available to download for free at openupresources org and used under the CC BY 4 0 license 199