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STUDENT EDITION Grade 8 VOLUME 3 Mission 7 Exponents and Scientific Notation Mission 8 Pythagorean Theorem and Irrational Numbers Mission 9 Putting It All Together NAME

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 used under the CC BY 4 0 license Download the original for free at openupresources org Zearn is a registered trademark Printed in the U S A ISBN 979 8 88868 881 6

Table of Contents Mission 7 Lesson 1 Exponent Review 3 Lesson 2 Multiplying Powers of Ten 9 Lesson 3 Powers of Powers of 10 15 Lesson 4 Dividing Powers of 10 21 Lesson 5 Negative Exponents with Powers of 10 27 Lesson 6 What about Other Bases 35 Lesson 7 Practice with Rational Bases 41 Lesson 8 Combining Bases 47 Lesson 9 Describing Large and Small Numbers Using Powers of 10 53 Lesson 10 Representing Large Numbers on the Number Line 61 Lesson 11 Representing Small Numbers on the Number Line 67 Lesson 12 Applications of Arithmetic with Powers of 10 73 Lesson 13 Definition of Scientific Notation 79 Lesson 14 Multiplying Dividing and Estimating with Scientific Notation 85 Lesson 15 Adding and Subtracting with Scientific Notation 91 Lesson 16 Is a Smartphone Smart Enough to Go to the Moon 97 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

Mission 8 Lesson 1 The Areas of Squares and Their Side Lengths 103 Lesson 2 Side Lengths and Areas 111 Lesson 3 Rational and Irrational Numbers 117 Lesson 4 Square Roots on the Number Line 123 Lesson 5 Reasoning about Square Roots 129 Lesson 6 Finding Side Lengths of Triangles 135 Lesson 7 A Proof of the Pythagorean Theorem 143 Lesson 8 Finding Unknown Side Lengths 149 Lesson 9 The Converse 155 Lesson 10 Applications of the Pythagorean Theorem 161 Lesson 11 Finding Distances in the Coordinate Plane 167 Lesson 12 Edge Lengths and Volumes 173 Lesson 13 Cube Roots 177 Lesson 14 Decimal Representations of Rational Numbers 181 Lesson 15 Infinite Decimal Expansions 187 Mission 9 Lesson 1 Mathematical Modeling 195 Lesson 2 Tessellations of the Plane 201 Terminology 209 iv 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

Grade 8 Mission 7 Exponents and Scientific Notation

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ZEARN MATH STUDENT EDITION G8M7 LESSON 1 Lesson 1 Exponent Review Let s review exponents Warm Up 1 Which expression does not belong Be prepared to share your reasoning 23 8 32 22 21 Concept Exploration ACTIVITY 1 2 Mai and Andre found an old brass bottle that contained a magical genie They freed the genie and it offered them each a magical 1 coin as thanks The magic coin turned into 2 coins on the first day The 2 coins turned into 4 coins on the second day The 4 coins turned into 8 coins on the third day This doubling pattern continued for 28 days Mai was trying to calculate how many coins she would have and remembered that instead of writing 1 2 2 2 2 2 2 for the number of coins on the 6th day she could just write 26 1 The number of coins Mai had on the 28th day is very very large Write an expression to represent this number without computing its value 2 Andre s coins lost their magic on the 25th day so Mai has a lot more coins than he does How many times more coins does Mai have than Andre 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

G8M7 LESSON 1 ZEARN MATH STUDENT EDITION ACTIVITY 2 4 3 After a while Jada picks up a coin that seems different than the others She notices that the next day only half of the coin is left On the second day only 14 of the coin is left On the third day 18 of the coin remains 1 What fraction of the coin remains after 6 days 2 What fraction of the coin remains after 28 days Write an expression to describe this without computing its value 3 Does the coin disappear completely If so after how many days 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

ZEARN MATH STUDENT EDITION G8M7 LESSON 1 Lesson Summary Exponents make it easy to show repeated multiplication For example 26 2 2 2 2 2 2 One advantage to writing 26 is that we can see right away that this is 2 to the sixth power When this is written out using multiplication 2 2 2 2 2 2 we need to count the number of factors Imagine writing out 2100 using multiplication Let s say you start out with one grain of rice and that each day the number of grains of rice you have doubles So on day one you have 2 grains on day two you have 4 grains and so on When we write 225 we can see from the expression that the rice has doubled 25 times So this notation is not only convenient but it also helps us see structure in this case we can see right away that it is on the 25th day that the number of grains of rice has doubled That s a lot of rice more than a cubic meter 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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ZEARN MATH STUDENT EDITION Name G8M7 LESSON 1 Date GRADE 8 MISSION 7 LESSON 1 Exit Ticket 1 What is the value of 34 2 How many times bigger is 315 compared to 312 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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ZEARN MATH STUDENT EDITION G8M7 LESSON 2 Lesson 2 Multiplying Powers of Ten Let s explore patterns with exponents when we multiply powers of 10 Warm Up 1 1 Clare says she sees 100 Tyler says he sees 1 Mai says she sees 100 Who do you agree with 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

G8M7 LESSON 2 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 10 In the diagram the medium rectangle is made up of 10 small squares The large square is made up of 10 medium rectangles 1 How could you represent the large square as a power of 10 2 If each small square represents 102 then what does the medium rectangle represent The large square 3 If the medium rectangle represents 105 then what does the large square represent The small square 4 If the large square represents 10100 then what does the medium rectangle represent The small square 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

ZEARN MATH STUDENT EDITION G8M7 LESSON 2 ACTIVITY 2 3 Solve the problems below 1 a Complete the table to explore patterns in the exponents when multiplying powers of 10 You may skip a single box in the table but if you do be prepared to explain why you skipped it Expression Expanded Single power of 10 102 103 10 10 10 10 10 105 104 103 104 104 10 10 10 10 10 10 10 10 1018 1023 b If you chose to skip one entry in the table which entry did you skip Why 2 a Use the patterns you found in the table to rewrite 10n 10m as an equivalent expression with a single exponent like 10 b Use your rule to write 104 100 with a single exponent What does this tell you about the value of 100 3 The state of Georgia has roughly 107 human residents Each human has roughly 1013 bacteria cells in his or her digestive tract How many bacteria cells are there in the digestive tracts of all the humans in Georgia 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

G8M7 LESSON 2 ZEARN MATH STUDENT EDITION Lesson Summary In this lesson we developed a rule for multiplying powers of 10 multiplying powers of 10 corresponds to adding the exponents together To see this multiply 105 and 102 We know that 105 has five factors that are 10 and 102 has two factors that are 10 That means that 105 102 has 7 factors that are 10 105 102 10 10 10 10 10 10 10 107 This will work for other powers of 10 too So 1014 1047 1061 This rule makes it easier to understand and work with expressions that have exponents 12 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

ZEARN MATH STUDENT EDITION Name G8M7 LESSON 2 Date GRADE 8 MISSION 7 LESSON 2 Exit Ticket 1 Rewrite 1032 1036 using a single exponent 2 Each year roughly 106 computer programmers each make about 105 How much money is this all together Express your answer both as a power of 10 and as a dollar amount 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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ZEARN MATH STUDENT EDITION G8M7 LESSON 3 Lesson 3 Powers of Powers of 10 Let s look at powers of powers of 10 Warm Up 1 What is the volume of a giant cube that measures 10 000 km on each side 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

G8M7 LESSON 3 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 1 Answer the questions about the table below a Complete the table to explore patterns in the exponents when raising a power of 10 to a power You may skip a single box in the table but if you do be prepared to explain why you skipped it expression expanded single power of 10 103 2 10 10 10 10 10 10 106 102 5 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 104 2 108 11 b If you chose to skip one entry in the table which entry did you skip Why 16 2 Use the patterns you found in the table to rewrite 10m n as an equivalent expression with a single exponent like 10 3 If you took the amount of oil consumed in 2 months in 2013 worldwide you could make a cube of oil that measures 103 meters on each side How many cubic meters of oil is this Do you think this would be enough to fill a pond a lake or an ocean 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

ZEARN MATH STUDENT EDITION G8M7 LESSON 3 ACTIVITY 2 3 Andre and Elena want to write 102 102 102 with a single exponent Andre says When you multiply powers with the same base it just means you add the exponents so 102 102 102 102 2 2 106 Elena says 102 is multiplied by itself 3 times so 102 102 102 102 3 102 3 105 Do you agree with either of them 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 17

G8M7 LESSON 3 ZEARN MATH STUDENT EDITION Lesson Summary In this lesson we developed a rule for taking a power of 10 to another power Taking a power of 10 and raising it to another power is the same as multiplying the exponents See what happens when raising 104 to the power of 3 104 3 104 104 104 1012 This works for any power of powers of 10 For example 106 11 1066 This is another rule that will make it easier to work with and make sense of expressions with exponents 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

ZEARN MATH STUDENT EDITION Name G8M7 LESSON 3 Date GRADE 8 MISSION 7 LESSON 3 Exit Ticket Here are some equivalent ways of writing 104 10 000 10 103 102 2 Write as many expressions as you can that have the same value as 106 Focus on using exponents and multiplication 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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ZEARN MATH STUDENT EDITION G8M7 LESSON 4 Lesson 4 Dividing Powers of 10 Let s patterns with exponents when we divide powers of 10 Warm Up 1 What is the value of the expression 25 34 32 2 36 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 21

G8M7 LESSON 4 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 Complete the table and answer the questions below 1 a Complete the table to explore patterns in the exponents when dividing powers of 10 Use the expanded column to show why the given expression is equal to the single power of 10 You may skip a single box in the table but if you do be prepared to explain why you skipped it Expression Expanded Single Power 104 102 10 10 10 10 10 10 10 10 1 10 10 10 10 10 10 102 10 10 10 10 10 10 10 10 10 10 1 10 10 10 10 10 10 10 106 103 1043 1017 b If you chose to skip one entry in the table which entry did you skip Why 22 2 10 as an equivalent expression of the form 10 Use the patterns you found in the table to rewrite 10 3 It is predicted that by 2050 there will be 1010 people living on Earth At that time it is predicted there will be approximately 1012 trees How many trees will there be for each person n m 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

ZEARN MATH STUDENT EDITION G8M7 LESSON 4 ACTIVITY 2 3 1 So far we have looked at powers of 10 with exponents greater than 0 What would happen to our patterns if we included 0 as a possible exponent Write 1012 100 with a power of 10 with a single exponent using the appropriate exponent rule Explain or show your reasoning a What number could you multiply 1012 by to get this same answer 2 Write 10 with a single power of 10 using the appropriate exponent rule Explain or show your 10 reasoning 8 0 a What number could you divide 108 by to get this same answer 3 If we want the exponent rules we found to work even when the exponent is 0 then what does the value of 100 have to be 4 Noah says If I try to write 100 expanded it should have zero factors that are 10 so it must be equal to 0 Do you agree Discuss with your 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 23

G8M7 LESSON 4 ZEARN MATH STUDENT EDITION Lesson Summary In an earlier lesson we learned that when multiplying powers of 10 the exponents add together For example 106 103 109 because 6 factors that are 10 multiplied by 3 factors that are 10 makes 9 factors that are 10 all together We can also think of this multiplication equation as division 106 10 103 9 So when dividing powers of 10 the exponent in the denominator is subtracted from the exponent in the numerator This makes sense because 109 103 106 103 106 1 106 106 103 103 103 33 This rule works for other powers of 10 too For example 10 1023 10 because 23 factors that are 10 in the numerator and in the denominator are used to make 1 leaving 33 factors remaining 56 This gives us a new exponent rule 10n n m 10m 10 So far this only makes sense when n and m are positive exponents and n m but we can extend this 6 6 0 rule to include a new power of 10 100 If we look at 10 100 using the exponent rule gives 10 which is equal to 106 So dividing 106 by 100 doesn t change its value That means that if we want the rule to work when the exponent is 0 then it must be that 100 1 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

ZEARN MATH STUDENT EDITION Name G8M7 LESSON 4 Date GRADE 8 MISSION 7 LESSON 4 Exit Ticket 11 Why is 10 10 equal to 10 Explain or show your thinking 15 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 25

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ZEARN MATH STUDENT EDITION G8M7 LESSON 5 Lesson 5 Negative Exponents with Powers of 10 Let s see what happens when exponents are negative Warm Up 1 Solve each equation mentally 100 x 1 10 100 1 x 10 x 0 100 10 100 x 1 000 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 27

G8M7 LESSON 5 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 Complete the table to explore what negative exponents mean Then solve the problems below 10 Using exponents 103 As a decimal 1 000 0 10 102 10 10 0 01 1 1 100 1 10 101 1 0 As a fraction 28 10 1 1 000 1 As you move toward the left each number is being multiplied by 10 What is the multiplier as you move right 2 How do each of these multipliers affect the placement of the decimal 3 Use the patterns you found in the table to write 10 7 as a fraction 4 Use the patterns you found in the table to write 10 5 as a decimal 5 1 Write 100 000 000 using a single exponent 6 Use the patterns in the table to write 10 n as a fraction 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

ZEARN MATH STUDENT EDITION G8M7 LESSON 5 ACTIVITY 2 3 For each table match the expressions that describe repeated multiplication in the same way Then solve the problems 1 a Match the expressions that describe repeated multiplication in the same way 102 3 1 1 1 10 10 10 10 10 10 102 3 1 1 1 1 1 1 10 10 10 10 10 10 10 2 3 10 2 3 1 10 1 101 1 10 1 101 1 10 1 101 10 10 10 10 10 10 b Write 102 3 as a power of 10 with a single exponent Be prepared to explain your reasoning 2 a Match the expressions that describe repeated multiplication in the same way 102 105 101 1 1 1 1 1 10 10 10 10 10 102 10 5 10 10 10 10 10 10 10 10 2 105 10 2 10 5 1 10 1 10 1 10 10 10 10 10 10 1 10 10 10 101 101 101 101 b Write 10 10 5 as a power of 10 with a single exponent Be prepared to explain your reasoning 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 29

G8M7 LESSON 5 ZEARN MATH STUDENT EDITION 3 a Match the expressions that describe repeated multiplication in the same way 104 103 1 1 1 10 10 10 10 10 10 10 104 10 3 1 1 1 1 1 1 1 10 10 10 10 10 10 10 10 4 103 1 1 1 1 10 10 10 10 10 10 10 10 4 10 3 10 10 10 10 10 10 10 b Write 10 4 103 as a power of 10 with a single exponent Be prepared to explain your reasoning 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

ZEARN MATH STUDENT EDITION G8M7 LESSON 5 Lesson Summary 1 When we multiply a positive power of 10 by 10 the exponent decreases by 1 1 108 10 107 This is true for any positive power of 10 We can reason in a similar way that multiplying by 2 factors that 1 are 10 decreases the exponent by 2 1 2 10 108 106 1 2 That means we can extend the rules to use negative exponents if we make 10 2 10 Just as 102 is two 1 factors that are 10 10 2 is two factors that are 10 More generally the exponent rules we have developed are true for any integers n and m if we make 1 n 10 n 10 101 n 10 n m Here is an example of extending the rule 10 to use negative exponents m 10 n 103 3 5 2 105 10 10 To see why notice that 103 103 103 1 1 105 103 102 103 102 102 which is equal to 10 2 Here is an example of extending the rule 10m n 10mn to use negative exponents 10 2 3 10 2 3 10 6 1 1 To see why notice that 10 2 10 10 This means that 1 1 3 1 1 1 1 1 1 10 2 3 10 10 10 10 10 10 10 10 101 6 10 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 31

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ZEARN MATH STUDENT EDITION Name G8M7 LESSON 5 Date GRADE 8 MISSION 7 LESSON 5 Exit Ticket Mark each of the following equations as true or false Explain or show your reasoning 1 10 5 105 2 102 3 10 2 3 3 103 11 1014 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 33

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ZEARN MATH STUDENT EDITION G8M7 LESSON 6 Lesson 6 What about Other Bases Let s explore exponent patterns with bases other than 10 Warm Up 1 Is each statement true or false Be prepared to explain your reasoning 1 35 46 2 3 2 32 3 3 3 33 4 5 2 52 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

G8M7 LESSON 6 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 Complete the table to show what it means to have an exponent of zero or a negative exponent Then answer the questions 2 Value 16 Exponent form 24 2 2 2 2 2 2 1 2 36 2 1 As you move toward the left each number is being multiplied by 2 What is the multiplier as you move toward the right 2 Use the patterns you found in the table to write 2 6 as a fraction 3 1 Write 32 as a power of 2 with a single exponent 4 What is the value of 20 5 From the work you have done with negative exponents how would you write 5 3 as a fraction 6 How would you write 3 4 as a fraction 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

ZEARN MATH STUDENT EDITION G8M7 LESSON 6 ACTIVITY 2 3 Lin Noah Diego and Elena decide to test each other s knowledge of exponents with bases other than 10 Follow the directions below Be prepared to explain your reasoning They each chose an expression to start with and then came up with a new list of expressions some of which are equivalent to the original and some of which are not Choose 2 lists to analyze For each list of expressions you choose to analyze decide which expressions are not equivalent to the original 1 Lin s original expression is 5 9 and her list is 53 3 2 5 6 53 59 53 2 5 4 5 5 Noah s original expression is 310 and his list is 35 32 35 2 3 3 3 3 3 3 3 3 3 3 37 33 320 310 320 32 3 x x x x x 4 x 8 x2 2 4 x x x3 1 13 10 Diego s original expression is x4 and his list is x8 x4 4 5 4 5 5 x 4 x8 Elena s original expression is 80 and her list is 0 83 8 3 82 82 100 110 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 37

G8M7 LESSON 6 ZEARN MATH STUDENT EDITION Lesson Summary Earlier we focused on powers of 10 because 10 plays a special role in the decimal number system But the exponent rules that we developed for 10 also work for other bases For example if 20 1 and 2 n 21n then 2m 2n 2m n 2m n 2m n 2m m n 2n 2 These rules also work for powers of numbers less than 1 For example 13 2 13 13 and 13 4 13 31 13 13 We can also check that 13 2 31 4 13 2 4 Using a variable x helps to see this structure Since x2 x5 x7 both sides have 7 factors that are x if we let x 4 we can see that 42 45 47 Similarly we could let x 23 or x 11 or any other positive value and show that these relationships still hold 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

ZEARN MATH STUDENT EDITION G8M7 LESSON 6 Name Date GRADE 8 MISSION 7 LESSON 6 Exit Ticket 1 Diego was trying to write 23 22 with a single exponent and wrote 23 22 23 2 26 Explain to Diego what his mistake was and what the answer should be 2 Andre was trying to write 77 3 with a single exponent and wrote 77 3 74 3 71 Explain to Andre what his mistake was and what the answer should be 4 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 39

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ZEARN MATH STUDENT EDITION G8M7 LESSON 7 Lesson 7 Practice with Rational Bases Let s practice with exponents Warm Up 1 Which expression doesn t belong 28 25 43 5 34 8 4 5 8 108 55 Concept Exploration ACTIVITY 1 2 Choose 6 of the following to write using a single exponent a 75 76 e 35 328 i 72 3 b 3 3 38 f 2 5 24 j 43 3 c g 65 6 8 k 2 8 4 h 10 12 10 20 l 2 4 2 3 d 56 4 56 5 6 3 5 Which problems did you want to skip Explain your 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 41

G8M7 LESSON 7 3 ZEARN MATH STUDENT EDITION Choose 3 of the following to write using a single positive exponent a 2 7 d 4 9 b 3 23 e 2 32 c 11 8 f 43 Choose 3 of the following to evaluate a 105 105 8 3 d 54 2 b 23 3 e 34 0 c f 28 2 8 72 2 ACTIVITY 2 53 42 Mark each equation as true or false What could you change about the false equations to make them true 1 13 2 13 4 13 6 2 32 53 155 3 54 55 59 4 12 4 103 57 5 32 52 152 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

ZEARN MATH STUDENT EDITION G8M7 LESSON 7 Lesson Summary In the past few lessons we found rules to more easily keep track of repeated factors when using exponents We also extended these rules to make sense of negative exponents as repeated factors of the reciprocal of the base as well as defining a number to the power of 0 to have a value of 1 These rules can be written symbolically as xn xm xn m xn m xn m xn n m xm x x n x1n and x0 1 where the base x can be any positive number In this lesson we practiced using these exponent rules for different bases and exponents 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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ZEARN MATH STUDENT EDITION Name G8M7 LESSON 7 Date GRADE 8 MISSION 7 LESSON 7 Exit Ticket 1 Rewrite each expression using a single positive exponent a 93 99 b 14 3 1412 2 Diego wrote 64 83 487 Explain what Diego s mistake was and how you know the equation is not true 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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ZEARN MATH STUDENT EDITION G8M7 LESSON 8 Lesson 8 Combining Bases Let s multiply expressions with different bases Warm Up 1 Evaluate the expressions below 1 Evaluate 53 23 2 Evaluate 103 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

G8M7 LESSON 8 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 1 Solve the problems below The table contains products of expressions with different bases and the same exponent Complete the table to see how we can rewrite them Use the expanded column to work out how to combine the factors into a new base expression expanded exponent 53 23 5 5 5 2 2 2 2 5 2 5 2 5 10 10 10 103 32 72 212 24 34 153 304 24 x4 an bn 74 24 54 2 48 What happens if neither the exponents nor the bases are the same Can you write 23 34 with a single exponent Explain or show 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

ZEARN MATH STUDENT EDITION G8M7 LESSON 8 ACTIVITY 2 3 Create a visual display to play a game Your teacher will give your group tools for creating a visual display to play a game Divide the display into 3 columns with these headers an am an m an n m am a an bn a b n How to play When the time starts you and your group will write as many expressions as you can that equal a specific number using one of the exponent rules on your board When the time is up compare your expressions with another group to see how many points you earn Your group gets 1 point for every unique expression you write that is equal to the number and follows the exponent rule you claimed If an expression uses negative exponents you get 2 points instead of just 1 You can challenge the other group s expression if you think it is not equal to the number or if it does not follow one of the three exponent rules 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

G8M7 LESSON 8 ZEARN MATH STUDENT EDITION Lesson Summary Before this lesson we made rules for multiplying and dividing expressions with exponents that only work when the expressions have the same base For example 103 102 105 or 26 22 24 In this lesson we studied how to combine expressions with the same exponent but different bases For example we can write 23 53 as 2 2 2 5 5 5 Regrouping this as 2 5 2 5 2 5 shows that 23 53 2 5 3 103 Notice that the 2 and 5 in the previous example could be replaced with different numbers or even variables For example if a and b are variables then a3 b3 a b 3 More generally for a positive number n an bn a b n because both sides have exactly n factors that are a and n factors that are b 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

ZEARN MATH STUDENT EDITION Name G8M7 LESSON 8 Date GRADE 8 MISSION 7 LESSON 8 Exit Ticket Using words and equations explain what you learned about exponents in this lesson so that someone who was absent could read what you wrote and understand the lesson Consider using an example like 24 34 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 51

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ZEARN MATH STUDENT EDITION G8M7 LESSON 9 Lesson 9 Describing Large and Small Numbers Using Powers of 10 Let s find out how to use powers of 10 to write large or small numbers Warm Up 1 1 Solve the problems below Use the value and word banks to match each expression in the table to its corresponding value and word Values Words a 1 000 000 000 000 a billion b 1 100 b milli c 1 000 c million d 1 000 000 000 d thousand e 1 000 000 e centi f 1 1000 f Expression Value trillion Word 10 3 106 109 10 2 1012 103 2 For each of the numbers think of something in the world that is described by that number 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

G8M7 LESSON 9 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 1 Solve the problems using the diagrams below Match each expression to one or more diagrams that could represent it For each match explain what the value of a single small square would have to be A 54 B C a 2 10 1 4 10 2 c 2 103 4 101 b 2 10 1 4 10 3 d 2 103 4 102 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

ZEARN MATH STUDENT EDITION G8M7 LESSON 9 2 a Write an expression to describe the base ten diagram if each small square represents 10 4 What is the value of this expression b How does changing the value of the small square change the value of the expression Explain or show your thinking c Select at least two different powers of 10 for the small square and write the corresponding expressions to describe the base ten diagram What is the value of each of your expressions 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

G8M7 LESSON 9 ZEARN MATH STUDENT EDITION ACTIVITY 2 3 Your teacher will give you a card that tells you whether you are Partner A or B and gives you the information that is missing from your partner s statements Do not show your card to your partner Read each statement assigned to you ask your partner for the missing information and write the number your partner tells you Partner A s statements 1 Around the world about 2 The mass of a proton is 3 The population of Russia is about 4 The diameter of a bacteria cell is about pencils are made each year kilograms people meters Partner B s statements 56 1 Light waves travel through space at a speed of 2 The population of India is about 3 The wavelength of a gamma ray is 4 The tardigrade water bear is meters per second people meters meters long 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

ZEARN MATH STUDENT EDITION G8M7 LESSON 9 Lesson Summary Sometimes powers of 10 are helpful for expressing quantities especially very large or very small quantities For example the United States Mint has made over 500 000 000 000 pennies In order to understand this number we have to count all the zeros Since there are 11 of them this means there are 500 billion pennies Using powers of 10 we can write this as 500 109 five hundred times a billion or even as 5 1011 The advantage to using powers of 10 to write a large number is that they help us see right away how large the number is by looking at the exponent The same is true for small quantities For example a single atom of carbon weighs about 0 0000000000000000000000199 grams We can write this using powers of 10 as 199 10 25 or equivalently 1 99 10 23 Not only do powers of 10 make it easier to write this number but they also help us avoid errors since it would be very easy to write an extra zero or leave one out when writing out the decimal because there are so many to keep track of 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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ZEARN MATH STUDENT EDITION Name G8M7 LESSON 9 Date GRADE 8 MISSION 7 LESSON 9 Exit Ticket 1 Write 0 000000123 as a multiple of a power of 10 2 Write 123 000 000 as a multiple of a power of 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 59

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ZEARN MATH STUDENT EDITION G8M7 LESSON 10 Lesson 10 Representing Large Numbers on the Number Line Let s visualize large numbers on the number line using powers of 10 Warm Up 1 Label the tick marks on the number line Be prepared to explain your reasoning 0 107 Concept Exploration ACTIVITY 1 2 Answer the questions and follow the directions below 0 1 107 Place the numbers on the number line Be prepared to explain your reasoning a 4 000 000 b 5 106 c 5 105 d 75 105 e 0 6 107 2 Trade number lines with a partner and check each other s work How did your partner decide how to place the numbers If you disagree about a placement work to reach an agreement 3 Which is larger 4 000 000 or 75 105 Estimate how many times larger 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

G8M7 LESSON 10 ZEARN MATH STUDENT EDITION ACTIVITY 2 3 1 The table shows how fast light waves or electricity can travel through different materials Material Speed meters per second space 300 000 000 water 2 25 108 copper wire electricity 280 000 000 diamond 124 106 ice 2 3 108 olive oil 200 000 000 Which is faster light through diamond or light through ice How can you tell from the expressions for speed Let s zoom in to highlight the values between 2 0 108 and 3 0 108 0 1 108 2 108 3 108 4 108 5 108 6 108 2 0 108 62 7 108 8 108 9 108 109 3 0 108 2 Label the tick marks between 2 0 108 and 3 0 108 3 Plot a point for each speed on both number lines and label it with the corresponding material 4 There is one speed that you cannot plot on the bottom number line Which is it Plot it on the top number line instead 5 Which is faster light through ice or light through diamond How can you tell from 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

ZEARN MATH STUDENT EDITION G8M7 LESSON 10 Lesson Summary There are many ways to compare two quantities Suppose we want to compare the world population about 7 4 billion to the number of pennies the U S made in 2015 about 8 900 000 000 There are many ways to do this We could write 7 4 billion as a decimal 7 400 000 000 and then we can tell that there were more pennies made in 2015 than there are people in the world Or we could use powers of 10 to write these numbers 7 4 109 for people in the world and 8 9 109 for the number of pennies For a visual representation we could plot these two numbers on a number line We need to carefully choose our end points to make sure that the numbers can both be plotted Since they both lie between 109 and 1010 if we make a number line with tick marks that increase by one billion or 109 we start the number line with 0 and end it with 10 109 or 1010 Here is a number line with the number of pennies and world population plotted People 0 Pennies 1 109 2 109 3 109 4 109 5 109 6 109 7 109 8 109 9 109 1010 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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ZEARN MATH STUDENT EDITION G8M7 LESSON 10 Name Date GRADE 8 MISSION 7 LESSON 10 Exit Ticket We described numbers in this lesson using both powers of 10 and using standard decimal value For example the speed of light through ice can be written as a multiple of a power of 10 such as 2 3 108 meters per second or as a value such as 230 000 000 meters per second Use the number line to answer questions about points A and B B 0 109 A 1 Describe point B as a A multiple of a power of 10 b A value 3 Describe point A as a A multiple of a power of 10 b A value 3 Plot a point C that is greater than B and less than A Describe point C as a A multiple of a power of 10 b A 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 65

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ZEARN MATH STUDENT EDITION G8M7 LESSON 11 Lesson 11 Representing Small Numbers on the Number Line Let s visualize small numbers on the number line using powers of 10 Warm Up 1 Kiran drew the following number line 0 1 10 4 2 10 4 3 10 4 4 10 4 5 10 4 6 10 4 7 10 4 8 10 4 9 10 4 10 5 Andre said That doesn t look right to me Explain why Kiran is correct or explain how he can fix the number line Concept Exploration ACTIVITY 1 2 Use the number line to solve the problems below 0 1 Label the tick marks on the number line 2 Plot the following numbers on the number line a 6 10 6 b 6 10 7 c 10 5 29 10 7 d 0 7 10 5 3 Which is larger 29 10 7 or 6 10 6 Estimate how many times larger 4 Which is larger 7 10 8 or 3 10 9 Estimate how many times larger 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

G8M7 LESSON 11 ZEARN MATH STUDENT EDITION ACTIVITY 2 3 The radius of an electron is about 0 0000000000003 cm Write this number as a multiple of a power of 10 Then solve the problems below a Decide what power of 10 to put on the right side of this number line and label it b Label each tick mark as a multiple of a power of 10 0 c 10 Plot the radius of the electron in centimeters on the number line 43 The mass of a proton is about 0 0000000000000000000000017 grams Write this number as a multiple of a power of 10 Then solve the problems below a Decide what power of 10 to put on the right side of this number line and label it b Label each tick mark as a multiple of a power of 10 0 c 10 Plot the mass of the proton in grams on the number line 53 Point A on the zoomed in number line describes the wavelength of a certain X ray in meters 0 1 10 12 2 10 12 3 10 12 4 10 12 5 10 12 6 10 12 7 10 12 8 10 12 9 10 12 10 11 A 6 10 12 7 10 12 a Write the wavelength of the X ray as a multiple of a power of 10 b Write the wavelength of the X ray as a decimal 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

ZEARN MATH STUDENT EDITION G8M7 LESSON 11 Lesson Summary The width of a bacterial cell is about 2 10 6 meters If we want to plot this on a number line we need to find which two powers of 10 it lies between We can see that 2 10 6 is a multiple of 10 6 So our number line will be labeled with multiples of 10 6 0 1 10 6 2 10 6 3 10 6 4 10 6 5 10 6 6 10 6 7 10 6 8 10 6 9 10 6 10 5 Note that the right side is labeled 10 10 6 10 5 The power of ten on the right side of the number line is always greater than the power on the left This is true for positive or negative powers of ten 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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ZEARN MATH STUDENT EDITION Name G8M7 LESSON 11 Date GRADE 8 MISSION 7 LESSON 11 Exit Ticket 1 Write 0 00034 as a multiple of a power of 10 2 Write 5 64 10 7 as a decimal 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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ZEARN MATH STUDENT EDITION G8M7 LESSON 12 Lesson 12 Applications of Arithmetic with Powers of 10 Let s use powers of 10 to help us make calculations with large and small numbers Warm Up 1 What information would you need to answer the following question 1 How many meter sticks does it take to equal the mass of the Moon 2 If all of these meter sticks were lined up end to end would they reach the Moon 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

G8M7 LESSON 12 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 Answer the questions below 1 How many meter sticks does it take to equal the mass of the Moon Explain or show your reasoning 2 Label the number line and plot your answer for the number of meter sticks 0 3 If you took all the meter sticks from the last question and lined them up end to end will they reach the Moon Will they reach beyond the Moon If yes how many times farther will they reach Explain your reasoning 4 One light year is approximately 1016 meters How many light years away would the meter sticks reach Label the number line and plot your answer 0 74 10 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

ZEARN MATH STUDENT EDITION G8M7 LESSON 12 Lesson Summary Powers of 10 can be helpful for making calculations with large or small numbers For example in 2014 the United States had 318 586 495 people who used the equivalent of 2 203 799 778 107 kilograms of oil in energy The amount of energy per person is the total energy divided by the total number of people We can use powers of 10 to estimate the total energy as 2 1012 and the population as 3 108 So the amount of energy per person in the U S is roughly 2 1012 3 108 That is the equivalent of 2 4 3 10 kilograms of oil in energy That s a lot of energy the equivalent of almost 7 000 kilograms of oil per person In general when we want to perform arithmetic with very large or small quantities estimating with powers of 10 and using exponent rules can help simplify the process If we wanted to find the exact quotient of 2 203 799 778 107 by 318 586 495 then using powers of 10 would not simplify the calculation 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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ZEARN MATH STUDENT EDITION Name G8M7 LESSON 12 Date GRADE 8 MISSION 7 LESSON 12 Exit Ticket What is a mistake you would expect to see others make when doing problems like the ones in this lesson Give an example of what such a mistake looks like 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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ZEARN MATH STUDENT EDITION G8M7 LESSON 13 Lesson 13 Definition of Scientific Notation Let s use scientific notation to describe large and small numbers Warm Up 1 Find the value of each expression mentally 123 10 000 3 4 1 000 0 6 100 7 3 0 01 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

G8M7 LESSON 13 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 The table shows the speed of light or electricity through different materials Circle the speeds that are written in scientific notation Write the others using scientific notation Material Speed meters per second space 300 000 000 water 2 25 108 copper electricity 280 000 000 diamond 124 106 ice 2 3 108 olive oil 0 2 109 Diamond 0 Olive Oil 2 108 80 1 108 2 108 3 108 4 108 5 108 6 108 7 108 8 108 9 108 Water 109 Space Ice Copper Wire 3 108 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

ZEARN MATH STUDENT EDITION G8M7 LESSON 13 ACTIVITY 2 3 You and your partner will receive a set of cards Some of the cards show numbers in scientific notation and other cards show numbers that are not in scientific notation Read the directions below 1 Shuffle the cards and lay them facedown 2 Players take turns trying to match cards with the same value 3 On your turn choose two cards to turn faceup for everyone to see Then a If the two cards have the same value and one of them is written in scientific notation whoever says Science first gets to keep the cards and it becomes that player s turn If it s already your turn when you call Science that means you get to go again If you say Science when the cards do not match or one is not in scientific notation then your opponent gets a point b If both partners agree the two cards have the same value then remove them from the board and keep them You get a point for each card you keep c 4 If the two cards do not have the same value then set them facedown in the same position and end your turn If it is not your turn a If the two cards have the same value and one of them is written in scientific notation then whoever says Science first gets to keep the cards and it becomes that player s turn If you call Science when the cards do not match or one is not in scientific notation then your opponent gets a point b Make sure both of you agree the cards have the same value If you disagree work to reach an agreement 5 Whoever has the most points at the end wins 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

G8M7 LESSON 13 ZEARN MATH STUDENT EDITION Lesson Summary The total value of all the quarters made in 2014 is 400 million dollars There are many ways to express this using powers of 10 We could write this as 400 106 dollars 40 107 dollars 0 4 109 dollars or many other ways One special way to write this quantity is called scientific notation In scientific notation 400 million dollars would be written as 4 108 dollars For scientific notation the symbol is the standard way to show multiplication instead of the symbol Writing the number this way shows exactly where it lies between two consecutive powers of 10 The 108 shows us the number is between 108 and 109 The 4 shows us that the number is 4 tenths of the way to 109 Some other examples of scientific notation are 1 2 10 8 9 99 1016 and 7 1012 The first factor is a number greater than or equal to 1 but less than 10 The second factor is an integer power of 10 Thinking back to how we plotted these large or small numbers on a number line scientific notation tells us which powers of 10 to place on the left and right of the number line For example if we want to plot 3 4 1011 on a number line we know that the number is larger than 1011 but smaller than 1012 We can find this number by zooming in on the number line 0 3 1011 1 1011 2 1011 3 1011 4 1011 5 1011 6 1011 7 1011 8 1011 9 1011 1012 3 4 1011 4 1011 TERMINOLOGY Scientific notation 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

ZEARN MATH STUDENT EDITION Name G8M7 LESSON 13 Date GRADE 8 MISSION 7 LESSON 13 Exit Ticket State whether each of the following is in scientific notation If not write it in scientific notation 1 5 23 108 2 48 200 3 0 00099 4 36 105 5 8 7 10 12 6 0 78 10 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 83

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ZEARN MATH STUDENT EDITION G8M7 LESSON 14 Lesson 14 Multiplying Dividing and Estimating with Scientific Notation Let s multiply and divide with scientific notation to answer questions about animals careers and planets Warm Up 1 Is each equation true or false Explain your reasoning 4 105 4 104 4 1020 7 106 7 2 10 6 4 2 104 8 4 103 2 8 4 2 10 3 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 85

G8M7 LESSON 14 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 86 Use the table to answer questions about different creatures on the planet Be prepared to explain your reasoning Creature Number Mass of one individual kg Humans 7 5 109 6 2 101 Cows 1 3 109 4 102 Sheep 1 75 109 6 101 Chickens 2 4 1010 2 100 Ants 5 1016 3 10 6 Blue whales 4 7 103 1 9 105 Antarctic krill 7 8 1014 4 86 10 4 Zooplankton 1 1020 5 10 8 Bacteria 5 1030 1 10 12 1 Which creature is least numerous Estimate how many times more ants there are 2 Which creature is the least massive Estimate how many times more massive a human is 3 Which is more massive the total mass of all the humans or the total mass of all the ants About how many times more massive is it 4 Which is more massive the total mass of all the krill or the total mass of all the blue whales About how many times more massive is it 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

ZEARN MATH STUDENT EDITION G8M7 LESSON 14 ACTIVITY 2 3 Use the table to answer questions about the Sun and the planets of the solar system sorry Pluto Express each answer in scientific notation and as a decimal number using a calculator Object Distance to Earth km Diameter km Mass kg Sun 1 46 108 1 392 106 1 989 1030 Mercury 7 73 107 4 878 103 3 3 1023 Venus 4 107 1 21 104 4 87 1024 Earth N A 1 28 104 5 98 1024 Mars 5 46 107 6 785 103 6 4 1023 Jupiter 5 88 108 1 428 105 1 898 1027 Saturn 1 2 109 1 199 105 5 685 1026 Uranus 2 57 109 5 149 104 8 68 1025 Neptune 4 3 109 4 949 104 1 024 1026 1 Estimate how many Earths side by side would have the same width as the Sun 2 Estimate how many Earths it would take to equal the mass of the Sun 3 Calculate how many times as far away from Earth the planet Neptune is compared to Venus 4 Calculate how many Mercuries it would take to equal the mass of Neptune 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

G8M7 LESSON 14 ZEARN MATH STUDENT EDITION Lesson Summary Multiplying numbers in scientific notation extends what we do when we multiply regular decimal numbers For example one way to find 80 60 is to view 80 as 8 tens and to view 60 as 6 tens The product 80 60 is 48 hundreds or 4 800 Using scientific notation we can write this calculation as 8 101 6 101 48 102 To express the product in scientific notation we would rewrite it as 4 8 103 Calculating using scientific notation is especially useful when dealing with very large or very small numbers For example there are about 39 million or 3 9 107 residents in California Each Californian uses about 180 gallons of water a day To find how many gallons of water Californians use in a day we can find the product 180 3 9 107 702 107 which is equal to 7 02 109 That s about 7 billion gallons of water each day Comparing very large or very small numbers by estimation also becomes easier with scientific notation For example how many ants are there for every human There are 5 1016 ants and 7 109 humans 16 To find the number of ants per human look at 57 10 109 Rewriting the numerator to have the number 50 1015 This gives us 50 106 Since 50 is roughly equal to 7 there are about 7 106 or 7 instead of 5 we get 507 10 9 7 7 million ants per person 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

ZEARN MATH STUDENT EDITION Name G8M7 LESSON 14 Date GRADE 8 MISSION 7 LESSON 14 Exit Ticket 1 Estimate how many times larger 6 1 107 is than 2 1 10 4 2 Estimate how many times larger 1 9 10 8 is than 4 2 10 13 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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ZEARN MATH STUDENT EDITION G8M7 LESSON 15 Lesson 15 Adding and Subtracting with Scientific Notation Let s add and subtract using scientific notation to answer questions about animals and the solar system Warm Up 1 Mentally decide how many non zero digits each number will have 1 3 109 2 107 2 3 109 2 107 3 3 109 2 107 4 3 109 2 107 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

G8M7 LESSON 15 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 1 Diego Kiran and Clare were wondering If Neptune and Saturn were side by side would they be wider than Jupiter Neptune has a diameter of 4 7 104 km Saturn has a diameter of 1 2 105 km and Jupiter has a diameter of 43 105 km They try to add the diameters 4 7 104 km and 1 2 105 km Here are the ways they approached the problem Do you agree with any of them Explain your reasoning a Diego says When we add the distances we will get 4 7 1 2 5 9 The exponent will be 9 So the two planets are 5 9 109 km side by side b Kiran wrote 4 7 104 as 47 000 and 1 2 105 as 120 000 and added them 120 000 47 000 167 000 c 2 92 Clare says I think you can t add unless they are the same power of 10 She adds 4 7 104 km and 12 104 to get 16 7 104 Jupiter has a diameter of 1 43 105 Which is wider Neptune and Saturn put side by side or Jupiter 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

ZEARN MATH STUDENT EDITION G8M7 LESSON 15 ACTIVITY 2 3 Use the table to answer the questions below object diameter km distance from the Sun km Sun 1 392 106 0 100 Mercury 4 878 103 5 79 107 Venus 1 21 104 1 08 108 Earth 1 28 104 1 47 108 Mars 6 785 103 2 28 108 Jupiter 1 428 105 7 79 108 1 When you add the distances of Mercury Venus Earth and Mars from the Sun would you reach as far as Jupiter 2 Add all the diameters of all the planets except the Sun Which is wider all of these objects side by side or the Sun Draw a picture that is close to scale 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

G8M7 LESSON 15 ZEARN MATH STUDENT EDITION Lesson Summary When we add decimal numbers we need to pay close attention to place value For example when we calculate 13 25 6 7 we need to make sure to add hundredths to hundredths 5 and 0 tenths to tenths 2 and 7 ones to ones 3 and 6 and tens to tens 1and 0 The result is 19 95 We need to take the same care when we add or subtract numbers in scientific notation For example suppose we want to find how much further the Earth is from the Sun than Mercury The Earth is about 1 5 108 km from the Sun while Mercury is about 5 8 107 km In order to find 1 5 108 5 8 107 we can rewrite this as 1 5 108 0 58 108 Now that both numbers are written in terms of 108 we can subtract 0 58 from 1 5 to find 0 92 108 Rewriting this in scientific notation the Earth is 9 2 107 km further from the Sun than Mercury 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

ZEARN MATH STUDENT EDITION Name G8M7 LESSON 15 Date GRADE 8 MISSION 7 LESSON 15 Exit Ticket Elena wants to add 2 3 105 3 6 106 and writes 2 3 105 3 6 106 5 9 106 Explain to Elena what her mistake was and what the correct solution is 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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ZEARN MATH STUDENT EDITION G8M7 LESSON 16 Lesson 16 Is a Smartphone Smart Enough to Go to the Moon Let s compare digital media and computer hardware using scientific notation Warm Up 1 In 1966 the Apollo Guidance Computer was developed to make the calculations that would put humans on the Moon Your teacher will give you advertisements for different devices from 1966 to 2016 Choose one device and compare that device with the Apollo Guidance Computer If you get stuck consider using scientific notation to help you do your calculations Apollo 11 Saturn V lifting off on July 16 1969 via Wikimedia Commons Public Domain 1 Which one can store more information How many times more information 2 Which one has a faster processor How many times faster 3 Which one has more memory How many times more memory For reference storage is measured in bytes processor speed is measured in hertz and memory is measured in bytes Kilo stands for 1 000 mega stands for 1 000 000 giga stands for 1 000 000 000 and tera stands for 1 000 000 000 000 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 97

G8M7 LESSON 16 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 For each question below think about what information you would need to figure out an answer Your teacher may provide some of the information you ask for Give your answers using scientific notation 1 Mai found an 80 s computer magazine with an advertisement for a machine with hundreds of kilobytes of storage Mai was curious and asked How many kilobytes would my dad s new 2016 computer hold 2 The old magazine showed another ad for a 750 kilobyte floppy disk a device used in the past to store data How many gigabytes is this 3 Mai and her friends are actively involved on a social media service that limits each message to 140 characters She wonders about how the size of a message compares to other media Estimate how many messages it would take for Mai to fill up a floppy disk with her 140 character messages Explain or show your reasoning 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

ZEARN MATH STUDENT EDITION G8M7 LESSON 16 4 Estimate how many messages it would take for Mai to fill a floppy disk with messages that only use emojis each message being 140 emojis Explain or show your reasoning 5 Mai likes to go to the movies with her friends and knows that a high definition film takes up a lot of storage space on a computer Estimate how many floppy disks it would take to store a high definition movie Explain or show your reasoning 6 How many seconds of a high definition movie would one floppy disk be able to hold 7 If you fall asleep watching a movie streaming service and it streams movies all night while you sleep how many floppy disks of information would that be 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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Grade 8 Mission 8 Pythagorean Theorem and Irrational Numbers

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ZEARN MATH STUDENT EDITION G8M8 LESSON 1 Lesson 1 The Areas of Squares and Their Side Lengths Let s investigate squares and their side lengths Warm Up 1 Which shaded region is larger Explain your reasoning B A Concept Exploration ACTIVITY 1 2 Find the area of the shaded square on the grid in square units 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

G8M8 LESSON 1 3 43 ZEARN MATH STUDENT EDITION Here is another square Find the area of the shaded square in square units Here is a square not on a grid Find the area of the shaded square in square units 7 3 3 7 7 3 3 104 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

ZEARN MATH STUDENT EDITION G8M8 LESSON 1 ACTIVITY 2 53 Find the side lengths and areas of the squares below A B C 1 What is the side length of square A What is its area 2 What is the side length of square C What is its area 3 What is the area of square B What is its side length Use tracing paper to check your answer to this 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

G8M8 LESSON 1 63 ZEARN MATH STUDENT EDITION Find the areas of squares D E and F Which of these squares must have a side length that is greater than 5 but less than 6 Explain how you know D 106 E F 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

ZEARN MATH STUDENT EDITION G8M8 LESSON 1 Lesson Summary The area of a square with side length 12 units is 122 or 144 units2 The side length of a square with area 900 units2 is 30 units because 302 900 Sometimes we want to find the area of a square but we don t know the side length For example how can we find the area of square ABCD A D B C One way is to enclose it in a square whose side lengths we do know The outside square EFGH has side lengths of 11 units so its area is 121 units2 The area of each of the four triangles is 12 8 3 12 so the area of all four together is 4 12 48 units2 To get the area of the shaded square we can take the area of the outside square and subtract the areas of the 4 triangles So the area of the shaded square ABCD is 121 48 73 or 73 units2 A H E D B G C F 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 107

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ZEARN MATH STUDENT EDITION G8M8 LESSON 1 Name Date GRADE 8 MISSION 8 LESSON 1 Exit Ticket Find the area and side length of square ACEG B A 6 8 H 6 8 G C 8 6 D 8 E 6 F 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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ZEARN MATH STUDENT EDITION G8M8 LESSON 2 Lesson 2 Side Lengths and Areas Let s investigate some more squares Warm Up 1 What do you notice What do you wonder A C B Concept Exploration ACTIVITY 1 2 1 Use the diagrams below to estimate the area of the square Use the circle to estimate the area of the square shown here 7 6 5 4 3 2 1 6 5 4 3 2 1 1 1 2 3 4 5 6 2 3 4 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 111

G8M8 LESSON 2 2 ZEARN MATH STUDENT EDITION Use the grid to check your answer to the first problem 7 6 5 4 3 2 1 6 5 4 3 2 1 1 1 2 3 4 5 6 2 3 4 5 6 ACTIVITY 2 3 1 Determine the area and side lengths of different squares Find the area of each square and estimate the side lengths using your geometry toolkit Then write the exact lengths for the sides of each square A 112 B C 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

ZEARN MATH STUDENT EDITION 2 G8M8 LESSON 2 Complete the tables with the missing side lengths and areas side length s 0 5 1 5 area a side length s 1 3 5 4 4 5 5 5 area a 3 2 5 25 9 6 5 16 7 5 36 49 64 Plot the points s a on the coordinate plane shown here 50 Area square units 45 40 35 30 25 20 15 10 5 2 4 5 4 6 8 Side length units 10 Use this graph to estimate the side lengths of the squares in the first question How do your estimates from the graph compare to the estimates you made initially using your geometry toolkit Use the graph to approximate 45 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

G8M8 LESSON 2 ZEARN MATH STUDENT EDITION Lesson Summary The area of square ABCD is 73 units2 A What is the side length The area is between and 82 64 and 92 81 so the side length must be between 8 units and 9 units We can also use tracing paper to trace a side length and compare it to the grid which also shows the side length is between 8 units and 9 units But we want to be able to talk about its exact length In order to write the side length of a square whose area is 73 square units we use the square root symbol The square root of 73 is written 73 and it means the length of a side of a square whose area is 73 square units We say the side length of a square with area 73 units2 is 73 units This means that All of these statements are also true D B C 73 2 73 9 3 because 32 9 16 4 because 42 16 10 units is the side length of a square whose area is 10 units2 and 10 2 10 9 3 10 10 16 4 TERMINOLOGY Square root 114 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

ZEARN MATH STUDENT EDITION Name G8M8 LESSON 2 Date GRADE 8 MISSION 8 LESSON 2 Exit Ticket Write the exact value of the side length of a square with the following areas If the exact value is not a whole number estimate the length 1 100 square units 2 95 square units 3 36 square units 4 30 square units 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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ZEARN MATH STUDENT EDITION G8M8 LESSON 3 Lesson 3 Rational and Irrational Numbers Let s learn about irrational numbers Warm Up 1 Find a positive solution to each equation 1 x2 36 2 x2 94 3 x2 14 4 x2 49 25 Concept Exploration ACTIVITY 1 2 Use the grids below to answer the questions 1 Draw 3 squares of different sizes with vertices aligned to the vertices of the grid 2 For each square a Label the area b Label the side length c Write an equation that shows the relationship between the side length and the area 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

G8M8 LESSON 3 ZEARN MATH STUDENT EDITION ACTIVITY 2 3 1 1 2 1 2 3 3 2 4 7 5 Determine if the number is a solution to the equation x2 2 Explain your reasoning for each ACTIVITY 3 43 118 A rational number is a fraction or its opposite or any number equivalent to a fraction or its opposite Let s think about the value of 2 1 Find some more rational numbers that are close to 2 2 Can you find a rational number that is exactly 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

ZEARN MATH STUDENT EDITION G8M8 LESSON 3 Lesson Summary In an earlier activity we learned that square root notation is used to write the length of a side of a square given its area For example a square whose area is 2 square units has a side length of 2 units which means that 2 2 2 A square whose area is 25 square units has a side length of 25 units which means that 25 25 25 2 2 Since 5 5 25 we know that 25 5 25 is an example of a rational number A rational number is a fraction or its opposite Remember that a fraction ab is a point on the number line found by dividing the segment from 0 to 1 into b equal intervals and going a of those intervals to the right of 0 We can always write a fraction in the form ab where a and b are whole numbers and b is not 0 but there are other ways to write them For example we can write 25 51 You first learned about fractions in earlier grades and at that time you probably didn t know about negative numbers Rational numbers are fractions but they can be positive or negative So 5 is also a rational number Because fractions and ratios are closely related ideas fractions and their opposites are called RATIOnal numbers Here are some examples of rational numbers 7 0 6 0 2 1 5 9 16 4 3 3 100 Can you see why they are each examples of a fraction or its opposite An irrational number is a number that is not rational That is it is a number that is not a fraction or its opposite 2 is an example of an irrational number It has a location on the number line and its location can be approximated by rational numbers it s a tiny bit to the right of 75 but 2 can not be found on a number line by dividing the segment from 0 to 1 into b equal parts and going a of those parts away from 0 if a and b are whole numbers 2 0 1 7 5 17 is also close to 2 because 17 2 289 289 is very close to 2 since 288 2 But we could keep looking 144 144 12 12 144 forever for solutions to x2 2 that are rational numbers and we would not find any 2 is not a rational number It is irrational 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

G8M8 LESSON 3 ZEARN MATH STUDENT EDITION In your future studies you may have opportunities to understand or write a proof that 2 is irrational but for now we just take it as a fact that 2 is irrational Similarly the square root of any whole number is either a whole number 36 6 64 8 etc or irrational 17 65 etc Here are some other examples of irrational numbers 10 3 5 2 TERMINOLOGY Irrational number Rational number 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

ZEARN MATH STUDENT EDITION Name G8M8 LESSON 3 Date GRADE 8 MISSION 8 LESSON 3 Exit Ticket 1 In your own words say what a rational number is Give at least three different examples of rational numbers 2 In your own words say what an irrational number is Give at least two examples 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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ZEARN MATH STUDENT EDITION G8M8 LESSON 4 Lesson 4 Square Roots on the Number Line Let s explore square roots Warm Up 1 Answer the questions about the diagram below y 1 0 0 1 What is the exact length of the line segment 2 Find a decimal approximation of the length 1 1 1 x 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

G8M8 LESSON 4 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 Estimate the length of the line segment to the nearest tenth of a unit each grid square is 1 unit Then find the exact length of the segment 0 1 ACTIVITY 2 3 Diego said that he thinks that 3 2 5 Answer the questions below 3 2 1 1 1 124 2 2 5 3 Use the square to explain why 2 5 is not a very good approximation for 3 Find a point on the number line that is closer to 3 Draw a new square on the axes and use it to explain how you know the point you plotted is a good approximation for 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

ZEARN MATH STUDENT EDITION 2 G8M8 LESSON 4 Use the fact that 3 is a solution to the equation x2 3 to find a decimal approximation of 3 whose square is between 2 9 and 3 1 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

G8M8 LESSON 4 ZEARN MATH STUDENT EDITION Lesson Summary Here is a line segment on a grid What is the length of this line segment By drawing some circles we can tell that it s longer than 2 units but shorter than 3 units To find an exact value for the length of the segment we can build a square on it using the segment as one of the sides of the square The area of this square is 5 square units Can you see why That means the exact value of the length of its side is 5 units Notice that 5 is greater than 4 but less than 9 That means that 5 is greater than 2 but less than 3 This makes sense because we already saw that the length of the segment is in between 2 and 3 With some arithmetic we can get an even more precise idea of where 5 is on the number line The image with the circles shows that 5 is closer to 2 than 3 so let s find the value of 2 12 and 2 22 and see how close they are to 5 It turns out that 2 12 4 41 and 2 22 4 84 so we need to try a larger number If we increase our search by a tenth we find that 2 32 5 29 This means that 5 is greater than 2 2 but less than 2 3 If we wanted to keep going we could try 2 252 and eventually narrow the value of 5 to the hundredths place Calculators do this same process to many decimal places giving an approximation like 5 2 2360679775 Even though this is a lot of decimal places it is still not exact because 5 is irrational 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

ZEARN MATH STUDENT EDITION G8M8 LESSON 4 Name Date GRADE 8 MISSION 8 LESSON 4 Exit Ticket Plot an approximation of 18 on the x axis Consider using the grid to help y 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10 x 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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ZEARN MATH STUDENT EDITION G8M8 LESSON 5 Lesson 5 Reasoning about Square Roots Let s approximate square roots Warm Up 1 Decide if each statement is true or false 5 2 5 10 2 100 9 2 3 16 22 7 7 2 Concept Exploration ACTIVITY 1 2 1 7 2 23 3 50 4 98 What two whole numbers does each square root lie between Be prepared to 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 129

G8M8 LESSON 5 ZEARN MATH STUDENT EDITION ACTIVITY 2 3 3 1 2 130 The numbers x y and z are positive and x2 3 y2 16 and z2 30 Answer the questions below 2 1 0 1 2 3 4 5 6 7 8 9 Plot the approximate locations of x y and z on the number line Be prepared to share your reasoning with the class Plot the approximate location of 2 on 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

ZEARN MATH STUDENT EDITION G8M8 LESSON 5 Lesson Summary In general we can approximate the values of square roots by observing the whole numbers around it and remembering the relationship between square roots and squares Here are some examples 64 8 65 is a little more than 8 because 65 is a little more than 64 and 64 8 80 is a little less than 9 because 80 is a little less than 81 and 81 9 75 is between 8 and 9 it s 8 point something because 75 is between 64 and 81 75 is approximately 8 67 because 8 672 75 1689 65 8 1 8 2 8 3 8 4 8 5 8 6 75 80 8 7 8 8 81 8 9 9 If we want to find a square root between two whole numbers we can work in the other direction For example since 222 484 and 232 529 then we know that 500 to pick one possibility is between 22 and 23 Many calculators have a square root command which makes it simple to find an approximate value of a square root 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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ZEARN MATH STUDENT EDITION Name G8M8 LESSON 5 Date GRADE 8 MISSION 8 LESSON 5 Exit Ticket Which of the following numbers are greater than 6 and less than 8 Explain how you know 7 60 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 133

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ZEARN MATH STUDENT EDITION G8M8 LESSON 6 Lesson 6 Finding Side Lengths of Triangles Let s find triangle side lengths Warm Up 1 Which triangle doesn t belong 5 33 A 50 5 B 5 2 5 1 C 65 8 4 D 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 135

G8M8 LESSON 6 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 Complete the tables below for triangles D F Then answer the following question 2 D E b Triangle a F c b a b c b c a c a Triangle D D E E F F a2 b2 c2 What do you notice about the values in the table for Triangle E but not for Triangles D and F 136 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

ZEARN MATH STUDENT EDITION G8M8 LESSON 6 Complete the tables below for triangles P R Then answer the following questions 3 P Q b b c a Triangle a R b c b a c c a Triangle P P Q Q R R a2 b2 c2 1 What do you notice about the values in the table for Triangle Q but not for Triangles P and R 2 What do Triangle E and Triangle Q have in common 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

G8M8 LESSON 6 ZEARN MATH STUDENT EDITION ACTIVITY 2 43 Find the missing side lengths Be prepared to explain your reasoning For which triangles does a2 b2 c2 b 40 a 10 a 8 P b Q c c b 17 a R c 5 138 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

ZEARN MATH STUDENT EDITION G8M8 LESSON 6 Lesson Summary se po ten us Leg e Leg nu ote Hyp Leg Hy A right triangle is a triangle with a right angle In a right triangle the side opposite the right angle is called the hypotenuse and the two other sides are called its legs Here are some right triangles with the hypotenuse and legs labeled Leg Leg We often use the letters a and b to represent the lengths of the shorter sides of a triangle and c to represent the length of the longest side of a right triangle If the triangle is a right triangle then a and b are used to represent the lengths of the legs and c is used to represent the length of the hypotenuse since the hypotenuse is always the longest side of a right triangle For example in this right triangle a 20 b 5 and c 5 Leg Hypotenuse Leg Leg Hypotenuse c b a Here are some right triangles c2 25 c2 17 b2 9 c2 18 b2 9 b2 1 a2 16 a2 16 a2 9 Notice that for these examples of right triangles the square of the hypotenuse is equal to the sum of the squares of the legs In the first right triangle in the diagram 9 16 25 in the second 1 16 17 and in the third 9 9 18 Expressed another way we have a2 b2 c2 This is a property of all right triangles not just these examples and is often known as the Pythagorean Theorem The name comes from a mathematician named Pythagoras who lived in ancient Greece around 2 500 BCE but this property of right triangles was also discovered independently by mathematicians in other ancient cultures including Babylon India and China In China a name for the same relationship is the Shang Gao Theorem In future lessons you will learn some ways to explain why the Pythagorean Theorem is true for any right triangle It is important to note that this relationship does not hold for all triangles Here are some triangles that are not right triangles and notice that the lengths of their sides do not have the special relationship a2 b2 c2 That is 16 10 does not equal 18 and 2 10 does not equal 16 c2 18 b2 10 b2 2 a2 16 a2 10 c2 16 TERMINOLOGY Hypotenuse Legs Pythagorean Theorem 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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ZEARN MATH STUDENT EDITION G8M8 LESSON 6 Name Date GRADE 8 MISSION 8 LESSON 6 Exit Ticket For each of the following triangles determine if a2 b2 c2 where a b and c are side lengths of the triangle and assuming that c is the longest side of the triangle Explain how you know 4 A 2 45 B 5 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 141

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ZEARN MATH STUDENT EDITION G8M8 LESSON 7 Lesson 7 A Proof of the Pythagorean Theorem Let s prove the Pythagorean Theorem Warm Up 1 What do you notice What do you wonder 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

G8M8 LESSON 7 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 Both figures shown here are squares with a side length of a b Let s call the hypotenuse of these triangles c Notice that the first figure is divided into two squares and two rectangles The second figure is divided into a square and four right triangles with legs of lengths a and b Answer the questions below F a a a b b b a a b a 144 G b b b a a b 1 What is the total area of each figure 2 Find the area of each of the 9 smaller regions shown the figures and label them 3 Add up the area of the four regions in Figure F and set this expression equal to the sum of the areas of the five regions in Figure G If you rewrite this equation using as few terms as possible what do you have 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

ZEARN MATH STUDENT EDITION G8M8 LESSON 7 ACTIVITY 2 3 Find the unknown side lengths in these right triangles x 2 5 8 y 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 145

G8M8 LESSON 7 ZEARN MATH STUDENT EDITION Lesson Summary The figures shown here can be used to see why the Pythagorean Theorem is true Both large squares have the same area but they are broken up in different ways Can you see where the triangles in Square G are located in Square F What does that mean about the smaller squares in F and H When the sum of the four areas in Square F are set equal to the sum of the 5 areas in Square G the result is a2 b2 c2 where c is the hypotenuse of the triangles in Square G and also the side length of the square in the middle Give it a try F a G b a a b b b a a b a b a a b b This is true for any right triangle If the legs are a and b the hypotenuse is c then a2 b2 c2 This property can be used any time we can make a right triangle For example to find the length of this line segment The grid can be used to create a right triangle where the line segment is the hypotenuse and the legs measure 24 units and 7 units 24 c 7 Since this is a right triangle 242 72 c2 The solution to this equation and the length of the line segment is c 25 146 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

ZEARN MATH STUDENT EDITION Name G8M8 LESSON 7 Date GRADE 8 MISSION 8 LESSON 7 Exit Ticket The Pythagorean Theorem is a True for all triangles b True for all right triangles c True for some right triangles d Never true 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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ZEARN MATH STUDENT EDITION G8M8 LESSON 8 Lesson 8 Finding Unknown Side Lengths Let s find missing side lengths of right triangles Warm Up 1 Which one doesn t belong 1 32 b2 52 2 b2 52 32 3 32 52 b2 4 32 42 52 Concept Exploration ACTIVITY 1 2 Label all the hypotenuses with c A B D C E F 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

G8M8 LESSON 8 ZEARN MATH STUDENT EDITION ACTIVITY 2 3 Find c a 10 b 40 Q c 43 Find b 8 P b 26 150 53 A right triangle has sides of length 2 4 cm and 6 5 cm What is the length of the hypotenuse 63 A right triangle has a side length of 4 and a hypotenuse of length 3 What is the length of the other side 1 1 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

ZEARN MATH STUDENT EDITION 73 G8M8 LESSON 8 Find the value of x in the figure 34 5 18 x 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

G8M8 LESSON 8 ZEARN MATH STUDENT EDITION Lesson Summary There are many examples where the lengths of two legs of a right triangle are known and can be used to find the length of the hypotenuse with the Pythagorean Theorem The Pythagorean Theorem can also be used if the length of the hypotenuse and one leg is known and we want to find the length of the other leg Here is a right triangle where one leg has a length of 5 units the hypotenuse has a length of 10 units and the length of the other leg is represented by g g 5 10 Start with a2 b2 c2 make substitutions and solve for the unknown value Remember that c represents the hypotenuse the side opposite the right angle For this triangle the hypotenuse is 10 a2 b2 c2 52 g2 102 g2 102 52 g2 100 25 g2 75 g 75 Use estimation strategies to know that the length of the other leg is between 8 and 9 units since 75 is between 64 and 81 A calculator with a square root function gives 75 8 66 152 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

ZEARN MATH STUDENT EDITION Name G8M8 LESSON 8 Date GRADE 8 MISSION 8 LESSON 8 Exit Ticket A right triangle has sides of length 3 4 and x 1 Find x if it is the hypotenuse 2 Find x if it is one of the legs 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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ZEARN MATH STUDENT EDITION G8M8 LESSON 9 Lesson 9 The Converse Let s figure out if a triangle is a right triangle Warm Up 1 Consider the tips of the hands of an analog clock that has an hour hand that is 3 centimeters long and a minute hand that is 4 centimeters long Answer the questions below 4 3 Over the course of a day 1 What is the farthest apart the two tips get 2 What is the closest the two tips get 3 Are the two tips ever exactly five centimeters apart 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 155

G8M8 LESSON 9 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 Here are three triangles with two side lengths measuring 3 and 4 units and the third side of unknown length x 3 y 3 4 z 3 4 4 Sort the following six numbers from smallest to largest Put an equal sign between any you know to be equal Be ready to explain your reasoning 1 5 7 x y z ACTIVITY 2 3 Given the information provided for the right triangles shown here find the unknown leg lengths to the nearest tenth 2 a 7 156 x x 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

ZEARN MATH STUDENT EDITION 43 G8M8 LESSON 9 The triangle shown here is not a right triangle What are two different ways you can change one of the values so it would be a right triangle Sketch these new right triangles and clearly label the right angle 7 4 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 157

G8M8 LESSON 9 ZEARN MATH STUDENT EDITION Lesson Summary What if it isn t clear whether a triangle is a right triangle or not Here is a triangle 17 15 8 Is it a right triangle It s hard to tell just by looking and it may be that the sides aren t drawn to scale If we have a triangle with side lengths a b and c with c being the longest of the three then the converse of the Pythagorean Theorem tells us that any time we have a2 b2 c2 we must have a right triangle Since 82 152 64 225 289 172 any triangle with side lengths 8 15 and 17 must be a right triangle Together the Pythagorean Theorem and its converse provide a one step test for checking to see if a triangle is a right triangle just using its side lengths If a2 b2 c2 it is a right triangle If a2 b2 c2 it is not a right triangle 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

ZEARN MATH STUDENT EDITION Name G8M8 LESSON 9 Date GRADE 8 MISSION 8 LESSON 9 Exit Ticket A triangle has side lengths 7 10 and 12 Is it a right triangle 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 159

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ZEARN MATH STUDENT EDITION G8M8 LESSON 10 Lesson 10 Applications of the Pythagorean Theorem Let s explore some applications of the Pythagorean Theorem Warm Up 1 1 2 3 4 24 Which estimate is closest to the actual value of the expression Explain your reasoning 4 4 5 5 7 2 2 5 3 42 6 6 5 7 10 97 13 13 25 13 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 161

G8M8 LESSON 10 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 Mai and Tyler were standing at one corner of a large rectangular field and decided to race to the opposite corner Since Mai had a bike and Tyler did not they thought it would be a fairer race if Mai rode along the sidewalk that surrounds the field while Tyler ran the shorter distance directly across the field The field is 100 meters long and 80 meters wide Tyler can run at around 5 meters per second and Mai can ride her bike at around 7 5 meters per second 162 1 Before making any calculations who do you think will win By how much Explain your thinking 2 Who wins Show 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

ZEARN MATH STUDENT EDITION G8M8 LESSON 10 ACTIVITY 2 3 Here are two rectangular prisms K L 6 5 4 5 5 5 1 Which figure do you think has the longer diagonal Note that the figures are not drawn to scale 2 Calculate the lengths of both diagonals Which one is actually longer 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

G8M8 LESSON 10 ZEARN MATH STUDENT EDITION Lesson Summary The Pythagorean Theorem can be used to solve any problem that can be modeled with a right triangle where the lengths of two sides are known and the length of the other side needs to be found For example let s say a cable is being placed on level ground to support a tower It s a 17 foot cable and the cable should be connected 15 feet up the tower How far away from the bottom of the tower should the other end of the cable connect to the ground It is often very helpful to draw a diagram of a situation such as the one shown here It s assumed that the tower makes a right angle with the ground Since this is a right triangle the cab le tower 17 15 a relationship between its sides is a2 b2 c2 where c represents the length of the hypotenuse and a and b represent the lengths of the other two sides The hypotenuse is the side opposite the right angle Making substitutions gives a2 152 172 Solving this for a gives a 8 So the other end of the cable should connect to the ground 8 feet away from the bottom of the tower 164 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

ZEARN MATH STUDENT EDITION G8M8 LESSON 10 Name Date GRADE 8 MISSION 8 LESSON 10 Exit Ticket 9 75 m 10 24 m 3 45 m Sails come in many shapes and sizes This drawing shows a sail that is a triangle Is it a right triangle 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 165

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ZEARN MATH STUDENT EDITION G8M8 LESSON 11 Lesson 11 Finding Distances in the Coordinate Plane Let s find distances in the coordinate plane Warm Up 1 1 Use the coordinate points below to answer the following questions Order the following pairs of coordinates from closest to farthest apart Be prepared to explain your reasoning a 2 4 and 2 10 b 3 6 and 5 6 c 12 12 and 12 1 d 7 0 and 7 9 e 1 10 and 4 10 2 Name another pair of coordinates that would be closer together than the first pair on your list 3 Name another pair of coordinates that would be farther apart than the last pair on your list 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

G8M8 LESSON 11 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 Find the distances between the three points shown y 10 14 9 8 6 4 2 16 14 12 10 8 14 3 6 4 2 2 2 4 6 8 10 4 12 14 16 18 20 x 16 3 ACTIVITY 2 3 168 Have each person in your group select one of the sets of coordinate pairs shown here Then calculate the length of the line segment between those two coordinates Once the values are calculated have each person in the group briefly share how they did their calculations 3 1 and 5 7 1 6 and 5 2 1 2 and 5 6 2 5 and 6 1 1 How does the value you found compare to the rest of your group 2 In your own words write an explanation to another student for how to find the distance between any two coordinate pairs 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

ZEARN MATH STUDENT EDITION G8M8 LESSON 11 Lesson Summary We can use the Pythagorean Theorem to find the distance between any two points on the coordinate plane For example if the coordinates of point A are 2 3 and the coordinates of point B are 8 4 let s find the distance between them This distance is also the length of line segment AB It is a good idea to plot the points first y B 8 4 4 3 2 1 8 7 6 5 4 3 2 1 x 1 2 Think of the distance between A and B or the length of segment AB as the hypotenuse of a right triangle The lengths of the legs can be deduced from the coordinates of the points 3 y B 8 4 4 3 2 4 3 7 The length of the horizontal leg is 6 which can be seen in the diagram but it is also the distance between the x coordinates of A and B since 8 2 6 The length of the vertical leg is 7 which can be seen in the diagram but this is also the distance between the y coordinates of A and B since 4 3 7 A 2 3 Once the lengths of the legs are known we use the Pythagorean Theorem to find the length of the hypotenuse AB which we can represent with c Since c is a positive number there is only one value it can take 1 8 7 6 5 4 3 2 1 x 1 2 8 2 6 A 2 3 3 62 72 c2 36 49 c2 85 c2 85 c This length is a little longer than 9 since 85 is a little longer than 81 Using a calculator gives a more precise answer 85 9 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 169

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ZEARN MATH STUDENT EDITION G8M8 LESSON 11 Name Date GRADE 8 MISSION 8 LESSON 11 Exit Ticket Here are two line segments with lengths e and f Calculate the exact values of e and f Which is larger 2 3 2 2 e f 1 1 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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ZEARN MATH STUDENT EDITION G8M8 LESSON 12 Lesson 12 Edge Lengths and Volumes Let s explore the relationship between the volumes and edge lengths of cubes Warm Up 1 Let a b c d e and f be positive numbers Given these equations arrange a b c d e and f from least to greatest Explain your reasoning a2 9 b3 8 c2 10 d3 9 e2 8 f3 7 Concept Exploration ACTIVITY 1 2 Fill in the missing values using the information provided Edges 5 Volume Volume equation 27 in3 16 3 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 173

G8M8 LESSON 12 ZEARN MATH STUDENT EDITION ACTIVITY 2 3 Your group will get a set of cards For each card with a letter and value find the two other cards that match One shows the location on a number line where the value exists and the other shows an equation that the value satisfies Be prepared to explain your reasoning Lesson Summary To review the side length of a square is the square root of its area In this diagram the square has an area of 16 units and a side length of 4 units These equations are both true 42 16 16 4 Now think about a solid cube The cube has a volume and the edge length of the cube is called the cube root of its volume In this diagram the cube has a volume of 64 units and an edge length of 4 units 4 These equations are both true 43 64 64 4 64 is pronounced The cube root of 64 Here are some other values of cube roots 8 2 because 23 8 27 3 because 33 27 125 5 because 53 125 TERMINOLOGY Cube root 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

ZEARN MATH STUDENT EDITION G8M8 LESSON 12 Name Date GRADE 8 MISSION 8 LESSON 12 Exit Ticket Plot 36 and 36 on the number line 0 1 2 3 4 5 6 7 8 9 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 175

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ZEARN MATH STUDENT EDITION G8M8 LESSON 13 Lesson 13 Cube Roots Let s compare cube roots Warm Up 1 Decide if each statement is true or false 5 3 5 27 3 3 7 7 3 10 3 1 000 64 23 Concept Exploration ACTIVITY 1 2 1 2 3 4 What two whole numbers does each cube root lie between Be prepared to explain your reasoning 5 23 81 999 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

G8M8 LESSON 13 ZEARN MATH STUDENT EDITION ACTIVITY 2 The numbers x y and z are positive and 3 x3 5 3 2 1 y3 27 0 1 2 3 z3 700 4 5 6 7 8 9 1 Plot the approximate locations of x y and z on the number line Be prepared to share your reasoning with the class 2 Plot the approximate location of 2 on the number line Lesson Summary Remember that square roots of whole numbers are defined as side lengths of squares For example 17 is the side length of a square whose area is 17 We define cube roots similarly but using cubes instead of squares The number 17 pronounced the cube root of 17 is the edge length of a cube which has a volume of 17 We can approximate the values of cube roots by observing the whole numbers around it and remembering the relationship between cube roots and cubes For example 20 is between 2 and 3 since 23 8 and 33 27 and 20 is between 8 and 27 Similarly since 100 is between 43 and 53 we know 100 is between 4 and 5 Many calculators have a cube root function which can be used to approximate the value of a cube root more precisely Using our numbers from before a calculator will show that 20 2 7144 and that 100 4 6416 Also like square roots most cube roots of whole numbers are irrational The only time the cube root of a number is a whole number is when the original number is a perfect cube 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

ZEARN MATH STUDENT EDITION G8M8 LESSON 13 Name Date GRADE 8 MISSION 8 LESSON 13 Exit Ticket Lin is asked to place a point on a number line to represent the value of 49 and she writes 0 1 2 3 4 5 6 7 8 9 10 Where should 49 actually be on the number line How do you think Lin got the answer she did 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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ZEARN MATH STUDENT EDITION G8M8 LESSON 14 Lesson 14 Decimal Representations of Rational Numbers Let s learn more about how rational numbers can be represented Warm Up 1 What do you notice What do you wonder Concept Exploration ACTIVITY 1 2 Rational numbers are fractions and their opposites All of these numbers are rational numbers Show that they are rational by writing them in the form ab or ab a 0 2 b 4 c 0 333 d 1000 e 1 000001 f 19 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

G8M8 LESSON 14 3 182 a 3 8 b 7 5 c 999 1000 d 111 2 e 81 ZEARN MATH STUDENT EDITION All rational numbers have decimal representations too Find the decimal representation of each of these rational numbers 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

ZEARN MATH STUDENT EDITION G8M8 LESSON 14 ACTIVITY 2 43 Answer the questions about the number lines below 0 1 1 2 On the topmost number line label the tick marks Next find the first decimal place of 11 using long 2 division and estimate where 11 should be placed on the top number line 2 2 Label the tick marks of the second number line Find the next decimal place of 11 by continuing the 2 long division and estimate where 11 should be placed on the second number line Add arrows from 2 the second to the third number line to zoom in on the location of 11 3 Repeat the earlier step for the remaining number lines 4 2 What do you think the decimal expansion of 11 is 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

G8M8 LESSON 14 ZEARN MATH STUDENT EDITION Lesson Summary We learned earlier that rational numbers are a fraction or the opposite of a fraction For example 34 and 52 are both rational numbers A complicated looking numerical expression can also be a rational number as long as the value of the expression is a positive or negative fraction For example 64 and 3 1 3 1 are rational numbers because 64 8 and 12 8 8 Rational numbers can also be written using decimal notation Some have finite decimal expansions like 0 75 2 5 or 0 5 Other rational numbers have infinite decimal expansions like 0 7434343 where the 43s repeat forever To avoid writing the repeating part over and over we use the notation 0 743 for this number The bar over part of the expansion tells us the part which is to repeat forever A decimal expansion of a number helps us plot it accurately on a number line divided into tenths For example 0 743 should be between 0 7 and 0 8 Each further decimal digit increases the accuracy of our plotting For example the number 0 743 is between 0 743 and 0 744 184 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

ZEARN MATH STUDENT EDITION Name G8M8 LESSON 14 Date GRADE 8 MISSION 8 LESSON 14 Exit Ticket Explain how you know that 3 4 is a rational number 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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ZEARN MATH STUDENT EDITION G8M8 LESSON 15 Lesson 15 Infinite Decimal Expansions Let s think about infinite decimals Warm Up 1 The first 3 digits after the decimal for the decimal expansion of 37 have been calculated Find the next 4 digits 0 428 7 3 0 0 0 2 8 20 14 60 56 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 187

G8M8 LESSON 15 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 Your group will receive a set of cards Each card will have a calculations side and an explanation side 1 The cards show Noah s work calculating the fraction representation of 0 485 Arrange these in 481 order to see how he figured out that 0 485 990 without needing a calculator 2 Use Noah s method to calculate the fraction representation of a 0 186 b 0 788 188 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

ZEARN MATH STUDENT EDITION G8M8 LESSON 15 ACTIVITY 2 3 Answer the questions below about 2 a Why is 2 between 1 and 2 on the number line b Why is 2 between 1 4 and 1 5 on the number line c Label all of the tick marks Plot 2 on all three number lines Make sure to add arrows from the second to the third number lines 1 2 d How can you figure out an approximation for 2 accurate to 3 decimal places 43 Answer the questions below about a Elena notices a beaker in science class says it has a diameter of 9 cm and measures its circumference to be 28 3 cm What value do you get for using these values and the equation for circumference C 2 r 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

G8M8 LESSON 15 ZEARN MATH STUDENT EDITION b Diego learned that one of the space shuttle fuel tanks had a diameter of 840 cm and a circumference of 2 639 cm What value do you get for using these values and the equation for circumference C 2 r c Label all of the tick marks on the number lines Use a calculator to get a very accurate approximation of and plot that number on all three number lines 3 4 3 1 3 2 d How can you explain the differences between these calculations of Lesson Summary Not every number is rational Earlier we tried to find a fraction whose square is equal to 2 That turns out to be impossible although we can get pretty close try squaring 57 Since there is no fraction equal to 2 it is not a rational number which is why we call it an irrational number Another well known irrational number is Any number rational or irrational has a decimal expansion Sometimes it goes on forever For example 2 has the decimal expansion 0 181818 with the 18s repeating forever Every the rational number 11 rational number has a decimal expansion that either stops at some point or ends up in a repeating 2 Irrational numbers also have infinite decimal expansions but they don t end up in a pattern like 11 repeating pattern From the decimal point of view we can see that rational numbers are pretty special Most numbers are irrational even though the numbers we use on a daily basis are more frequently rational 190 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

ZEARN MATH STUDENT EDITION Name G8M8 LESSON 15 Date GRADE 8 MISSION 8 LESSON 15 Exit Ticket Let x 0 147 and y 0 147 Is x a rational number Is y a rational number Which is larger x or y 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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ZEARN MATH STUDENT EDITION G8M9 LESSON 1 Lesson 1 Mathematical Modeling Let s see if we can predict the weather Warm Up 1 What factors or variables can influence the outside temperature Make a list of different factors Write a sentence for each factor describing how changing it could change the temperature Example one factor is time of day Often after sunrise the temperature increases reaches a peak in the early afternoon and then decreases 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

G8M9 LESSON 1 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 1 Andre and Lin are wondering if temperature is a function of latitude Answer the questions below to assess their thinking Andre says I think temperature is a function of latitude as long as we fix the time when we are measuring the temperature Lin says But what if you have two places with the same latitude Look at this weather map for Washington State Seattle and Spokane have the same latitude but different temperatures right now What do Andre and Lin mean Map of Washington by United States Census Bureau via American Fact Finder Public Domain 2 Andre and Lin are discussing whether it is possible to define latitude and temperature in a way that makes sense to talk about temperature as a function of latitude They are considering different options What are some advantages and disadvantages of each option Here are the options a Finding the temperature right now in cities with different latitudes b Finding the daily high temperature from cities that have different latitudes c Finding the average high temperature in a specific month e g September from cities that have different latitudes d Finding the average yearly temperature from cities that have different latitudes 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

ZEARN MATH STUDENT EDITION G8M9 LESSON 1 ACTIVITY 2 3 Lin and Andre decided that modeling temperature as a function of latitude doesn t really make sense They realized that they can ask whether there is an association between latitude and temperature 1 What information could they gather to determine whether temperature is related to latitude 2 What should they do with that information to answer the question ACTIVITY 3 43 What do you notice What do you wonder Rainfall in inches 20 16 12 8 4 0 0 2 4 6 Month 8 10 12 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

G8M9 LESSON 1 ZEARN MATH STUDENT EDITION ACTIVITY 4 53 Examine the data in the table to answer the questions below Latitude Average high degrees temperature in North September degrees F Atlanta GA 33 38 82 Portland ME 43 38 69 Boston MA 42 22 73 Dallas TX 32 51 88 Denver CO 39 46 77 Edmonton AB 53 34 62 Fairbanks AK 64 48 55 Juneau AK 58 22 56 Kansas City MO 39 16 78 Lincoln NE 40 51 77 Miami FL 25 45 88 Minneapolis MN 44 53 71 New York City NY 40 38 75 Orlando FL 28 26 90 Philadelphia PA 39 53 78 San Antonio TX 29 32 89 San Francisco CA 37 37 74 Seattle WA 47 36 69 Tampa FL 27 57 89 Tucson AZ 32 13 93 Yellowknife NT 62 27 50 City 198 1 What information does each row contain 2 What is the range for each variable 3 Do you see an association between the two variables If so describe the association 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

ZEARN MATH STUDENT EDITION G8M9 LESSON 1 ACTIVITY 5 63 Use the grid to create a scatter plot of the data and draw a line that fits the data 1 Make a scatter plot of the data Describe any patterns of association that you notice 2 Draw a line that fits the data Write an equation for this line ACTIVITY 6 73 1 In the previous activity you found the equation of a line to represent the association between latitude and temperature This is a mathematical model Use your mathematical model to answer the questions below Use your model to predict the average high temperature in September of the following cities that were not included in the original data set a Detroit Lat 42 14 b Albuquerque Lat 35 2 c Nome Lat 64 5 d Your own city Lat 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

G8M9 LESSON 1 ZEARN MATH STUDENT EDITION 2 Draw points that represent the predicted temperatures for each city on the scatter plot from Activity 5 3 The actual average high temperature in September of these cities were Detroit 74 F Albuquerque 82 F Nome 49 F Your own city How well does your model predict the temperature Compare the predicted and actual temperatures 4 If you added the actual temperatures for these four cities to the scatter plot would you move your line 5 Are there any outliers in the data What might be the explanation ACTIVITY 7 83 200 Use your equation for the line that models the association between latitude and temperature of the cities to answer the questions below 1 What does the slope mean in the context of this situation 2 Find the vertical and horizontal intercepts and interpret them in the context of the situation 3 Can you think of a city or a location that could not be represented using this same model Explain your 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

ZEARN MATH STUDENT EDITION G8M9 LESSON 2 Lesson 2 Tessellations of the Plane Let s explore geometric patterns Warm Up 1 What do you notice What do you wonder 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 201

G8M9 LESSON 2 ZEARN MATH STUDENT EDITION Concept Exploration ACTIVITY 1 2 202 Pick one of the shapes below Create a tessellation by tracing copies of your shape Make sure to use the same shape as your partner Then answer the following questions 1 Compare your tessellation to your partner s How are they similar How are they different 2 If possible make a third tessellation of the plane with your shape different from the ones you and your partner already created If not possible explain why it is not possible 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

ZEARN MATH STUDENT EDITION G8M9 LESSON 2 ACTIVITY 2 3 Pick one of the figures and describe the tessellation Your partner will identify which tessellation you are describing Then trade roles so your partner describes the tessellation and you identify the figure 1 You and your partner each have a card with a tessellation Describe what is on your card so that your partner can produce the tessellation this should be done so that you cannot see your partner s work until it is complete 2 Check together to see if your partner s tessellation agrees with your card and discuss any differences 3 Change roles so your partner describes a tessellation which you attempt to produce 4 Check the accuracy of your construction and discuss any discrepancies 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 203

G8M9 LESSON 2 ZEARN MATH STUDENT EDITION ACTIVITY 3 43 For each shape triangle square pentagon hexagon and octagon decide if you can use that shape to make a regular tessellation of the plane Explain your reasoning For the polygons that do not work what goes wrong Explain your reasoning ACTIVITY 4 53 204 Refer to the regular polygons printed in the previous activity for a visual representation of an equilateral triangle Use the triangle to answer the questions below 1 What is the measure of each angle in an equilateral triangle How do you know 2 How many triangles can you fit together at one vertex Explain why there is no space between the triangles 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

ZEARN MATH STUDENT EDITION G8M9 LESSON 2 3 Explain why you can continue the pattern of triangles to tessellate the plane 4 How can you use your triangular tessellation of the plane to show that regular hexagons can be used to give a regular tessellation of the plane ACTIVITY 5 63 Can you make a regular tessellation of the plane using regular polygons with 7 sides What about 9 sides 10 sides 11 sides 12 sides Explain 1 How does the measure of each angle in a square compare to the measure of each angle in an equilateral triangle How does the measure of each angle in a regular 8 sided polygon compare to the measure of each angle in a regular 7 sided polygon 2 What happens to the angles in a regular polygon as you add more sides 3 Which polygons can be used to make regular tessellations of the plane 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 205

G8M9 LESSON 2 ZEARN MATH STUDENT EDITION ACTIVITY 6 73 You will be assigned one of the three triangles below You can use the picture to draw copies of the triangle on tracing paper Your goal is to find a tessellation of the plane with copies of the triangle ACTIVITY 7 83 206 Can you make a tessellation of the plane with copies of the quadrilateral Explain 1 Choose and trace a copy of one of the other two quadrilaterals Next trace images of the quadrilateral rotated 180 degrees around the midpoint of each side What do you notice 2 Can you make a tessellation of the plane with copies of the quadrilateral from the previous problem 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

ZEARN MATH STUDENT EDITION G8M9 LESSON 2 ACTIVITY 8 93 Can you tessellate the plane with copies of the pentagon Explain why or why not Note that the two sides making angle A are congruent A 120 80 100 120 120 Pause your work here 1 Take one pentagon and rotate it 120 degrees clockwise about the vertex at angle A and trace the new pentagon Next rotate the pentagon 240 degrees clockwise about the vertex at angle A and trace the new pentagon 2 Explain why the three pentagons make a full circle at the central vertex 3 Explain why the shape that the three pentagons make is a hexagon i e the sides that look like they are straight really are straight TERMINOLOGY Tessellation 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 207

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ZEARN MATH STUDENT EDITION G8V3 Terminology Cube root The cube root of a number n is the number whose cube is n It is also the edge length of a cube with a volume of n We write the cube root of n as n For example the cube root of 64 written as 64 is 4 because 43 is 64 64 is also the edge length of a cube that has a volume of 64 Leg Hy po ten Leg Leg Leg Hypotenuse Leg Leg Hypotenuse An irrational number is a number that is not a fraction or the opposite of a fraction Pi and 2 are examples of irrational numbers Legs Leg e Leg se us Leg nu ote Hyp po ten Irrational number The legs of a right triangle are the sides that make the right angle Here are some right triangles Each leg is labeled se nu ote Hyp Leg Hy The hypotenuse is the side of a right triangle that is opposite the right angle It is the longest side of a right triangle Here are some right triangles Each hypotenuse is labeled us Hypotenuse e Leg Leg Leg Leg Hypotenuse Leg Hypotenuse Leg Pythagorean Theorem The Pythagorean Theorem describes the relationship between the side lengths of right triangles The diagram shows a right triangle with squares built on each side If we add the areas of the two small squares we get the area of the larger square The square of the hypotenuse is equal to the sum of the squares of the legs This is written as a2 b2 c2 c2 25 b2 9 a2 16 Rational number A rational number is a fraction or the opposite of a fraction Some examples of rational numbers are 74 0 36 0 2 13 5 9 Scientific notation Scientific notation is a way to write very large or very small numbers We write these numbers by multiplying a number between 1 and 10 by a power of 10 For example the number 425 000 000 in scientific notation is 4 25 108 The number 0 0000000000783 in scientific notation is 7 83 10 11 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 209

G8V3 ZEARN MATH STUDENT EDITION Square root The square root of a positive number n is the positive number whose square is n It is also the side length of a square whose area is n We write the square root of n as n For example the square root of 16 written as 16 is 4 because 42 is 16 16 is also the side length of a square that has an area of 16 Tessellation A tessellation is a repeating pattern of one or more shapes The sides of the shapes fit together perfectly and do not overlap The pattern goes on forever in all directions This diagram shows part of a tessellation 210 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 org NAME Grade 8 Student Edition Vol 1 Mission 1 Rigid Transformations and Congruence Mission 2 Dilations Similarity and Introducing Slope Mission 3 Linear Relationships Mission 4 Linear Equations and Linear Systems Mission 5 Functions and Volume Mission 6 Associations in Data Student Edition Vol 2 Vol 3 Mission 7 Exponents and Scientific Notation Mission 8 Pythagorean Theorem and Irrational Numbers Mission 9 Putting It All Together G8 Vol 3 Zearnmath_SE_Grade8_Vol3 indd 1 Grade 8 Volume 3 MISSIONS 1 2 3 4 5 6 7 8 9 12 15 22 3 41 PM