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QCM52B Lesson Plan Assignment Ng Zi Xuan 1 ChatGPT Lesson Plan Critique I attempted to write my prompt as detailed as I can in hopes that the generated lesson plan can satisfy more of the assignment criteria. Generally, the lesson plan has a logical flow that can be referred to when planning a lesson. However, it is not very detailed and does not satisfy all the requirements stated in my prompt despite it being so detailed. Furthermore, it also does not ensure that all the SIOs stated are taught during the lesson, let alone assessed. Therefore, it is surely not a lesson plan that can be adopted directly and definitely require a substantial amount of modification. Firstly, the lesson plan does not satisfy all the technical requirements stated in my prompt. For instance, the lesson plan is for a 70-minute lesson instead of the 80-minute requirement stated in my prompt. Secondly, in the second column, it does not have subheadings as well. The contents in the last two columns are also mismatched with their headers. Besides technical requirements, some other requirements related to teaching that I prompted were also not met. The biggest issue is that the lesson plan does not include the teaching and assessment of SIO3 and SIO4. It is also not clear how the lesson plan is targeted at an average student (of middle readiness). Finally, it does not highlight the specific misconceptions or difficulties that students might face. While I expected that some of these requirements will not be met and tried to prompt it to mention them explicitly in the last paragraph of my prompt, it seemed to be overlooked as well. Even though the other requirements of my prompts are met, they generally lack depth. In the following, I will analyse each segment of the lesson in detail. For the Introduction, it suggested to start the lesson by recalling the standard form of a quadratic equation before providing an expression that can be factorised. While I do not find it necessary to recall the standard form of a quadratic equation as it is too basic for the students, I agree that we should introduce an expression that is factorisable to engage students’ prior knowledge. Here, it could have specified the use of the multiplication frame too. However, merely factorising the expression is insufficient as we need students to proceed to solving the equation as well, recapping on the correct application of the zero-product rule. This is to help students realise later that some equations cannot be solved by factorisation. In the next segment, it suggested to directly explain how to complete the square of a quadratic expression of the form . In my opinion, this should not be done so soon as students have yet to understand the need for this process. We should allow students to try solving an equation that cannot be solved by factorisation – the only method that students know to solve a quadratic equation, to provide a rationale for learning the new concept of completing the square, arousing their interest to the topic. This would be the trigger to the lesson, which is missing in the lesson plan generated. I would also not explain the concept of completing the square by directly doing so for an algebraic expression. Instead, it is a good opportunity to leverage on Bruner’s CPA approach, to guide students to learn about this concept through an Investigation activity involving some physical manipulatives. In the Guided Practice segment, the instructional moves seem to be very similar to that of the previous segment, apart from the additional element of encouraging students to identify the values needed to complete the square. This seems to me to be the “I do, We do, You do” approach that ChatGPT has adopted, with this segment being the “We do” portion. However, I am mindful of the time allocated to do so, as the total time of 20 minutes allocated for the previous and current segment is too long. With a shortage of time, if the worked example is clear enough and the solutions be kept on the board for students to refer during the class
QCM52B Lesson Plan Assignment Ng Zi Xuan 2 practice, I feel that the “We do” segment can be skipped. It is also worth noting that this lesson plan stopped at “We do” and did not proceed to allow students to practice on their own. Next, while the lesson plan stated to apply Bruner’s CPA approach by progressing from concrete examples to abstract concepts, the description of the instructional move does not tally with the approach. This suggests that ChatGPT might not fully understand what constitutes the CPA approach. Furthermore, the time allocated for this segment is also too long. However, one thing commendable in this segment is the incorporation of examples involving both integers and fractions in such problems. It is also commendable that it highlights the need to include the negative root when taking square roots on both sides of the equation. In the consolidation segment, it is good that it incorporated worked examples before tasking students to do class practice as I prompted, as well as providing concrete examples of such questions and their solutions which are accurate. It is also good that the assigned class practices are similar to the worked example. However, they are limited to having square numbers on the RHS and did not involve other non-square integers or non-integers that it has suggested in the previous segment. Similar to the previous segment, the number of questions generated are also insufficient to fill up the time allocated. In the last segment of Closure, with 10 minutes allocated to the segment, the teacher can probably do more than just summarising key concepts by direct instruction, but also ask students questions or reflect on their learning. However, if only 5 minutes are allocated, then it might be more effective for the teacher to quickly wrap up the lesson by summarising the lesson through direct instruction and also assign homework. For the questions generated to ask students, they are not really Key Questions as required in the assignment and also does not really check for students’ understandings. In conclusion, the lesson plan lacks depth and the positive aspects of the lesson plan generated does not provide much motivation to use AI to generate a lesson plan because they can easily be done without its use. In my opinion, the process of generating this lesson plan using AI has not helped in my lesson planning in any way, including generating of ideas. With regards to this assignment, I would still carry on with my lesson planning with the current ideas and lesson flow that I have in mind. Perhaps, the current stage of development of AI is still not at a stage where it can aid teachers in this aspect of lesson planning. Moving forward, I would still choose to rely on myself to design my lesson plan as I believe it is more efficient in coming up with quality lesson plans.
QCM52B Lesson Plan Assignment Ng Zi Xuan 3 Topic: Mathematics, Solving Quadratic Equations by Completing the Square Textbook: think! Mathematics 3A (8th Edition, Shinglee) Class Profile: Sec 3 Express, Low to Middle Readiness Suggested Lesson Duration: 80 minutes Prerequisite Knowledge Before the start of the lesson, the students should be able to: 1. Solve a quadratic equation in one variable by factorisation 2. State examples of numbers that are perfect squares 3. Use the algebraic identities and to solve problems 4. Recognise the pictorial representation of as a square Main Concepts 1. Completing the square of a quadratic expression Specific Instructional Objectives At the end of the lesson, the students should be able to: 1. Solve quadratic equations of the form 2. Describe the process of completing the square for quadratic expressions of the form 3. Express quadratic expressions of the form in the form 4. Express quadratic expressions of the form in the form (Contingency plan if there is time) Big Ideas in Mathematics 1. Equivalence (Page 2, 7 and 8) 2. Notations (Page 4) Time Teaching / Learning Activities Rationales Resources 10 min Introduction Access Handout 1A • Instruct students to open softcopy of the handout from Google Classroom or take out the hardcopy they printed • Project softcopy of student’s copy handout on the screen throughout the lesson to annotate and teach from there accordingly • Inform students that we are learning a new topic today that builds upon their prior knowledge of solving quadratic equations Recap (Section A) • Theory of Constructivism: Recapping prior knowledge allows students to construct the new knowledge later of solving a quadratic equation by completing the square Student’s Handout 1A (see Appendix II p.21)
QCM52B Lesson Plan Assignment Ng Zi Xuan 4 Time Teaching / Learning Activities Rationales Resources • Instruct students to attempt Example 1 in 3 minutes as recap. Start and project online timer on screen. • Walk around to supervise seatwork and render assistance especially to students of lower-readiness • Recap to students individually on how to use the multiplication frame if they are struggling to factorise the expression • Some students might apply the ‘Zero Product Principle’ incorrectly by subtracting 2 on both sides of the equation to obtain then factorise the LHS of the equation to get , before incorrectly concluding that or . Correct their mistake immediately if spotted. • When time is up or when all students have completed, annotate the solution step by step on the handout while highlighting following points. • Highlight that one side of the equation needs to be 0 before we can solve quadratic equations by factorisation using the multiplication frame • Highlight how to correctly apply the Zero Product Principle • Reemphasise that the basis of solving an equation is to convert it into a sequence of appropriate equivalent equations. (Big Idea: Equivalence) Trigger – Introductory Problem (IP) (Section B) • Instruct students to solve the Introductory Problem (Example 2) similarly in 3 minutes. Start and project online timer on screen. • If students ask for help, inform them to think and try for a while more. • At about 1 minute or when most students are ‘sufficiently stuck’, instruct students to stop trying and ask what is the difficulty that they faced in completing the task • Example 1 is a simple quadratic equation that can be solved easily by factorisation so students will not take too much time of the lesson • Allocated maximum 3 minutes for Example 1 considering the exam guideline of 1.5min per mark for this 2-mark question • The use of the online timer helps teacher and students to keep track of time better, especially for the IP later, so that not too much time is spent on Introduction • Rendering assistance to students to provide scaffolding for them to be able to follow the class • Highlight the respective points when going through the solutions to address common misconceptions and teach towards Big Idea Student’s Handout 1A (see Appendix II p.21) Online Timer (see Appendix II, p.17) Student’s Handout 1A (see Appendix II p.21) Online Timer (see Appendix II, p.17)
QCM52B Lesson Plan Assignment Ng Zi Xuan 5 Time Teaching / Learning Activities Rationales Resources • Elicit expected response if needed: Due to the expression being non-factorisable Key Question (Section B): • Does this mean that the equation is unsolvable? • Invite students to ponder on the question, practice wait time. • After wait time, ask students to raise their hands if they think it is unsolvable or still solvable respectively. Introduce Main Objective • Inform students that this equation indeed cannot be solved using the factorisation method since the expression cannot be factorised. Annotate under ‘Remarks’ for students to fill in the blank. • Inform students that it is still solvable through other methods. One of which is by completing the square, which is what we are learning in this lesson. • Write ‘Completing the Square’ on the left side of whiteboard • Get students to ponder and practice wait time to stimulate thinking and arouse interest to lesson topic • Write ‘Completing the Square’ on the whiteboard to remind students of the learning objective throughout the lesson Teacher’s Handout 1A (see Appendix II p.27) Whiteboard & Markers 05 min Development 1 (Skills: Solving equations of the form ) Introduction (Section C) • Inform students that when a quadratic equation cannot be solved using factorisation, such as the equation in the IP, it can be expressed in the form , where • Annotate on handout for students to fill in the blank • Inform students that we will see later in the chapter how the equation in the IP can be expressed in this form to be solved, but first we will learn how to solve equations of this form Worked Example 1 (WE1) • Write ‘Solve ’ on the whiteboard, guide students step by • Draw relevance to why we need to learn this skill • “I do, We do, You do” approach: Show worked example first as a Teacher’s Handout 1A (see Appendix II p.28) Whiteboard & Markers
QCM52B Lesson Plan Assignment Ng Zi Xuan 6 Time Teaching / Learning Activities Rationales Resources step to solve (see Appendix II, p.17), highlighting the following points. • Remind students the need to include both the positive and negative roots • Highlight that the notation is used to represent ‘+’ or ‘’ in a concise manner. Thus, means or (Big Idea: Notations) • Emphasise to students to adhere to my presentation • Students may ask if they can solve using the algebraic identity by first subtracting both sides of the equation by . Allow this method if student is comfortable but highlight the risk of making careless mistakes when manipulating the signs. • Highlight that for equations of this form, we do not expand the LHS guide for students to try on their own later. • Highlight to teach towards Big Idea • Presentation is different from textbook but easier for students, reducing careless mistakes • Do not teach both methods of solving due to students’ low readiness in working with algebraic identities. Teaching both methods might confuse students. Method in WE1 is likely to be easier. 10 min Consolidation 1 (Skills: Solving equations of the form ) Class Practice 1 (Section C) • Leaving WE1 on the whiteboard, instruct students to do Example 3(a) - (b) • Walk around to supervise seatwork and render assistance especially to students of lower-readiness. Assess students’ learning of SIO1 based on their ability to solve the questions. • After most students are done with 3(a), invite a student who has successfully solved it to quietly write his/her solution on the whiteboard • Direct attention of students to the solution on the whiteboard when needed, e.g., to provide scaffolding if they are stuck or to ask them to correct their mistake • After most students are done with 3(b), invite another student who has successfully solved 3(b) to quietly • Theory of Behaviourism: Students learn and master one skill at a time. • Leave WE1 on the whiteboard for students to refer if they need to • Included examples with integers and non-integers answers to prepare students for non-integer answers when solving the equation in the IP later. Part (b) Student’s Handout 1A (see Appendix II p.22) Whiteboard & Markers
QCM52B Lesson Plan Assignment Ng Zi Xuan 7 Time Teaching / Learning Activities Rationales Resources write his/her solution on the whiteboard • When most of the class has completed both questions, direct students’ attentions to the whiteboard to tally their answers and presentations. • Praise and give feedback accordingly to students’ work changes the variable and sets its coefficient . • Invite students to write their solutions quietly so that students who are not done can continue to attempt the questions on their own. 05 min Development 2 (Concept: Completing the Square for ) Recap on Perfect Squares (Section D) • Inform students that the first concept they need to remember before we can express the equation in the IP in the general form in Section C is the concept of perfect squares • Ask students to state some numbers that are perfect squares, raising their hands to answer. After 2-3 students have answered, instruct the class to fill in the first blank. • Inform students that algebraic expressions like those written in the 2nd paragraph are also perfect squares and can be represented as squares pictorially • Inform students what to fill in the rest of the blanks in the section • Referring to Figure 1, remind students of the pictorial representation of that they have come across last year • Highlight to students that and are perfect squares because they are the squares of and respectively, which are the respective lengths of each of the square in Figure 1. • Highlight to students the length and breadth of each coloured square and rectangle • Asking students questions to help recap their concept of perfect squares more effectively • Having some blanks in the handout for students to fill keeps them attentive in class • Highlight these facts to prepare students for the IBL activity later Teacher’s Handout 1A (see Appendix II p.28)
QCM52B Lesson Plan Assignment Ng Zi Xuan 8 Time Teaching / Learning Activities Rationales Resources 20 min • Highlight to students that the expressions in each coloured square or rectangle denotes the area of that shape Key Question (Section E): • What does it mean to complete the square and what is the process to do so? • Inform students that in Section E, we are doing an Investigation activity to learn what it means to complete the square and how to do so. IBL: Investigation (Section E) • Instruct students to form their usual designated groups of 4-5 while group leader collects a folder containing physical manipulatives from the teacher • Give students 15 minutes to complete the Investigation in Section E as group work • Inform students that some parts contain footnotes that provide hints if they require them • Walk around to supervise each group to render assistance or answer questions from students to ensure every group is on the right track. • Students might face some difficulties when abstracting the relationship in part (d). Guide them by directing their attention to , then express in terms of . Consolidation of Activity • After 15 minutes or when every group has completed, end the activity, and gather the class to go through the answers. • Project softcopy of teacher’s copy handout to show solutions of part (a) and (b) for students to tally their answers. • Highlight that their pictorial representations might differ in orientation but are still equivalent • Bruner’s guided-discovery approach: Investigation activity aims to guide students to discover the process of completing the squares by using Bruner’s CPA approach • The IBL activity adopts Bruner’s CPA approach, going from concrete physical pieces of paper to the pictorial drawing of algebraic expressions, to abstracting the process of completing the squares • Working in groups encourages students to communicate their ideas and learn from each other • Modifying the Investigation in the textbook to allow more concrete approach given students’ low-readiness. • In part (d), instead of using as in the textbook, Student’s Handout 1A (see Appendix II p.23) Folders containing physical manipulatives (see Appendix II p.19-20) Teacher’s Handout 1A (see Appendix II p.29-30)
QCM52B Lesson Plan Assignment Ng Zi Xuan 9 Time Teaching / Learning Activities Rationales Resources because they have the same lengths and area. • Highlight that pictorially, we can see that for all values of . Thus, they are not equivalent, and students cannot write The same applies for the other expressions. (Big Idea: Equivalence) • For part (b), highlight that since a square cannot be formed for the pictorial representations, they are not perfect squares • Project softcopy of teacher’s copy handout to show solutions of part (c) for students to tally their answers. • Highlight to students that this Investigation focuses on positive coefficients of but coefficients can be positive or negative, which will be seen later during class practice. • Invite each group to provide their answer to each sub-part of part (d). If a group happens to answer incorrectly, invite other groups if they have different answers and provide them Provide explanation for correct answer then check in with the group that obtained a wrong answer privately later. • Emphasise the relationships established in (d)(i) and (d)(ii) • Scroll down the page to reveal answer to each part as each group answers • Check for understanding (SIO2): Ask students what should be filled in the blanks for the main findings at the end of the Investigation to complete the square. Involve as many students from different groups as possible in contributing their thoughts. • Correct answer: and • Possible wrong answer: Instead of , students might say or , highlighting their misconception that was used so that students can be guided to express their answer in index form without using the terminology since they have not yet learnt it. This prevents students from evaluating the number and struggle to identify the relationship. • Optional hints are provided as a form of DI. • Part (d) contains the crucial learning point from the IBL Activity, thus important to ensure that students obtain the correct results • Consolidate activity to ensure that the learning points of the IBL activity is delivered to every student • Highlight because it is a common mistake that students make. At the same time, teaching towards Big Idea. • Inviting each group to provide answer for part (d) to ensure that
QCM52B Lesson Plan Assignment Ng Zi Xuan 10 Time Teaching / Learning Activities Rationales Resources the area of the ‘missing piece’ is or . Address by referring to the pictorial representation of the ‘missing’ area. • Emphasise the main result of the Investigation activity at the end of Section E, instruct students to fill in their blanks if they have not done so • Ask students to explain pictorially why this process is called completing the square. Students should explain that pictorially, we need to add a piece representing a number, to complete the algebraic expressions in part (b) into a square every group has obtained the crucial learning points, indicating that most students can achieve SIO2 to describe the process of completing the square • Involving as many students as possible when checking for understanding is to assess if most groups have achieved SIO2. 15 min Consolidation 2 (Concept: Completing the Square for ) Worked Example 2 (WE2) • Inform students that since , we need a way to express in terms of a perfect square. • Inform students that teacher will demonstrate this process using the example. • Write ‘Express in the form ’ on the whiteboard, guide students step by step to solve (see Appendix II, p.17). • Highlight to students that since we added to complete the square of the expression, we need to subtract to maintain equivalence of the expressions. (Big Idea: Equivalence) • Remind students of the algebraic identity before the final step. Class Practice 2 (Section F) • Leaving WE2 on the whiteboard, instruct students to attempt Example 4 part (a), following my presentation. • Some students might not understand the need to subtract . Also to teach towards Big Idea. • Lower-readiness students might not understand why the last two steps are equivalent if they do not Whiteboard & Markers Student’s Handout 1A (see Appendix II p.24)
QCM52B Lesson Plan Assignment Ng Zi Xuan 11 Time Teaching / Learning Activities Rationales Resources 10 min • Walk around to supervise seatwork and render assistance especially to students of lower-readiness. Assess students’ learning of SIO3 based on their ability to solve the questions. • After most students are done, project solutions from teacher’s copy for students to tally their answer. Worked Example 3 (WE3) • Write ‘Express in the form ’ on the whiteboard, guide students step by step to solve (see Appendix II, p.17) • Ask students what is , i.e., the coefficient of in this expression, and thus what is ? (Ans: , • Highlight that can be negative and also a non-integer as they will see in Example 4(c) and (d) • Highlight to students not to make the mistake of equating and . • Remind students of algebraic identity before the final step • Emphasise that the in is not always a positive integer Class Practice 3 (Section F) • Leaving WE3 on the whiteboard, instruct students to attempt Example 4 part (b)-(d), following my presentation • Walk around to supervise seatwork and render assistance especially to students of lower-readiness. Assess students’ learning of SIO3 based on their ability to solve the questions. • After most students are done with each part, project solutions from teacher’s copy for students to tally their answer for that part recognise the algebraic identity. • Ask students to get them to realise that the value of in is not always a positive integer, since can be negative • Highlight this as low-readiness students in the class might make this mistake • Lower-readiness students might not understand why the last two steps are equivalent if they do not recognise the algebraic identity. • Go through solutions part by part instead of all 3 parts together so that the lower-readiness students can spot any mistake they made earlier and Whiteboard & Markers Student’s Handout 1A (see Appendix II p.24)
QCM52B Lesson Plan Assignment Ng Zi Xuan 12 Time Teaching / Learning Activities Rationales Resources prevent making the same mistake in other parts 05 min Closure Summarise SIO • Teacher summarises each of the SIO from the lesson while scrolling to the relevant sections of the handout • Highlight that we observed that some quadratic equations cannot be solved by factorisation and thus, we need another method to solve (Section A-B) • Highlight that we have learnt how to solve quadratic equations of the form which we will revisit the next lesson again. (Section C) • Highlight that we discovered what it means to complete the square and express algebraic expressions in the completed square form (Section E-F) • Highlight that we have yet to solve the equation in the IP, but we will be able to do so by the end of the next lesson. Assign Homework • Task students to work on Textbook Exercise 1A: Q2 (d)(e), Q3(b)(d)(f) as homework • Closure is for teacher to consolidate the SIOs students have learnt in the lesson • Give students homework for them to consolidate what they have learnt in the lesson • Questions selected are similar to class practice questions. Handout 1A Textbook (see Appendix II, p.18) Solutions (see Appendix II, p.18) If there is still time, then proceed with the following: 05 min Development 3 (Skill: Completing the Square for ) Worked Example 4 (WE4) • Inform students that teacher will demonstrate how to complete the square if the equation is now of the form Whiteboard & Markers
QCM52B Lesson Plan Assignment Ng Zi Xuan 13 • Write ‘Express in the form ’ on the whiteboard, guide students step by step to solve (see Appendix p.17) • Highlight to students that we leave the constant term alone and complete the square for just like in WE3 05 min Consolidation 3 (Skill: Completing the Square for ) Class Practice 4 (Section F) • Leaving WE4 on the whiteboard, instruct students to attempt Example 5 part (a), following my presentation • Walk around to supervise seatwork and render assistance especially to students of lower-readiness. Assess students’ learning of SIO4 based on their ability to solve the questions. • After most students are done, project solutions from teacher’s copy for students to tally their answer. • Highlight that we leave the constant term alone and complete the square for Student’s Handout 1A (see Appendix II p.25) Remarks: The handout created is meant to be used to teach the entire topic on completing the squares and not just for this lesson plan. Thus, some parts of the handouts are not used in this lesson plan. Class practice questions are deliberately incorporated in the handout, rather than instructing students to attempt on foolscap paper, for ease of revision. All questions in worked examples and class practices in the handout are modified from the textbook. References 1. think! Mathematics 3A Teachers’ Resources (8th Edition, Shinglee) 2. Lesson Plan Slides, Dr Joseph Yeo Reflection after Lesson Implementation 1. Strengths: 2. Areas for Improvement:
QCM52B Lesson Plan Assignment Ng Zi Xuan 14 Appendix I – ChatGPT Prompt and Lesson Plan Prompt: Write a lesson plan with the following details: 80-minute lesson, average students, Singapore Express Stream, Secondary 3, Mathematics topic on Completing the Square, with the following specific instructional objectives (SIO): At the end of the lesson, the students should be able to: 1. Solve quadratic equations of the form (x+r)^2=s 2. Describe the process of completing the square for quadratic expressions of the form x^2+bx 3. Express quadratic expressions of the form x^2+bx in the form (x+r)^2+u 4. Express quadratic expressions of the form x^2+bx+c in the form (x+r)^2+u Write your lesson plan in a table with 4 columns, with each heading being: Time, Teaching / Learning Activities, Rationales, Resources. Write your time interval for each activity in multiples of 5minutes. Write your instructional moves in detail so that the reader who does not observe the actual lesson can picture the lesson in mind when reading (e.g., specifying that you will walk around to supervise seatwork). Write your rationale clearly for each of the instructional moves, incorporating theories like Bruner's CPA approach, Bruner's Guided Discovery Approach, Constructivism, Behaviourism. Make sure you give a worked example before tasking students to do class practice and for each worked example or class practice question you mention, write the specific question in the plan with its corresponding solutions. The header of your instructional moves should consist of the following headings: Introduction, Development of concepts/skills, Consolidation of the development (consisting of worked examples or class practice), Closure (summary of the lesson). Have subheadings in your 2nd column other than the headings I mentioned. Make sure you plan for questions to ask students and for each questions asked, write down the expected correct answer or the wrong answers that might surface from students. At each point of concept or skill development, highlight the difficulties that students might face or misconceptions that they may have. Ensure that you fulfil all the requirements above, if unable to, mention explicitly which criteria is not met.
QCM52B Lesson Plan Assignment Ng Zi Xuan 15 Lesson Plan Generated (Unable to remove black background formatting from ChatGPT): Time Teaching / Learning Activities Rationales Resources 5 min Introduction Start the lesson by asking students to recall the standard form of a quadratic equation. Provide a simple example, like , and ask students to identify its factors. Engage students' prior knowledge and activate schema related to quadratic equations. 10 min Explaining Completing the Square Present a quadratic expression like on the board. Explain the concept of completing the square step-by-step, emphasizing the need to create a perfect square trinomial. Provide a concrete example of how to complete the square, demonstrating the process with the expression given. Utilize Bruner's Guided Discovery Approach to allow students to construct their understanding through guided instruction. 10 min Guided Practice: Completing the Square Provide another quadratic expression and guide students through completing the square. Encourage students to identify the values needed to complete the square and guide them through the steps. Address any misconceptions or difficulties that arise during the practice. Offer scaffolded support to help students internalize the process of completing the square. 15 min Development of Concepts/Skills Introduce quadratic equations of the form s. Explain how to solve such equations using the square root property. Provide examples, both with integers and fractions, and work through them step-by-step. Encourage students to ask questions and clarify any doubts. Apply Bruner's CPA approach by progressing from concrete examples to abstract concepts. 20 min Consolidation of the Development Worked Examples: 1. Solve the equation for . Solution: , so or . 2. Solve the equation for . Solution: , so or . Assign class practice: Solve the equation for . Reinforce understanding through worked examples, allowing students to see the application of completing the square in problem-solving. 10 min Closure Summarize key concepts covered in the lesson: completing the square to solve quadratic equations, transforming quadratic expressions, and solving equations in vertex form. Encourage students to ask any final questions or seek clarification on any concepts. Provide closure to the lesson, reinforcing learning and allowing for any final inquiries.
QCM52B Lesson Plan Assignment Ng Zi Xuan 16 Questions to Ask: 1. What is the standard form of a quadratic equation? • Expected answer: ++=0 2. How do we complete the square for a quadratic expression? • Expected answer: By adding and subtracting a constant term to make a perfect square trinomial. 3. What are the steps to solve a quadratic equation by completing the square? • Expected steps: 1. Rewrite the equation in the form +=. 2. Move the constant term to the other side of the equation. 3. Add and subtract (/2) to both sides of the equation. 4. Factor the perfect square trinomial. 5. Solve for . 4. What is the square root property? • Expected answer: If , then =±
QCM52B Lesson Plan Assignment Ng Zi Xuan 17 Appendix II – Online Timer, Solutions and Homework Questions Online Timer: https://www.online-stopwatch.com/countdown-timer/ Figure 1: Screenshot of online timer Solutions: Worked Example 1 (WE1) Solutions: Worked Example 2 (WE2) Solutions: Worked Example 3 (WE3) Solutions: Worked Example 4 (WE4)
QCM52B Lesson Plan Assignment Ng Zi Xuan 18 Textbook - Homework Questions: Exercise 1A, Pg.8, Q2 (d)(e), Q3(b)(d)(f) Solutions: Homework Q2(d)(e) Solutions: Homework Q 3(b)(d)(f)
QCM52B Lesson Plan Assignment Ng Zi Xuan 19 Physical Manipulatives (1 Set) Pieces to be cut out individually and placed in a folder for each group.
QCM52B Lesson Plan Assignment Ng Zi Xuan 20
QCM52B Lesson Plan Assignment Ng Zi Xuan 21 SECONDARY 3 EXPRESS MATHEMATICS Name: ( ) Class: Date: 1.1 Solving Quadratic Equations by Completing the Square A. Recap: Solving Quadratic Equations by Factorisation Example 1 [For Revision: Refer to TB Pg 2 for similar examples] Solve the equation B. Introductory Problem Example 2 Solve the equation Remarks: Learning Objectives: • Solve quadratic equations of the form by completing the square Handout: 1A
QCM52B Lesson Plan Assignment Ng Zi Xuan 22 C. Solving Quadratic Equations of the Form We have seen from the Introductory Problem that some quadratic equations, e.g., , using factorisation. Such equations can be expressed in the form where and are real numbers (which will be covered later in this chapter) Example 3 [For Revision: Refer to TB Pg 3 for similar examples] Solve each of the following equations. (a) (b) D. Recap: Perfect Squares Recall that the numbers and are perfect squares because they can be expressed as the square of an integer: and . State two other examples of numbers that are perfect squares: Algebraic expressions like , , and are also perfect squares because they are the squares of the expressions , , and respectively. They can be represented as pictorially. For example, we have seen that the expression can be represented pictorially as a , with lengths and area . Similarly, the expressions and in Example 3 are also . • The expressions in each square or rectangle denotes the for that shape Figure 1: Pictorial representations of and
QCM52B Lesson Plan Assignment Ng Zi Xuan 23 E. Investigation Key Question: What does it mean to complete the square and what is the process to do so? (a) Complete the table below. Expand each of the algebraic expression. (Recall: Algebraic identity ) In your groups, form the squares that represent each of the expressions above using the pieces provided1. Lay each representation out on the table and draw them in the spaces provided. Indicate the dimensions and area of each piece, with the lengths and breadths labelled on the outside (unlike in the physical pieces). (b) Complete the table below. Remove a piece from each square in part (a) to represent the following algebraic expressions pictorially2, then draw them in the spaces provided. Indicate the dimensions and area of each piece, with the lengths and breadths labelled on the outside (unlike in the physical pieces). Are these expressions perfect squares3? Circle your answer: Yes / No (Flip over for part (c) and (d)) 1 Hint: Refer to the pictorial representation of on the previous page 2 Hint: What is the difference between the expressions in part (b) and the expanded forms of those in part (a)? 3 Hint: For each pictorial representation, are you able to form a square with the remaining pieces?
QCM52B Lesson Plan Assignment Ng Zi Xuan 24 (c) To make quadratic expressions of the form , such as those in part (b), into a perfect square , such as those in part (a), we need to add a number to . This process is called completing the square. Complete the table below. Quadratic Expressions Number, , to be added to complete the square (write in the form ) Half the coefficient of , i.e., Completed square (d) Fill in the blanks based on your answers in the previous parts. i. What is the relationship between and ? ii. What is the relationship between and ? iii. Based on your answer to (d)(i), what number must you add to to complete the square? iv. Based on your answer to (d)(ii), what is the completed square after completing the square for x? From the Investigation, we observe the following: F. Completing the Square Example 4 [For Revision: Refer to TB Pg 6 for similar examples] Express each of the following expressions in the form . (a) (b) (Flip over for part (c) and (d)) To complete the square for , we add the number so that the completed square is
QCM52B Lesson Plan Assignment Ng Zi Xuan 25 (c) (d) Example 5 [For Revision: Refer to TB Pg 6-7 for similar examples] Express (a) in the form of (b) in the form of
QCM52B Lesson Plan Assignment Ng Zi Xuan 26 G. Solving Quadratic Equations by Completing the Square Example 6 [Refer to TB Pg 7 for similar examples] Revisiting the Introductory Problem: Solve the equation . Example 7 [For Revision: Refer to TB Pg 7 for similar examples] Solve each of the following equations, giving your answers correct to 2 decimal places. (a) (b)
QCM52B Lesson Plan Assignment Ng Zi Xuan 27 SECONDARY 3 EXPRESS MATHEMATICS Name: ( ) Class: Date: 1.1 Solving Quadratic Equations by Completing the Square A. Recap: Solving Quadratic Equations by Factorisation Example 1 [For Revision: Refer to TB Pg 2 for similar examples] Solve the equation B. Introductory Problem Example 2 Solve the equation Remarks: The expression cannot be factorised. Therefore, the equation cannot be solved by factorisation. Learning Objectives: • Solve quadratic equations of the form by completing the square TEACHER’S COPY Handout: 1A
QCM52B Lesson Plan Assignment Ng Zi Xuan 28 C. Solving Quadratic Equations of the Form We have seen from the Introductory Problem that some quadratic equations, e.g., , cannot be solved using factorisation. Such equations can be expressed in the form where and are real numbers (which will be covered later in this chapter) Example 3 [For Revision: Refer to TB Pg 3 for similar examples] Solve each of the following equations. (c) (d) D. Recap: Perfect Squares Recall that the numbers and are perfect squares because they can be expressed as the square of an integer: and . State two other examples of numbers that are perfect squares: 36 and 81 (any square numbers) Algebraic expressions like , , and are also perfect squares because they are the squares of the expressions , , and respectively. They can be represented as squares pictorially. For example, we have seen that the expression can be represented pictorially as a square, with lengths and area . Similarly, the expressions and in Example 3 are also perfect squares. • The expressions in each square or rectangle denotes the area for that shape Figure 1: Pictorial representations of and
QCM52B Lesson Plan Assignment Ng Zi Xuan 29 E. Investigation Key Question: What does it mean to complete the square and what is the process to do so? (a) Complete the table below. Expand each of the algebraic expression. (Recall: Algebraic identity ) In your groups, form the squares that represent each of the expressions above using the pieces provided4. Lay each representation out on the table and draw them in the spaces provided. Indicate the dimensions and area of each piece, with the lengths and breadths labelled on the outside (unlike in the physical pieces). (b) Complete the table below. Remove a piece from each square in part (a) to represent the following algebraic expressions pictorially5, then draw them in the spaces provided. Indicate the dimensions and area of each piece, with the lengths and breadths labelled on the outside (unlike in the physical pieces). Are these expressions perfect squares6? Circle your answer: Yes / No (Flip over for part (c) and (d)) 4 Hint: Refer to the pictorial representation of on the previous page 5 Hint: What is the difference between the expressions in part (b) and the expanded forms of those in part (a)? 6 Hint: For each pictorial representation, are you able to form a square with the remaining pieces?
QCM52B Lesson Plan Assignment Ng Zi Xuan 30 (c) To make quadratic expressions of the form , such as those in part (b), into a perfect square , such as those in part (a), we need to add a number to . This process is called completing the square. Complete the table below. Quadratic Expressions Number, , to be added to complete the square (write in the form ) Half the coefficient of , i.e., Completed square 3 4 5 6 (d) Fill in the blanks based on your answers in the previous parts. i. What is the relationship between and ? ii. What is the relationship between and ? iii. Based on your answer to (d)(i), what number must you add to to complete the square? 81 or iv. Based on your answer to (d)(ii), what is the completed square after completing the square for x? From the Investigation, we observe the following: F. Completing the Square Example 4 [For Revision: Refer to TB Pg 6 for similar examples] Express each of the following expressions in the form . (a) (b) (Flip over for part (c) and (d)) To complete the square for , we add the number so that the completed square is
QCM52B Lesson Plan Assignment Ng Zi Xuan 31 (c) (d) Example 5 [For Revision: Refer to TB Pg 6-7 for similar examples] Express (a) in the form of (b) in the form of
QCM52B Lesson Plan Assignment Ng Zi Xuan 32 G. Solving Quadratic Equations by Completing the Square Example 6 [Refer to TB Pg 7 for similar examples] Revisiting the Introductory Problem: Solve the equation . Example 7 [For Revision: Refer to TB Pg 7 for similar examples] Solve each of the following equations, giving your answers correct to 2 decimal places. (a) (b)