Module Focus: Grade 9 – Module 4 Sequence of Sessions

Overarching Objectives of this February 2014 Network Team Institute Participants will develop a deeper understanding of the sequence of mathematical concepts within the specified modules and will be able to articulate

how these modules contribute to the accomplishment of the major work of the grade.

Participants will be able to articulate and model the instructional approaches that support implementation of specified modules (both as classroom teachers and school leaders), including an understanding of how this instruction exemplifies the shifts called for by the CCLS.

Participants will be able to articulate connections between the content of the specified module and content of grades above and below, understanding how the mathematical concepts that develop in the modules reflect the connections outlined in the progressions documents.

Participants will be able to articulate critical aspects of instruction that prepare students to express reasoning and/or conduct modeling required on the mid-module assessment and end-of-module assessment.

High-Level Purpose of this Session● Implementation: Participants will be able to articulate and model the instructional approaches to teaching the content of the first half of the lessons.● Standards alignment and focus: Participants will be able to articulate how the topics and lessons promote mastery of the focus standards and how the

module addresses the major work of the grade.● Coherence: Participants will be able to articulate connections from the content of previous grade levels to the content of this module.

Related Learning Experiences● This session is part of a sequence of Module Focus sessions examining the Grade 9 curriculum, A Story of Functions.

Key Points Topic A Consider having students come up with their own summaries for how they approach factoring /solving / graphing a quadratic. It’s better to study deeply a given application problem and the analysis of its graph’s features than to do multiple problems. Introduce concepts like domain, range, increasing, decreasing, average rate of change, etc. by using words that feel natural in the

context, and then repeat the statement or question using the more formal words. Scaffolds are a critical tool for successful implementation. In addition to those given in the module, consider the ones we explored in

this session. (Take time now to reflect and take note of them.)

Key Points Topic B

Completing the square has a geometric meaning. A scaffold for completing the square when the leading coefficient is not 1 involves multiplying the equation through first by the

leading coefficient (if not already a perfect square) and then by the factor 4 , if the coefficient of the x−¿term is not easily halved. This same scaffold used with the geometric model provides an alternative to the purely algebraic derivation of the quadratic formula. The final lesson should include a reflection on the student’s general strategy for graphing quadratic functions. Lessons 16-21 in Topics B and C provide a second opportunity for students to master transformations of functions.

Key Points Topic C Comparing features of functions provided in different forms deepens and consolidates student understanding of the relationship

between the structure of expressions and equations, the graphs of equations and functions, and the contexts they model. Students should walk away from quadratics understanding that a primary use of these functions is in modeling height over time of

projectile objects, that they are naturally related to rectangular area problems, and that there are also used in an early study of business applications.

Key Points Module 4 Students are called upon to Look for and make use of structure (MP.7) as they choose equivalent forms of quadratics to gain insight

into the function’s behavior and its graph. Students are called upon to reason abstractly and quantitatively (MP.2) as they decontextualize and work with quadratic equations

representing real-world contexts and then re-contextualize as they analyze and interpret the key features of the function and its graph in the context of the problem.

Note that the physics contexts have the same coefficients due to the mathematics of objects in motion.

Session Outcomes

What do we want participants to be able to do as a result of this session?

How will we know that they are able to do this?

Participants will develop a deeper understanding of the sequence of mathematical concepts within the specified modules and will be able to articulate how these modules contribute to the accomplishment of the major work of the grade.

Participants will be able to articulate and model the instructional approaches that support implementation of specified modules (both as classroom teachers and school leaders), including an understanding of how this

Participants will be able to articulate the key points listed above.

instruction exemplifies the shifts called for by the CCLS.

Participants will be able to articulate connections between the content of the specified module and content of grades above and below, understanding how the mathematical concepts that develop in the modules reflect the connections outlined in the progressions documents.

Participants will be able to articulate critical aspects of instruction that prepare students to express reasoning and/or conduct modeling required on the mid-module assessment and end-of-module assessment.

Session Overview

Section Time Overview Prepared Resources Facilitator Preparation

Introduction to Module 15 minsConduct an overview of module stucture, lesson types, and lesson components.

Grade 9 Module 4 Grade 9 Module 4 PPT

Review Grade 9 Module 4

Foundations for the Study of Quadratics

45 minsEstablish a foundation from which to begin the study of Module 4.

Grade 9 Module 4 Grade 9 Module 4 PPT

Topic A Lesson 100 minsExamine lessons in Grade 9 Module 4 Topic A.

Grade 9 Module 4 Grade 9 Module 4 PPT

Fluency Exercises 20 minsExperience rapid white board exchanges that support fluency in the skill of factoring .

Personal white boards.Practice leading a rapid white board exchange.

Mid-Module Assessment

45 minsComplete a portion of the assessment, score, and discuss.

Grade 9 Module 4 Mid- Module Assessment

Grade 9 Module 4 PPT

Review assessment, rubric, and sample solutions.

Topic B Lessons 115 minsExamine lessons in Grade 9 Module 4 Topic B.

Grade 9 Module 4 Grade 9 Module 4 PPT

Topic C Lessons 65 minsExamine lessons in Grade 9 Module 4 Topic C.

Grade 9 Module 4 Grade 9 Module 4 PPT

Module Summary 25 mins Grade 9 Module 4 Grade 9 Module 4 PPT

End-of-Module Assessment

60 minsComplete a portion of the assessment, score, and discuss.

Grade 9 Module 4 End-of Module Assessment

Grade 9 Module 4 PPT

Review assessment, rubric, and sample solutions.

Session Roadmap

Section: Grade 9 Module 4 Time: 512 minutes

TimeSlide #

Slide #/ Pic of Slide Script/ Activity directions GROUP

3 min 1.

5 min 2. Give participants 5 min to do the opening exercise.

2 min 3. During this session you will be actively engaged in unpacking the content of Grade 9 Module 4. You will be asked to interact with the materials from both the student’s and teacher’s perspective at various times during the session to deeply understand the content of the module.

We will revisit the opening exercise shortly. First let’s get to know each other a bit.

2 min 4. In order for us to better address your individual needs, it is helpful to know a little bit about you collectively.Pick one of these categories that you most identify with. As we go through these, feel free to look around the room and identify other folks in your same role that you may want to exchange ideas with over lunch or at breaks.

By a show of hands who in the room is a classroom teacher?Math trainer?Principal or school-level leaderDistrict-level leader?And who among you feel like none of these categories really fit for you. (Perhaps ask a few of these folks what their role is).

Regardless of your role, what you all have in common is the need to understand this curriculum well enough to make good decisions about implementing it. A good part of that will happen through experiencing pieces of this curriculum and then hearing the commentary that comes from the classroom teachers and others in the group.

2 min 5. We have three main objectives for this mornings work. Our main task will be experiencing lessons and assessments. As a secondary objective, you should walk away from the study of module 4 being able to articulate how these lessons promote mastery of the standards and how they address the major work of the grade. Lastly, you should be able to get a sense for the coherent connections to the content of earlier grade levels.

2 min 6. Here is our agenda for the day. If needed, we will start with orienting ourselves to what the materials consist of. Next I will walk you through some exercises I find lay an important foundation for teachers embarking on a module studying quadratics. After that brief introduction to quadratics, we’ll begin examining and experiencing some excerpts from lessons of each topic. After topic A, I’ll take time to model for you an fluency strategy that you can use as-needed while your students work their way through the module. At the mid-way point, we’ll stop and take a portion of the mid-module assessment, and then at the end we’ll take a portion of the end of module assessment.

(Click to advance animation.) Let’s begin with an orientation to the materials for those that are new to the materials (Skip if participants are already familiar with the materials).

4 min 7. (Not accounted for in the timing – these slides are optional if participants are new to the materials.)

Each module will be delivered in 3 main files per module. The teacher materials, the student materials and a pack of copy ready materials.

Teacher materials include a module overview, and topic overviews, along with daily lessons and a mid- and end-of-module assessment. (Note that shorter modules of 20 days or less do not include a mid-module assessment.)

Student materials are simply a package of daily lessons. Each daily lesson includes any materials the student needs for the classroom exercises and examples as well as a problem set that the teacher can select from for homework assignments.

The copy ready materials are a single file that one can easily pull from to make the necessary copies for the day of items like exit tickets, or fluency worksheets that wouldn’t be fitting to give the students ahead of time, as well as the assessments.

4 min 8. (Not accounted for in the timing – these slides are optional if participants are new to the materials.)

There are 4 general types of lessons in the 6-12 curriculum. There is no set formula for how many of each lesson type we included, we always use whichever type we feel is most appropriate for the content of the lesson. The types are merely a way of communicating to the teacher, what to expect from this lesson – nothing more. There are not rules or restrictions about what we put in a lesson based on the types, we’re just communicating a basic idea about the structure of the lesson.

Problem Set Lesson – Teacher and students work through a sequence of 4 to 7 examples and exercises to develop or reinforce a concept. Mostly teacher directed. Students work on exercises individually or in pairs in short time periods. The majority of time is spent alternating between the teacher working through examples with the students and the students completing exercises.

Exploration Lesson – Students are given 20 – 30 minutes to work independently or in small groups on one or more exploratory challenges followed by a debrief. This is typically a challenging problem or question that requires students to collaborate (in pairs or groups) but can be done individually. The lesson would normally conclude with a class discussion on the problem to draw conclusions and consolidate understandings.

Socratic Lesson – Teacher leads students in a conversation with the aim of developing a specific concept or proof. This lesson type is useful when conveying ideas that students cannot learn/discover on their own. The teacher asks guiding questions to make their point and engage students.

Modeling Cycle Lesson --Students are involved in practicing

all or part of the modeling cycle (see p. 62 of the CCLS, or 72 of the CCSSM). The problem students are working on is either a real-world or mathematical problem that could be described as an ill-defined task, that is, students will have to make some assumptions and document those assumptions as they work on the problem. Students are likely to work in groups on these types of problems, but teachers may want students to work for a period of time individually before collaborating with others.

5 min 9. (Not accounted for in the timing – these slides are optional if participants are new to the materials.)

Follow along with a lesson from the materials in your packet.

The teacher materials of each lesson all begin with the designation of the lesson type, lesson name, and then 1 or more student outcomes. Lesson notes are provided when appropriate, just after the student outcomes.

Classwork includes general guidance for leading students through the various examples, exercises, or explorations of the day, along with important discussion questions, each of which are designated by a solid square bullet. Anticipated student responses are included when relevant – these responses are below the questions; they use an empty square bullet and are italicized. Snapshots of the student materials are provided throughout the lesson along with solutions or expected responses. The snap shots appear in a box and are bold in font. Most lessons include a closing of some kind – typically a short discussion. Virtually every lesson includes a lesson ticket and a problem set.

What you won’t see is a standard associated with each

lesson. Standards are identified at the topic level, and often times are covered in more than one topic or even more than one module… the curriculum is designed to make coherent connections between standards, rather than following the notion that the standards are a checklist of items to cover.

Student materials for each lesson are broken into two sections, the classwork, which allows space for the student to work right there in the materials, and the problem set which does not include space – those are intended to be done on a separate sheet so they can be turned in. Some lessons also include a lesson summary that may serve to remind students of a definition or concept from the lesson.

2 min 10. That concludes the materials orientation. Let’s get started with our study of quadratics.

15 min

11. These next 3 slides were directly informed and inspired by the work of Dr. James Tanton and his posted course on quadratics found at www.Gdaymath.com. I find these exercises a highly desirable experience base for teachers embarking on a quadratics module.

Reflecting back at our experience with sequences from module 3. Consider the sequence that begins with, … What did your study in module 3 tell you about the next number in this sequence? (Some participants may answer X, hopefully someone will also suggest that one isn’t sure what it is unless one is given more information, like an explicit or recursive formula for the sequence, or perhaps one is told that it is arithmetic). So we learned not to always trust patterns, and we also spend a good deal of time studying arithmetic and geometric sequences and comparing and contrasting those particular types of sequences. So a valid conjecture for the next term in this sequence might be what? (X)

10 min

12. As usual, as presenter, my interactions with you will bounce back and forth between speaking one professional to another, and speaking as though I were a teacher and you are the student. Please indulge me in those times where I am modeling my own delivery of this in a classroom.

•Do heavier objects fall through the air faster than lighter objects of the same shape and size? Consider a real elephant and a life-sized paper Mache model of an elephant. Which do you suppose would hit the ground first? How do you think your students would answer this question?•What Galileo was thinking is that they would land at the same exact time. He recognized that as they fell their speed would be increasing all the while. Depending on your student population you may need to scaffold their understanding. Always through questioning, not through telling. So we might say, how fast do you think the

elephants are moving when they first get pushed off the edge? Give me an estimate? 5 mph? 20 mph? Do they keep going that same speed as they fall? What is causing the speed of the elephants to increase? Galileo was thinking that the speed was getting faster and faster as time passed, and he was thinking that they would land at the same exact time. Do you agree?•Imagine what experiment / data he would need to do to find out.•Suppose he got the following data, would this confirm or contradict his thinking? (0 sec, 100 m) (1 sec, 95 m) (2 sec, 80 m) (3 sec, 55m)•(4 sec, 20 m)•Since measuring the height of the elephant moving through the air over constant time intervals will give us speed, and measuring the change in speed over constant time intervals will give us acceleration, Galileo is suggesting that the formula that the formula that describes the height over time could be given by something of degree 2; a formula of the form: at2+bt+cSo why such fascination? The force of gravity is a huge part of life on earth! Quadratic formulas predict elevation over time of projectiles.

20 min

13. Each table has a chain of sorts, an extra large sheet of graph paper with an x and y- axis drawn on it, and some tape. Hold up the graph paper and tape the ends of the chain so that the bottom of the chain hangs down nicely at the origin of the graph. As a group, identify points that the chain goes through, and then come up with a quadratic formula to model the curve you’ve created.

(Give the participants time to work on this and then ask them to report their experiences. They will find that there is not a nice quadratic equation to model a hanging chain, and that not all u-shaped curves are representative of a quadratic function.)

3 min 14. (total elapsed time at the end of this slide is 1 hour)

We’re ready to begin examination of Topic A. First let’s read the standards we’ll be covering in this module. (Give participants time to read the standards listed on pages 4-7 of the teacher materials.)Now let’s go through an overview of the flow of the module as it addresses those standards.

2 min 15. (Go through the bullets to give an overview of the progression or flow of each topic and the module as a whole.)

2 min 16. (Go through the bullets to give an overview of the progression or flow of each topic and the module as a whole.)

2 min 17. (Go through the bullets to give an overview of the progression or flow of each topic and the module as a whole.)

3 min 18. (Review the bullet points with participants to remind them of the background students are coming in to this module with.)

15 min

19. The study of quadratic expressions, equations and functions is a common part of most any algebra course. In today’s session we will be focused on the following key ideas:-Studying quadratics with the shifts in mind: Understanding, Application and Fluency-Scaffolding to meet the needs of your specific students whether they be at grade level, below grade level, or above grade level.-Engaging students through questioning not telling.

Let’s get started just as the students would with the opening exercise of Lesson 1.Now go ahead and work Example 1. Here is a possible extension for advanced students. (see bullet)Page 19 contains mathematically accurate descriptions that lead up to describing what it means for a polynomial expression to be prime or irreducible over the integers. Let’s examine some scaffolds for making these descriptions digestible. First read through the page and give it some thought.Students who find the word factor too abstract, can benefit from thinking about what it means to count by 2’s, county by 3’s, count by 7’s. So 7 is prime because you can’t get to 7 when counting by any other number except 7 and 1. 4 is not prime because you can get to 4 when counting by 2’s, as well as by 1’s and 4’s.So next we need to ask, what do you think it means when my polynomial expression is prime? Let the students explore coming up with what it means to be prime in this new context of polynomials.In tackling the description given, have students come up with their own binomials and test it out with the criteria for prime. Try to write it as a product of two other polynomials with integer coefficients, get confident that there are some for which it can’t be done. Get the class to agree on some examples of prime polynomials.

Students spend the remainder of the lesson multiplying and then factoring a binomial of degree 1 times another binomial of degree 1, spending the bulk of their time practicing with the difference of two squares and then contrasting it with the product of (a + b)2.

10 min

20. Lesson 2 cites the term quadratic expression. Why are they called quadratic expressions? Why not something indicating that the degree is 2. We call a polynomial of degree 3, cubic, and a polynomial of degree 4 a quartic, and one of degree 5 a quintic, degree 6 a sextic or hexic. Why quadratic for degree 2?

The usefulness of quadratics in solving problems involving quadrangles, and the relationship between squares and solving quadratics is at the heart of this association. Hence the name of the Topic, and the term ‘completing the square’.

Note the scaffold box at the top of page 31, I strongly recommend using the tabular method to reinforce proper use of the distributive property and to help students’ thinking about factoring. Let’s work through that process together with Exercise 7. Then try the same process for Exercise 8.

15 min

21. Lesson 3 opens up with an application problem. Work the opening exercise now.Continue to use the tabular method as needed.Encourage students to verbalize their process of finding factors that work. This is preferable over telling them a procedure that isn’t originating from their own thinking.Work problem 3 from the problem set in lesson 4.

15 min

22. Lesson 5’s opening exercise and exercises 1-4 lead students to know and apply the zero factor property without real world context. Let’s work now on Example 1 which provides context for its application.

This capacity to discern and recall their new-found power to solve quadratic equations might be overlooked or underappreciated by students at this time, but it should be celebrated as huge. One suggestion for emphasis is to open the exercise with the challenge to solve an general quadratic equation (one that could be set to zero, factored, and solved with integer answers). Students may be able to use trial and error to find one or both solutions from the get go, but they might appreciate their newfound skill more readily as they contrast their trial and error approach to this one.

Lesson 6 encourages students to recall that in module 1, they used reasoning to solve some quadratic equations. For example, given 6x2 = 24, students reasoned that x2 = 4 and that x could equal 2 or -2. This is still a viable option!

15 min

23. In working the Extension question, remind participants that average rate of change is defined on closed intervals only, encourage participants to be consistent with the closed interval requirement when presenting such questions to students.

15 min

24. Lesson 9 is on graphing quadratic equations from factored form. Let’s go through the opening exercise from the students perspective.Example 2 – reading example 2 asks students to take leap. Pose the question, how do you suppose they came up with this formula? Does it seem reasonable that the science class was able to create this formula?So from here they get to use the entire graph to help make sense of some questions about the context including questions about increasing and decreasing and domain and maximum height and such. So these exercises really dig in to the graph and identifying its key features.Scaffold in formal terms by using contextual everyday language and then repeating with more formal words. For example, ask over what time period is this graph relevant? What’s the domain of this function in this context?

Lesson 10: Interpreting quadratic functions from graphs and tables.Dolphin height vs time graph. This graph bears discussion, is it reasonable that a single force sets the vertical motion in

place after the dolphin has descended to the lowest point? Is it reasonable that gravity is the main force once the dolphin hits the water? Do we think the dolphin might also be engaging in a swimming motion? Should the curve appear steeper or wider under water than above water.

Example 2 uses data from a table and asks the leading question of “What kind of function do you think this table represents and how do you know?” Really spend some time here… how do you know?

6 min 25. (Total elapsed time 2 hour and 40 min + 20 min of time budgeted for Q&A at any point during the process = 3 hours.)

(Go through each point listed.)

2 min 26. Let’s experience now an exercise to build students fluency with the skills in this module.

15 min

27. Factoring is a skill that students need to develop fluency with. It is a great example of a skill for which a rapid white board exchange is a fitting fluency exercise.How to conduct a white board exchange:All students will need a personal white board, white board marker, and a means of erasing their work. An economical recommendation is to place card stock inside sheet protectors to use as the personal white boards, and to cut sheets of felt into small squares to use as erasers. You have these materials at your tables today.It is best to prepare the questions in a way that allows you to reveal them to the class one at a time. A flip chart, or Powerpoint presentation can be used, or one can write the problems on the board and either cover some with paper or simply write only one on the board at a time.

2 min 28. We’re ready to have a look at the mid-module assessment

25 min

29. Have participants locate the assessment. Give them approximately 25 min to take the assessment with their partner. After 20 minutes have passed give a verbal warning for them to scan any remaining questions that they have not yet attempted. If everyone finishes early, stop this part and start the next portion of this session.

20 min

30. Again, work with a partner to examine your work against the rubric and exemplar. If you have any questions or concerns, jot them down on a post-it note and we will address those before we move on.

After 6 minutes or so have passed, call the group together and address any questions or concerns that participants noted on their post-it notes.

2 min 31. We’re ready to move on to Topic B.

20 min

32. Topic B covers the method of completing the square and the derivation of the quadratic formula.

A valuable scaffold even with the opening exercise is to use a geometric model of a square. Knowing that that you are attempting to factor it such that you are creating a perfect square develops students capacity to do so.

Let’s work Example 1 together using the geometric model.

…Getting to the last example. So my model says (x + 4)(x + 4) produces x2 + 8x + 16. But what I wanted was x2 + 8x + 3. Let’s just make it so, If what I want is x2 + 8x + 3, I can use my model and make an adjustment. I’ll write my x2 + 8x + 16 then I’ll subtract 13 so that I have the + 3 that I want.

Then I can write (x+4)2 – 13. Is this useful? Why was I bothering to re-write these things anyhow? Remember all that work I made us do with factoring… I wanted to do it to help solve quadratic equations.

Is this form useful in solving quadratic equations? If I have the equation x2 + 8x + 3 and I write it as (x+4)2 – 13 = 0, is that helpful for solving the equation? (Participants are likely to see this readily, but ask them, ) How are your students likely to respond to my question?

Notice the Lesson Summary suggests exactly what we’ve indicated, “Just as factoring a quadratic expression can be useful for solving a quadratic equation, completing the square also provides a form that facilitates solving a quadratic equation.” (See the bottom of page 130.)

If your students are not likely to see the possibility here. You might scaffold this lesson by starting right off the bat solving simple quadratics by inspection.

•Solve by inspection: 𝑥2 = 16, 𝑥2 – 25 = 0, (𝑥 – 1)2 = 9,(𝑥 + 2)2 −36 = 0Discourage use of the ± sign. Instead model less abstract ‘or’ that emulates our thinking. “If something squared is 9, then either that something equals 3 or that something equals -3. “ This scaffold helps students follow their own thinking all the way through to a final answer.It is vital that throughout this process, students feel a sense of why am I doing these things. It is far too easy to get lost in a procedural shuffle here, so draw them back to purpose whenever you sense they are losing connection.

35 min

33. So what happens when things get messy? What are some strategies we use to complete the square for something like 2𝑥2 + 16𝑥 + 3.

(Participants will likely suggest factoring out the GCF.)

Let’s try Lesson 12 Example 1, then. (Take time to make sure participants understand the mechanics of this approach, working it together if needed, or asking folks to share with their neighbor.)

Now let’s do Lesson 12 Example 2. (Give participants time to work it through on their own and then discuss.)

Do your students struggle with this approach?

So here is an uncomfortable fork in the road. For students ready to do this work of reasoning with abstract algebraic equations, this approach of completing the square without the context of an equation has advantages.

I’m going to take you through a scaffold that relies on the

advantage of putting this into the perspective of an equation. This an alternative approach that helps us make the best of the geometric model and what students already know about equations. But, be forewarned it does not get us all the way to our goal of completing the square in the context of a quadratic function.

Suppose consider that we are solving the equation 2x2 + 16 x + 3 = 0I would like to make a perfect square from this quadratic expression that will help me solve my equation.That 2 in front is making things difficult. Would I change my solution set if I multiplied both sides of my equation through by 2?

So now I have 4 x2 + 32x + 6 = 0, which I can solve by creating a perfect square expression. Here’s where I think the geometric model is so helpful in making sure a struggling student keeps things straight. (2x + ___)(2x + ___) What will go in these blanks if I want these two to combine to give me 32? 8. Let’s double check. To make the scaffold of the geometric method be a true help to your students you must establish a routine of always checking. Yep, I get 4x2 + 32 x + 64. Hmmm, that gives me a perfect square but it’s not exactly what I want, I wanted + 6. So I’ll subtract off 58 to get exactly what I want. (2x + 8)2 -58 = 0.

Now suppose I have something like 3x2 + 5x – 12 = 0. I can multiply through by 3.

This scaffold option I’ve gone through with you here requires a significant deviation from the basic layout of these lessons. So I encourage you think carefully about this choice. You know your students best. What approach will work the best for them. Make the decision ahead of time.

This scaffold has the added advantage of providing a

substantial scaffold to deriving the quadratic formula as we shall see. But it has a downside as well.

Eventually your students will need to be able to take an equation in 2 variables (or a function) of the form f(x) = ax2 + bx + c and reason through getting it into vertex form. One doesn’t have the luxury of simply multiplying through by constants now.

So if you choose to take this particular scaffolded approach, you won’t be able to rely on completing the square as an approach to putting something into vertex form. What can students do instead? In topic A, they learned how to graph the quadratic function by finding the intercepts and then using symmetry to identify the vertex. Their understanding of transformations of functions can get them into vertex form. Let’s continue to reflect on this as we progress through the module.

15 min

34. (Depending on the audience either allow them to work this through independently, walk them through it together, or agree that neither is necessary, then demonstrate using the square geometric model to get to the same place.)

20 min

35. Lesson 15: Look with me at Exercises 1-5 and the discussion that follows (allow a couple of minutes, or depending on the needs of the participants ask them to do the exercises).Using the quadratic formula, students solve using the quadratic formula and find that sometimes there might be two solutions, one solution, or no (real) solutions, and that sometimes there is a radical and sometimes not. The term discriminant is introduced. Students use the discriminant to categorize the solutions to a quadratic without finding them, they relate their findings to what the graphs must look like.

Lesson 16: Graphing quadratic functions from the vertex form: students graph 3 equations with horizontal shifts and are asked to make a conjecture about where a vertex is for another such function without graphing; then students are asked to create the equation with a given vertex (jumped here to include vertical shifts – not such a big jump though because of module 3 exposure). Then a pretty big jump that in a discussion just before the closing, students are asked what is the effect on the graph to have a leading coefficient other than 1. Though the lesson does not include a≠1 examples, the exit ticket does. Consider the scaffold on the bottom of page 172 and explore this deeply if students still need time to study the effect of 𝑎 on the ‘steepness’ of the graph.

20 min

36. Lesson 17: Graphing from standard form. This is a put it all together, examine these forms in context of another gravity (baseball) situation. Let’s work this Opening Exericse.

Example 1 provides another opportunity to ask, ‘How do you suppose the math class was able to determine this formula?’

It is not explicitly asked or stated, but is suggested, ‘How can I put a function into vertex form?’ What strategies do students have for doing that?•Using intercepts or point plotting and symmetry to find the vertex.•How do they identify the leading coefficient a? Will it be the same as if it were in general form?•Completing the square is an option for students ready.

Have students consolidate their thinking by asking them to come up with a general approach to graphing a quadratic equation on their own before considering what is outlined for them.

5 min 37. (Review the points outlined.)

2 min 38. Now let’s have a look at Topic C.

20 min

39. Allow participants to complete exercises 1 – 3. Then challenge them to describe privately with a neighbor how the graphs of these functions in Exercise 2 and in Exercise 3 relate to each other.

Discuss the suggestion. This is a real opportunity for reflection, coherence (bringing in their experiences in Grade 8) and students practicing their articulation of abstract ideas.

We use the term symmetric about the origin once in this lesson, but this idea should not be forced on students who are not ready, reserve it only as an extension for high level students.

20 min

40. Lesson 20: Stretching and shrinking functions – who can identify what is wrong with the name of this lesson?Lesson 21: Transformations of the quadratic parent function. Did we introduce the term “parent function?” we just start using it and define it in that first instance, like “the quadratic parent function f(x) = x^2Lesson 22: Comparing Quadratic, Square root and cube root functions represented in different ways. Students answer questions that compare two functions where for one we only have a graph and for another we only have an equation, then in another case one has a table and one has a graph.Lesson 23: Modeling with Quadratic Functions – Students and teachers both get notes about objects in motion. Makes students capacity so very deliberate to decipher out the physics and relationships and units and such. Then works with business applications for both teacher and students.Lesson 24: L23 continued. Given 2 points, how many quadratic graphs can you draw? (this raises the question how do I know that the graph I’ve drawn is even quadratic). There are infinitely many. Given 3 points how many quadratic graphs can you draw (that go through the

points)? Check out that scaffold box on page 247. Gives data and asks, how do we know this data “will” (uggh) be represented by a quadratic function? Students use 3 points to come up with the quadratic function that models the data. But you have to know!

20 min

41. Lesson 22: Comparing Quadratic, Square root and cube root functions represented in different ways. In these exercises students answer questions that compare two functions where for one we only have a graph and for another we only have an equation, then in another case one has a table and one has a graph. Let’s do these exercises now.Lesson 23 opens with a discussion of the mathematics of objects in motion. Note the following points. (Go through the bullet points). One can not make up ones own quadratic equation to use in these situations. The coefficient must be one of the two options or an equivalent option given the unit of measurement used for height.Lesson 23: Try Example 1.The physics-based exercises throughout this module makes students’ capacity to make sense of these contexts a beneficial and inseparable outcome of their study of quadratics.Lesson 23 Example 2 then works with business applications with notes for both the teacher and students, let’s do this example now.Lesson 24: Opening Exercise: Given 2 points, how many quadratic graphs can you draw? (this raises the question how do I know that the graph I’ve drawn is even quadratic). There are infinitely many. Given 3 points how many quadratic graphs can you draw (that go through the points)? Notice the scaffold box on page 247. Given data, how do we know this data will be represented by a quadratic function? Students use 3 points to come up with the quadratic function that models the data. But you don’t overlook the fact that 3 points does not assume in and of

itself a quadratic graph, there could be a hexic function going through those points, or something not even polynomial!

4 min 42. Go through each point

5 min 43. Let’s revisit the answers to the questions in the opening exercise.

Students study quadratics because of their relevance to science, specifically the physics of projectiles. There are also business applications and applications in geometric problems involving quadrangles.

The word quadratic is derived from the Latin word quadratus for square.

Not all u-shaped graphs represent parabolas / quadratic functions. The classic counter example is that of the catenary curve.

20 min

44. Review each key point one at a time.

Take a moment now to re-read the standards that this module covers… Can you think back to moments in the lessons that get students to arrive at those understandings? What things stand out to you now that did not stand out early on?

2 min 45. Finally let’s examine the end-of-module assessment for G9-M4.

25 min

46. Have participants locate the assessment. Give them approximately 25 min to take the assessment with their partner. After 20 minutes have passed give a verbal warning for them to scan any remaining questions that they have not yet attempted. If everyone finishes early, stop this part and start the next portion of this session.

20 min

47. Again, work with a partner to examine your work against the rubric and exemplar. If you have any questions or concerns, jot them down on a post-it note and we will address those before we move on.

After 6 minutes or so have passed, call the group together and address any questions or concerns that participants noted on their post-it notes.

4 min 48. (Review each key point one at a time.)

10 min

49. Take a few minutes to reflect on this session. You can jot your thoughts on your copy of the powerpoint. What are your biggest takeaways? (pause while participants reflect then click to advance to the next question). Now, consider specifically how you can support successful implementation of these materials at your schools given your role as a teacher, school leader, administrator or other representative.

Use the following icons in the script to indicate different learning modes.

Video Reflect on a prompt Active learning Turn and talk

Turnkey Materials Provided

Grade 9 Module 4 Grade 9 Module 4 PPT

Additional Suggested Resources

A Story of Functions Curriculum Overview