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Module Focus: Grade 9 – Module 3 Sequence of Sessions

Overarching Objectives of this November 2013 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 PointsTOPIC A

Function notation is introduced simply as a shorthand for ‘the formula for the n th term of a sequence’. This interpretation will later be extended to serve as shorthand for ‘a formula for the function value for a given input value’.

Seeing structure in the formulas for arithmetic and geometric sequences is a crucial part of meeting both the content standards and the MP standards.

It is not accurate to say simply that geometric sequences “grow faster” than arithmetic sequences.

TOPIC B

When referring to a function, we use the letter of the function only, e.g. the graph of f .

The graph of f is the same set of points as the graph of the equation y=f (x ).

In either case, the axes are labeled as x and y .

TOPIC C

Use technology to facilitate understanding that the intersection of the graphs of y=f (x ) and y=g (x) provide solution set to the equation f (x)=g (x).

Relate transformations of functions to the already familiar transformations of graphs while making a clear distinction.

Use language accurate to the transformation being described:

o Shift, stretch, reflect are used when describing transformations of graphs

o Function transformations are described by talking about the values of the inputs and outputs

TOPIC D

Summarizes the key ideas and concepts from Topics A – C.

Brings together the concepts of linear and exponential growth, transformations of functions, and using key features of a graph to solve a problem.

Applies the functions learned (exponential, piecewise, step) to real world situations.

Allows students the opportunity to go through the modeling cycle outlined in the Standards of Mathematical Practices.

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 instruction exemplifies the shifts called for by the CCLS.

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

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

Module Overview32 minutes

Overview of the instructional focus of Grade 9 Module 3.

Grade 9 Module 3 Grade 9 Module 3 PPT

Review module overview.

Topic A: Lessons 1-3, 5

64 minutes

Examination of the conceptual understandings that are developed in Topic A.


Review Topic A.

Topic B: Lessons 8, 9-10, 11-12

86 minutes

Examination of the conceptual understandings that are developed in Topic B.


Review Topic B.

Mid-Module Assessment

37 minutes

Exploration of the Mid-Module Assessment.

Grade 9 Module 3 Grade 9 Module 3 PPT Student version of mid-

module assessment

Review the Mid-Module Assessment.

Topic C: Lessons 16, 18-19

36 minutes

Examination of the conceptual understandings that are developed in Topic C.


Review Topic C.

End of Module Assessment

35 minutes

Exploration of the End-of-Module Assessment.

Grade 9 Module 3 Grade 9 Module 3 PPT Student version of end-of-

module assessment

Review the End-of Module Assessment.

Making Instructional Decisions

Strategies to support teaching decisions to help students be more successful in learning the material, while maintaining the rigor of the instruction

Grade 9 Module 3 Grade 9 Module 3 PPT Curriculum Overviews

Review the curriculum overview for A Story of Ratios and A Story of Units.

Session Roadmap

Section: Orientation to Materials Time: 32 minutes

[32 minutes] In this section, you will…Establish the instructional focus and structure of Grade 9 Module 3.

Materials used include:

Time Slide # Slide #/ Pic of Slide Script/ Activity directions GROUP

3 1.

52.

2 3. During this morning’s session you will be actively engaged in unpacking the content of Grade 9 Module 3. 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.

3 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 5. We have three main objectives for this mornings work. Our main task will be experiencing lessons and assessments. Much of this will be done as I model the delivery of the lesson to you as you play the role of the students. As a secondary objective, you should walk away from the study of module 3 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 6. Here is our agenda for the day. We will start with orienting ourselves to what the materials consist of and then we’ll dig in and begin examining and experiencing some excerpts from lessons from each of the first 3 topics. 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 the orientation to the materials.

4 7. 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 8. 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 9. 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 10. That wraps up our orientation to the materials.(Click to advance animation.)Now let’s have a look at the module overview for grade 9 module 3.

Section: Topic A: Lessons 1-3, 5 Time: 64 minutes

[64 minutes] In this section, you will…Examine the conceptual understandings that are developed inTopic A.



5 11.

3 12.

8 13.

3 14.

4 15. In Module 3, students take a look at sequences starting from a different angle… look at the Opening Exercise in the student materials for Lesson 1.

Note: sequences don’t have to follow a simple pattern, and often do not.

5 16. Example 1 is intended to lead students into a discussion about the initial term of a sequence, is it the 0th term or is it the 1st term?, students are invited to create an expression that shows the nth number for a sequence showing powers of two.

Again, there is a chance your students are not yet even feeling comfortable with the notion of the nth term of a sequence. If this is the case I suggest going through a discussion like the one on the next slide first and then coming back to this.

Raise your hand if you think your more than ½ of your students are not able to give a good effort at persevering to attempt to create and

more importantly at least understand when a peer comes up with an expression for the nth term of the sequence. If that is the case I would adjust this discussion, I would go through the next slide first, and approach this discussion in a less challenging way. We’ll go over that option in a minute for now, let’s walk -through this example and discussion.

Looking at your student materials, it asks, write an expression for the nth number of Jerry’s sequence. Do that now.

Advance bullets through the discussion.

For bullet 3: “If I asked you what is the first term of this sequence what would you say?” 1, and how does the number 1 relate to powers of 2? OK, so I can find the first term by taking 2 to the 0 power.

What is the 2nd term? 2, and how does it relate to a power of 2? 2 to the 1;Then the third term is found by taking 2 to the 2. So the pattern of raising 2 to a power one less than my term felt natural… nobody thought to say that 1 was the 0th term and the 1st term was 2 right?

Advance.

For now, let’s make it a habit that when we write expressions or formulas for sequences, we assume n = 1 to find the first term in the sequence.Sometimes there might be a compelling reason to use 0 instead. We know it is ok to start a sequence with n = 0 – we did that in the double and add 5 game of module 1, where we called a, the result of round 1, and then we wanted to call a0 the number we started with. But for now we agree to use n = 1 for computing our first term, if we ever have an exception we will make it very clear that we want to do it differently… in reality we can start with n = any integer so long as we make it really clear to the others that are using the formula.

Ignore last bullet.

3 17. So what is the notion behind the nth term then?Let’s create a table of values for the sequence.The first term is 1The second term is 2The third term is 4The fourth term is 8What would be an appropriate heading for each column in my table?Term Number, Term Itself or the Value of the TermGood, and so this table is nice, but what if I wanted to know the 100th term?Click to remove questionIt is even nicer to have a formula we could use that tells us that easily.Remember how we related the first term to powers of 2? the first term (advance bullets) was 2 to the 0, the second was 2 to the 1, …So we said we are curious what the 100th term was, how will the 100th term relate to powers of 2?

Let’s make it even more generic than that… no matter what term number I have over here… let’s call it n, the nth term can be computed by what expression?

So n is the term number, what do I mean by the nth term? Do I mean the term number? No the nth term is referring to the value of the term… the term itself. N is a place-holder for the term number. When I say what is the nth term? I mean what is the value of term number n.

So we like this generalization here that lets me pick any term number and find the term value, right?

Let’s pause here and talk again about adjusting this progression we just went through for students who were not ready to conceptualize the expression 2 ^ n or 2 ^ n-1 and how that expression generates the sequence of numbers shown in this Example regarding a sequence showing powers of 2.I might not ask them yet to try to write the expression. I would start

only with reading the description and using the first several numbers of the sequence to build a table of values… then I would say but wait, the problem says these are powers of 2… do you see any powers of 2 here? What do we mean by powers of 2? Continue through this same line of questioning and then just mention that we think it makes sense to start our list with 1, but sometimes people start with 0 or some other number, for now we are going to always start with 1.

Why do it the way the lesson presents it? Why not just go with what I just said…. Engaging with the MP’s at the level that is appropriate for your students.

3 18. In this case, what formula would work?A formula for the nth term = 2^ n-1.Would it be ok if I wrote f(n) to stand for a formula for the nth term?So in this case a formula for the nth term, f(n) = 2^ n-1. Ok?

So we don’t make a big fuss about this notation at this time… it is merely a convenient shorthand for “a formula for the nth term”

3 19. AFTER using this notation with the accompanying language of a formula for the nth term, clarify that we are not using the parentheses to mean multiplication, right, we chose to use this notation to mean a formula for the nth term.

In the coming exercises we will use other letters like A(n) to stand for Akeli’s sequence.

Students begin now (with Example 2) to practice writing their own formulas for situations where the pattern is already described, specifically they begin to be asked to do this for patterns that they will later learn are arithmetic (showing a linear growth pattern in the values of the terms) and for patterns that are geometric (as in Exercise 4) (showing an exponential growth pattern in the values of the terms)

And, as in Exercise 1, they begin to practice evaluating a formula for a given value of n.

The closing and lesson summary for Lesson 1 are well written and deserve your attention, let’s go through it now.(You can follow along on page 21 of the teacher materials.)

3 20. For the question, “Can one sequence have two different formulas,” add the idea that given

1) A finite sequence {1,2,3,4,5} can have infinitely many formulas that satisfy the sequence.

2) However, for infinite sequences, we are usually looking at one formula up to equivalent expressions used to define the formula.

5 21. The exercises asking students to make a conjecture about a sequence and then create a formula for the nth term based on their conjecture, steer quickly towards examples of arithmetic and geometric sequences, but we want students to spend a lot of time working with them before we give them a formal name.

So let’s take a look at Lesson 2: Recursive Formulas for SequencesStarting with Example 1Read Example 1 and answer part a.(I’ll now take you through the teacher led demonstration you can find on page 26 of the teacher materials.)

This sequence was made up by Akelia, She decide to call this sequence “Akelia’s sequence”Her teacher asked Akelia to find a formula for the sequence and so she wrote the following on a piece of paper….(Advance until you get to the …)

Can you use her reasoning to help you write formula for Akelia’s

sequence writing your formula as A(n) where the A stands for Akelia?Record the formula in part b of your student materials.

What does the A(n) represent. (some people might not have used A(n)… just say.. When Akelia wrote it she wrote A(n) instead of f(n) what does the A(n) mean? The nth term of Akelia’s sequence.Who can epxlain the formula and why it works?(Go through with bullets if appropriate)

Record how each part of Akelia’s formula works in part c of your student materials

Akelia’s formula like many of the formula’s we wrote in lesson 1 is an explicit formula - you can use it to find the value of any term you want without having to know the value of the term before it.

3 22. Go through the bulletsAgain students are asked to record how Johnny’s formula works.

3 23. RECURSIVE SEQUENCE (description). An example of a recursive sequence is a sequence that (1) is defined by specifying the values of one or more initial terms and (2) has the property that the remaining terms satisfy a recursive formula that describes the value of a term based upon an expression in numbers, previous terms, or the index of the term.

An explicit formula specifies the 𝒏th term of a sequence as an expression in 𝒏.A recursive formula specifies the 𝒏th term of a sequence as an

expression in the previous term (or previous couple of terms).

4 24. The exercises at this lesson can be an opportunity to persevere… if one or two kids come up with one… you could even extend the chance another day and start a collection of them on the wall.

Example: Yearly fee for movie service of $14.98 plus $3.99 charge per movie downloaded.

Example: Amount of money in a bank after n with an initial $10,000 invested and a 2% interest rate per year.By the end of the module, students should have been exposed to enough examples and contexts that they will find the task more readily available to them.

5 25. 0.01, 0.02, 0.04, 0.08, 0.160.32, 0.64, 1.28, 2.56, 5.1210.24, 20.48, 40.96, 81.92, 163.84327.68, 655.36, 1310.72, 2621.44, 5242.8810485.76, 20971.52, 41943.04, 83886.08, 167772.16335544.32 $671,088.64 $1,342,177.28 $2,684,354.56 $5,368,709.12Grand total: $10,737,418.23

Read through the opening exercise.Do the work to Complete as much of Exercises 1 and 2 as you can in next 3 minutes.

4 26. Reveal each key point one at a time.

After 2nd bullet – the MP’s will be assessed and related to the content of the grade.

Section: Topic B: Lessons 8, 9-10, 11-12 Time: 86 minutes

[86 minutes] In this section, you will…Examine of the conceptual understandings that are developed in Topic B.


Time Slide #

Slide #/ Pic of Slide Script/ Activity directions GROUP

2 27.

10 28. Students have already been studying functions by studying sequences; up to this point, the domain has been a subset of the integers.Bullet 1 – ask it and then,Refer to the OpeningWhat are triangular numbers?Advance animation and ask it good

Advance animation again and ask it I would allow a bit of creativity in the discussion, strictly speaking they don’t have meaning in this context, could we imagine a revised context in which they would?

Advance to next bullet and ask itReview teacher materials for Exercises 5-8 on p. 84.What do you expect your students to say for number 7? Number 8?In this way we take students naturally into situations where we can write formulas for contexts that have non whole-number inputs.

Now have a look at exercises 9-12.… show me your graphs… good… distinguish between 12 and 13 via 14.

5 29. Give 5 minutes for the audience to make up its own definition of rule. Then project some of the definitions on the screen.

Regarding the 8.F.1 definition: Ask: What is wrong with using the word rule? Consider: f:{sentences} {true, false} and f(“This sentence is false.”)?Ask the audience if they can name examples of functions in high school that are not rules. Examples: Transformations of the plane, transformations of functions, derivatives, integrals, etc.

The Common Core State Standards expect students to recognize when there is a functional relationship between two sets and build a function that models that relationship. For that, they need to understand the full definition.

10 30. Give 5 minutes for the audience to make up its own definition of rule. Then project some of the definitions on the screen.Regarding the 8.F.1 definition: Ask: What is wrong with using the word rule? Consider: f:{sentences} {true, false} and f(“This sentence is false.”)?Ask the audience if they can name examples of functions in high school that are not rules. Examples: Transformations of the plane, transformations of functions, derivatives, integrals, etc.The Common Core State Standards expect students to recognize when there is a functional relationship between two sets and build a function that models that relationship. For that, they need to understand the full definition.

5 31. Give 5 minutes for the audience to make up its own definition of rule. Then project some of the definitions on the screen.The Common Core State Standards expect students to recognize when there is a functional relationship between two sets and build a function that models that relationship. For that, they need to understand the full definition.

3 32. All of the correspondences above are examples of functions, but not all correspondences are functions in general. Functions satisfy an extra property that makes the correspondence predictive in the sense that once we model a real-life situation with a function, we can often use that function to make predictions about its future behavior. Thus, for students to recognize a functional relationship, they need to recognize that there is a correspondence and see that the correspondence matches each element of the first set with an element of the second set. Once they know the relationship is functional in nature, they can search for the rule that describes the functional relationship.

3 33.

3 34. The word “matched” in the definition of function can be replaced with “assigned,” after students understand that assigned is a synonym for “matched” or “corresponds.” When assigned is used instead, this definition is just a slight rephrasing of the CCSS F-IF.A.1 definition.Point out that the information about the meaning of f(x) should help the audience analyze the opening exercise.

10 35. This slide is to here to justify the use of an equation to define a

function (two different objects entirely). It follows a similar reasoning that is behind using the statement, “Let x=3,” to substitute 3 for x.The equation, f (x)=2x, is about truth values and solution sets. Ask: what is a solution to this equation? Give a couple of minutes for each table to discuss the implications. Then do Exercise 3.After they do Exercise 3, ask:

1) How is the left hand side of the equation different from the right hand side in the exercise?

2) Is the equation always true? (Yes, for numbers in the domain it is true.)

3) Did we lose any information using the equation “f ( x )=2x?” No. We are stating that the value of the function f (x) is the same as 2x for all values of x. So we can use an equation to name a function in this way.

Not only is this notation more convenient for defining functions with real domains and ranges, the statement f (x)=2xcan be thought of as a formula and can be thought of as an equation as well.Why bother? If it checks out okay anyway, why should we care? Answer: because the “=“ is the most commonly used symbol in mathematics and the use of it should always refer back to the same concept.

10 36. Ask audience to “un-compact” the set-builder notation.

In giving a conceptual image of how to build these sets using pseudo code, we are also trying to give meaning to:

1) The universal quantifier “for all.”2) Variable as a placeholder.3) Highlight the differences between the graph of f and

the graph of y=f ( x ) .4) How graphing calculators generate graphs.

5) Explore algorithms.

5 37. BIG Lead in here… create an actual experience that distinguishes between what is technically different and what is technically the same between these two phrases that we seem to use interchangeably “graph of a function” and the “graph of an equation y = some function of or expression in x” and get students to be precise.This is a fairly abstract thing to talk about with students at all… these lessons make that discussion feel experiential and substantial helps them internalize the thinking instead of just listening to you ramble on about it for a couple of minutes which they will promptly most likely forget you ever mentioned.

3 38. Ask:1) What is the domain? (x real, 2≤x ≤8)2) What does x take on multiple values at the same time?

(No. For each loop, x is just one value.)3) What is G? (The set of points in the plane—the

geometric figure of a line.)

3 39. How would you program a computer to do the second part… take a minute to think about it, brainstorm with a partner if it helps.

10 40. How is this procedurally different from the process we went through before…

Before we took all the x values in the domain and for each one we constructed the point (x, f(x)) and added it to the set G, then plotted all the points in G.

What are we doing that is different now?

2 41.

2 42. Reveal each key point one at a time. Ask a participant to read each one as you reveal them. Ask if they have any other key points that they would like to add.

Section: Mid-Module Assessment Time: 37 minutes

[37 minutes] In this section, you will…Explore the mid-module assessment.

Materials used include: Student version of mid-module assessment


2 43.

25 44. Have participants locate the assessment. Give them approximately 20 min to take the assessment with their partner. After 15 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.

8 45. Locate the scoring rubric in your materials. This four-point rubric shows the progression towards mastery. The general idea behind the steps is as follows: Step 1 – I don’t get it. Step 2 – I’m beginning to get it. Step 3 – I got it. Step 4 – I could teach it. These steps are in no way intended to place a point system for grading. Those decisions are left to the district, school, or teacher.

Each question including its parts are broken down with a descriptor of each step. The standards assessed are also noted on the rubric. 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.

2 46. (Review the key points with participants.)

Section: Topic C: Lessons 16, 18-19 Time: 36 minutes

[36 minutes] In this section, you will…Examine the conceptual understandings that are developed in Topic C.



0 47.

5 48. What standard am I working towards by posing this question?

Was there a similar standard in the state of NY before common core?Has anyone ever taught this approach deliberately before?(Looking at a 1 variable equation as though it is an equation of the form f(x) = g(x)?)

Solve the problem in a strictly algebraic approach first. Let’s use the idea behind the standard to solve this problem.

4 49.

5 50.

20 51.

2 52.

Section: Topic D: Lessons 21, 23 Time:

[minutes] In this section, you will…Examine the conceptual understandings that are developed in Topic D.


Time Slide #Slide #/ Pic of Slide Script/ Activity directions GROUP

2 53.

2 54.

12 55.

2 56.

4 57.

5 58.

4 59.

10 60.

2 61.

6 62.

4 63. 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?

Section: End of Module Assessment Time: 35 minutes

[35 minutes] In this section, you will…Explore the End-of-Module Assessment



2 64.

25 65. Have participants locate the assessment. Give them approximately 20 min to take the assessment with their partner. After 15 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.

8 66. 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 15 minutes or so have passed, call the group together and address any questions or concerns that participants noted on their post-it notes.

6 67. (Review each key point one at a time.)

5 68. 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.

Section: Making Instructional Decisions Time: 35 minutes

[35 minutes] In this section, you will…• Find ways to study the mathematics of the modules and the

way it develops in order to pinpoint where the gaps occur in student learning.

• Determine strategies to support teaching decisions to help students be more successful in learning the material, while maintaining the rigor of the instruction


2 69. Welcome! One of the big concerns as a teacher is how to bridge the gaps in student knowledge to make the learning most effective. During this session you will be actively engaged in analyzing the content of a module from the perspective of making instructional decisions.In particular, you will explore how a particular concept or component fits into the development of the mathematics topic and how deep understanding of that mathematics can assist you in identifying the steps to take.

2 70. There are two main objectives in this session. You will:• Find ways to study the mathematics of the modules and the

way it develops in order to pinpoint where the gaps occur in student learning as well as

• Determine strategies to support teaching decisions to help students be more successful in learning the material, while maintaining the rigor of the instruction

1 71.

3 72. When assessing what skills or concept students need more experience with, check the Foundational Standards in the Module Overview.The Foundational Standards can be a map of sorts, highlighting what students need to be successful in the module.Obviously, the Grade 8 standards direct you to grade 8 modules, while the algebra standards point to earlier modules in grade 9.

1 73.

3 74.

2 75. Translations, reflections, and rotations are covered in Lessons 1-6 of Grade 8, Module 2.

2 76. Dilations are addressed in the first three lessons of Grade 8, Module 3.

2 77. To give students a snapshot study of the material covered in Grade 8 regarding transformations, decide on the vocabulary they must be familiar with.

5 78. Then select key exercises where they will have to articulate the vocabulary to make sense of the exercise.

1 79.

2 80. There will be times when instructional decisions must be made based on time available. Always honor the objective of the lesson and maintain the rigor of the lesson.Honor the objective of the lesson – What is the main idea? Does your problem selection focus on those student outcomes?Maintain the rigor of the lesson – Keep the balance of the fluency (procedural skills), conceptual understanding, and application. Do not skip an entire section on a regular basis. Look for other places to incorporate these components

2 81. In general, the strategy to bridge the gaps is to:- Student outcomes and assessments – identify the big ideas. This enables you to select the most appropriate problems based on what your students need. You will know their backgrounds.- Focus Standards linked to Foundational Standards from previous material – gives a location to get foundational problems from earlier grades and modules- Lessons Grouped within Topics – gives several days for big ideas to be revisited and make connections- Terminology, Tools and Representations – do any of these need to be taught explicitly? Are there earlier ones that need to be reviewed or taught?

3 82.

3 83. Take two minutes to turn and talk with others at your table. During this session, what information was particularly helpful and/or insightful? What new questions do you have?Allow 2 minutes for participants to turn and talk. Bring the group to order and advance to the next slide.

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 3● Grade 9 Module 3 PPT

Additional Suggested Resources● A Story of Ratios Curriculum Overview

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