Grade 9 – Module 3 Module Focus Session

Grade 9 – Module 3 Module Focus Session
November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 3 minutes MATERIALS NEEDED: Powerpoint Projector Clicker Laptop Document camera Participant Binder A Story of Functions A Close Look at Grade 9 Module 3

November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 5 minutes Opening Exercise Discuss with a neighbor – which of the following phrases contain incorrect usage of language or symbols? The graph of 𝑓(𝑥) has an average rate of change of 3 on the interval (0, 1). The function 𝑔(𝑥) = 𝑥2 + 3 is increasing on the interval (0, ∞). The function 𝑔(𝑥) defined above is the function 𝑓(𝑥) = 𝑥2 shifted to the left 3 units. The terms of the arithmetic sequence, 𝑓(𝑛) = 2 + 3𝑛, is a straight line.

November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 2 mimutes A Story of Functions A Close Look at Grade 9 Module 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.

November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 3 minutes MATERIALS NEEDED: none Participant Poll Classroom teacher Math trainer Principal or school leader District representative / leader Other 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 leader District-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.

November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 2 minutes MATERIALS NEEDED: none Session Objectives Experience and model the instructional approaches to teaching the content of Grade 9 Module 3 lessons. Articulate how the lessons promote mastery of the focus standards and how the module addresses the major work of the grade. Make connections from the content of previous modules and grade levels to the content of this module. 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.

November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 2 minutes MATERIALS NEEDED: Agenda Orientation to Materials Examine and experience excerpts from: Topic A: Lessons 1-3, 5 Topic B: Lessons 8, 9-10, 11-12 Mid-Module Assessment Topic C: Lessons 16, 18-19 Topic D: Lessons 21, 23 End of Module Assessment 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.

November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 4 minutes MATERIALS NEEDED: None What’s In a Module? Teacher Materials Module Overview Topic Overviews Daily Lessons Assessments Student Materials Daily Lessons with Problem Sets Copy Ready Materials Exit Tickets Fluency Worksheets / Sprints 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.

November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 4 minutes MATERIALS NEEDED: None Types of Lessons Problem Set Students and teachers work through examples and complete exercises to develop or reinforce a concept. Socratic Teacher leads students in a conversation to develop a specific concept or proof. Exploration Independent or small group work on a challenging problem followed by debrief to clarify, expand or develop math knowledge. Modeling Students practice all or part of the modeling cycle with real-world or mathematical problems that are ill-defined. 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.

November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 5 minutes MATERIALS NEEDED: None What’s In a Lesson? Teacher Materials Lessons Student Outcomes and Lesson Notes (in select lessons) Classwork General directions and guidance, including timing guidance Bulleted discussion points with expected student responses Student classwork with solutions (boxed) Exit Ticket with Solutions Problem Set with Solutions Student Materials Problem Set 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.

November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 2 minutes MATERIALS NEEDED: None Agenda Orientation to Materials Examine and experience excerpts from: Topic A: Lessons 1-3, 5 Topic B: Lessons 8, 9-10, 11-12 Mid-Module Assessment Topic C: Lessons 16, 18-19 Topic D: Lessons 21, 23 End of Module Assessment 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.

Mathematical Themes of Module 3
Grade 9 – Module 3 Module Focus Session November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 5 minutes MATERIALS NEEDED: None Mathematical Themes of Module 3 Functional relationships Graphs and transformational geometry Linear functions versus exponential functions

November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 3 minutes MATERIALS NEEDED: None Flow of Module 3 Arithmetic and geometric sequences: function notation (Topic A) Precise definition of function and function notation (Topic B) Graphs of functions (Topic B) Transformations of functions (Topic C) Applications of functions and their graphs (Topic D)

Prior Experience with Sequences
Grade 9 – Module 3 Module Focus Session November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 8 minutes MATERIALS NEEDED: None Prior Experience with Sequences G9-M1 Lesson 26-27: Recursive Challenge Problem The Double and Add 5 Game Make 3 more entries into the table. What is the smallest starting whole number that produces a result of 100 or greater in 3 rounds or less? If we call 𝒂𝟏 the result of the first round and 𝒂𝟐 the result of the second round, what should we call our starting number? Write a recursive formula for 𝒂𝒏.

Examples of Other Recursive Definitions
Grade 9 – Module 3 Module Focus Session November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 3 minutes MATERIALS NEEDED: None Examples of Other Recursive Definitions

Lesson 1: Integer Sequences – Should You Believe in Patterns?
Grade 9 – Module 3 Module Focus Session November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 4 minutes MATERIALS NEEDED: None Lesson 1: Integer Sequences – Should You Believe in Patterns? What is the next number in the sequence? 2, 4, 6, 8, … Is it 17? Yes, if the formula for the sequence was: 𝒇 𝒏 = 𝟕 𝟐𝟒 𝒏−𝟏 𝟒 – 𝟕 𝟒 𝒏−𝟏 𝟑+ 𝟕𝟕 𝟐𝟒 (𝒏−𝟏) 𝟐 + 𝟏 𝟒 𝒏−𝟏 +𝟐 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.

Lesson 1 – Example 1 Start with n = 0 or with n = 1?
Grade 9 – Module 3 Module Focus Session November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 5 minutes MATERIALS NEEDED: None Lesson 1 – Example 1 Start with n = 0 or with n = 1? Some of you have written 2n and some have written 2n-1. Who is correct? Is there a way that both could be correct? What is the 1st term of the sequence? The 2nd? It feels more natural in this case to start with 𝑛 = 1. Let’s agree to do that for now. If we start with 𝑛 = 1, which formula should we use for finding the 𝑛𝑡ℎ term? 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.

Terms, Term Numbers, and “the 𝑛𝑡ℎ term”
Grade 9 – Module 3 Module Focus Session November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 3 minutes MATERIALS NEEDED: None Terms, Term Numbers, and “the 𝑛𝑡ℎ term” Let’s clarify - what do I mean by “the nth term”? Let’s create a table of the terms of the sequence. Term Number Term or Value of the Term 1 2 3 4 What would be an appropriate heading for each of our columns? = 20 = 21 = 22 = 23 … 100 n 1 2 4 8 = 299 = 2n-1 So what is the notion behind the nth term then? Let’s create a table of values for the sequence. The first term is 1 The second term is 2 The third term is 4 The fourth term is 8 What would be an appropriate heading for each column in my table? Term Number, Term Itself or the Value of the Term Good, and so this table is nice, but what if I wanted to know the 100th term? Click to remove question It 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.

Introducing the 𝑓(𝑛) Notation
Grade 9 – Module 3 Module Focus Session November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 3 minutes MATERIALS NEEDED: None Introducing the 𝑓(𝑛) Notation I’d like to have a formula that works like this: I pick any term number I want and plug it into the formula, and it will give me the value of that term. I’d like a formula for the 𝑛𝑡ℎ term, where I pick what 𝑛 is. In this case: 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”? 𝑓(𝑛) =2 𝑛−1 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”

Introducing the 𝑓(𝑛) Notation
Grade 9 – Module 3 Module Focus Session November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 3 minutes MATERIALS NEEDED: None Introducing the 𝑓(𝑛) Notation AFTER using the notation with accompanying language of “formula for the nth term”; take the chance to make very explicit to the students that 𝑓(𝑛) does not mean 𝑓 times 𝑛. We agreed, we will use it to mean, “a formula for the nth term.” In exercises to come students will use 𝐴(𝑛) for example to be a formula for Akelia’s sequence. Beginning with Example 2, students practice writing their own formulas for situations where the pattern is given verbally. 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.)

Lesson 1: Closing & Lesson Summary
Grade 9 – Module 3 Module Focus Session November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 3 minutes MATERIALS NEEDED: None Lesson 1: Closing & Lesson Summary Closing: Why is it important to have a formula to represent a sequence? Can one sequence have two different formulas? What does f(n) represent? How is it read aloud? Lesson Summary: A sequence can be thought of as an ordered list of elements. To define the pattern of the sequence, an explicit formula is often given, and unless specified otherwise, the first term is found by substituting 1 into the formula. For the question, “Can one sequence have two different formulas,” add the idea that given A finite sequence {1,2,3,4,5} can have infinitely many formulas that satisfy the sequence. However, for infinite sequences, we are usually looking at one formula up to equivalent expressions used to define the formula.

Lesson 2: (Explicit and) Recursive Formulas for Sequences
Grade 9 – Module 3 Module Focus Session November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 5 minutes MATERIALS NEEDED: None Lesson 2: (Explicit and) Recursive Formulas for Sequences Example 1 Term 1: 5 Term 2: = Term 3: = Term 4: = Term 5: = ... Term n: = x ? = x 1 = x 2 = x 3 = x 4 Usual TSM: They give the formula first, and then ask students to make sense of it. Here we not going to give the formula and ask students to come up with the formula themselves. (Most teachers put the formula on the board right away.) This is an instructional shift!! 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 Sequences Starting with Example 1 Read 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.

Lesson 2: Recursive Formulas for Sequences
Grade 9 – Module 3 Module Focus Session November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 3 minutes MATERIALS NEEDED: None Lesson 2: Recursive Formulas for Sequences When Johnny saw Akeila’s sequence he wrote the following: 𝐴(𝑛 + 1) = 𝐴(𝑛) + 3 for 𝑛≥1 and 𝐴 1 = 5 Why do you suppose he would write that? Can you make sense of what he is trying to convey? What does the 𝐴(𝑛 + 1) part mean? Still in teacher mode (modeling what we are saying to our students), go through the bullets in this slide. Again students are asked to record how Johnny’s formula works.

Lesson 2: Closing & Lesson Summary
Grade 9 – Module 3 Module Focus Session November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 3 minutes MATERIALS NEEDED: None Lesson 2: Closing & Lesson Summary Closing: What are two types of formulas that can be used to represent a sequence? What information besides the formula equation do you need to provide when using these types of formulas? List the first 5 terms of the sequence: 𝑓 𝑛+1 =5𝑓 𝑛 −3. Lesson Summary: Provides a description of a what a recursively defined sequence is. See page 32 of the teacher materials Third bullet: purposely is missing f(1)=1 to encourage audience to demand for the f(1) term. 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).

November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 4 minutes MATERIALS NEEDED: None Lesson 3 Two types of sequences are studied: ARITHMETIC SEQUENCE - described as follows: A sequence is called arithmetic if there is a real number 𝑑 such that each term in the sequence is the sum of the previous term and 𝑑. GEOMETRIC SEQUENCE - described as follows: A sequence is called geometric if there is a real number 𝑟 such that each term in the sequence is a product of the previous term and 𝑟. Exercise: Think of a real-world example of an arithmetic sequence. Describe it and write its formula. Exercise: Think of a real-world example of an geometric sequence. Describe it and write its formula. 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.

November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 5 minutes MATERIALS NEEDED: None Lesson 5 Which is better? Getting paid $33, every day for 30 days (for a total of just over $ 1 million dollars), OR Getting paid $0.01 today and getting paid double the previous day’s pay for the 29 days that follow? Why does the 2nd option turn out to be better? What if the experiment only went on for 15 days? Is it fair to say that the values of the geometric sequence grow faster than the values of the arithmetic sequence? Review the Opening Exercise and Examples 1 and 2 0.01, 0.02, 0.04, 0.08, 0.16 0.32, 0.64, 1.28, 2.56, 5.12 10.24, 20.48, 40.96, 81.92, 327.68, , , , , , , , $671, $1,342, $2,684, $5,368,709.12 Grand 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.

November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 4 minutes MATERIALS NEEDED: None Key Points – Topic A Function notation is introduced simply as a shorthand for ‘the formula for the 𝑛𝑡ℎ 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. 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.

November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 2 minutes MATERIALS NEEDED: None Agenda Orientation to Materials Examine and experience excerpts from: Topic A: Lessons 1-3, 5 Topic B: Lessons 8, 9-10, 11-12 Mid-Module Assessment Topic C: Lessons 16, 18-19 Topic D: Lessons 21, 23 End of Module Assessment

Lesson 8 Why Stay With Whole Numbers?
Grade 9 – Module 3 Module Focus Session November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 10 minutes MATERIALS NEEDED: None Lesson 8 Why Stay With Whole Numbers? Why are square numbers called square numbers? If 𝑆(𝑛) denotes the 𝑛𝑡ℎ square number, what is a formula for 𝑆(𝑛)? In this context what would be the meaning of 𝑆 0 , 𝑆 𝜋 , 𝑆 −1 ? Exercises 5-8: Suppose we extend our thinking to consider squares of side-length 𝑥 cm… Create a formula for the area, 𝐴(𝑥) cm2 of a square of side length 𝑥 cm. Review Exercises 9-12, taking time to do #10 and #12 Do Exercises 13-14 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 Opening What 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 it Review 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.

Lesson 9-10: Definition of Function
Grade 9 – Module 3 Module Focus Session November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 5 minutes MATERIALS NEEDED: None Lesson 9-10: Definition of Function On a piece of paper, write down a definition for the word function. CCSS 8.F.A.1: A function is a rule that assigns to each input exactly one output. This description doesn’t cover every example of a function. To see why: Write down all functions from the set {1,2,3} to the set {0,1,2}, and write a concise linear rule if there is one. Give 5 minutes for the audience to make up its own definition of function. 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 simple rules. Examples: Transformations of the plane, transformations of functions, derivatives, integrals, etc.

Grade 9 – Module 3 Module Focus Session November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 10 minutes MATERIALS NEEDED: None Lesson 9-10: Definition of Function Function Rule {1,2,3}{0,1,2} f(x)=x-1 {1,2,3}{0,2,1} {1,2,3}{1,0,2} {1,2,3}{1,2,0} {1,2,3}{2,0,1} {1,2,3}{2,1,0} {1,2,3}{0,0,0} f(x)=0 {1,2,3}{1,1,1} f(x)=1 {1,2,3}{2,2,2} f(x)=2 {1,2,3}{0,0,1} Function Rule {1,2,3}{0,1,0} {1,2,3}{1,0,0} {1,2,3}{0,0,2} {1,2,3}{0,2,0} {1,2,3}{2,0,0} {1,2,3}{1,1,0} {1,2,3}{1,0,1} {1,2,3}{0,1,1} {1,2,3}{1,1,2} {1,2,3}{1,2,1} Function Rule {1,2,3}{2,1,1} {1,2,3}{2,2,0} {1,2,3}{2,0,2} {1,2,3}{0,2,2} {1,2,3}{2,2,1} {1,2,3}{2,1,2} {1,2,3}{1,2,2} 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.

Grade 9 – Module 3 Module Focus Session November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 5 minutes MATERIALS NEEDED: None Lesson 9-10: Definition of Function A rule, like, “Let 𝑓 𝑥 =𝑥−1,” only describes a subset of the types of all functions. How might you describe the distinguishing features of the examples on the previous slide (regardless of whether there is a rule or not)? For every input there is one and only one output. They all involve correspondences. 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.

Grade 9 – Module 3 Module Focus Session November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 3 minutes MATERIALS NEEDED: None Lesson 9-10: Definition of Function Fortunately, students have been studying correspondences since Kindergarten: Kindergarten: Matching Exercises 6th & 7th Grade: Proportional Relationships 7th Grade: Scale drawings 8th Grade: Transformations 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.

Grade 9 – Module 3 Module Focus Session November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 3 minutes MATERIALS NEEDED: None Lesson 9-10: Definition of Function Why do correspondences matter? F-BF.A and MP4 (modeling): Students build functions that models relationships between two types of quantities To recognize a functional relationship, students first need to be able to recognize correspondences.

Grade 9 – Module 3 Module Focus Session November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 3 minutes MATERIALS NEEDED: None Lesson 9-10: Definition of Function CCSS F-IF.A.1. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. FUNCTION. A function is a correspondence between two sets, 𝑋 and 𝑌, in which each element of 𝑋 is matched (assigned) to one and only one element of 𝑌. The set 𝑋 is called the domain of the function. If 𝑓 is a function and 𝑥 is an element of its domain, then 𝑓(𝑥) denotes the output of 𝑓 corresponding to the input 𝑥. If 𝑓: 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 →{𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠}, then 𝑓(𝑥) stands for a real number, not the function itself. To refer to a function, we use its name: 𝑓. For example, “graph of 𝑓(𝑥)” doesn’t make sense, while the “graph of 𝑓” does. 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.

Grade 9 – Module 3 Module Focus Session November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 10 minutes MATERIALS NEEDED: None Lesson 9-10: Definition of Function Recall that an equation is a statement of equality between two expressions. When do you explicitly link functions with equations? For example, what does, “Let 𝑓 𝑥 = 2 𝑥 ,” mean, and why is it okay to use it? Read the Lesson Notes to Lesson 10 and do Exercise 3. The exercise shows that the definition of the exponential function with base 2, 𝑓: 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 → 𝑥 | 𝑥>0 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝑥↦ 2 𝑥 is equivalent to, “Let 𝑓 𝑥 = 2 𝑥 , where 𝑥 can be any real number.” We get the best of both worlds: The equal sign “=“ still means equal and we can use it in a formula to define a function. 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)= 2 𝑥 , 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: How is the left hand side of the equation different from the right hand side in the exercise? Is the equation always true? (Yes, for numbers in the domain it is true.) Did we lose any information using the equation “𝑓 𝑥 = 2 𝑥 ?” No. We are stating that the value of the function 𝑓(𝑥) is the same as 2 𝑥 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)= 2 𝑥 can 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.

November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 10 minutes MATERIALS NEEDED: None Lesson 11-12: Graphs Let 𝑓 𝑥 = 𝑥 2 for 𝑥 any real number. Discuss the meaning of 𝑥,𝑓 𝑥 | 𝑥 𝑟𝑒𝑎𝑙 Now discuss the meaning of 𝑥,𝑦 | 𝑥 𝑟𝑒𝑎𝑙 and 𝑦=𝑓(𝑥) How are they the same? Different? Both set-builder notations describe every single element in the their respective set. The first “constructs each point” while the second “tests every point in the plane.” We need to help students develop a “conceptual image” of how these sets can be generated. 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: The universal quantifier “for all.” Variable as a placeholder. Highlight the differences between the graph of 𝑓 and the graph of 𝑦=𝑓 𝑥 . How graphing calculators generate graphs. Explore algorithms.

Lesson 11 The Graph of a Function
Grade 9 – Module 3 Module Focus Session November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 5 minutes MATERIALS NEEDED: None Lesson 11 The Graph of a Function To make these lessons work, it is important that teachers spend time getting comfortable with pseudo code. Consider this set of pseudo-code: Declare 𝒙 integer For all 𝒙 from 1 to 5 Print 𝟐 𝒙 Next 𝒙 What would be printed out if this code were executed? Work through Exercise 1 2 4 8 16 32 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. Then continue through student pages for Examples and Exercises (through Example 3) to arrive at Discussion

Lesson 11 The Graph of a Function
Grade 9 – Module 3 Module Focus Session November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 3 minutes MATERIALS NEEDED: None Lesson 11 The Graph of a Function Consider this pseudo code: Declare 𝒙 real Let 𝒇(𝒙) = 𝟐𝒙 + 𝟑 Initialize G as {} For all 𝒙 such that 𝟐≤𝒙≤ 𝟖 Append (𝒙, 𝒇(𝒙)) to G Next 𝒙 Plot G The Graph of f: Given a function f whose domain D and range are subsets of the real numbers, the graph of 𝑓 is the set of ordered pairs in the Cartesian plane given by 𝑥, 𝑓(𝑥) |𝑥∈𝐷 Ask: What is the domain? (x real, 2≤𝑥≤8) What does x take on multiple values at the same time? (No. For each loop, x is just one value.) What is G? (The set of points in the plane—the geometric figure of a line.)

The Graph of a Function vs. The Graph of an Equation in 2 Variables
Grade 9 – Module 3 Module Focus Session November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 3 minutes MATERIALS NEEDED: None The Graph of a Function vs. The Graph of an Equation in 2 Variables The Graph of 𝑓: Given a function 𝑓 whose domain D and range are subsets of the real numbers, the graph of 𝑓 is the set of ordered pairs in the Cartesian plane given by 𝑥, 𝑓(𝑥) |𝑥∈𝐷 The Graph of an equation in two variables: The set of all its solutions, plotted in the coordinate plane, often forming a curve (which could be a line) 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.

Lesson 12: The graph of the equation 𝑦 = 𝑓(𝑥)
Grade 9 – Module 3 Module Focus Session November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 10 minutes MATERIALS NEEDED: None Lesson 12: The graph of the equation 𝑦 = 𝑓(𝑥) Declare 𝒙 and 𝒚 real Let 𝒇 𝒙 = 𝒙 𝒙−𝟐 𝒙+𝟐 Initialize G as { } For all 𝒙 in the real numbers For all 𝒚 in the real numbers If 𝒚 = 𝒇(𝒙) then Append (𝒙, 𝒚) to G else Do NOT append (𝒙, 𝒚) to G End If Next 𝒚 Next 𝒙 Plot G 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?

Lessons 11-12 The Graph of a Function
Grade 9 – Module 3 Module Focus Session November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 2 minutes MATERIALS NEEDED: None Lessons The Graph of a Function The Graph of 𝑓: Given a function 𝑓 whose domain D and range are subsets of the real numbers, the graph of 𝑓 is the set of ordered pairs in the Cartesian plane given by 𝑥, 𝑓(𝑥) |𝑥∈𝐷 The Graph of y= 𝑓(𝑥): Given a function 𝑓 whose domain D and range are subsets of the real numbers, the graph of 𝑦=𝑓(𝑥) is the set of ordered pairs (𝑥, 𝑦) in the Cartesian plane given by 𝑥, 𝑦 |𝑥∈𝐷 𝑎𝑛𝑑 𝑦=𝑓(𝑥) The Graph of 𝑓 is the same as the graph of the equation 𝑦 = 𝑓(𝑥).

November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 2 minutes MATERIALS NEEDED: None Key Points – 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. 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.

November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 2 minutes MATERIALS NEEDED: None Agenda Orientation to Materials Examine and experience excerpts from: Topic A: Lessons 1-3, 5 Topic B: Lessons 8, 9-10, 11-12 Mid-Module Assessment Topic C: Lessons 16, 18-19 Topic D: Lessons 21, 23 End of Module Assessment

Mid-Module Assessment
Grade 9 – Module 3 Module Focus Session November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 25 minutes MATERIALS NEEDED: Student version of assessment Mid-Module Assessment Work with a partner on this assessment 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.

Scoring the Assessment
Grade 9 – Module 3 Module Focus Session November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 8 minutes MATERIALS NEEDED: Scoring rubric and exemplar Post-its Scoring the Assessment 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.

Key Points – Mid-Module Assessment
Grade 9 – Module 3 Module Focus Session November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 2 minutes MATERIALS NEEDED: Key Points – Mid-Module Assessment As much as possible, assessment items are designed to asses the standards while emulating PARCC Type 2 and Type 3 tasks. Rubrics are designed to inform each district / school / teacher as they make decisions about the use of assessments in the assignment of grades. (Review the key points with participants.)

November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 2 minutes MATERIALS NEEDED: Agenda Orientation to Materials Examine and experience excerpts from: Topic A: Lessons 1-3, 5 Topic B: Lessons 8, 9-10, 11-12 Mid-Module Assessment Topic C: Lessons 16, 18-19 Topic D: Lessons 21, 23 End of Module Assessment

Lesson 16 Graphs Can Solve Equations Too
Grade 9 – Module 3 Module Focus Session November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 5 minutes MATERIALS NEEDED: None Lesson 16 Graphs Can Solve Equations Too Solve for x in the following equation: 𝑥+2 −3=0.5𝑥+1 A-REI.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.★ Use technology in this lesson! 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.

November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 4 minutes MATERIALS NEEDED: None Explain why…

Lessons 17-20: Transformations
Grade 9 – Module 3 Module Focus Session November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 5 minutes MATERIALS NEEDED: None Lessons 17-20: Transformations Can you translate a function 3 units up? We don’t translate a function up, down, left, right, or stretch and shrink functions. This language applies to graphs of functions. We can, however, use the transformation of the graph of a function to give meaning to the transformation of a function. We can describe transformed functions using language that refers to the values of the function inputs and outputs. For example: “For the same inputs, the values of the transformed function are two times as large as the values of the original function.” This is summarized for you in the lesson notes at the beginning of L17 – my apologies for not including that lesson in your printed materials… this slide can serve as a reminder.

Lessons 17-20: Transformations
Grade 9 – Module 3 Module Focus Session November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 20 minutes MATERIALS NEEDED: None Lessons 17-20: Transformations Lesson 18: Complete Example 1 Continue working through the Exercises and Examples. Lesson 19: Horizontal scaling with a scale factor 𝑘 of the graph of 𝑦 = 𝑓(𝑥) corresponds to changing the equation from 𝑦 = 𝑓(𝑥) to 𝑦 = 𝑓 1 𝑘 𝑥

November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 2 minutes MATERIALS NEEDED: None Key Points – 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: Shift, stretch, reflect are used when describing transformations of graphs Function transformations are described by talking about the values of the inputs and outputs

Lesson 21: Comparing Linear and Exponential Functions Again
Grade 9 – Module 3 Module Focus Session November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 2 minutes MATERIALS NEEDED: None Lesson 21: Comparing Linear and Exponential Functions Again Student Outcomes from Lesson 14 Students compare linear and exponential models by focusing on how the models change over intervals of equal length. Students observe from tables that a function that grows exponentially will eventually exceed a function that grows linearly. Student Outcomes from Lesson 21 Students create models and understand the differences between linear and exponential models that are represented in different ways. Lesson 14 introduces students to the difference between linear and exponential growth rates. This is a topic of huge importance in the standards and which students need to have a solid understanding. CCSS.Math.Content.HSF-LE.A.1 Distinguish between situations that can be modeled with linear functions and with exponential functions. CCSS.Math.Content.HSF-LE.A.1a Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. CCSS.Math.Content.HSF-LE.A.1b Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. CCSS.Math.Content.HSF-LE.A.1c Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. The topic is re-addressed in more detail in lesson 21.

Lesson 21: Delve more deeply
Grade 9 – Module 3 Module Focus Session November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 12 minutes MATERIALS NEEDED: None Lesson 21: Delve more deeply Moral from Lesson 14: “Linear functions grow additively while exponential functions grow multiplicatively.” Do Exercises 1 and 2 from Lesson 21. The main points that students should have drawn from lesson 14: linear functions grow additively while exponential functions grow multiplicatively given an increasing linear function and an increasing exponential function, the exponential will eventually exceed the linear function for a sufficiently large input value. These ideas are seen again in lesson 21. Students have spent several lessons in this module studying linear and exponential functions. This lesson has students compare and contrast the two functions to solidify their understanding of each. Complete exercises 1 and 2.

November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 2 minutes MATERIALS NEEDED: None Exercise 2 Is Terrance correct or incorrect? If students claim he is correct, ask them for the formula. Examine the formula as a class. Does this formula satisfy all of the values in the table? What mistake did Terrance make? Have students find the average rate of change between each pair of input-output values. What type of formula could be used to model this data? What is the formula? Exercise 1 is fairly straightforward. What has changed in exercise 2? Will some students want to claim that Terrance is correct? Advanced to next point What formula did you write for the data? Some may say f(x) = 3^x. Does this work when x = 0 is input? Yes. X = 1? Yes. X = 4? No. Advance He did not pay attention to the input. It is not changing by a constant value. It is important to examine both the input and output values before jumping to conclusions. Based on this, write a formula that could be used to model the data. What is the formula? f(x) = 2x + 1 The point of exercise 2 is not to trick kids but to encourage them not to treat mathematics too formulaically. Instead use logic and examine all of the facts before making a conclusion.

Lesson 23: Newton’s Law of Cooling
Grade 9 – Module 3 Module Focus Session November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 4 minutes MATERIALS NEEDED: None Lesson 23: Newton’s Law of Cooling Students are now applying knowledge of exponential functions and the transformations studied in Topic C to a modeling problem. This formula will be addressed again in subsequent math courses once students have learned about the number e. For now, we are using its approximate value of 𝑻 𝒕 = 𝑻 𝒂 + 𝑻 𝟎 – 𝑻 𝒂 ∙𝟐.𝟕𝟏 𝟖 −𝒌𝒕 𝑻(𝒕) is the temperature of the object after a time of t hours has elapsed, 𝑻 𝒂 is the ambient temperature (the temperature of the surroundings), assumed to be constant, not impacted by the cooling process, 𝑻 𝟎 is the initial temperature of the object, and 𝒌 is the decay constant. The focus of the lesson is to apply the knowledge of transformations learned in topic C to a modeling problem. Newton’s Law of Cooling is something that students would study Physics or Calculus. The formula contains notation that students have studied in this unit, both the function notation and the subscripts. Point out that the subscripts help us to distinguish between the different T values in the formula just like they do when working with sequences.

November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 5 minutes MATERIALS NEEDED: None Opening Exercise A detective is called to the scene of a crime where a dead body has just been found. He arrives at the scene and measures the temperature of the dead body at 9:30 p.m. After investigating the scene, he declares that the person died 10 hours prior at approximately 11:30 a.m. A crime scene investigator arrives a little later and declares that the detective is wrong. She says that the person died at approximately 6:00 a.m., 15.5 hours prior to the measurement of the body temperature. She claims she can prove it by using Newton’s Law of Cooling. 𝑇 𝑎 =68˚F (the temperature of the room) 𝑇 0 =98.6˚F (the initial temperature of the body) 𝑘= (13.35 % per hour - calculated by the investigator from the data collected) The purpose of the opening exercise is to introduce the formula and have an opportunity to go over what each parameter in the formula represents. It also immediately gives students a compelling reason for needing such a formula. Take a moment to read through and work the opening exercise. A logarithm is required to find k. If students ask how the investigator found k, you could explain that it requires a topic that will be covered in Algebra II (logarithms). T(15.5) = 68 + (98.6 – 68)*2.718^( *15.5) = degrees F T(10) = 76 degrees F

Math Modeling Exercise
Grade 9 – Module 3 Module Focus Session November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 4 minutes MATERIALS NEEDED: None Math Modeling Exercise Students will explore the effect of each parameter in Newton’s Law of Cooling by using a demonstration on Wolfram Alpha. This can be done as a whole class. At what type of graph are we looking? Why is it still an exponential decay function when the base is greater than 1? Using the demonstration on Wolfram Alpha, have students examine the graph by using the sliders. Advance This is an exponential graph. It is decreasing (a decay graph) because the base is being raised to a negative power which reflected the graph across the y-axis.

Math Modeling Exercise
Grade 9 – Module 3 Module Focus Session November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 10 minutes MATERIALS NEEDED: None Math Modeling Exercise Work through the modeling exercise. If it is 42˚F outside, can a cup of coffee ever cool to below 42˚F? How did you find the answer for part c of number 1? Which should you do to drink the coffee sooner: walk outside in the 42˚F temperature or pour milk into your coffee? How does changing the initial coffee temperature affect the graph? How does changing the ambient temperature of the coffee affect the graph? Work through the modeling exercise. (give them 5 – 7 minutes) Advance No. The temperature of the cup will decrease until it is roughly the same temperature as its ambient surroundings, which the graph shows. Coffee is safe to drink when its temperature is below 140˚F. Estimate how much time elapses before each cup is safe to drink. Cup 1: Approximately 2 min. Cup 2: Approximately 1.5 min. Students may have used the graph or used trial and error by filling values into the equation. At first, decreasing the initial temperature with milk has the greater impact because the coffee cools enough to drink more quickly. But, as time elapses, decreasing the ambient temperature has the greater impact because the coffee continues to cool and levels out at a lower temperature. The graph had a lower “starting point” (𝑦-intercept). The cold milk cools the coffee quickly at first, but, compared to the hotter cup, takes longer to cool to the same temperature. This is because the cooler cup has a smaller temperature difference with the ambiance, which leads to the slower cooling rate. At a lower ambient temperature, the coffee cools more quickly and levels off to a lower temperature because of the cooler surrounding temperature.

November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 2 minutes MATERIALS NEEDED: None Key Points – 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. Go through each point emphasizing the opportunity to deepen understanding and to incorporate the SMP, especially MP 4 (modeling)

November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 6 minutes MATERIALS NEEDED: None Opening Exercise Discuss with a neighbor – which of the following phrases contain incorrect usage of language or symbols? The graph of 𝑓(𝑥) has an average rate of change of 3 on the interval (0, 1). The function 𝑔(𝑥) = 𝑥2 + 3 is increasing on the interval (0, ∞). The function 𝑔(𝑥) defined above is the function 𝑓(𝑥) = 𝑥2 shifted to the left 3 units. The terms of the arithmetic sequence, 𝑓(𝑛) = 2 + 3𝑛, is a straight line.

Key Points – Module 3 Lessons
Grade 9 – Module 3 Module Focus Session November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 4 minutes MATERIALS NEEDED: X Key Points – Module 3 Lessons Lessons emphasize a freedom to ask questions, experiment, observe, look for structure, reason and communicate. Timing of lessons cannot possibly meet the needs of all student populations. Teachers should preview the lesson and make conscious choices about how much time to devote to each portion. While many exercises support the mathematical practices in and of themselves, the discussions and dialog points are often critical for both their content and for enacting the mathematical practice standards. 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?

End-of-Module Assessment
Grade 9 – Module 3 Module Focus Session November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 25 minutes MATERIALS NEEDED: Student version of assessment End-of-Module Assessment Work with a partner on this assessment 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.

Scoring the Assessment
Grade 9 – Module 3 Module Focus Session November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 8 minutes MATERIALS NEEDED: Scoring rubric and exemplar Post-its Scoring the Assessment 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.

Key Points – End-of-Module Assessment
Grade 9 – Module 3 Module Focus Session November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 6 minutes(90) MATERIALS NEEDED: Key Points – End-of-Module Assessment End of Module assessment are designed to assess all standards of the module (at least at the cluster level) with an emphasis on assessing thoroughly those presented in the second half of the module. Recall, as much as possible, assessment items are designed to asses the standards while emulating PARCC Type 2 and Type 3 tasks. Recall, rubrics are designed to inform each district / school / teacher as they make decisions about the use of assessments in the assignment of grades. (Review each key point one at a time.)

November 2013 NTI Institute TIME ALLOTTED FOR THIS SLIDE: 5 minutes(175) MATERIALS NEEDED: Biggest Takeaway What are your biggest takeaways from the study of Module 3? How can you support successful implementation of these materials at your schools given your role as a teacher, trainer, school or district leader, administrator or other representative? 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.

Grade X – Module X Module Focus Session
October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 2 minutes MATERIALS NEEDED: X A Story of Functions Making Instructional Decisions Grade 9 – Module 3 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.

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 2 minutes MATERIALS NEEDED: X Session Objectives To find ways to bridge gaps in student understanding for Grade 9, Module 3. Determine strategies to support teaching decisions while maintaining the rigor of the instruction 2 mins 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 Our objectives for this session are to: Provide ideas of how to present information from earlier modules/grades to bring students up to speed Identify strategies to support teaching decisions while maintaining the rigor of the instruction

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 1 minutes MATERIALS NEEDED: X Agenda Foundational Standards Topics to Consider during Transition Years Making Instructional Decisions

G9-M3 Module Overview  Foundational Standards
Grade X – Module X Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 3 minutes MATERIALS NEEDED: X G9-M3 Module Overview  Foundational Standards Grade 8 8.EE.A.1 8.EE.A.2 8.F.A.1 8.F.A.2 8.F.A.3 8.F.B.4 8.F.B.5 Grade 9 3 mins 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. N-Q.A.1 N-Q.A.2 N-Q.A.3 A-SSE.A.1 A-SSE.A.2 A-CED.A.2 A-CED.A.3 A-CED.A.4 A-REI.A.1 A-REI.B.3 A-REI.C.6 A-REI.D.10

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 1 minute MATERIALS NEEDED: X Agenda Foundational Standards Topics to Consider during Transition Years Making Instructional Decisions

Topics to Consider during Transition Years
Grade X – Module X Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 3 minutes MATERIALS NEEDED: X Topics to Consider during Transition Years As you look through the Grade 9 curriculum, you will discern what content and skills that your students need to be successful with the Grade 9 material. Grade 9, Module 3,Topic C: Transformations of Functions The topic of Transformation is not commonly covered in current Grade 8 material 3 mins Lesson 17 Lesson Notes

Grade X – Module X Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 2 minutes MATERIALS NEEDED: X Topics to Consider during Transition Years Grade 8 Module 2 Overview: 2 mins Translations, reflections, and rotations are covered in Lessons 1-6 of Grade 8, Module 2.

Grade X – Module X Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 2 minutes MATERIALS NEEDED: X Topics to Consider during Transition Years Grade 8 Module 3 Overview: 2 mins Dilations are addressed in the first three lessons of Grade 8, Module 3.

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 2 minutes MATERIALS NEEDED: X Strategies to efficiently deliver the key information in Grade 8 regarding Transformations: Present key vocabulary to students (complete list in Module Overviews) Transformation Reflection Image of a Point Basic Rigid Motions Translation Dilation Distance-preserving Rotation Scale factor 2 mins - To give students a snapshot study of the material covered in Grade 8 regarding transformations, decide on the vocabulary they must be familiar with.

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 5 minutes MATERIALS NEEDED: X Strategies to efficiently deliver the key information in Grade 8 regarding Transformations: Select exercises from the relevant lessons that require students to articulate vocabulary and highlights understanding needed for G9-M3 Topic C. Examples: 1. To illustrate what makes a translation a rigid motion, have students complete G8-M2-L2, Example 3. The hands on nature of the exercise will help cement why the graph of the equation of y = f(x - 4) translates all the points of y = f(x) to the right 4 units. 2. To provide context for reflections, have students complete G8-M2-L4, Exercises Students know that a reflection is rigid, that there must be a line to reflect over, and that any point on the line of reflection will be unmoved by the reflection. 3. To highlight how dilations are related to vertical and horizontal scaling, have students complete G8-M3-L2, Example 1. Students will distinguish how scaling on the coordinate plane is a limited form of dilation in the plane. 5 mins - Then select key exercises where they will have to articulate the vocabulary to make sense of the exercise.

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 1 minutes MATERIALS NEEDED: X Agenda Foundational Standards Topics to Consider during Transition Years Making Instructional Decisions

Making Instructional Decisions
Grade X – Module X Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 2 minutes MATERIALS NEEDED: X Making Instructional Decisions Key Ideas Honor the objective of the lesson Maintain the rigor of the lesson 2 mins 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

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 2 minutes MATERIALS NEEDED: X Bridging the Gaps Student Outcomes and Assessments Focus Standards linked to Foundational Standards from previous material Lessons Grouped within Topics Terminology, Tools and Representations 2 mins 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?

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 3 minutes MATERIALS NEEDED: X Turn and Talk What are other gaps in understanding you foresee students having? How can you use the days built into each module for review to address these issues? How will planning change for the following school year? 3 mins

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 3 minutes MATERIALS NEEDED: X Biggest Takeaway Turn and Talk: What questions were answered for you? What new questions have surfaced? 3 mins 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.

Grade 9 – Module 3 Module Focus Session

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Grade 9 – Module 3 Module Focus Session

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