Grade 9 – Module 4 Module Focus Session

Grade 9 – Module 4 Module Focus Session
February 2014 Network Team 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 11 Module 1 and Grade 11 and 12 Overview 2 min [Note: This session is 9 hours in length.] Introduce myself and talk about the session. Mostly we will be working on the student pages located at the front of your binder. Avoid looking at the teacher pages for the most part. I mainly want you to experience the module from the student perspective.

February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 5 minutes Opening Exercise Use the following algebraic identities to answer each of the following without using a calculator. 𝑥 2 − 𝑎 2 = 𝑥+𝑎 𝑥 −𝑎 𝑥 𝑛 − 𝑎 𝑛 =(𝑥 −𝑎)( 𝑥 𝑛−1 +𝑎 𝑥 𝑛−2 + 𝑎 2 𝑥 𝑛−3 +…+ 𝑎 𝑛−2 𝑥+ 𝑎 𝑛−1 ) Compute each of the following. 112 2 − ∙1,007 2) Prove that −1 is divisible by 9. 3) Explain why no number written in the form 𝑥 𝑛 − 𝑎 𝑛 can be prime. 5 min Start here. Then back up to opening slide. Give participants 5 min to do the opening exercise.

February 2014 Network Team 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 2 min. 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.

February 2014 Network Team 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 11 Module 1 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. 2 min. We have three main objectives for this mornings work. Our main task will be experiencing lessons and assessments. As a secondary objective, you should walk away from the study of module 4 being able to articulate how these lessons promote mastery of the standards and how they address the major work of the grade. Lastly, you should be able to get a sense for the coherent connections to the content of earlier grade levels.

February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 2 minutes MATERIALS NEEDED: Agenda Orientation to Materials (if needed) Examination and exploration of Module 1: Topic A Topic B Mid-Module Assessment Topic C Topic D End of Module Assessment Preview the rest of Grade 11 and Grade 12 2 min. Here is our agenda for the day. If needed, we will start with orienting ourselves to what the materials consist of. We will spend most of our time on G11 M1. As we go through the module, I will talk about foundational skills developed in prior grades, particularly those developed in G9. We will discuss some fluency drills and other scaffolds that can be used to address possible gaps in content knowledge. At the end of the session, we will preview the rest of the G11 curriculum and the beginning of G12. (Click to advance animation.) Let’s begin with an orientation to the materials for those that are new to the materials (Skip if participants are already familiar with the materials).

February 2014 Network Team 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 (Not accounted for in the timing – these slides are optional if participants are new to the materials.) Each module will be delivered in 3 main files per module. The teacher materials, the student materials and a pack of copy ready materials. Teacher materials include a module overview, and topic overviews, along with daily lessons and a mid- and end-of-module assessment. (Note that shorter modules of 20 days or less do not include a mid-module assessment.) Student materials are simply a package of daily lessons. Each daily lesson includes any materials the student needs for the classroom exercises and examples as well as a problem set that the teacher can select from for homework assignments. The copy ready materials are a single file that one can easily pull from to make the necessary copies for the day of items like exit tickets, or fluency worksheets that wouldn’t be fitting to give the students ahead of time, as well as the assessments.

February 2014 Network Team 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. (Not accounted for in the timing – these slides are optional if participants are new to the materials.) There are 4 general types of lessons in the 6-12 curriculum. There is no set formula for how many of each lesson type we included, we always use whichever type we feel is most appropriate for the content of the lesson. The types are merely a way of communicating to the teacher, what to expect from this lesson – nothing more. There are not rules or restrictions about what we put in a lesson based on the types, we’re just communicating a basic idea about the structure of the lesson. Problem Set Lesson – Teacher and students work through a sequence of 4 to 7 examples and exercises to develop or reinforce a concept. Mostly teacher directed. Students work on exercises individually or in pairs in short time periods. The majority of time is spent alternating between the teacher working through examples with the students and the students completing exercises. Exploration Lesson – Students are given 20 – 30 minutes to work independently or in small groups on one or more exploratory challenges followed by a debrief. This is typically a challenging problem or question that requires students to collaborate (in pairs or groups) but can be done individually. The lesson would normally conclude with a class discussion on the problem to draw conclusions and consolidate understandings. Socratic Lesson – Teacher leads students in a conversation with the aim of developing a specific concept or proof. This lesson type is useful when conveying ideas that students cannot learn/discover on their own. The teacher asks guiding questions to make their point and engage students. Modeling Cycle Lesson --Students are involved in practicing all or part of the modeling cycle (see p. 62 of the CCLS, or 72 of the CCSSM). The problem students are working on is either a real-world or mathematical problem that could be described as an ill-defined task, that is, students will have to make some assumptions and document those assumptions as they work on the problem. Students are likely to work in groups on these types of problems, but teachers may want students to work for a period of time individually before collaborating with others.

February 2014 Network Team 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 (Not accounted for in the timing – these slides are optional if participants are new to the materials.) Follow along with a lesson from the materials in your packet. The teacher materials of each lesson all begin with the designation of the lesson type, lesson name, and then 1 or more student outcomes. Lesson notes are provided when appropriate, just after the student outcomes. Classwork includes general guidance for leading students through the various examples, exercises, or explorations of the day, along with important discussion questions, each of which are designated by a solid square bullet. Anticipated student responses are included when relevant – these responses are below the questions; they use an empty square bullet and are italicized. Snapshots of the student materials are provided throughout the lesson along with solutions or expected responses. The snap shots appear in a box and are bold in font. Most lessons include a closing of some kind – typically a short discussion. Virtually every lesson includes a lesson ticket and a problem set. What you won’t see is a standard associated with each lesson. Standards are identified at the topic level, and often times are covered in more than one topic or even more than one module… the curriculum is designed to make coherent connections between standards, rather than following the notion that the standards are a checklist of items to cover. Student materials for each lesson are broken into two sections, the classwork, which allows space for the student to work right there in the materials, and the problem set which does not include space – those are intended to be done on a separate sheet so they can be turned in. Some lessons also include a lesson summary that may serve to remind students of a definition or concept from the lesson.

February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 2 minutes MATERIALS NEEDED: Agenda Orientation to Materials (if needed) Overview of Module 1 Examination and exploration of Module 1: Topic A Topic B Mid-Module Assessment Topic C Topic D End of Module Assessment Preview the rest of Grade 11 and Grade 12 (no time allotted)

Mathematical Themes of Module 1
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute Mathematical Themes of Module 1 Being able to Look for and make use of structure (MP.7) when working with a polynomial function allows us to gain insight into the function’s behavior and its graph. The connection between zeros, factors, and x-intercepts The arithmetic of polynomials is governed by the same rules as the arithmetic of integers, and the arithmetic of rational expressions is governed by the same rules as the arithmetic of rational numbers. 5 minutes Display and have participants read the Module overview, scan the standards and the new terminology

February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 3 minutes MATERIALS NEEDED: None Flow of Module 1 Topic A: Polynomial – From Base Ten to Base X Topic B: Factoring – Its Use and Its Obstacles Topic C: Solving and Applying Equations – Polynomial, Rational, and Radical Topic D: A Surprise from Geometry – Complex Numbers Overcome All Obstacles 2 min (Go through the bullets to give an overview of the progression or flow of each topic and the module as a whole.)

What are students coming in with?
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 8 minutes MATERIALS NEEDED: None What are students coming in with? Experience multiplying with polynomials using the distributive property (G9-M1) Experience relating the distributive property to an area model or an modified area model (the tabular method) (G9-M1) Experience with solving and graphing quadratics (G9-M4) 2 min (Review the bullet points with participants to remind them of the background students are coming in to this module with.) We will be discussing in more detail as we go through the module what students’ previous experiences have been

February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 2 minutes MATERIALS NEEDED: Agenda Orientation to Materials (if needed) Overview of Module 1 Examination and exploration of Module 1: Topic A Topic B Mid-Module Assessment Topic C Topic D End of Module Assessment Preview the rest of Grade 11 and Grade 12 (no time allottted)

Topic A: Polynomial – From Base Ten to Base X
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute Topic A: Polynomial – From Base Ten to Base X Expanding and building on experiences from grade 9 Polynomial multiplication and division MP 7: Look for and make use of structure The power of algebra – why is factoring useful? 5 minutes Read the topic A opener (p. 12 – 13)

February 2014 Network Team Institute From Base Ten to Base X Why do humans have a predilection for the number 10? What do you suppose some societies use base-20? Base-60? 2 min

February 2014 Network Team Institute From Base Ten to Base X How would we write the number 113 in base 5? 1 3 5 min 4 2 3

February 2014 Network Team Institute From Base 10 to Base X Let’s be as general as possible and write a number in base x 𝟏 × 𝒙𝟑 𝟐×𝒙𝟐 𝟕×𝒙 𝟑×𝟏 A variable is a placeholder for a number. If 𝑥 is 10, what is the number? If 𝑥 is 5, what is the number? 5 min In this way, we can think of a polynomial as a number in base x where the x is a placeholder for some number yet to be determined. Is it ok to have a coefficient of 7 if we decide we are in base 5?

A Foundation for the Study of Polynomials: Differences and Diagonals
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 8 minutes MATERIALS NEEDED: None A Foundation for the Study of Polynomials: Differences and Diagonals What is the next number in the sequence? 4, 7, 10, 13, 16, … 4, 5, 8, 13, 20, 29, … 0, 2, 20, 72, 176, 350, 612, … 1, 2, 4, 8, 16, 32, 64, 128, 256, … 1, 4, 9, 16, 25, 36, 49, 64, 81, … 1, 8, 27, 64, 125, 216, 343, … 5 min. These next 3 slides were directly informed and inspired by the work of Dr. James Tanton and his posted course on quadratics found at I find these exercises a highly desirable experience base for teachers embarking on a polynomial module. Students studied sequences in grade 9 (M3). Based on the apparent pattern, what might the next number in the sequence be? How did you decide it would be 19? You noticed that the difference between each term was constant; it was 3. Some people describe a sequence as linear if the difference from term to term is constant; if it has a constant set of first differences. If we imagine our sequence as a set of data values and plot them on a graph we see why they might call this a linear sequence. How about the next sequence. What might you be inclined to believe is the next number in this sequence? (40). How did you get that? Did anyone notice that the second differences are constant? Show that for the following sequence, the 3rd differences are constant. How many differences must one complete in the sequence below, the powers of two, to first see a row of constant differences? The square numbers begin, 1, 4, 9, 16, … is there a row of their difference table that is constant?

Differences and Diagonals
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 8 minutes MATERIALS NEEDED: None Differences and Diagonals If I only gave you the leading diagonal (and it eventually led to a row of zero’s) could you find every number in the sequence? What does the leading diagonal look like for each of the following: 1: 1, 1, 1, 1, 1, …. n: 1, 2, 3, 4, 5, 6, … n2: 1, 4, 9, 16, 25, 36, … n3: 1, 8, 27, 64, 125, 216, … What do you suppose the leading diagonal would look like for n2 + n? 10 minutes Switch to document camera and do an example where given leading diagonal. Write out lead diagonals for 1, n, n^2, n^ 3. Do you think the leading diagonal for n^2 + n is the sum of the leading diagonal of n^2 and n? Let’s see.

Differences and Diagonals
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute Differences and Diagonals If I gave you only the leading diagonal could you find an explicit formula for the sequence? 5 minutes Document camera Give them a leading diagonal and ask them to write an explicit formula for the sequence.

Lesson 1: Successive Differences in Polynomials
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 8 minutes MATERIALS NEEDED: None Lesson 1: Successive Differences in Polynomials Example 3 Exit ticket 10 min Give participants time to work and then address any questions or concerns. The exit ticket is in the teacher pages. Note that the exit ticket is material students would be familiar with from grade 9 (m4)

Lesson 1: Successive Differences in Polynomials
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 8 minutes MATERIALS NEEDED: None Lesson 1: Successive Differences in Polynomials Numerical Symbol: A numerical symbol is a symbol that represents a specific number. Examples: 1, 2,3,4, 𝜋, −3.2 Variable Symbol: A variable symbol is a symbol that is a placeholder for a number from a specified set of numbers. The set of numbers is called the domain of the variable. Examples: 𝑎, 𝑦, 𝑧 Monomial: A monomial is an algebraic expression generated using only the multiplication operator (__×__). Examples: 𝑥 3 , 3𝑥 Polynomial Expression: A polynomial expression is a monomial or sum of two or more monomials. Examples: 𝑥 3 , 𝑥 3 +3𝑥, 𝑎 2 −6𝑎−2 5 min Review of some vocabulary. A variable should be thought of as a place holder for a number whose value has not been determined.

Lesson 2: The Multiplication of Polynomials
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 8 minutes MATERIALS NEEDED: None Lesson 2: The Multiplication of Polynomials Multiply (x + 8)(x + 7) using: The tabular method The distributive property 5 min Students use the area model to multiply in elementary school. We continue using this idea is high school but switch the terminology to the “tabular method” to allow the inclusion of negative values. Students learned both ways of multiplying polynomials in grade 9 (m1) Is everyone familiar with the area model or tabular method? Do we need to see another example?

Lessons 3-4: Division of Polynomials – Comparing Methods
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 8 minutes MATERIALS NEEDED: None Lessons 3-4: Division of Polynomials – Comparing Methods Reverse tabular method – Find the quotient 2 𝑥 2 +𝑥−10 𝑥 −2 . Lesson 3 Problem Set # 1 and 12 Lesson 4 Example 1 10 minutes Students learn a variety of ways of dividing polynomials within the module – reverse tabular method, long division, reducing a common factor, inspection. Complete the example on the ppt. Switch to document camera and work another example using the reverse tabular method. Work on L3 problem set #1 and 12 The long division algorithm to divide polynomials is analogous to the long division algorithm for integers. The long division algorithm to divide polynomials produces the same results as the reverse tabular method. L4 Example 1 – We use a base 10 number system. A polynomial can be thought of as a number in “base x” an idea we explored in grade 9.

Lessons 6: Dividing by x – a and x + a
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute Lessons 6: Dividing by x – a and x + a Find each quotient: 𝑥 2 −4 𝑥− 𝑥 3 −8 𝑥− 𝑥 4 −16 𝑥−2 Based on your results, predict the quotient: 𝑥 5 −32 𝑥−2 From this lesson, we can identify some patterns: 10 min Give students the freedom here to select the appropriate method for finding the quotients. If students see the pattern (MP 8) here, it will be much easier for them to remember how to factor various polynomials based on its structure (MP 7).

February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 5 minutes Opening Exercise 𝑥 2 − 𝑎 2 = 𝑥+𝑎 𝑥 −𝑎 𝑥 𝑛 − 𝑎 𝑛 =(𝑥 −𝑎)( 𝑥 𝑛−1 +𝑎 𝑥 𝑛−2 + 𝑎 2 𝑥 𝑛−3 +…+ 𝑎 𝑛−2 𝑥+ 𝑎 𝑛−1 ) Compute each of the following. 112 2 − ∙1,007 2) Prove that −1 is divisible by 9. 3) Explain why no number written in the form 𝑥 𝑛 − 𝑎 𝑛 can be prime. 4) 2 3 −1, 2 5 −1, and 2 7 −1 are all prime numbers. Is 2 9 −1 also prime? 5 min The opening exercise came from lesson 7 on mental math. Add question 4. Ask for participant responses. – Do we believe in patterns? (a theme in G9) Questions on any of the other exercises?

Lessons 8: The Power of Algebra – Finding Primes
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute Lessons 8: The Power of Algebra – Finding Primes A Mersenne prime is a prime number of the form 2 𝑛 −1. As of May 2014, 48 Mersenne primes are known. The largest known prime number is 257,885,161 − 1. The question “Are there infinitely many Mersenne primes?” is an unsolved math problem. Connection to primes and data encryption: 5 min This is a very interesting topic that has an abundance of information that can be found. Spark student curiosity by having them research this topic or by showing the video.

Lessons 10: The Power of Algebra – Finding Pythagorean Triples
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute Lessons 10: The Power of Algebra – Finding Pythagorean Triples A-APR.C.4: Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x2 + y2)2 = (x2 - y2)2 + (2xy)2 can be used to generate Pythagorean triples. Example 1 and 2 10 min Connects Algebra to Geometry. Work through examples 1 and 2. Discuss. Share responses.

February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 2 minutes MATERIALS NEEDED: None Key Points – Topic A Seeing structure in expressions requires a dynamic view in which potential rearrangements and manipulations are possible. Algebra is a powerful and useful tool in a variety of situations. Multiplication and division of polynomials follows the same principles as multiplication and division of integers. 2 min. Total time elapsed: 121 min + 9 min for questions/discussion = 130 min (2 hr 10 min) (Go through each point listed.)

Topic B: Factoring – Its Uses and Its Obstacles
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute Topic B: Factoring – Its Uses and Its Obstacles From Grade 9, students should be able to solve a quadratic equation by factoring, completing the square, or using the quadratic formula. Students may have used the tabular method to assist in factoring or completing the square. 15 minutes Read through the topic opener for topic B (p. 128 in the teacher pages) I want to examine students’ experiences from grade 9. You may have to provide some scaffolding for next year’s students. Switch over to the document camera and show factoring (tabular method) and completing the square from grade 9. Show both the technique of working on one side of the equation and also multiplying by 4 or a multiple of 4.

Rapid White Board Exchange
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 25 minutes MATERIALS NEEDED: Student version of assessment Rapid White Board Exchange Factoring trinomials 15 min Supplies: personal board, dry erase markers, felt Factoring is a skill that students need to develop fluency with. It is a great example of a skill for which a rapid white board exchange is a fitting fluency exercise. How to conduct a white board exchange: All students will need a personal white board, white board marker, and a means of erasing their work. An economical recommendation is to place card stock inside sheet protectors to use as the personal white boards, and to cut sheets of felt into small squares to use as erasers. You have these materials at your tables today. It is best to prepare the questions in a way that allows you to reveal them to the class one at a time. A flip chart, or Powerpoint presentation can be used, or one can write the problems on the board and either cover some with paper or simply write only one on the board at a time. Prepare problems that progress in difficulty. The best way to get the feel is for us to do one ourselves. I’ll reveal the problem, you work it as fast as you can and still do accurate work and then hold it up for me to see. (Reveal the first problem in the list and announce, “Go”. Give immediate feedback to each participant, pointing and/or making eye contact with the participant and responding with an affirmation for correct work such as, “Good job!”, “Yes!”, or “Correct!”, or guidance for incorrect work such as “Look again,” “Try again,” “Check your work,” etc. Do several to demonstrate the progression of problems.) If many students have struggled to get the answer correct, go through the solution of that problem as a class before moving on to the next problem in the sequence. Fluency in the skill has been established when the class is able to go through each problem in quick succession without pausing to go through the solution of each problem individually. If only one or two students have not been able to get a given problem correct when the rest of the students are finished, it is appropriate to move the class forward to the next problem without further delay; in this case find a time to provide remediation to that student before the next fluency exercise on this skill is given.

Lesson 12: Overcoming Obstacles in Factoring
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 2 minutes MATERIALS NEEDED: None Lesson 12: Overcoming Obstacles in Factoring Exercise 1 Example 2 10 min Try some of the techniques I’ve shown you to work exercise 1. Switch to document camera. For example 2, let students be stumped for a minute. Factoring is not always clear or obvious (or possible). What if I told you that one factor was x – 3? Would that be helpful? How? Then, work it by grouping.

Lesson 14: Graphing Factored Polynomials
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 2 minutes MATERIALS NEEDED: None Lesson 14: Graphing Factored Polynomials Why is it useful to have the polynomial in factored form when trying to graph? Consider the function f(x) = (x – 4)(x – 4)(x + 4). What are the values of x such that f(x) = 0? What are the zeros of f? What are the x-intercepts of the graph of f? What is the degree of f? How does the degree compare with the number of factors? How does the degree compare with the number of x-intercepts? Is this enough information to sketch a graph of f? What else do we need to know? 10 minutes It is not correct to refer to the graph of f(x) or the zeros of f(x). It is ok to say the graph of f or the graph of the equation y = f(x). Work the example out on chart paper. How could we know which direction the graph should go? Use test values How do we know when the graph changes directions? We don’t exactly. We really need Calculus to find the points where the graph changes directions. In lesson 15, students will study the structure of the graphs more closely and look at end behavior, Look at the Discussion (S.71)

February 2014 Network Team Institute Lesson 18: Overcoming a Second Obstacle in Factoring— What If There Is a Remainder? From the CCSS Algebra Progressions document: The analogy between polynomials and integers carries over to the idea of division with remainder. Just as in Grade 4 students find quotients and remainders of integers (4.NBT.6) in high school they find quotients and remainders of polynomials (A-APR.6). The method of polynomial long division is analogous to, and simpler than, the method of integer long division. Opening Exercise Example 1 10 min Ask about familiarity with progressions documents. If you have not read thru the algebra (and functions) progessions document, I recommend you do so. It really helps to clarify how students skills should develop throughout high school. With this idea in mind, look at the opening exercise. Work out the opening exercise on chart paper or document camera. How does understanding this problem help to lead students to the idea of example 1 (b) where we have a remainder for the first time?

February 2014 Network Team Institute Lesson 18: Overcoming a Second Obstacle in Factoring— What If There Is a Remainder? A-APR.D.6 Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system. Exercises 1 - 9 5 min Look at the standard. Work through a selection of the exercises from the lesson. Notice students are using inspection, reverse tabular system, and long division. Also notice what is missing…synthetic division.

Lesson 19: The Remainder Theorem
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute Lesson 19: The Remainder Theorem From the CCSS Algebra Progressions document: A particularly important application of polynomial division is the case where a polynomial p(x) is divided by a linear factor of the form x - a, for a real number a. In this case the remainder is the value p(a) of the polynomial at x = a (A-APR.2). It is a pity to see this topic reduced to “synthetic division,” which reduced the method to a matter of carrying numbers between registers, something easily done by a computer, while obscuring the reasoning that makes the result evident. It is important to regard the Remainder Theorem as a theorem, not a technique. 5 min Read through the excerpt. The remainder theorem is a hugely important topic. Students will apply it in a later lesson to a modeling problem. We will use it again to develop the FTA at the end of the module.

Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute Lesson 19: The Remainder Theorem A-APR.B.2 Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x - a is p(a), so p(a) = 0 if and only if (x - a) is a factor of p(x). From the CCSS Algebra Progressions document: A consequence of the Remainder Theorem is to establish the equivalence between linear factors and zeros that is the basis of much work with polynomials in high school: the fact that p(a) = 0 if and only if x - a is a factor of p(x). It is easy to see if x -a is a factor then p(a) = 0. 5 min It is easy to get students to understand that if p(a) = 0, then x = a is a zero of p, and x – a is a factor. Make sure you don’t leave off the next part.

Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute Lesson 19: The Remainder Theorem From the CCSS Algebra Progressions document: But the Remainder Theorem tells us that we can write 𝑃(𝑥) = 𝑞(𝑥)(𝑥 −𝑎) + 𝑃(𝑎) for some polynomial q(x). In particular, if P(a) = 0 then P(x) = (x – a)q(x) so x – a is a factor of P. Exercise 1 Continue to draw the analogy to integer operations. Think about the quotient We could write this as 13 = 4∙3+1, where 4 is the quotient and 1 is the remainder. So, exercise 1 could be rewritten as 𝑓(𝑥)=(3𝑥+14)(𝑥−2) What is f(2)? Exercise 6 10 min This is the part that we sometimes don’t emphasize or that we lose kids with. Look at exercise 1. Discuss the analogy. Work exercise 6. Topic B closes with a 2 day modeling problem pertaining to a river bed. It incorporates the Remainder Theorem and the other topics covered in this part of the module.

February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 2 minutes MATERIALS NEEDED: None Key Points – Topic B Polynomial functions are easy to graph when written in factored form (though not always easy to get into factored form). Polynomials can still be divided just like integers even if there is a remainder. When a polynomial is divided by (𝑥 −𝑎), the remainder is the value of the polynomial at x = 𝑎. When 𝑎 is a zero of polynomial p, then 𝑥 −𝑎 is a factor of p. 2 min. (Go through each point listed.)

Mid-Module Assessment
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 25 minutes MATERIALS NEEDED: Student version of assessment Mid-Module Assessment Work with a partner on this assessment 20 min 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 4 Module Focus Session February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 8 minutes MATERIALS NEEDED: Scoring rubric and exemplar Post-its Scoring the Assessment 15 min Total time elapsed: 252 min + 8 min questions / discussion = 260 min (4 hr 20 min) 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 10 minutes or so have passed, call the group together and address any questions or concerns that participants noted on their post-it notes.

February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 2 minutes MATERIALS NEEDED: Agenda Orientation to Materials (if needed) Overview of Module 1 Examination and exploration of Module 1: Topic A Topic B Mid-Module Assessment Topic C Topic D End of Module Assessment Where do we go from here – a look at the rest of Grade 11 and Grade 12 (no time allotted)

February 2014 Network Team Institute Topic C: Solving and Applying Equations—Polynomial, Rational, and Radical Continue to build fluency in solving equations and factoring polynomials. Connect operations with rational numbers to operations on rational expressions . Systems of equations (including systems in three variables) The geometric definition of a parabola 5 minutes Read topic opener for topic C (p. 237 in teacher pages)

Lessons 22: Equivalent Rational Expressions
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 2 minutes MATERIALS NEEDED: None Lessons 22: Equivalent Rational Expressions Rational expressions form a system analogous to the rational numbers. Just as is equivalent to or , 𝑥 𝑥−5 𝑥−5 is equivalent to x except when x = 5. It is important that any restriction on the values of the variable accompany a rational expression. 𝑥 2 −4 𝑥−2 =𝑥+2, for 𝑥≠2 5 min In grade 12, we will learn that the graphs of y = (x^2-4)/(x – 2) and y = x + 2 look identical except at x = 2. In calculus, we will call that a removable discontinuity. The next 2 lessons are fairly straightforward, students are working with operations on rational expressions.

Lessons 26 and 27: Rational Equations
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute Lessons 26 and 27: Rational Equations From the CCSS Algebra Progressions document: Understanding solving equations as a process of reasoning demystifies “extraneous” solutions that can arise under certain solution procedures. The reasoning begins from the assumption that x is a number that satisfies the equation and ends with a list of possibilities for x. But not all the steps are necessarily reversible, and so it is not necessarily true that every number in the list satisfies the equation. 5 min Applying the properties of Equality is guaranteed to preserve the solution set. Applying the Distributive, Associative, and Commutative Properties or the properties of rational exponents to either side is also guaranteed to preserve the solution set. You can do anything that’s useful, but it is not guaranteed to preserve your solution set! So if we do anything outside of these properties, we MUST check the solutions! Give participants time to work on the problems and then discuss.

Lessons 26 and 27: Rational Equations
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute Lessons 26 and 27: Rational Equations Multiplicative Property of Equality: Whenever 𝑎 = 𝑏 is true, then 𝑎𝑐 = 𝑏𝑐 will also be true, (and when 𝑎 = 𝑏, is false, then 𝑎𝑐 = 𝑏𝑐 will also be false) for all non-zero, real numbers 𝑐. So why does multiplying each side by a factor potentially change the solution set? Lesson 26 Problem Set 3 and 4 Lesson 27 Example 1 15 min Discuss the multiplicative property of equality. (students learn this in G7) Pose the equation and look at Exercise 4 from lesson 26 as an example. Work on the L26 problem set questions. Discuss extension. Work through Lesson 27 Example 1 – use tape diagram?

February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 25 minutes MATERIALS NEEDED: Student version of assessment Sprint Simplifying radicals 15 min Supplies: sprints A and B A sprint is another fluency exercise that can help to bridge gaps in knowledge and increase automaticity of a skill. They are fast-paced and don’t take up too much class time. Students worked with radicals to some extent in G9 and G10. They also covered them in Lesson 9 of this module. But this is still an area where there could be potential gaps in content knowledge. This might be a good place to do a rapid white board exchange as we saw earlier or a sprint. We are going to do a sprint now. Conduct sprint.

Lesson 28: A Focus on Square Roots
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute Lesson 28: A Focus on Square Roots The discussion of extraneous solutions and why they occur should be one of the focal points of the lesson. Problem set 1, 2, 23, 24 10 min As we saw with rational eqns, the idea of potential extraneous solutions should be highlighted. Scan through the problem set and work a few of the problems.

Lesson 30: Linear Systems in Three Variables
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute Lesson 30: Linear Systems in Three Variables Students solved linear systems in two variables in Grade 8 (M4) and Grade 9 (M1 and 4). In Algebra I, students modeled quadratic data both with and without technology. Algebra I problem: Write the equation for the quadratic whose graph contains the three points (𝟎,𝟒), (𝟏,𝟗), and (−𝟑,𝟏). Algebra II problem: Find the equation of the form 𝑦=𝑎𝑥2 + 𝑏x + c that satisfies the points (1,6), (3,20), and (−2,15). 5 min Students were introduced to this idea to some degree in G9 when writing equations of quadratics. However, they were always given the y-intercept as one of the points. In Algebra II, students must solve systems in three variables.

Lesson 31: Systems of Equations
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute Lesson 31: Systems of Equations A-REI.C.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = -3x and the circle x2 + y2 = 3. Exit ticket 10 min Students should be able to find the intersection point(s) between a line and parabola or a line and a circle both algebraically and graphically. In grade 9, students graph parabolas. In grade 10, students graph circles (including completing the square to write the equation in standard form). These skills are reviewed in this lesson and the lesson 32 Work on the exit ticket (Teacher p. 336) Stop here day 1?

U-Shaped Symmetric Graphs
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute U-Shaped Symmetric Graphs Can a hanging chain be modeled precisely by a quadratic function? Tape the ends of the chain so that the lowest part of the chain falls right at the origin. Mark off evenly spaced 𝑥-intervals. Find the 𝑦-value for each 𝑥-value marked by measuring its length with the ruler. Does the data support the idea that the curve is quadratic? 15 min Supplies: chains, chart paper, ruler, tape Had participants come up with a sequence using a ruler. For equal changes in x, collect data. Measure length of the y-value. Made a table of values. Analyze the data together looking at first and second differences. So…are all u-shaped curves quadratics? Hanging chains are not modeled by quadratics. They are catenary curves.

U-Shaped Symmetric Graphs
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute U-Shaped Symmetric Graphs Not all u-shaped curves are parabolas. These are called catenary curves. This is a parabola. 5 min Catenary is derived from the latin word chain. A free hanging chain or wire forms a catenary curve not a parabola. Nice to be prepared for student questions. May spark their curiosity. While students are not responsible for knowing what a catenary curve is, they should realize that not all u shaped graphs can be represented by a quadratic. A quadratic produces one special type of u shaped graph called a parabola.

Lesson 33: The Definition of a Parabola
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute Lesson 33: The Definition of a Parabola G-GPE.A.2 Derive the equation of a parabola given a focus and directrix. The term parabola is never used in Grade 9. The graph is simply called the graph of a quadratic and is described as a u-shaped curve. A parabola is a specific type of a u-shaped curve. A parabola with directrix line 𝐿 and focus point 𝐹 is the set of all points in the plane that are equidistant from the point 𝐹 and line 𝐿 . Exercises 1 and 2 15 min Supply: ruler This is the only geometry standard in G11. Ask about familiarity with the definition of a parabola. Discuss the definition of a parabola. Complete Exercise 1. Work through Exercise 2 together if necessary. Do we need to see another example? If yes, work problem set #12.

Lesson 34: Are All Parabolas Congruent?
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute Lesson 34: Are All Parabolas Congruent? A congruence is a sequence of basic rigid motions (translations, reflections, or rotations) that maps one figure onto another. A two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations. Based on this definition, are all parabolas congruent? 10 min Supplies: Graph paper, transparencies, dry erase markers Students learned these facts in grade 8 and solidified this knowledge in grade 10. In grade 8 students perform rigid motions and in grade 9 students apply these rigid motions to transformations of graphs of functions on the coordinate plane. Demonstrate with transparencies.

Lesson 35: Are All Parabolas Similar?
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute Lesson 35: Are All Parabolas Similar? Two figures are said to be similar if you can map one onto another by a dilation followed by a congruence. Based on this definition, are all parabolas similar? These three lessons draw connections between geometry, algebra, and functions. Solving geometric problems using algebra and functions is one of the most powerful techniques we have to solve science, engineering, and technology problems. Exit ticket 10 min Supplies: Graph paper, transparencies, dry erase markers Students have performed dilations about a point in grade 8 and 10. dilations about a line in grade 9 (stretching/shrinking the graph of a function on a coordinate plane) Demonstrate with personal board and transparencies. Work on exit ticket. (Teacher p. 398)

February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 2 minutes MATERIALS NEEDED: None Key Points – Topic C Rational expressions are analogous to rational numbers. When solving a rational or radical equation, it is necessary to check for extraneous solutions because steps may have been taken that are not guaranteed to preserve the solution set. A parabola is a specific type of u-shaped curve. There is a connection between algebra, geometry, and functions. 2 min total time elapsed: 387 min + 8 min questions / discussion = 395 min (6 hr 35 min) (Review the points outlined.)

February 2014 Network Team Institute Topic D: A Surprise from Geometry—Complex Numbers Overcome All Obstacles Students encounter complex numbers for the first time. Complex numbers are introduced through their relationship to geometric transformations. Every polynomial can be rewritten as the product of linear factors, which is not possible without complex numbers. 5 min Read topic D topic opener (teacher p. 403)

Lesson 37: A Surprising Boost from Geometry
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 2 minutes MATERIALS NEEDED: None Lesson 37: A Surprising Boost from Geometry Complex numbers are neither imaginary, as in make believe, nor complex, as in complicated. Students develop an understanding of 𝑖 through geometry. Students will begin to see just how important complex numbers are to geometry and computer science in Modules 1 and 2 in Precalculus. Problem set 5, 6, and 11 15 min Supply: Graph paper Students are immediately suspicious of complex numbers mainly because of the unfortunate terms of “complex” and “imaginary.” Historically people have always been suspicious about numbers they don’t understand. Negative numbers were not fully embraced until the 18th century. The Greeks did not believe x^2 = 2 had a solution. We define numbers as we need them. For example, we need negative numbers to explain the solution of x + a = b when a > b. We need an irrational number to explain the hypotenuse of an isosceles right triangle. We need a complex number to explain the solution of x^2 = -1 so i^2 is defined to be -1. By the way, the unfortunate term imaginary with exactly such a connotation has been coined by Descartes. He also called the negative roots of an equation false which, fortunately, did not stuck. In Grade 12, students will see how complex numbers and trigonometry are connected. Switch to document camera and demonstrate. Work on problem set 5, 6, and 11.

Lessons 40: Obstacles Resolved—A Surprising Result
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 2 minutes MATERIALS NEEDED: None Lessons 40: Obstacles Resolved—A Surprising Result Opening exercise Can every quadratic be written in terms of linear factors? How many linear factors? Does this mean that every quadratic has two x-intercepts? Is it possible for a cubic function to have no real zeros? The real zero can be written as a linear factor (Remainder Theorem) What will be left if we remove the linear factor from the cubic function? So every cubic polynomial can be written in terms of how many linear factors? 15 min Work opening exercise. Discuss each bullet point. Work through example 2

Lessons 40: Obstacles Resolved—A Surprising Result
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute Lessons 40: Obstacles Resolved—A Surprising Result 5 min Why would this knowledge be considered “fundamental” to mathematicians? The Fundamental Theorem says that the complex number system contains every zero of every polynomial function. We do not need to look anywhere else to find zeros to these types of function The Fundamental Theorem of Algebra ensures that there are as many zeros as we’d expect for a polynomial function, and that factoring will always (in theory) work to find solutions to polynomial equations. This is not trivial. The FTA was published in That is about 100 years after the Fundamental Theorem of Calculus. One reason we spend so much time studying polynomials in algebra is because mathematicians love to work with polynomials. They are predictable, continuous, and easy to work with. In Calculus II, students will learn to write approximate different types of functions as a polynomial function (Taylor series).

February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 2 minutes MATERIALS NEEDED: None Key Points – Topic D Complex numbers are neither imaginary nor complex. Complex numbers have a geometric meaning. The inclusion of complex numbers in our number system means that every polynomial of degree n can be written in terms of n linear factors. 2 min Go through each point.

Key Points – Module 1 Lessons
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 4 minutes MATERIALS NEEDED: X Key Points – Module 1 Lessons 10 min Ask participants to list key points or ideas from module 1. Ex. Students are called upon to Look for and make use of structure (MP.7) as they work with polynomials to gain insight into the function’s behavior and its graph. Polynomials are analogous to integers; rational expressions are analogous to rational numbers. Students see algebra as a powerful tool that can assist in solving a wide range of mathematical problems. 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 4 Module Focus Session February 2014 Network Team 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 20 min 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 4 Module Focus Session February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 8 minutes MATERIALS NEEDED: Scoring rubric and exemplar Post-its Scoring the Assessment 15 min 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 10 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 4 Module Focus Session February 2014 Network Team 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. 2 min total time elapsed: 484 min + 11 min questions / discussion = 495 min (8 hr 15 min) (Review each key point one at a time.)

Grade 11 Module 2: Trigonometric Functions
Module Focus Session February 2014 Network Team Institute Grade 11 Module 2: Trigonometric Functions Students will: Understand the six trigonometric functions as they relate to circle-ometry. Build fluency with values of sine, cosine, and tangent at 𝜋 6 , 𝜋 4 , 𝜋 3 , 𝜋, etc. Graph the sine and cosine functions. Use the graphs of sine and cosine to derive simple trigonometric identities. Model periodic data with trigonometric functions. 10 min

Grade 11 Module 3: Exponential and Logarithmic Functions
Module Focus Session February 2014 Network Team Institute Grade 11 Module 3: Exponential and Logarithmic Functions Students will: Extend the domain of exponential functions. (In grade 9, exponential functions are limited to those with domains in the integers.) Expand their ability to solve exponential equations through the use of logarithms. Explore logarithmic properties and their uses. Graph exponential and logarithmic functions by using transformations of functions. Select and revise an appropriate model given a set of data or real-world context. 10 min

Grade 11 Module 4: Inferences and Conclusions from Data
Module Focus Session February 2014 Network Team Institute Grade 11 Module 4: Inferences and Conclusions from Data Students will: Build a more formal understanding of probability (continuing work from grade 7). Distinguish between data distributions for which it would be reasonable to use normal distribution as a model and those for which it would not. Use data from a random sample to estimate a population mean or a population proportion. Draw conclusions based on data from a statistical experiment. 10 min

Grade 12 Module 1: Complex Numbers and Transformations
Module Focus Session February 2014 Network Team Institute Grade 12 Module 1: Complex Numbers and Transformations Students will Return to the study of complex numbers that began in grade 11 . Study of linear transformations in the complex plane. Understand that complex number multiplication has the geometric effect. Use the connection to trigonometry to simplify complex arithmetic. Develop the 2×2 matrix notation for planar transformations represented by complex number arithmetic. 10 min Discuss what else is covered in G12?

February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 5 minutes(175) MATERIALS NEEDED: Biggest Takeaway What are your biggest takeaways from the study of Module 1? What are your biggest takeaways from the preview of Grade 11 and 12? 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? 5 min Total time: 540 minutes (9 hr) 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 9 – Module 4 Module Focus Session

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Grade 9 – Module 4 Module Focus Session

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