Grade 8 – Module 3 Module Focus Session

Grade 8 – Module 3 Module Focus Session
October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X A Story of Ratios Grade 8 – Module 7 (1 min) Welcome! In this module focus session, we will examine Grade 8 – Module 7.

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 1 minute MATERIALS NEEDED: X Session Objectives Examination of the development of mathematical understanding across the module using a focus on Concept Development within the lessons. Introduction to mathematical models and instructional strategies to support implementation of A Story of Ratios. (1 min) Our objectives for this session are: Examination of the development of mathematical understanding across the module using a focus on Concept Development within the lessons. Introduction to mathematical models and instructional strategies to support implementation of A Story of Ratios. As an overall theme of this NTI, we’ve been asked to pay special attention to the ways in which we can provide scaffolds to support specific student needs. Before we begin our examination of the mathematics in this module, let’s take a few minutes to review some of the principles we can use to support learning.

A Story of Ratios Scaffolding Mathematics Instruction
Grade 3 – Module 5 Module Focus Session February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 1 minute MATERIALS NEEDED: X A Story of Ratios Scaffolding Mathematics Instruction Key Points Amplify Language Move from Concrete to Representation to Abstract Give Specific Guidelines for Speaking, Reading, Writing, or Listening The mathematics modules were created based on the premise that scaffolding must be folded into the curriculum in such a way that it is part of its very DNA. The instruction in these modules is intentionally designed to provide multiple entry points for students at all levels. Teachers are encouraged to pay particular attention to the manner in which knowledge is sequenced in the curriculum and to capitalize on that sequence when working with special student populations. Most lessons move from simple to complex allowing teachers to locate specific steps where students are struggling or need a challenge. That said, there are specific resources to highlight and enhance strategies that can provide critical access for all students. In developing the scaffolds already contained in the curriculum, Universal Design for Learning (UDL) has provided a structure for thinking about how to meet the needs of diverse learners. Broadly speaking, that structure asks teachers to consider multiple means of representation; multiple means of action and expression; and multiple means of engagement. These dimensions promote engagement of students and provide multiple approaches to the same content. Individual lessons contain marginal notes to teachers (in text boxes) highlighting specific UDL information about scaffolds that might be employed with particular intentionality when working with students. These tips are strategically placed in the lesson where the teacher might use the strategy to the best advantage. Let’s now examine additional strategies that can be considered. In this module study, we will focus on three key ideas for developing scaffolds that can be adapted for your classroom to meet the needs of your students. Explicit focus on the language of mathematics, using the development from concrete to representation to abstract in the building of concepts, and communicating clear expectations in instructions are areas that can provide multiple entry points for students and can be used to promote student learning.

February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 1 minute MATERIALS NEEDED: X Amplify Language Give clear mathematical definitions Explain multiple meanings Maintain consistency and point out interchangeable terminology Much of what we share in the mathematics classroom with students is embedded in language that is specific. Students learn casual language before academic language. This means they may sound comfortable and fluent, but may need additional support in their writing and speaking in an academic environment. Presenters should stress that academic language is an essential component of closing the achievement gap and providing access to grade level content and beyond. Students may have a preconceived or informal idea of the meaning of a mathematical term. Be specific in the definition or meaning that will be used. Be cautions of words with multiple meanings that might be confusing a garden plot and the request to plot points on a coordinate plane Words with multiple meaning must be anticipated and then addressed, and teachers must also be prepared to pause and provide explanations when students identify words the teacher has not anticipated. Whenever possible, words with multiple means should be avoided on assessments, particularly when the meanings may be close enough to be confusing. Make sure that Language is internally consistent (if practice problems ask students to solve, the assessments should use the same term). If language is not internally consistent, then different terms are highlighted and taught. add, plus, sum, combine, all mean the same thing prism, a rectangular prism, box, package all reference the same figure in G6M5_L11

Move from Concrete to Representation to Abstract
Grade 3 – Module 5 Module Focus Session February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 1 minute MATERIALS NEEDED: X Move from Concrete to Representation to Abstract Use familiar contexts Visually organize thinking Provide multiple representations The more concrete and visual these ideas can be in foundational stages, the better! Use contexts that are familiar to students in your classroom. Use graphic organizers or other means for students to visually organize thinking. Note: Teachers should be thoughtful and purposeful about which graphic organizers they select. Are teachers introducing a new concept with a need to organize notes or are they connecting ideas comparing and contrasting? The goal is always to help students make those connections and not use a graphic organizer just for the novelty of it. Consider using non-verbal displays of mathematical relationships in your scaffolding. Use multiple representations and multiple approaches in explaining problems and allowing students to express solutions. Use pictures/ visuals/ illustrations are used to make content clearer.

Give Guidelines for Speaking, Reading, Writing, or Listening
Grade 3 – Module 5 Module Focus Session February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 1 minute MATERIALS NEEDED: X Give Guidelines for Speaking, Reading, Writing, or Listening Provide structured opportunities to speak and write in English Give explicit instructions in student-friendly language Use visuals or examples in giving instructions. Each day needs structured opportunities for students to speak and write in English. Students can chorally repeat key vocabulary or phrases Have them “turn to a neighbor and explain” Clearly set expectations by the explicit instructions in student-friendly language. Use visuals in your instructions. Be direct about language. Pause to discuss a vocabulary term and discuss how it may be used in the lesson. Have students repeat the word chorally so that they can all hear and practice. Provide sentence frames for anyone who may benefit. “The volume of my prism is ___units cubed. I found this by ______. “My idea is similar to _____’s because ____.” Generic/ universal sentence frames may remain posted in the classroom throughout the year. These might include: “I agree with ____ because ___” or “I think the answer is _____ because...”

February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 2 minutes MATERIALS NEEDED: X Key Points Amplify Language Move from Concrete to Representation to Abstract Give Guidelines for Speaking, Reading, Writing, or Listening Let’s review some key points of scaffolding instruction. As we study the module for this session, be thinking about specific scaffolds that might be most helpful for your classroom. We will pause at various points in the session to intentionally examine and discuss suggestions for scaffolds.

February 2014 Network Team Institute Turn and Talk What difficulties would you anticipate with student understanding of the mathematics in this section? What scaffolds would be effective for addressing those difficulties? Note to presenter: Insert this slide at appropriate points in the module study for an in-depth look at scaffolds. You may highlight a scaffold that already exists and discuss it or locate a point where a student might encounter difficulty and explore options. Delete the slide from this current sequence after you’ve inserted it in appropriate places throughout your session. Note to presenter: When you have inserted the slide, list several suggestions for scaffolds that would address the situation. Possible scaffolds:

February 2014 Network Team Institute Turn and Talk How does this task align with the Universal Design for Learning (UDL) in providing multiple options for: Representation: the “what of learning” Action/Expression: the “how” of learning Engagement: the “why” of learning Note to presenter: If applicable, insert this slide at an appropriate point in the module study for an in-depth examination of a problem or task for multiple entry points through the principles of the Universal Design for Learning (UDL). Delete this slide from this current sequence after you’ve used it elsewhere as needed. REPRESENTATION: The “what” of learning. How does the task present information and content in different ways? How students gather facts and categorize what they see, hear, and read. How are they identifying letters, words, or an author's style? In this task, teachers can ... Pre-teach vocabulary and symbols, especially in ways that build a connection to the learners’ experience and prior knowledge by providing text based examples and illustrations of fields. Integrate numbers and symbols into word problems. ACTION/EXPRESSION: The “how” of learning. How does the task differentiate the ways that students can express what they know? How do they plan and perform tasks? How do students organize and express their ideas? In this task, teachers can... Anchor instruction by pre-teaching critical prerequisite concepts through demonstration or models (i.e. use of two dimensional representations of space and geometric models). ENGAGEMENT: The “why” of learning. How does the task stimulate interest and motivation for learning? How do students get engaged? How are they challenged, excited, or interested? Optimize relevance, value and authenticity by designing activities so that learning outcomes are authentic, communicate to real audiences, and reflect a purpose that is clear to the participants. If available, reviewing student work would provide participants with the opportunity to deeply understand the benefits of students sharing their thinking in working the problem. Assessments in the module have rubrics that clearly outline expectations and could be used in the discussion.

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Agenda Introduction to the Module Concept Development Module Review (1 min) We will begin by exploring the module overview to understand the purpose of this module. Then we will dig in to the math of the module. We’ll lead you through the teaching sequence, one concept at a time. Along the way, we’ll also examine the other lesson components and how they function in collaboration with the concept development. Finally, we’ll take a look back at the module, reflecting on all the parts as one cohesive whole. Let’s get started with the module overview.

Curriculum Overview of A Story of Ratios
Grade 8 – Module 3 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Curriculum Overview of A Story of Ratios (3 min) The seventh module in Grade 8 is called Introduction to Irrational Numbers Using Geometry. The module is allotted 35 instructional days. The module motivates the need to learn about irrational numbers by requiring a precise length of one of the sides of a right triangle-a triangle that does not have integer side lengths. Students learn about square and cube roots and how to estimate their values when they are not perfect squares or cubes. That is only the beginning. Students learn and review how to determine the decimal expansion of numbers, both rational and irrational. Then apply all of their knowledge to finding the volume of cones, pyramids and spheres as well as composite solids. Average and constant rates are revisited in a more rigorous context in the last few lessons of the module.

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Module 7 Overview Table of Contents Overview Focus Standards Foundational Standards Focus Standards for Mathematical Practice Terminology Tools Assessment Summary (12 min) “I want to give you some time to familiarize yourself with the content of Module 7 by reading the Module Overview. Please take about 10 minutes to quietly read through the following sections (point to sections on slide). We will look closely at the assessment portions of the module today and tomorrow, so focus mainly on the overview.”

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Agenda Introduction to the Module Concept Development Module Review

Topic A: Square and Cube Roots
Grade 8 – Module 4 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Topic A: Square and Cube Roots Learning is motivated by the Pythagorean Theorem and need to get a precise length of a side of a right triangle. Square roots are defined. Square and cube roots exist and are unique. Students simplify square roots (optional). Students solve equations using roots. (3 min) Read the bullet points on the slide, “These are the basic concepts in Topic A. Next we will look at the specific lessons within this topic.”

Before Beginning the Module:
Grade 8 – Module 4 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Before Beginning the Module: Pythagorean Theorem Lessons from Modules 2 and 3 must be taught: Module 2, Lesson 15 Module 2, Lesson 16 Module 3, Lesson 13 Module 3, Lesson 14 Lessons contain proofs and practice. (2 min) “As teachers made their way through the modules there were several lessons related to the Pythagorean Theorem that at the time were identified as optional lessons. Now that we are beginning our investigation into square roots and irrational numbers, it is imperative that students are taught the content of these lessons. The lessons contain proofs and practice using the Pythagorean Theorem that students need to be successful in this module.”

Lesson 1: The Pythagorean Theorem
Grade 8 – Module 3 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Lesson 1: The Pythagorean Theorem Students use the Pythagorean Theorem to find unknown lengths of right triangles. Students estimate lengths when they are not equal to an integer. (1 min) Read the bullet points on the slide.

October, 2013 Common Core Institute Example 1 TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Write an equation that will allow you to determine the length of the unknown side of the right triangle. The length is 12 cm. (1 min) Allow time for participants to write the equation on the white boards, say “show me”, then allow participants to solve the equation. Next, select a participant to share their answer with the group. Remind participants that they must define their variables, as learned in Module 4.

October, 2013 Common Core Institute Example 2 TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Write an equation that will allow you to determine the length of the unknown side of the right triangle. How do we figure out what c is? (2 min) Allow time for participants to write the equation on the white boards, say “show me”. Instruct participants to begin solving the equation, but stop when they get to a point limited by the knowledge of a grade 8 student. That point should be when c^2=97. “It is this kind of problem that leads us into the need to learn about square roots. For now though, we learn to estimate the unknown length.”

October, 2013 Common Core Institute Example 2 TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Since 97 is not a perfect square, we can estimate the length of c by figuring out which two perfect squares 97 falls between. At this point, we know c must be between 9 cm and 10 cm. Which is it closer to? Since 97 is closer to 100 than 81, then c is closer to 10 cm. (3 min) Read through the points on the slide. “This is how we want students to begin their estimates of square roots. We return to this method when we discuss how to approximate the value of an irrational number.”

October, 2013 Common Core Institute Example 4 TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X In the figure below we have an equilateral triangle with a height of 10 inches. Determine the approximate length of the side of the triangle. What we actually have are two congruent right triangles. How can we prove this fact to students? (3 min) “Example 4 is slightly more complicated than the first three examples.” Read the prompt, then ask the question near the bottom of the slide. If necessary, allow participants to discuss the answer at their tables or with a partner before sharing their responses with the whole group. The grade appropriate response is to trace one of the right triangles and use reflection to prove congruence.

October, 2013 Common Core Institute Example 4 TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Since the triangles are congruent, we can look at just one of them and use the Pythagorean Theorem to help us determine the length of one side. In your handout: What is the length of the base? Explain. Write the equation and solve to determine the length of the side of the equilateral triangle. What we actually have are two congruent right triangles. How can we prove this fact to students? (2 min) Read the first bullet. Then instruct participants to work in their handout to solve the problem.

October, 2013 Common Core Institute Example 4 TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Consider the math that students needed to know in order to answer this question: Knowledge/Properties of Equilateral Triangle Understanding of Congruence (M2) Triangle Sum Theorem (M2) Laws of Exponents (M1) Properties of Equality/Solving Equations (M4) Estimating Square Roots (M7) (2 min) Read through the bullets on the slide. Stress the fact that we are in Module 7 and using skills learned earlier in the year. “Students may need this pointed out to them as well. We learn the math that we learn so that we can better understand more sophisticated concepts.”

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Lesson 2: Square Roots Students know that for most integers n, n is not a perfect square. Students know the notation, , and find the square root of small perfect squares. Students approximate the location of square root of non perfect square integers on the number line. (1 min) Read the bullet points on the slide.

October, 2013 Common Core Institute Discussion TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Why learn about square roots? Consider the length of the diagonal, s, of a unit square. Our method of estimating doesn’t yield much information. (2 min) Read through the points on the slide. “Again, we want to note that we need to learn about more numbers in order to find the unknown length of one side of a right triangle. We no longer want to estimate the length, we want a precise answer.”

October, 2013 Common Core Institute Discussion TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X The number s must be between 1 and 2. We can see it graphically: Consider a circle with center O and radius equal to the length of the diagonal. Again, we see that the answer must be between 1 and 2, but exactly what? There are many numbers on the number line between the integers. (2 min) Read through the points on the slide. “Ask students, what numbers exist between the integers on the number line? They should respond with decimals, fractions, mixed numbers. These are all the numbers they know about, now we will learn about other numbers that exist there.”

October, 2013 Common Core Institute Definitions: Revised TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X For now, we focus on square roots. A SQUARE ROOT OF A NUMBER. A square root of a number x is a number whose square is x. In symbols, a square root of x is a number a such that Negative numbers do not have any square roots, zero has exactly one square root, and positive numbers have two square roots. Again, we see that the answer must be between 1 and 2, but exactly what? (2 min) Read the first bullet. “Currently the overview defines square root as (read the definition). However, in the next edition of the curriculum the definition will be made more clear, as shown in the second bullet point.”

October, 2013 Common Core Institute Definitions: New TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X THE SQUARE ROOT OF A NUMBER. Every positive real number x has a unique positive square root called the square root or principle square root of x; it is denoted (The square root of zero is zero.) Altogether, every positive real number has two square roots: and It is common for teachers to refer to the principle square root as “the” square root. In common language, “the square root of 4” is 2, while “the square roots of 4” are 2 and -2. Again, we see that the answer must be between 1 and 2, but exactly what? (1 min) “We also plan to add this definition to the overview.”

October, 2013 Common Core Institute Definitions: TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X What’s the big deal? “a square root” versus “the square root” A square root of 4 can be 2 or -2 The square root of 4 is 2. The focus in Grade 8 is on knowing that the square root symbol automatically denotes a positive number. The context of finding the unknown length of a right triangle makes it clear that the number must be positive, but it is not as clear when students work with square roots abstractly. Students are asked to find only the positive square root of a number. (3 min) Read through the points on the slide.

October, 2013 Common Core Institute Discussion: TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Take a number line from 0 to 4. (2 min) “Once square root is defined, we want students estimating the location of square roots on the number line, yet another skill that will be useful when students begin estimating the value of irrational numbers. Use the number line in your handout to place the numbers on the number line.”

October, 2013 Common Core Institute Discussion: TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X (3 min) “Now place these numbers on the same number line.” Read the question and ask participants to respond. “We assume students will divide the unit from 1 to 2 into 3 equal parts and place the sqrt 2 and sqrt 3 on the first and second division.” Click to advance the animation. “Our goal is to have students estimate the locations. Being precise is great, but not right now. Students will learn how to be precise later on.” What strategy did you use to place the numbers? The goal is to estimate, not be precise.

October, 2013 Common Core Institute Discussion: TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X (2 min) “Continue placing numbers on the number line. Did you use a different strategy this time?”

October, 2013 Common Core Institute Discussion: TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X (3 min) “Continue placing the numbers on the number line.” Click to advance the animations. “The purpose of the activity is for students to identify the approximate location of square roots on the number line, but also for students to see the structure of our number system. The square roots follow the same number order as our whole number system. Lastly, even though we all know we are placing mainly irrational numbers here, we are not calling them irrational because that word has not yet been defined for students. Hold off on calling these numbers irrational until it is defined in Lessons 10 and 11.” We want students to see that the structure is the same for whole numbers and square roots.

Lesson 3: Existence and Uniqueness of Square Roots
Grade 8 – Module 3 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Lesson 3: Existence and Uniqueness of Square Roots Students know that the positive square root and cube root of a number exist and are unique. Students solve simple equations that require them to find square and cube roots of numbers. Option 1: Discuss existence and uniqueness via the Trichotomy Law. Option 2: “Find the Rule” game. (1 min) Read the bullet points on the slide. “We will focus our time on Option 1.”

Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Option 1: We want to convince students that there is only one square root of a number so they can be sure they have found all of the solutions to an equation that requires the use of square roots to solve. To prove uniqueness (and existence), we say there is only one number b so that If n = 2, then c is the square root of b. Consider this with concrete numbers: , then the square root of 25 is 5. (2 min) Read through the points on the slide. “Throughout the discussion we will try to relate the symbols back to concrete numbers. With students, teachers may consider demonstrating with half of the board in symbols and the other half with concrete numbers.”

Option 1: We can show uniqueness by showing that if
Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Option 1: We can show uniqueness by showing that if This proves uniqueness because both and must be equal to the same number b (because ). To show this, we use the Trichotomy Law. Given two numbers c and d, only one of the following can be true: (2 min) Read through the points on the slide. “With the goal clear in our mind, we have to take a slight detour into inequalities in order to show that II and III cannot be true.”

Option 1: The “Basic Inequality”
Header July 2013 Network Team Institute Option 1: The “Basic Inequality” TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X To show that c < d and c > d cannot be true, we use the Basic Inequality. If x, y, w, z are positive numbers so that x < y and w < z, is it true that xw < yz? (3 min) Read through the points on the slide. “The Basic Inequality isn’t something you’d find in textbook glossaries, but we needed to call this fact something because we use it in various places throughout the module. For that reason, we gave it a name.”

Option 1: Assume c < d. Then and
Header July 2013 Network Team Institute Option 1: TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Assume c < d. Then and We can make these claims because of the Basic Inequality. But this contradicts our original statement that Similar reasoning is used with the assumption that if c > d, then Again contradicting our original statement. Therefore, by the Trichotomy Law, c = d and the square root of a number is unique and therefore exists. (2 min) Read the first two points on the slide. “To produce the statements c^2 < d^2 and c^3 < d^3 we use the Basic Inequality. Since we know c<d, we multiply the left side of the inequality by c and the right side of the inequality by d to get c^2 < d^2. We repeat that step to get c^3 < d^3 .” Read through the remaining bullet points on the slide. “By showing that the square root of a number is unique, we also prove that it exists. That is, we do not need a separate proof of existence.”

February 2014 Network Team Institute Turn and Talk What difficulties would you anticipate with student understanding of the mathematics in this section? What scaffolds would be effective for addressing those difficulties? Possible scaffolds: Explain the Trichotomy Law by showing a number line, selecting a number and describing the three possibilities for another selected number. Explain the proof using concrete numbers first, then using symbols. Activate prior knowledge about inequalities.

Header July 2013 Network Team Institute Option 2: TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Students are shown two tables, fill in the blanks, and explain their reasoning. (2 min) “The second option simply has students filling in blanks in each table, one at a time. Following the completion of the tables, the teacher leads a discussion where students are challenged to find another number to go in the blank, which they cannot do, therefore showing (but not proving) uniqueness and existence.”

Solving Equations with Square Roots
Header July 2013 Network Team Institute Solving Equations with Square Roots TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X We ask students for the positive solution to the equations: At this point students don’t know how to find all of the solutions to the equation without using the square root symbol: (2 min) Read the bullets on the slide. “We know how to solve quadratics without using the square root symbol, but Grade 8 students do not. Since they must use the square root symbol, then they can only find the positive solution. It will not be until Algebra I that they learn how to find all of the solutions to equations like these.”

Solving Equations with Square Roots
Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Solving Equations with Square Roots Complete Exercises 1-9 in the handout. (6 min) Provide time for participants to work through the exercises. Discuss if necessary.

Lesson 4: Simplifying Square Roots
Grade 8 – Module 3 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Lesson 4: Simplifying Square Roots Students use factors of a number to simply a square root. Optional Lesson Will better prepare students for simplifying solutions to quadratic equations in Grade 9 Algebra. (1 min) Read the bullet points on the slide.

October, 2013 Common Core Institute Discussion TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Use what we know about square roots of perfect squares to simplify square roots of non-perfect squares. Show that the square root of a number can be expressed as a product of its factors. for positive numbers C and D, and positive integer n. If we can show (3 min) “As before, using concrete numbers along with the symbolic work will aide understanding.” Read through the bullets on the slide.

October, 2013 Common Core Institute Discussion TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Based on what we know about exponential notation: (2 min) “We raise both C and D to the nth power and simplify to show that they are the same number. This proof can be simplified by replacing all of the n’s with 2’s. The goal is for students to understand that the the square root of the factors of a radicand is equal to the product of the radicand.”

October, 2013 Common Core Institute Discussion TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Make sense of the abstract using concrete numbers: Which matches our expectation for the square root of 36. Now we apply this knowledge to non-perfect squares. (1 min) “Here we apply what we just learned to something familiar, the square root of 36. We show the factors of 36 in order to convince students that what we just proved really is true. Once convinced we can begin work with non-perfect squares.”

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Example 1 What do you do when students say 50 = 10 x 5? (3 min) “In this first example we ask students to find the square root of 50. There’s a good chance that when you ask for the factors of 50 they will not immediately say 2 x 5^2 and that’s ok. Whatever factors they give you can ask them to continue factoring until we have at least two of the same factor. That is when we know we can simplify the radical. Another option is to tell students that we always try to find the largest perfect square factor of a number. Either way, we want students to see that squared numbers are what allow us to simplify. Also, with respect to the notation, it may seem laborious at first but we must show students that 5 root 2 is actually 5 times the square root of 2. Remember, this is new notation for them. We must be explicit and clear. You may even need to have a discussion about why we put the 5 in front of the root 2. It’s by convention. Just like we do with variable expressions like 5x. In fact, it will be beneficial for students to know this because in Geometry they will learn to add radical expressions like 5 root root 2 which is (5+6) root 2. Highlight the structure. We treat these numbers just like any other numbers we’ve worked with in the past. Following this example is another simple example and an opportunity for students to practice this skill.”

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Example 3 We assume that students will not recognize 288 as 144 x 2. (3 min) “Next we move on to a slightly more complicated example. Again, students may not immediately factor 288 as shown here. Have them keep factoring until they see the perfect squares. We are reaching back to Module 1 with respect to the work with exponents. You can see here that again the work seems laborious but it is purposeful.” Point to the perfect square factors and how we show that when simplified we multiply the perfect squares that we were able to simplify in order to reach the simplified expression 12 root 2.

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Problem Set #6 and #7 Complete the problems in the handout. (3 min) “Try these problems in your handout.” Provide time for participants to work, then show the solutions. “In subsequent lessons two answers are given, one with a simplified square root and one without, just in case you decide to skip this lesson. Just remember that this is the first work that students do with square roots and it will really help out students’ understanding of solving quadratic equations in Algebra I.”

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Problem Set #6 and #7 (1 min) Show solution, discuss if necessary.

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X (1 min) Someone posted this on Facebook. They obviously need to be reminded of the definition of square.

Lesson 5: Solving Radical Equations
Grade 8 – Module 3 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Lesson 5: Solving Radical Equations Students transform equations until they are in the form of Students find the positive solutions for equations that require them to use square and cube roots. (2 min) Read the bullet points on the slide. “We titled this Solving Radical Equations because students have to use radicals to solve the equations.”

October, 2013 Common Core Institute Example 1 TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Complete Example 1 in your handout. (2 min) “In this example we want students to practice using the Distributive Property and the properties of equality to transform the equation into the form of x^3=27. Now is when students have to take the cube root of both sides of the equation. Remind them about the properties of equality: whatever you do to one side of the equal sign you have to do the same thing to other side. It’s in line with what we already know about solving equations, but now we have a new tool.”

October, 2013 Common Core Institute Grade 8 Expectation TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Students are shown that an equation could yield two solutions and even check that both solutions are correct. From that point on they are instructed to find just the positive solution to a problem. (1 min) “The expectation in grade 8 is that students find the positive solution to equations. However, we want to tell the whole story to students. That story includes both solutions. But recall, we have defined the symbol to mean positive numbers only. That is why we require students to give only the positive solution.”

October, 2013 Common Core Institute Example 2 TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X (1 min) “In this example we show students that the solution can be positive or negative 8.”

October, 2013 Common Core Institute Example 2: Check TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X (1 min) “Then we have students check to verify that both solutions are in fact correct. Make sure that they check one solution at a time. It’s happened in the past where students will use both 8 and -8 in one equation.”

Exercises 1-8 (really only 1-7)
Grade 8 – Module 2 Module Focus Session October, 2013 Common Core Institute Exercises 1-8 (really only 1-7) TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X “Find the positive value of x that makes each equation true, then verify your solution is correct.” As an option, we have included both positive and negative solutions where applicable. Mistake on Exercise 3, should only be positive: (2 min) “First, the header for this section of the lesson says “Exercises 1-8” but there are really only 7 exercises. The instructions for all problems is to find the positive value of x that makes the equation true, however, in case you wanted to push the kids a bit we have included both positive and negative solutions where applicable. But we did make a mistake on exercise 3. Because the context is length it makes sense to only consider the positive solution. These errors are currently being corrected and the next edition won’t have them.”

October, 2013 Common Core Institute Exercise 7: Challenge TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Complete Exercise 7 in your handout. We need to determine the value of x so that its square root, multiplied by 4 satisfies: (4 min) Instruct participants to complete exercise 7. “Exercise 7 is challenging. Notice what we are asking students to determine. Not the length of a leg of the triangle, but the number whose square root multiplied by 4 represents a length that with the other two, satisfies the Pythagorean Theorem.”

Topic B: Decimal Expansion of Numbers
Grade 8 – Module 4 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Topic B: Decimal Expansion of Numbers Rational numbers are defined as numbers whose decimal expansions eventually repeat. Irrational numbers are defined as numbers that are not rational. Focus of Topic B is writing decimal expansions of numbers. Students learn why the long division algorithm leads to the decimal expansion of a number. Students learn how to determine the decimal expansion of irrational numbers, including pi. (1 min) Read the bullet points on the slide, “These are the basic concepts in Topic B. We are going to discuss the topic a bit before we get into the lessons.”

October, 2013 Common Core Institute Partner Talk What’s the difference between expanded form of a number and decimal expansion of a number? (3 min) Allow 1 minute for partner/table talk, then have participants share their responses. Explanation of the difference is in the next slide.

Grade 8 – Module 4 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Topic B: Decimal Expansion of Numbers Expanded form: numbers written as a sum Decimal expansion: the value of the number written as a decimal All numbers have a decimal expansion that either terminates or is infinite. (1 min) Read the bullets on the slide. “We want to make this distinction clear because our focus in this topic is writing the decimal expansions of numbers and in some cases we begin with the expanded form of a number.”

October, 2013 Common Core Institute Table Talk How have you seen irrational numbers defined? A number that cannot be written in the form a/b where a and b are both integers. Nonrepeating, nonterminating decimals. Irrational numbers cannot be expressed as fractions. (5 min) Allow time for participants to discuss irrational numbers at their table. Then share their responses with the whole group. The definitions shown are taken from textbooks. “In the past, irrational numbers have been defined in this manner, which is ok, but causes confusion for students. For example, we say that pi is irrational, but frequently use 22/7 to represent pi. So, by definition (the first and third bullet points), pi must be rational. For this reason, we focus a great deal of work in Topic B on writing the decimal expansions of numbers, then use the decimal expansions to classify numbers as rational or irrational.”

Grade 8 – Module 4 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Topic B: Decimal Expansion of Numbers Rational numbers have decimal expansions that eventually repeat. Irrational numbers are those that cannot be expressed as a rational number. (1 min) Read the bullet points on the slide.

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Topic B: Every number has a decimal expansion that is finite or infinite. Investigate strategies that lead to decimal expansions: Place value/denominator Long division Rational approximation Rewrite repeating decimals as fractions (rational numbers) Use expansions to classify numbers as rational or irrational. (2 min) “This is a brief outline of the work in Topic B.” Read the points on the slide.

Lesson 6: Finite and Infinite Decimals
Grade 8 – Module 3 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Lesson 6: Finite and Infinite Decimals Students know that every number has a decimal expansion that is finite or infinite. Students know that when a fraction has a denominator that can be expressed as a product of 2’s and/or 5’s, then the decimal expansion of the number is finite. (1 min) Read the bullet points on the slide.

October, 2013 Common Core Institute Opening Exercises 1-5 TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Complete the exercises in your handout. Discuss: How would you classify the decimal expansions in Exercises 1 and 2? How would you classify the decimal expansions in Exercises 3 and 4? What do you notice about the denominators in Exercises 1 and 2 compared to the denominators in Exercises 3 and 4? (4 min) Instruct participants to complete the exercises in the handout. “Following the opening exercises, we ask questions that focus students’ attention on the decimal expansions and the denominators. The goal is for students to recognize that the denominator a fraction is what dictates the decimal expansion.”

October, 2013 Common Core Institute Example 1 TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Fractions whose denominators are a product of 2’s and/or 5’s have finite decimal expansions. Because 8 is a product of 2’s (2x2x2) we know that the fraction is equal to a finite decimal. (2 min) Read the bullets on the slide.

October, 2013 Common Core Institute Example 1 TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X How do we find k and n? n must equal 3. k must equal 625. (2 min) “We want to find an equivalent fraction whose denominator is a power of ten because we know that a fraction whose denominator is a power of 10 can easily be written as a decimal. No long division required.” Read through the points on the slide. “Notice that we reach back to Module 1 content with respect to the Laws of Exponents to help us with our work.”

October, 2013 Common Core Institute Example 1 TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Students use what they know about powers of 10 to write the decimal expansion of the fraction: (1 min) Show the solution. Discuss if necessary.

October, 2013 Common Core Institute Exercise 7 TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Complete exercise 7 in your handout. (3 min) Instruct participants to complete the exercise in their handout. Show the solution. Discuss if necessary.

February 2014 Network Team Institute Turn and Talk What difficulties would you anticipate with student understanding of the mathematics in this section? What scaffolds would be effective for addressing those difficulties? Possible scaffolds: Put a poster in the room reminding students of the Laws of Exponents prior to the lesson. Inserting a few simple problems where students have to determine the exponent, n, in problems like 2^6 x 5^n=(2x5)^6 and 2^n x 5^7=(2x5)^7.

October, 2013 Common Core Institute Example 3 TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Check the denominator: Determine the value of k and n so that: n must be 4 (3 min) “Now we use this strategy for writing the decimal expansion with a slightly more complicated problem, but the procedure is the same.” Read through the example.

October, 2013 Common Core Institute Exercise 12 TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Complete Exercise 12 in your handout. (2 min) Instruct participants to complete exercise 12 in their handout. Show the solution and discuss if necessary.

Lesson 7: Infinite Decimals
Grade 8 – Module 3 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Lesson 7: Infinite Decimals Students know the intuitive meaning of an infinite decimal. Students will be able to explain why the infinite decimal … = 1. (1 min) Read the bullet points on the slide.

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Example 1 Each decimal digit of a decimal expansion shows another division of a power of 10. Consider the finite decimal 0.253: (4 min) “Our goal is for students to develop an intuitive sense of what an infinite decimal is. To make this clear, we begin with a finite decimal. There are a finite number of steps that it takes to represent the decimal The blue lines between the number lines show a magnification of that particular interval. First, we identify the tenth where the number belongs. Then we magnify the interval between 0.2 and 0.3 and locate the hundredth interval where the number belongs. We repeat this process until we represent each decimal digit of the number We do something similar to this when we find the decimal expansions of irrational numbers but the work is more computational. That is why it is important that students literally see what we are doing here with rational numbers.”

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Example 2 Examine the infinite decimal … on the number line: When does it end? (2 min) “Now we show the steps that represent the infinite decimal … Writing the expanded form of the number will help students identify the number of steps necessary to show the decimal. After 5 steps we ask students, when does it end? Students should say that it will never end because the digit 3 in the decimal continues, infinitely.”

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Exercise 4 Do you think it is acceptable to write that = …? Discuss. Consider the hundredth (or thousandth) step in the number line diagram. The value that is added becomes increasing smaller, almost zero, as the diagram approaches one. We treat infinite decimals as finite decimals for computational purposes. (3 min) Read the bullet points on the slide. “Knowing that it is acceptable to write that …=1 is not part of any standard, but we want students to understand that we cannot completely represent an infinite decimal and that it is ok to approximate the value of the infinite decimal. We cannot compute with decimals in this form. We approximate the value of the number so that we can compute. We also want students to know that approximations are good enough for computations. That is why we want students to be clear that as the number of decimal digits we use increases, the smaller the value it is that we are adding to the number.”

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Exercises 8-10 Complete the exercises in your handout. (7 min) Instruct participants to complete the exercises in their handout. Show solutions (on this and the next slide) and discuss if necessary.

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Exercises 8-10

Lesson 8: The Long Division Algorithm
Grade 8 – Module 3 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Lesson 8: The Long Division Algorithm Students relate the long division algorithm to determining the decimal expansion of a number and division with remainder. Students know why digits of a decimal expansion repeat in terms of the algorithm. Students know that every rational number has a decimal expansion that repeats eventually. (1 min) Read the bullet points on the slide.

October, 2013 Common Core Institute Example 1 TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Recognize that division is just another form of multiplication. (2 min) “Students learned in grade 7 how to use the long division algorithm to write the decimal expansion of a number. The goal now is to deepen students’ understanding of the long division algorithm and explain why we can use it to get the decimal expansion. For those reasons, we have students look at division as another form of multiplication. A secondary goal of this work is to develop fluency in the manipulation of rational numbers. In this one example students are working with fractions in ways they likely haven’t before.”

October, 2013 Common Core Institute Exercise 3 TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X (1 min) “After the first example, students complete a set of exercises that requires them to find the decimal expansion of a fraction using the long division algorithm as well as the method we just saw in Example 1.”

October, 2013 Common Core Institute Exercise 3 TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Exercise 1: Exercise 2: Exercise 4: Note that the numerator does not change in order to draw attention to the denominator and the impact that number has on the decimal expansion. (3 min) “Of the four exercises, 2 have finite decimal expansions and 2 do not. We again focus on the denominator of the fraction to make sense of why the decimal expansions are finite or infinite.”

October, 2013 Common Core Institute Discussion TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X is equal to 35.5 What decimal digits can we include to the right of without changing the value of the number? … is equal to … How did you know when you could stop dividing? The steps of the long division algorithm kept repeating; 10 and then 100. (3 min) “The discussion highlights the fact that rational numbers have decimal expansions that eventually repeat. It is very likely that when students wrote the decimal expansion of 142/4 they just wrote We want to display clearly that the 5 could be followed by an infinite number of zeroes and the value remains unchanged. We are closing in on the definition of a rational number with this part of the discussion. We also want students to connect the fact that the repeating decimal is a result of the work they do using the algorithm. That is, when they notice the work repeating, then they know that the decimal digits will repeat.”

October, 2013 Common Core Institute Discussion TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Rational numbers are defined as those numbers that have a decimal expansion that repeats eventually. This definition aligns with students’ existing understanding that rational numbers can be expressed as a ratio of integers a/b. (2 min) Read the points on the slide. “The discussion that follows the exercises reveals the formal definition of a rational number which is in line with students’ current understanding.”

October, 2013 Common Core Institute Problem Set 6-8 TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Students apply definition of rational number to classify: (1 min) “In this set of problems we do not yet ask students to identify numbers as rational or irrational, simply to use the definition of rational numbers to state whether or not a number fits that definition.”

Lesson 9: Decimal Expansions of Fractions, Part 1
Grade 8 – Module 3 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Lesson 9: Decimal Expansions of Fractions, Part 1 Students use equivalent fractions, long division and the Distributive Property to write the decimal expansion of fractions. Students recognize that the remainder, usually truncated, adds little value to number. (1 min) Read the bullet points on the slide.

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Example 1 Why is it ok to put extra zeroes to the right of the decimal point when using the long division algorithm? Make sense of this using what is known about equivalent fractions. (1 min) Read the bullets on the slide. “We pose this question to students because it is something that we do frequently without really thinking about. We want students to not just know what to do, but to understand why it works.”

October, 2013 Common Core Institute Example 1 TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Multiplying by is similar to putting extra zeroes when using the algorithm. (3 min) “We begin with the fraction 5/8. We know by the denominator that it has a finite decimal expansion. We use what we know about equivalent fractions to change the fraction to /8. Kids will likely feel that doing the division of /8 is easier than 5/8. Again we are trying to develop students’ fluency with numbers, but more importantly explain why putting extra zeroes when we use the long division algorithm is ok. We also point out that the number of zeroes we include is unimportant. In this example we multiplied by 10^-5, but we could have used 10^-3 or 10^-10. Too many is always better than too few.”

October, 2013 Common Core Institute Example 3 TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Apply this strategy to fractions we know do not have finite decimal expansions. (1 min) “Now we try out this strategy with a number we know has an infinite decimal expansion. The goal here is to show students why it is ok to approximate the value of an infinite decimal. We do all of this work in order to focus on the “remainder”.”

October, 2013 Common Core Institute Example 3 TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X At this point, we have a good estimate: Will the remaining portion greatly effect the value of the number? (2 min) Read the points on the slide. “In investigating the value of the remainder we use The Basic Inequality. It then becomes clear that the portion of the value of the number that we are not including in our estimate is so small that it doesn’t really affect the value of the number. Again pointing out that approximations of infinite decimals are reasonable.” The value of the remaining portion is less than

October, 2013 Common Core Institute Exercise 3 TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Complete the Exercise in your handout. (3 min) Instruct participants to complete the exercise in their handout. Show solution and discuss if necessary.

Lesson 10: Converting Repeating Decimals to Fractions
Grade 8 – Module 3 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Lesson 10: Converting Repeating Decimals to Fractions Numbers with decimal expansions that repeat are written as fractions. Numbers that cannot be expressed by a rational number are irrational. (1 min) Read the bullet points on the slide.

October, 2013 Common Core Institute Example 1 TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X (3 min) “Now we want students to know that infinite decimals that repeat can be expressed as a fraction. To find that fraction we use skills related to solving equations that were learned in Module 4. After the fourth step we’d normally divide both sides by We ask students why that is not a good idea. Ideally they will respond that dividing by 100 still leaves the infinite decimal in the numerator. For that reason we have to do something else. That something else is rewriting … as 81 + x. Then we continue to solve as usual.”

October, 2013 Common Core Institute Exercise 1 TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Complete Exercise 1 in your handout. (4 min) Instruct participants to complete the exercise in their handout. Show the solution and discuss if necessary.

October, 2013 Common Core Institute Example 2 TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X (2 min) “After students practice this method we show a slightly more complicated example. In Example 2 there is just one digit of the decimal expansion that is repeating. That means that we will have to do something slightly different. We now treat … as a separate problem.” Last time we subtracted x from both sides. Will that help now? Treat the repeating decimal as a separate, mini- problem.

October, 2013 Common Core Institute Example 2 TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X (1 min) “Students now go back to what they just practiced to find the fraction equal to …” Now back to the original problem.

October, 2013 Common Core Institute Example 2 TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X (1 min) “Now we finish the problem as before. Notice how the fluency with rational numbers developed in the last few lessons will be utilized here.”

October, 2013 Common Core Institute Exercise 3 TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Complete Exercise 3 in your handout. (5 min) Instruct participants to complete the exercise. Show solution and discuss if necessary.

October, 2013 Common Core Institute Discussion TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Obviously numbers with decimal expansions that repeat are rational numbers. Do you think infinite decimals that do not repeat are rational as well? The method for writing the fraction that represents an infinite repeating decimal expansion will not work for decimal expansions that do not repeat. Infinite decimals that do not repeat are irrational numbers, that is, when a number is not equal to a rational number, it is irrational. (2 min) Read the points on the slide. “It is at the end of lesson 10 we define irrational numbers. We explore their decimal expansions in the next lesson.”

Lesson 11: The Decimal Expansion of Some Irrational Numbers
Grade 8 – Module 3 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Lesson 11: The Decimal Expansion of Some Irrational Numbers Students use rational approximation to determine the approximate decimal expansion of numbers. Students classify numbers as rational or irrational based on their decimal expansions. (1 min) Read the bullet points on the slide.

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Example 1 Between which two integers will the number lie? This is the first approximation. We formalize the estimation process that we began in Lesson 1 by using the Basic Inequality: To write the decimal expansion of the number we use a method called rational approximation: Using a sequence of rational numbers to get closer and closer to the given number in order to estimate the value of the number. (2 min) Read the points on the slide. “Now we come to writing the decimal expansion of irrational numbers, but students don’t know that they are irrational quite yet. Only after they write the decimal expansion can they state that a number is irrational.”

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Example 1 Between which two tenths will the number lie? Again, use what we learned about estimating in Lesson 1. The number should be closer to 2 than 1. Use the Basic Inequality: (2 min) Read the points on the slide.

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Example 1 Between which two hundredths will the number lie? This method is called rational approximation because with each new interval, we are approximating the value of the number by focusing on which two rational numbers it must be between. (2 min) Read the points on the slide.

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Example 2 Discussion Students approximate the value of using the method of rational approximation, then are shown: How is this infinite decimal different from others we have worked with? Can we rewrite it like we did with numbers in the last lesson? Since the number cannot be expressed as a rational number, we say it is irrational. (3 min) Read the points on the slide. “We want students to recognize that the decimal expansions of these numbers do not repeat. That is what makes these numbers different. We can’t express these numbers as fractions like we did in the previous lesson, therefore these kinds of numbers are irrational.”

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Problem Set 2, 6-8 Complete the problems in your handout. (6 min) Instruct participants to complete the exercises in their handout. Discuss if necessary.

Lesson 12: Decimal Expansions of Fractions, Part 2
Grade 8 – Module 3 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Lesson 12: Decimal Expansions of Fractions, Part 2 Students use rational approximation to determine the decimal expansion of fractions. Students relate the method of rational approximation to the long division algorithm. (1 min) Read the bullet points on the slide.

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Example 1 (1 min) “We use a method similar to rational approximation with numbers that we know are rational. We know that the decimal expansion of 35/11 begins with the whole number 3. Now we need to examine the interval in which 2/11 belongs.” In which interval of tenths will we find

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Example 1 In other words, between which two consecutive integers, m and m + 1, would the fraction lie when the intervals are tenths? (1 min) Read the bullets on the slide.

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Example 1 Scaffold if necessary. (1 min) Read the point on the slide.

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Example 1 Multiply through by 10: This implies that m = 1 and m + 1 = 2, therefore the tenths digit is 1. (2 min) Read through the points on the slide.

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Example 1 Two things we know: What is the difference between these two numbers? In which interval of hundredths is (2 min) “We use what we know in order to determine the next decimal digit of the number.”

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Example 1 Multiply through by 100: (1 min) “We repeat the process, but now for the hundredths place.”

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Example 1 Two things we know: What is the difference between these two numbers? In which interval of thousandths is (1 min) “Again using what we know to determine the next decimal digit.”

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Example 1 Ideally, students will notice that and that the reappearance of the fraction means that decimal digits will begin to repeat. (1 min) Read the bullet points on the slide.

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Exercise 2 Complete Exercise 2 in your handout. (4 min) Instruct participants to complete the exercise in their handout. Show the answer and discuss if necessary.

October, 2013 Common Core Institute Fluency Exercise TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X (10 min) If time, have the participants actually do the fluency activity. Give them less than a minute per problem. Then discuss at the end how each pair of exercises was related. The first of each pair is a calculation needed to determine the volume of the figure in the second problem of each pair.

October, 2013 Common Core Institute Fluency Exercise TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X

Lesson 13: Comparison of Irrational Numbers
Grade 8 – Module 3 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Lesson 13: Comparison of Irrational Numbers Students use rational approximations of irrational numbers to compare the size of numbers. Students place irrational numbers in their approximate location on a number line. (1 min) Read the bullet points on the slide.

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Exercises 1-11 Complete Exercises 1-11 in your handout. Which would you expect all students to complete? Which would students who are on grade level be able to complete? Which would students be challenged by? (12 min) Instruct participants to complete the exercises in their handout. When they have finished, tell them to discuss at their tables the answers to the questions on the slide.

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Discussion How do we know if a number is rational or irrational? Is the number rational or irrational? Explain. Is the number rational or irrational? Explain. Which strategy did you use to write the decimal expansion of a fraction? Which did you use to write the decimal expansion of square and cube roots? (1 min) “Following the work of the exercises is a discussion that debriefs the activity.”

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Lesson 14: The Decimal of Students use area to calculate the decimal expansion of Students estimate the value of expressions such as (1 min) Read the bullet points on the slide. “Now we come to the last lesson about decimal expansions.”

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Discussion The number is defined as the ratio of the circumference to the diameter of a circle. The number is also the area of a unit circle. A unit circle is a circle with a radius of 1 unit. We will use the latter definition to determine the decimal expansion of . Two options: Have students draw a quarter circle on graph paper. Use copy ready 10 x 10 and 20 x 20 grids. (2 min) Read the first two points on the slide. “We do not expect students to memorize and use the latter definition except for during this lesson.” Read the remaining points on the slide.

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Discussion All of the squares on the graph/grid paper are congruent and therefore equal in area. Trace the border of squares contained completely within the circle. Trace the border of squares that are outside, but near the arc of the circle. (2 min) Read the instructions on the slide. “We will call the region within the circle r_2 and the region outside of the circle s_2. Then the actual area of the circle must fall within those two values.”

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Discussion (2 min) “Now we count the number of squares for each region and use the numbers to approximate the decimal expansion of pi.” Then ask the question. Participants should respond that we can improve the accuracy of the estimation by including the partial squares. Have them How can we improve the accuracy of our estimation?

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Discussion (12 min) “You can see that by including the partial squares we are getting closer to the real value of pi.” Ask the question. Participants should respond that if we had smaller squares to work with we could get an even better approximation. Instruct participants to try this using the 20 x 20 grid in their handout. How can we improve the accuracy of our estimation?

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Exercise 5 (1 min) “One of the other objectives in this lesson is to estimate the value of irrational expressions. Notice that the same strategy for estimating pi is used to estimate the value of this expression.”

Topic C: The Pythagorean Theorem
Grade 8 – Module 4 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Topic C: The Pythagorean Theorem Students learn another proof of the Pythagorean Theorem, using area. Students learn a proof of the converse of the Pythagorean Theorem. Entire lesson dedicated to finding distance on the coordinate plane. Students apply the Pythagorean Theorem to finding area and perimeter of triangles, determining length of diagonal of a rectangle as it relates to TV sizes, etc. (2 min) Read the bullet points on the slide, “These are the basic concepts in Topic C. Next we will look at the lessons of the topic.”

Lesson 15: The Pythagorean Theorem, Revisited
Grade 8 – Module 3 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Lesson 15: The Pythagorean Theorem, Revisited Students review the proof of the Pythagorean Theorem using Similarity, then use parts of that proof to make sense of the Pythagorean Theorem in terms of area. Students practice explaining a proof of the Pythagorean Theorem. (1 min) Read the bullet points on the slide.

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Discussion Review proof of Pythagorean Theorem using similarity. The triangle shown is a right triangle. If we drop a perpendicular from vertex C to the opposite side, we generate three similar triangles. (2 min) Read the points on the slide.

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Discussion Use a notecard to help students see the three triangles. (4 min) Have participants label and cut the notecard to show the three similar triangles.

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Discussion Because the three triangles are similar, they have corresponding sides that are equal in ratio. (4 min) “How do we know these triangles are similar?” Participants should state that the triangles are similar by the AA criterion and the fact that similarity is transitive. Then click through the animations.

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Discussion Adding the two equations and together, we have (2 min) Read through the points on the slide. “This is a review of the proof from Module 3. We will use parts of this proof to learn another one.”

October, 2013 Common Core Institute Discussion TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Take the same right triangle and construct squares off of each side. (2 min) Read the bullet point, then ask the question. Participants should say that geometrically, the sum of the area of the squares of a and b should be equal to the area of c squared.

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Video Demonstration New link that has been added to the lesson: (1 min) “This link was found after the materials had been finalized. It has been added for the second edition.” Click link to show video.

October, 2013 Common Core Institute Discussion TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Using the facts from the similarity proof, we can discuss area. (9 min) Click through the animations. If time, show the youtube video (about 6 minutes). If you do not have time, let the participants know it is available and included in the teacher materials.

Lesson 16: The Converse of the Pythagorean Theorem
Grade 8 – Module 3 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Lesson 16: The Converse of the Pythagorean Theorem Students learn another proof of the converse of the Pythagorean Theorem. Students apply the theorem and its converse to solve problems. (1 min) Read the bullet points on the slide.

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Discussion Assume we are given a triangle ABC so that sides a, b, c satisfy the Pythagorean Theorem. (1 min) Read through the information on the slide.

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Discussion (2 min) Click through the animations on the slide. By substitution: and then and because congruence is degree preserving.

Lesson 17: Distance on the Coordinate Plane
Grade 8 – Module 3 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Lesson 17: Distance on the Coordinate Plane Students determine the distance between two points on the coordinate plane using the Pythagorean Theorem. (1 min) Read the bullet points on the slide.

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Example 2 Given two points A, B on the coordinate plane, determine the distance between them. First estimate, then try to find a more precise answer. (1 min) Read the bullet. Then ask participants what they think students will give as their estimate.

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Example 2 (2 min) “We tell students to connect A to B, then draw the horizontal and vertical lines (as they did in Module 4) to generate a right triangle. Now we can use the Pythagorean Theorem. We count to find the length of the legs and use the theorem to determine the distance between the two points.”

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Example 3 (3 min) “Now we present a more challenging problem. How can we determine if these points form a right triangle?” Have participants share their thoughts. “What we need to do is find the lengths between each pair of points using the Pythagorean Theorem, then use those lengths with the converse of the Pythagorean Theorem to see if it’s a right triangle.”

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Example 3 (1 min) “As before, we draw horizontal and vertical lines to produce right triangles.”

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Example 3 (1 min) “We repeat this process to find the length of BC.”

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Example 3 (1 min) “Now that we know all three lengths we can use the converse of the Pythagorean Theorem and conclude that the three points form a right triangle.”

Lesson 18: Applications of the Pythagorean Theorem
Grade 8 – Module 3 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Lesson 18: Applications of the Pythagorean Theorem Students apply the Pythagorean Theorem to real world and mathematical problems in two dimensions. Review Exercises 1-5 and Problem Set 2 in your handout. (8 min) Read the bullet point on the slide. Instruct participants to review the exercises in their handout. They can work them if they want, but they should at least read through them. Discuss if necessary.

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Fluency Exercise Another white board exchange. (1 min) “Included in this lesson is another white board exchange similar to the one we looked at earlier.”

Topic D: Applications of Radicals and Roots
Grade 8 – Module 4 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Topic D: Applications of Radicals and Roots Students use the Pythagorean Theorem to determine the slant height or radius of a cone. Students use the Pythagorean Theorem to determine the radius of a sphere, then volume. Students learn how to find the volume of a truncated cone (and truncated pyramid) using what they know about similar triangles. Students find the volume of composite solids. Optional lessons about average rate of change and nonlinear motion. (2 min) Read the bullet points on the slide, “These are the basic concepts in Topic D. Next we will look at the lessons of the topic.”

Lesson 19: Cones and Spheres
Grade 8 – Module 3 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Lesson 19: Cones and Spheres Students the Pythagorean Theorem to determine an unknown dimension of a cone or a sphere. Students know that a pyramid is a special type of cone with triangular faces and a rectangular base. Students know how to use the lateral length (slant height) of a cone and the length of a chord of a sphere to solve volume problems. (1 min) Read the bullet points on the slide.

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Example 1 Students are asked to state as many facts as they know about a cone. Teacher identifies the lateral length (slant height). (2 min) “Students are shown the cone on the left and asked to state as many things as they can about the cone. Then the teacher identifies the lateral length as one part of the cone that has not yet been named, so we give it one. Next, students use the Pythagorean Theorem to determine the lateral length, height, or radius of the cone.”

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Discussion Let O be the center of a circle. Let P, Q be two points on the circle. The segment PQ is called a chord of the circle. What do you notice about the lengths OP and OQ? When is a right angle, then the Pythagorean Theorem can be used to find the length of the radius. Teacher identifies the lateral length (slant height). (1 min) “Now we define a chord of a circle and show students how they can use the Pythagorean Theorem to find the length of the chord or the radius of the sphere.”

Lesson 20: Truncated Cones
Grade 8 – Module 3 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Lesson 20: Truncated Cones Students know that truncated cones and pyramids are solids obtained by removing the top portion above a plane parallel to the base. Students find the volume of truncated cones. (1 min) Read the bullet points on the slide.

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Discussion Explanation of truncated cone: Are the triangles in the rightmost figure similar? (3 min) “We explain that a truncated cone is just a cone where the top portion has been removed. Then we ask students if the triangles shown in the rightmost figure are similar. Are they?” Let participants respond. They are similar by the AA criterion. Each triangle has a right angle and they share the angle at the top of the cone. We also know that the bases of each cone are parallel, then the angles along the lateral length are equal, corresponding angles.”

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Example 1 Find the volume of the truncated cone by drawing in the portion of the cone that has been removed. (1 min) Read the bullet on the slide.

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Example 1 Determine the height of top portion of the cone, x. (1 min) “Because we know the triangles are similar, we know that their corresponding sides are equal in ratio (learned in Module 3). Then students solve to find x (learned in Module 4).

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Example 1 The volume of the truncated cone is equal to the volume of the entire cone, with the top portions’ volume removed. (3 min) Read through the information on the slide.

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Exercise 2 Complete Exercise 2 in your handout. (4 min) Instruct participants to complete the exercise in their handout. Show solution and discuss if necessary.

Lesson 21: Volume of Composite Solids
Grade 8 – Module 3 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Lesson 21: Volume of Composite Solids Students compute the volume of a figure composed of combinations of cylinders, cones, and spheres. Review Exercises 1-4 in your handout. (8 min) Read the bullet points on the slide. Provide time for the participants to review/complete the exercises in the handout. Discuss if necessary.

Lesson 22: Average Rate of Change
Grade 8 – Module 3 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Lesson 22: Average Rate of Change Students know how to compute the average rate of change in the height of water level when water is poured into a conical container at a constant rate. (1 min) Read the bullet points on the slide.

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Opening Discussion Convince students that the water level does not rise at a constant rate when being filled at a constant rate. Youtube video link included. Ask students to explain in their own words what they observe. (1 min) Read the points on the slide. Students should observe that the narrow portion of the cone fills faster than the wider part.

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Discussion Calculate the volume of the cone: If the water flows into the container at a constant rate of 6 ft3 per minute, how long will it take to fill? (2 min) “To compute the average rate of change of the height of the water level we first determine the total volume of the cone. Then use what we know about the rate at which the cone is being filled to determine how long it would take to fill the cone.”

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Discussion How many minutes will it take to reach a water level of 1 ft? Using what we know about truncated cones: and it takes minutes to fill the cone to a height of 1 ft. (1 min) “To investigate the rate of change, we compute the volume of the water level when it is 1 ft high, then compute how long it would take to fill that volume.”

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Discussion Students repeat this process for various water levels: (1 min) Read the point on the slide.

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Discussion Students inspect the rate of change, graph the data, and conclude that the water level of the cone does not rise at a constant rate. (1 min) Read the point on the slide.

Lesson 23: Nonlinear Motion
Grade 8 – Module 3 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Lesson 23: Nonlinear Motion Using square roots, students determine the position of the bottom of a ladder as its top slides down a wall at a constant rate. Optional Lesson (1 min) Read the bullet points on the slide.

Mathematical Modeling Exercise
Grade 8 – Module 2 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Mathematical Modeling Exercise (2 min) Read the prompt for the problem. “Before we go any further we want to make sure students know what all of the symbols on the diagram mean.” Click to the next slide.

Mathematical Modeling Exercise
Grade 8 – Module 2 Module Focus Session October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Mathematical Modeling Exercise (3 min) “Once students are clear what each symbol means we can move ahead in trying to determine whether or not the motion of the bottom of the ladder is linear.”

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Why is the distance from A to A’ equal to v? We are given that the ladder is sliding at a constant rate of v ft per second. We are also given that it takes 1 second for the ladder to go from L1 to L2. By definition of constant rate if d is the distance the ladder slid in one second: In that same 1 second interval, the bottom of the ladder moves from B to B’, which is a distance of h ft. Does the bottom of the ladder move at a constant rate away from point O? (2 min) “We need to make sense of why the distance from A to A’ is equal to v. We use what we know about average speed to show that the distance between A and A’ is equal to v. Then we investigate the movement of the ladder.”

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X It is easier to answer this question by investigating the last second of the sliding ladder. If the top of the ladder was v feet from the floor, it would reach O in one second. Then the foot of the ladder would be at the point where , the length of the ladder. (1 min) Read through the points on the slide.

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X But we know that in one second, the bottom of the ladder moves h feet, from D to D’ as shown. This contradiction means that the bottom of the ladder is not moving at a constant rate. (1 min) Read the points on the slide.

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X How do we make sense of this for the kids? (2 min) “We rely on students knowledge of the Pythagorean Theorem to make sense of this. In each right triangle we can relate the hypotenuse to the ladder, the height of the triangle to the height that the ladder is as it slides down the wall. Then the base of the triangle represents the movement of the bottom of the ladder. By comparing the difference in those lengths we can clearly see that as the ladder slides down the wall at a constant rate, the movement at the bottom of the ladder is not constant.”

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Now that we have an idea that the movement at the bottom of the ladder is nonlinear, how can we describe it? Let y be the rule of the function that describes the distance of the bottom of the ladder from O over time t. We want to determine the length of BO, which by definition is the rule for the function y. (1 min) Read the points on the slide.

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Find |BO| using the Pythagorean Theorem. (2 min) Ask participants to share at their tables the 3 lengths of the right triangle. Then show how to solve for y.

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Simplify our equation for y using the Distributive Property: (1 min) “We use the Distributive Property with (L-vt)^2. Then we simplify the expression and use the Distributive Property again.”

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Obviously our function to describe the distance is nonlinear. Investigate the movement using concrete numbers. Examine the rate of change. Further proof that the motion is nonlinear. (10 min) Read the first two points. “As students did in Module 5 with functions, they inspect the rate of change to determine if it is linear or nonlinear. Thus providing further evidence that the motion is nonlinear.”

End-of-Module Assessment
Grade 8 – Module 2 Module Focus Session October, 2013 Common Core Institute End-of-Module Assessment TIME ALLOTTED FOR THIS SLIDE: 0 minutes MATERIALS NEEDED: X Take about 20 minutes of quiet time to complete the assessment. Once finished working, read through the rubric. Discuss at your table: Language/vocab issues Challenging problems Lessons that are critical for success on assessment Strategies for scaffolding content and/or ways to remediate (40 minutes total) Ask participants to observe 20 minutes of quiet time while everyone works on the assessment. Then have a discussion at their tables.

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 min) 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.

October, 2013 Common Core Institute TIME ALLOTTED FOR THIS SLIDE: 2 minutes MATERIALS NEEDED: X Key Points The learning related to rational and irrational numbers is motivated by the need to find precise lengths using the Pythagorean Theorem. Much of the work in Module 7 connects learning of previous modules: Writing and Solving Equations (Module 4) Working with Integer Exponents (Module 1) Congruence, related to area (Module 2) Similarity, specifically Similar Triangles (Module 3) Computation of Volume of Cylinders, Cones, and Spheres (Module 5) Constant Rate, Rate of Change (Modules 4 and 5) (2 min) Let’s review some key points of this session. Read through the points on the slide.

Grade 8 – Module 3 Module Focus Session

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Grade 8 – Module 3 Module Focus Session

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