mc-14193-39844713.us-east-1.elb.amazonaws.commc-14193-39844713.us-east-1.elb.amazonaws.com/...ma… · Web viewModule Focus: Grade 8 – Module 7. Sequence of Sessions. Overarching

Module Focus: Grade 8 – Module 7 Sequence of Sessions

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

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

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

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

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

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

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

Related Learning Experiences● This session is part of a sequence of Module Focus sessions examining the Grade 8 curriculum, A Story of Ratios.

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)

• Scaffolding Focused: Amplify Language

• Scaffolding Focused: Move from Concrete to Representation to Abstract• Scaffolding Focused: Give Specific Guidelines for Speaking, Reading, Writing, or Listening

Session Outcomes

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

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

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

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

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

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

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

Session Overview

Section Time Overview Prepared Resources Facilitator Preparation

Introduction 24 min Introduces Grade 8 Module 7• Grade 8 Module 7 PPT• Facilitator Guide

Review Grade 8 Module 7

Topic A: Square and Cube Roots

91 min

Explores the Pythagorean Theorem and the need to get a precise length of a side of a right triangle and solving equations using roots

• Grade 8 Module 7 PPT• Facilitator Guide

Review Topic A

Topic B: Decimal Expansion of Numbers

166 minExplores decimal expansion of numbers


Review Topic B

Topic C: The Pythagorean Theorem

50 min

Explores applying the Pythagorean Theorem to find area and perimeter, learning another proof of the Pythagorean Theorem, and learning a proof of the converse of the Pythagorean


Review Topic C

Topic D: Applications of Radicals and Roots

105 min

Explores using the Pythagorean Theorem to determine measurements and volume of objects


Review Topic D

Session Roadmap

Section: Introduction Time: 24 minutes

In this section, you will be introduced to Grade 8 Module 7. Materials used include:• Grade 8 Module 7 PPT• Grade 8 Module 7 Facilitator Guide• Grade 8 Module 7 Module Overview

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

1 min 1. Welcome! In this module focus session, we will examine Grade 8 – Module 7.

1 min 2. 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.

1 min 3. 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.

1 min 4. 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

1 min 5. 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.

1 min 6. 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...”

2 min 7. 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.

8. 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.

9. 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?

In this task, teachers can...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.

1 min 10. 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.

3 min 11. 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.

12 min 12. “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.”

13.

Section: Topic A: Square and Cube Roots Time: 91 minutes

In this section, you will focus on the Pythagorean Theorem and need to get a precise length of a side of a right triangle and solving

Materials used include:• Grade 8 Module 7 PPT

equations using roots. • Grade 8 Module 7 Facilitator Guide• Grade 8 Module 7 Topic Opener A


3 min 14. 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.”

2 min 15. “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.”

1 min 16. Read the bullet points on the slide.

1 min 17. 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.

2 min 18. 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.”

3 min 19. 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.”

3 min 20. “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.

2 min 21. Read the first bullet. Then instruct participants to work in their handout to solve the problem.

2 min 22. 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.”


2 min 24. 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.”

2 min 25. 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.”

2 min 26. 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.”

1 min 27. “We also plan to add this definition to the overview.”

3 min 28. Read through the points on the slide.

2 min 29. “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.”

3 min 30. “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.”

2 min 31. “Continue placing numbers on the number line. Did you use a different strategy this time?”

3 min 32. “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.”

1 min 33. Read the bullet points on the slide. “We will focus our time on Option 1.”

2 min 34. 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.”

2 min 35. 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.”

3 min 36. 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.”

2 min 37. 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.”

38. 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.

2 min 39. “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.”

2 min 40. 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.”

6 min 41. Provide time for participants to work through the exercises. Discuss if necessary.


3 min 43. “As before, using concrete numbers along with the symbolic work will aide understanding.” Read through the bullets on the slide.

2 min 44. “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.”

1 min 45. “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.”

3 min 46. “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 2 + 6 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.”

3 min 47. “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.

3 min 48. “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.”

1 min 49. Show solution, discuss if necessary.

1 min 50. Someone posted this on Facebook. They obviously need to be reminded of the definition of square.

2 min 51. Read the bullet points on the slide. “We titled this Solving Radical Equations because students have to use radicals to solve the equations.”

2 min 52. “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.”

1 min 53. “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.”

1 min 54. “In this example we show students that the solution can be positive or negative 8.”

1 min 55. “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.”

2 min 56. “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.”

4 min 57. 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.”

Section: Topic B: Decimal Expansion of Numbers Time: 166 minutes

In this section, you will focus on decimal expansion of numbers. Materials used include:• Grade 8 Module 7 PPT• Grade 8 Module 7 Facilitator Guide• Grade 8 Module 7 Topic Opener B


1 min 58. 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.”

3 min 59. Allow 1 minute for partner/table talk, then have participants share their responses. Explanation of the difference is in the next slide.

1 min 60. 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.”

5 min 61. 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.”


2 min 63. “This is a brief outline of the work in Topic B.” Read the points on the slide.


4 min 65. 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.”

2 min 66. Read the bullets on the slide.

2 min 67. “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.”

1 min 68. Show the solution. Discuss if necessary.

3 min 69. Instruct participants to complete the exercise in their handout. Show the solution. Discuss if necessary.

70. 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.

3 min 71. “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.

2 min 72. Instruct participants to complete exercise 12 in their handout. Show the solution and discuss if necessary.


4 min 74. “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 0.253. 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 0.253. 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.”

2 min 75. “Now we show the steps that represent the infinite decimal 0.8333… 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.”

3 min 76. Read the bullet points on the slide. “Knowing that it is acceptable to write that 0.99999999…=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.”

7 min 77. Instruct participants to complete the exercises in their handout. Show solutions (on this and the next slide) and discuss if necessary.

78.


2 min 80. “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.”

1 min 81. “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.”

3 min 82. “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.”

3 min 83. “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 35.5. 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.”

2 min 84. 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.”

1 min 85. “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.”


1 min 87. 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.”

3 min 88. “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 500000/8. Kids will likely feel that doing the division of 500000/8 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.”

1 min 89. “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”.”

2 min 90. 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.”

3 min 91. Instruct participants to complete the exercise in their handout. Show solution and discuss if necessary.


3 min 93. “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 100. 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 81.81818181… as 81 + x. Then we continue to solve as usual.”

4 min 94. Instruct participants to complete the exercise in their handout. Show the solution and discuss if necessary.

2 min 95. “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 0.88888… as a separate problem.”

1 min 96. “Students now go back to what they just practiced to find the fraction equal to 0.8888…”

1 min 97. “Now we finish the problem as before. Notice how the fluency with rational numbers developed in the last few lessons will be utilized here.”

5 min 98. Instruct participants to complete the exercise. Show solution and discuss if necessary.

2 min 99. 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.”


2 min 101. 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.”

2 min 102. Read the points on the slide.


3 min 104. 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.”

6 min 105. Instruct participants to complete the exercises in their handout. Discuss if necessary.


1 min 107. “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.”

1 min 108. Read the bullets on the slide.

1 min 109. Read the point on the slide.


2 min 111. “We use what we know in order to determine the next decimal digit of the number.”

1 min 112. “We repeat the process, but now for the hundredths place.”

1 min 113. “Again using what we know to determine the next decimal digit.”


4 min 115. Instruct participants to complete the exercise in their handout. Show the answer and discuss if necessary.

10 min 116. 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.

117.

118.

119.

120.


12 min 122. 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.

1 min 123. “Following the work of the exercises is a discussion that debriefs the activity.”

1 min 124. Read the bullet points on the slide. “Now we come to the last lesson about decimal expansions.”

2 min 125. 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.

2 min 126. 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.”

2 min 127. “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

12 min 128. “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.

1 min 129. “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.”

Section: Topic C: The Pythagorean Theorem Time: 50 minutes

In this section, you will focus on applying the Pythagorean Theorem to find area and perimeter, learn another proof of the Pythagorean Theorem, and learn a proof of the converse of the Pythagorean

Materials used include:• Grade 8 Module 7 PPT• Grade 8 Module 7 Facilitator Guide

Theorem. • Grade 8 Module 7 Topic Opener C


2 min 130. 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.”



4 min 133. Have participants label and cut the notecard to show the three similar triangles.

4 min 134. “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.

2 min 135. 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.”

2 min 136. 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.

1 min 137. “This link was found after the materials had been finalized. It has been added for the second edition.” Click link to show video.

9 min 138. 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.


1 min 140. Read through the information on the slide.

2 min 141. Click through the animations on the slide.


1 min 143. Read the bullet. Then ask participants what they think students will give as their estimate.

2 min 144. “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.”

3 min 145. “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.”

1 min 146. “As before, we draw horizontal and vertical lines to produce right triangles.”

1 min 147. “We repeat this process to find the length of BC.”

1 min 148. “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.”

8 min 149. 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.

1 min 150. “Included in this lesson is another white board exchange similar to the one we looked at earlier.”

Section: Topic D: Applications of Radicals and Roots Time: 105 minutes

In this section, you will focus on using the Pythagorean Theorem to determine measurements and volume of objects.

Materials used include:• Grade 8 Module 7 PPT• Grade 8 Module 7 Facilitator Guide

• Grade 8 Module 7 Topic Opener D• Grade 8 Module 7 End-of-Module Assessment


2 min 151. 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.”


2 min 153. “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.”

1 min 154. “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.”


3 min 156. “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.”

1 min 157. Read the bullet on the slide.

1 min 158. “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).

3 min 159. Read through the information on the slide.

4 min 160. Instruct participants to complete the exercise in their handout. Show solution and discuss if necessary.

8 min 161. Read the bullet points on the slide. Provide time for the participants to review/complete the exercises in the handout. Discuss if necessary.


1 min 163. Read the points on the slide. Students should observe that the narrow portion of the cone fills faster than the wider part.

2 min 164. “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.”

1 min 165. “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.”




2 min 169. 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.

3 min 170. “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.”

2 min 171. “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.”



2 min 174. “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.”


2 min 176. Ask participants to share at their tables the 3 lengths of the right triangle. Then show how to solve for y.

1 min 177. “We use the Distributive Property with (L-vt)^2. Then we simplify the expression and use the Distributive Property again.”

10 min 178. 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.”

40 min 179. Ask participants to observe 20 minutes of quiet time while everyone works on the assessment. Then have a discussion at their tables.

3 min 180. 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.

2 min 181. Let’s review some key points of this session. Read through the points on the slide.

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

Video Reflect on a prompt Active learning Turn and talk

Turnkey Materials Provided

● Grade 8 Module 7 PPT● Grade 8 Module 7 Facilitator Guide● Grade 8 Module 7 Module Overview● Grade 8 Module 7 Topic Opener A● Grade 8 Module 7 Topic Opener B● Grade 8 Module 7 Topic Opener C● Grade 8 Module 7 Topic Opener D● Grade 8 Module 7 End-of-Module Assessment

Additional Suggested Resources

● How to Implement A Story of Ratios● A Story of Ratios Year Long Curriculum Overview● A Story of Ratios CCLS Checklist

mc-14193-39844713.us-east-1.elb.amazonaws.commc-14193-39844713.us-east-1.elb.amazonaws.com/...ma… · Web viewModule Focus: Grade 8 – Module 7. Sequence of Sessions. Overarching

Documents