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## A Story of Ratios

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**A Story of Ratios**Grade 8 – Module 7**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.**A Story of RatiosScaffolding Mathematics Instruction**Key Points • Amplify Language • Move from Concrete to Representation to Abstract • Give Specific Guidelines for Speaking, Reading, Writing, or Listening**Amplify Language**• Give clear mathematical definitions • Explain multiple meanings • Maintain consistency and point out interchangeable terminology**Move from Concrete to Representation to Abstract**• Use familiar contexts • Visually organize thinking • Provide multiple representations**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.**Key Points**• Amplify Language • Move from Concrete to Representation to Abstract • Give Guidelines for Speaking, Reading, Writing, or Listening**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?**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**Agenda**Introduction to the Module Concept Development Module Review**Module 7 Overview**• Table of Contents • Overview • Focus Standards • Foundational Standards • Focus Standards for Mathematical Practice • Terminology • Tools • Assessment Summary**Agenda**Introduction to the Module Concept Development Module Review**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.**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.**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.**Example 1**Write an equation that will allow you to determine the length of the unknown side of the right triangle. The length is 12 cm.**Example 2**• 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?**Example 2**• 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.**Example 4**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?**Example 4**• 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?**Example 4**• 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)**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.**Discussion**• 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.**Discussion**• 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.**Definitions: Revised**• 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?**Definitions: New**• 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?**Definitions:**• 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.**Discussion:**• Take a number line from 0 to 4.**Discussion:**• What strategy did you use to place the numbers? • The goal is to estimate, not be precise.**Discussion:**• 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**• 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.**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.**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:**Option 1: The “Basic Inequality”**• 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?**Option 1:**• 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.**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?**Option 2:**• Students are shown two tables, fill in the blanks, and explain their reasoning.**Solving Equations with Square Roots**• 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:**Solving Equations with Square Roots**• Complete Exercises 1-9 in the handout.**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.**Discussion**• 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**Discussion**• Based on what we know about exponential notation:**Discussion**• 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.**Example 1**• What do you do when students say 50 = 10 x 5?**Example 3**We assume that students will not recognize 288 as 144 x 2.**Problem Set #6 and #7**Complete the problems in the handout.