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A Story of Ratios

A Story of Ratios. Grade 7 – Module 3 Session 2. Session Objectives. Examine the development of mathematical understanding across the module using a focus on Concept Development within the lessons.

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A Story of Ratios

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  1. A Story of Ratios Grade 7 – Module 3 Session 2

  2. Session Objectives • Examine the development of mathematical understanding across the module using a focus on Concept Development within the lessons. • Introduce mathematical models and instructional strategies to support implementation of A Story of Ratios.

  3. Agenda • Introduction to the Module • Concept Development • Module Review • Introduction to the Module • Concept Development • Module Review Introduction to the Module Concept Development Module Review

  4. Progressions for the Common Core State Standards in Mathematics Expressions and Equations Grades 6-8 • What to expect in Grade 7: • More complex expressions; containing signed numbers • Various forms of equivalent expressions to model contextual relationships • Relating numeric solution process to algebraic steps • Using mental math and estimation to check answers to complex problems • Using variables and equation models to represent real-world problems • Extending the process of solving equations to inequalities, while realizing that multiplying or dividing both sides of an inequality by a negative number reverses the comparison order.

  5. Curriculum Overview of A Story of Ratios

  6. Grade 7 Module 3 Overview • Lessons and topics • Narrative • Focus Standards • Builds upon Foundations • Mathematical Practices • Terminology (old and new) • Tools and Representations

  7. Grade 7 Module 3 Overview • Lessons and topics • Narrative • Focus Standards • Builds upon Foundations • Mathematical Practices • Terminology (old and new) • Tools and Representations

  8. Topic AUse Properties of Operations to Generate Equivalent Expressions • Take a moment to read through: • Module Overview paragraphs 2-3 • Topic A Opener • What do you expect to see conceptually as Topic A unfolds?

  9. Important Terminology Equivalent Expressions An Expression in Expanded Form An Expression in Standard Form An Expression in Factored Form

  10. Equivalent Expressions Lesson 1 / Example 1(a) Example 1 (a): Rewrite and by combining like terms. Write the original expressions and expand each term using addition. What are the new expressions equivalent to?

  11. Equivalent Expressions Lesson 1 / Example 1 (a) • Example 1 (a): (cont’d) • Because both terms have the common factor of , we can use the distributive property to create an equivalent expression. • Expressions and Equations Progression Document (p. 6)

  12. Equivalent Expressions • Example 1 (b): Find the sum of and . • Any order, any grouping property • Combined like terms (distributive property) Lesson 1 / Example 1 (b) Example 1 (b): Find the sum of and . Associative property of addition Commutative property of addition Associative property of addition Combined like terms (distributive property) Expressions and Equations Progression Document (p. 5)

  13. Equivalent Expressions Can the any order, any grouping property be used with other operations? Subtraction? No. However a difference can be rewritten as a sum using inverse relationships: Multiplication? Yes! Multiplication is both associative and commutative. Division? No. However a quotient can be rewritten as a product using inverse relationships:

  14. Equivalent Expressions Lesson 1 / Example 4 (f) Example 4 (f): Alexander says that is equivalent to because of any order, any grouping. Is he correct? Why or why not? Sample response: If I let and let , then:

  15. Writing Products as Sums Lesson 3 / Example 1 3(3 + 2) = 9 + 6 Example 1: Draw a tape diagram representing the sum using square units. Redraw the diagram without squares, as two rectangles that are 1 unit high. Draw a rectangular array representing .

  16. Writing Products as Sums Lesson 3 / Example 1 3(x + 2) = 3x + 6 (continued) Draw a tape diagram representing the sum using square units. Redraw the diagram without squares, as two rectangles that are 1 unit high. Draw a rectangular array representing .

  17. Writing Products as Sums Lesson 3 / Example 6 Example 6 A square fountain area with side length is bordered by a single row of square tiles as shown. Express the total number of tiles needed in terms of three different ways.

  18. Writing Products as Sums 4(s + 1) = 4s + 4 = 2s + 2(s + 2) Lesson 3 / Example 6

  19. Writing Sums as Products 2x + 12y + 8 = 2(x + 6y + 4) Lesson 4 / Example 2 Example 2: and are positive integers. , , and stand for the number of unit squares in a rectangular array. What does the large rectangle that contains the three smaller rectangles represent? 2x + 12y + 8 square units How many rows are there in this rectangular array? 2 rows What are the missing values and how do you know?

  20. Using the Identity and Inverse to Write Equivalent Expressions • Lesson Highlights: • Students come to recognize the identity properties of 0 and 1. • Students come to recognize the existence of inverses (opposites and reciprocals) that together make 0’s and 1’s in expressions. • Why do we emphasize student awareness of these properties?

  21. Collecting Rational Number Like Terms Lesson 6 / Example 4 Example 4: Model how to write the expression in standard form using rules of rational numbers.

  22. Topic BSolve Problems Using Expressions, Equations, and Inequalities • Take a moment to read through: • Module Overview paragraph 4 • Topic B Opener • What do you expect to see conceptually as Topic B unfolds?

  23. Understanding Equations Where does an equation come from? Expressions and Equations Progression p. 4 How will we approach equations? Lots and lots of wonderful word problems!

  24. Understanding Equations Youngest Sister 45 1 Middle Sister 1 Oldest Sister 1 Lesson 7 / Example 1 Example 1: The ages of three sisters are consecutive integers. The sum of their ages is 45. Find their ages. Use a tape diagram to find their ages. , and The sisters are 14, 15, and 16.

  25. Understanding Equations Lesson 7 / Example 1 (continued) If the youngest sister is 𝒙 years old, describe the ages of the other two sisters in terms of 𝒙, write an expression for the sum of their ages in terms of 𝒙, and use that expression to write an equation that can be used to find their ages Youngest sister’s age in years: Middle sister’s age in years: Oldest sister’s age in years: Sum of their ages: Sum of their ages is 45:

  26. Understanding Equations Lesson 7 / Example 1 (continued) Determine if your answer from part (a) is a solution to the equation you wrote in part (b). TRUE!

  27. Using If-Then Moves in Solving Equations • What is an If-Then move? • Any order, any grouping If , then • Why if-then moves? • If-then moves promote reasoning about the equality of expressions.

  28. Using If-Then Moves in Solving Equations Lesson 8 / Example 1 Example 1: Julia, Keller, and Israel are volunteer firefighters. On Saturday the volunteer fire department held its annual coin drop fundraiser at a streetlight. After one hour, Keller had collected $42.50 more than Julia and Israel had collected $15 less than Keller. Altogether, the three firefighters collected $125.95. How much did each person collect? Find the solution using a tape diagram.

  29. Using If-Then Moves in Solving Equations Lesson 8 / Example 1 The amount that Keller collected in dollars: The amount Isreal collected in dollars: or Write an equation in terms of j that can be used to find the amount each person collected. The sum of the amounts collected by all three people equals .

  30. Using If-Then Moves in Solving Equations Lesson 8 / Example 1 any order, any grouping If a=b, then to make 0 any order, any grouping and additiveinverse additive identity property of 0 If a=b, then ; to make 1 any order, any grouping multiplicative inverse

  31. Using If-Then Moves in Solving Equations i - 27.50 i+ 15 k- 27.50 k - 15 Lesson 8 / Example 1

  32. Using If-Then Moves in Solving Equations x + 5 7x + 5 Bonnie’s present age is 11 years old. Lesson 9 / Example 2 Example 2: Shelby is seven times as old as Bonnie. If in 5 years, the sum of Bonnie and Shelby’s ages is 98, find Bonnie’s present age.

  33. Lessons 1 through 9 Lessons 1- 9/ Concepts and Representations

  34. A Story of Ratios Grade 7 – Module 3 Session 3

  35. Angle Problems and Solving Equations Acute angle: , Obtuse angle: • 2xᵒ • xᵒ Lesson 10 / Example 4 Example 4: Two line intersect in the following figure. The ratio of the measurements of the obtuse angle to the acute angle in any adjacent angle pair is 2:1.

  36. Angle Problems and Solving Equations • Angles and are complimentary and sum to ; angles and are complimentary and sum to . Lesson 11 / Exercise 2 Exercise 2 In a complete sentence, describe the angle relationships in the diagram. Write an equation for the angle relationship shown in the figure and solve for x and y. Confirm your answers by measuring with a protractor.

  37. Fluency Sprint Lesson 12 / Opening Exercise • Opening Exercise: • Round 1: You have 1 minute to complete as many as possible…do not skip around, go in order! You only need to provide the answer. • If you have the answer correct, clap once! • Finish the remaining problems.

  38. Properties of Inequalities Lesson 12 / Example 1 Example 1: Using the number cubes at your tables, complete the station exercises by recording each trial of integers in the first and third columns. Station 1: If c. Station 2: If then . Station 3: If and , then . Station 4: If and , then .

  39. Inequalities Lesson 13 / Opening Exercise Opening Exercise: Tarik is trying to save to buy a new tablet. Right now he has and can save a week from his allowance. Write and evaluate an expression to represent the amount of money saved after: 2 weeks: 3 weeks: 4 weeks: 5 weeks: 6 weeks: 7 weeks: 8 weeks:

  40. Solving Inequalities The camp may take up to 77 campers. Lesson 14 / Example 1 Example 1 A youth summer camp has budgeted $𝟐𝟎𝟎𝟎 for the campers to attend a carnival. The cost for each camper is $𝟏𝟕.𝟗𝟓 which includes general admission to the carnival and 𝟐 meals. The youth summer camp must also pay $𝟐𝟓𝟎 for the chaperones to attend and $𝟑𝟓𝟎 for transportation to and from. What is the greatest number of campers that can attend the carnival if the camp must stay within their budgeted amount?

  41. Graphing Solutions to Inequalities Lesson 15 / Example 1 Example 1 A local car dealership is trying to sell all of the cars that are on the lot. Currently it has 𝟓𝟐𝟓 cars on the lot and the general manager estimates that they will consistently sell 𝟓𝟎 cars per week. Estimate how many weeks it will take for the number of cars on the lot to be less than 𝟕𝟓? let represent the number of complete weeks of sales It will take 10 weeks or more.

  42. Mid-Module AssessmentTopics A and B Mid Module Assessment / Problem #6

  43. Topic CUse Equations and Inequalities to Solve Geometry Problems • Take a moment to read through: • Module Overview paragraphs 5-7 • Topic C Opener • What do you expect to see conceptually as Topic C unfolds?

  44. The Most Famous Ratio of All Lesson 16 / Opening Exercise • Opening Exercise • Draw a circle using a compass. • Write your own definition for the term circle. • Circle: Given a point in the plane and a number , the circle with center and radius is the set of all points in the plane that are a distance from the point .

  45. The Most Famous Ratio of All Lesson 16 / Example 1 Example 1: Mark a point on the wheel with a piece of masking tape or chalk. Mark a starting point on the floor, align it with the mark on the wheel, and carefully roll the wheel so that is rolls one complete revolution. Mark the endpoint on the floor with a piece of masking tape or chalk.

  46. The Most Famous Ratio of All From the Educator Guide to the 2013 Grade 7 Common Core Mathematics Test: • This curriculum utilizes the exact value of pi as well as three approximations of the value of pi, including: • Calculator approximation

  47. The Area of a Circle Lesson 17 / Discussion Discussion: Cut a circle into 16 equal size sectors. Arrange the “triangles” by alternating their directions and sliding them together to form a parallelogram.

  48. The Area of a Circle Lesson 17 / Discussion The circumference is , where is the radius. Therefore, half of the circumference is . What is the area of the rectangle using the side lengths shown?

  49. The Area of a Circle Lesson 17 / Example 1 Example 1 Michael is laying tile in the floor of a square walk-in shower. He wants a circle pattern as shown. He needs to determine the area of the circle so that he can order the special tile for the shower project. Michael drew the tile pattern on graph paper so that he could have a visual model to help guide his thinking.

  50. The Area of a Circle Lesson 17 / Problem Set #10 Problem Set #10 If the diameter of a circle is double the diameter of a second circle, what is the ratio of area of the first circle to the area of the second?

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