A Story of Ratios

1. A Story of Ratios Examining Module 1 for Grades 6 – 8

2. 2-Day Agenda • Day 1: A Story of Ratios • G6-M1 & G7-M1 • G8-M1 • Day 2: A Story of Functions • Geometry: G10-M1 • Algebra & Precalculus: G9-M1, G11-M1 & G12-M1

3. Key Areas of Focus in Mathematics

4. Grapes and Peaches

5. Exemplar Module Analysis Grade 6 – Module 1 Grade 7 – Module 1

6. Agenda • What’s in a Module? • A Study of Grade 6 Module 1 • A Study of Grade 7 Module 1 • 6–7 Ratios and Proportional RelationshipsProgressions • Exercise: Assessments & Scoring Rubrics

7. What’s in a Module? • Teacher Packet • Table of Contents • Module Overview • Topic Overviews • Daily Lessons • Grading Rubrics and Student Work Samples • Student Packet • Daily Lesson Materials and Problem Sets • Exit Ticket and Assessment Packet (copy machine ready pages) • Exit Tickets • Assessments (Mid-Module and End-of-Module) • Project Assignments

8. What’s in a Module Overview? • Module overview • Narrative • Focus Standards • Foundational Standards • Focus Mathematical Practice Standards • New Terms and Symbols • Familiar Terms and Symbols • Suggested Tools and Representations • Scaffolds

9. Agenda • What is in a Module? • A Study of Grade 6 Module 1 • A Study of Grade 7 Module 1 • 6 – 7 Ratios and Proportional RelationshipsProgressions • Exercise: Assessments & Scoring Rubrics

10. Grade 6 Module 1 Ratios and Unit Rates

12. Write downthree key mathematical ideas from the entire narrative. • While you are reading, think about how the concept is represented and related between the four topics.

14. Foundational Standards • 4.OA.2Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison. • 5.NF.3 Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. • 5.MD.1 Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems. • 5.G.1 Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., x-axis and x-coordinate, y-axis and y-coordinate). • 5.G.2 Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.

15. Focus Standards • 6.RP.1Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.” • 6.RP.2 Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.” Expectations for unit rates in this grade are limited to non-complex fractions. • 6.RP.3Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. • a.Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios. • b. Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed? • c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent. • d. Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.

18. Reflect Upon the Module Overview • How do you see this document valuable to you? • How would you use the module overview in preparing to teach this module?

19. Agenda • What is in a Module? • A Study of Grade 6 Module 1 • A Study of Grade 7 Module 1 • 6 – 7 Ratios and Proportional RelationshipsProgressions • Exercise: Assessments & Scoring Rubrics

20. Grade 7 Module 1 Ratios and Proportional Relationships `

26. Agenda • What is in a Module? • A Study of Grade 6 Module 1 • A Study of Grade 7 Module 1 • 6 – 7 Ratios and Proportional RelationshipsProgressions • Exercise: Assessments & Scoring Rubrics

27. 6-7 Ratios and Proportional Relationships

28. Ratio Definitions Ratio - a pair of non-negative numbers, A:B, which are both not zero. They are used to indicate that there is a relationship between two quantities such that when there are A units of one quantity, there are B units of the second quantity Value of a Ratio - for the ratio A:B, the value of the ratio is the quotient A/B as long as B is not zero. Equivalent Ratios – two ratios A:B and C:D are equivalent if there is a positive number, c, such that C=cA and D=cB. They are ratios that have the same value.

29. Equivalent Ratios Take 3 minutes to locate and highlight all of the multiple representations of equivalent ratios in pages 2 – 7 in the progressions document.

31. Equivalent Ratios: Tape Diagrams vs. Double Number Lines Are the units being compared the same or different? Monique walks 3 miles in 25 minutes. vs. Sean spends 5 minutes watching television for every 2 minutes he spends on homework.

32. Terms Rate –If I traveled 180 miles in 3 hours; my average speed is 60 mph. The quantity, 60 mph, is an example of a rate Unit Rate-The numeric value of the rate, e.g. in the rate 2.5 mph, the unit rate is 2.5 Rate’s Unit – The unit of measurement for the rate, e.g. mph

33. Transitions to Grade 7 What are the connections between Grade 6 terms and concepts and Grade 7 terms and concepts?

34. Constant of Proportionality Song Downloads cost $3 each. (unit rate) Where and how is the constant of proportionality represented (pages 8-9)?

35. Agenda • What is in a Module? • A Study of Grade 6 Module 1 • A Study of Grade 7 Module 1 • 6 – 7 Ratios and Proportional RelationshipsProgressions • Exercise: Assessments & Scoring Rubrics

36. Assessment Exercise: G7-M1 End-of-Module Assessment Problem #1: It is a Saturday morning and Jeremy has discovered he has a leak coming from the water heater in his attic. Since plumbers charge extra to come out on weekends, Jeremy is planning to use buckets to catch the dripping water. He places a bucket under the drip and steps outside to walk the dog. In half an hour, the bucket is 1/5 of the way full. (7.RP.1, 7.RP.2.c, 7.EE.4.a) What is the rate at which the water is leaking in buckets per hour? What is the unit rate? Write an equation that represents the relationship between the number of buckets filled, y, in x hours. What is the longest that Jeremy can be away from the house before the bucket will overflow?

40. Closing Question: How does the study of unit rate and constant of proportionality in seventh grade prepare students for the study of linear equations in grade eight?