7 th Grade Unit 2: Ratios and Proportions (4 Weeks)
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1 7 th Grade Unit 2: Ratios and Proportions (4 Weeks) Stage 1 Desired Results Established Goals Unit Description This unit focuses on analyzing proportional relationships and using them to solve real-world problems. The unit deepens students understanding and use of unit rates; develops their understanding of proportional relationships as represented in words, tables, equations, graphs and diagrams with a focus on using fractions and decimal numbers. Common Core Learning Standards 7.RP.1 Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction 1/2/1/4 miles per hour, equivalently 2 miles per hour. 7.RP.2 Recognize and represent proportional relationships between quantities. a. Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. c. Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t=pn. d. Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate. Common Core Standards of Mathematical Practice 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. ESL Language Standards Standard 1: Students will listen, read, and write in English for information and understanding 1.1 Identify and use reading and listening strategies to make text comprehensible and meaningful Present information clearly in a variety of oral and written forms for different audiences and purposes related to all academic content areas Convey information and ideas through spoken and written language, using conventions and features of American English appropriate to audience and purpose Apply self-monitoring and self-correcting strategies for accurate language production and oral and written presentation, using established criteria for effective presentation of information Apply learning strategies to acquire information and make texts comprehensible and meaningful Apply learning strategies to examine and interpret a variety of materials. Standard 4: Student will listen, speak, read, and write in English for classroom and social interaction Explain actions, choices, and decisions in social and academic situations.
2 Big Ideas 1. Some situations that involve comparison of quantitative information are a form of proportional relationships Essential Questions 1. What makes a relationship proportional? How can I tell if a proportional relationship exists? 2. One representation may sometimes be more helpful than another. Multiple representations give a fuller understanding of a problem. Content (Students will know.) A. Unit rates in relation to ratios using fractions and decimal numbers. (7.RP.1 ) Unit prices Time to distance Speed as a rate Similarity in geometric shapes Measurement unit conversions Currency exchange 2. How can representing mathematical ideas in different ways (graphs, tables, equations, diagrams, words) help me solve problems? 2. How can I use tables, graphs or equations to determine whether a relationship is proportional? Skills (Students will be able to ) A1. Apply concepts of rate and ratio to problems solving tasks and situations. Unit prices Time to distance Speed as a rate Similarity in geometric shapes Measurement unit conversions Currency exchange A2.Compute unit rates associated with ratios of fractions. A3. Find rate of change using Ordered Pairs in the coordinate plane B. Proportional and non-proportional relationships between quantities (in tables, graphs, equations, diagrams) (7.RP.2 ) Ratio tables Graphing in the coordinate plane Graph of a line C. The constant of proportionality (unit rate) in different representations. (7.RP.2) Tables Graphs Equations Diagrams Verbal description of proportional relationships B1.Analyze whether two quantities are in a proportional relationship by testing for equivalent ratios in a table. B2. Analyze whether two quantities are in a proportional relationship by observing if a straight line passes through the origin. B3.Analyze whether the points plotted on the coordinate plane if connected, can they make a line that passes through the origin. B4. Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate. C1. Display proportional relationships through equations. C2. Display proportional relationships through graphing. C3. Display proportional relationships through diagrams. C4. Display proportional relationships through words. D. Represent proportional relationships using equations derived from: (7.RP.2) Tables
3 Graphs Equations Diagrams Verbal description of proportional relationships D1. Write and solve equations that represent proportional relationships Terms/ Vocabulary Linear equations, rate, complex fraction, unit rate, per, proportion, proportional relationships, scale, ratio, composed unit, constant rate of change, multiplicative relationship, coordinate plane, origin. Stage 2 Assessment Evidence Initial Task: Diagnostic Assessment (adapted from NYCDOE Bundle Proportional Reasoning) Final Performance Task: Final Assessment (adapted from NYCDOE Bundle Proportional Relationships) *Scoring Guidance: pp in Bundle Other Evidence Teacher observation, conferencing, teacher designed assessment pieces, student work, exit slips, journal entries Stage 3 Learning Plan Everyday Mathematics /Impact Mathematic Lessons The following lessons will support some of the CCLS & essential questions outlined in this unit map: 7.RP.1 Impact Lesson 10.1 Ratios pgs Impact Lesson 10.4 Rates pgs RP.2 Impact Lesson 8.1 Rates pgs Impact Lesson 10.1 Ratios pg 495 Impact Lesson 10.2 Proportions and Similarity pgs Impact Lesson 10.4 Rates pgs Impact 7th Grade Teacher Guide-Chapter 10 Resources pg 494B (Ratios) Impact 7th Grade Teacher Guide-Chapter 10 Resources pg505b ( Proportion and similarity) Impact 7th Grade Teacher Guide-Chapter 10 Resources pg540b (Rates) Impact 7th Grade Teacher Guide-Chapter 8 Resources pg368b (Rates in graphs and tables) At the end of this unit-sample activities Additional Resources: What is Unit Rate? pg10 Orange Fizz Experiment pg 15 Creating A Scale Map pg 17 Which Is The Better Deal? pg 20 Patterns & Percents pg 22 Nate & Natalie s Walk pg 26
4 Grade 7 Unit 2 Initial Assessment: Diagnostic 1. Julia made observations about the selling price of a new coffee that sold in three different-sized bags. (7RP1, 7RP2a, 7RP2b) She recorded those observations in the following table: Ounces of Coffee Price in Dollars $2.40 $3.20 $6.40 a) Is there a proportional relationship between the amount of coffee and the price? Why or why not? b) Find the unit rates associated with this problem. c) Explain in writing what the unit rates mean in the context of this problem.
5 d) Explain in writing why it is helpful for Julia to determine if the relationship between the amount of coffee and the price is proportional before she buys a bag of the new coffee. 2. EZ Electric Company sells light bulbs to major outlets like Home Depot, Sears, Wal-mart and other big chain stores. They sample their bulbs for defects routinely. (7RP1, 7RP2c) A sample of 96 light bulbs included 4 defective ones. Assume that today s batch of 6,000 light bulbs has the same proportion of defective bulbs as the sample. Determine the total number of defective bulbs made today.
6 Grade 7 Unit 2 Final Assessment: Ratios and Proportions 1. Amy and her family were traveling during their vacation. She looked at her watch at Point 1 in the diagram below, and then again at Point 2 in the diagram below. Her mom told her how far they traveled in that time, as noted below. (7.RP.1) Point 1 Point 2 a. Based on this information, what is the unit rate of the car? Explain in writing what that unit rate means in the context of the problem. b. Amy s dad said that the entire trip was 1200 miles. How many hours will it take to complete the trip? Explain how you know.
7 2. On a map of the United States, 24 centimeters represents 18 miles. (7.RP.1) 0 24 centimeters A B 18 miles a. How many centimeters represent one mile? b. How long is the line segment between A and B in centimeters? c. If A and B represent two cities, what is the actual distance between the two cities?
8 3. Jack and Jill raced cross-country on motor bikes. Jack drove 325 miles in 5 hours; Jill took 6 ½ hours to travel the same distance as Jack. (7.RP.1, 7.RP.2, c) a. Compute the unit rates that describe Jack s average driving speed and Jill s average driving speed. Show how you made your decisions. b. A portion of the graph of Jack and Jill s race appears below. Identify which line segment belongs to Jack and which belongs to Jill. Explain in writing how you made your decisions.
9 4. Reynaldo is planning to drive from New York to San Francisco in his car. Reynaldo started to fill out the table below showing how far in miles he can travel for each gallon of gas he uses. (7.RP.2) Use the information in Reynaldo s table to answer the questions below. a. Complete the table for Reynaldo. Assume the relationship in the table is proportional. b. Based on the table, how many miles per gallon did Reynaldo s car get? Explain in writing how you know. c. Write an equation that Reynaldo can use to find the distance (d) he can drive on any number of gallons of gas (g). d. When Reynaldo s tank is full, it holds 20 gallons. How far can Reynaldo drive on a full tank of gas?
10 5. The monthly cost of Jazmine s cell phone plan is graphed on the grid below. Her friend Kiara selected a plan that charges $0.25 per text, with no monthly fee, because she only uses her phone for texting. (7.RP.2) a. Write an equation to represent the monthly cost of Kiara s plan for any number of texts. b. Graph the monthly cost of Kiara s plan on the grid above. c. Using the graphs above, explain the meaning of the following coordinate pairs: i. (0, 20): ii. (0, 0): iii. (10, 2.5): iv. (100, 25): d. When one of the girls doubles the number of texts she sends, the cost doubles as well. Who is it? Explain in writing how you know.
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