UNIT 2: RATIOS AND PROPORTIONAL RELATIONSHIPS Unit Description/ Topic Length: This 3-week unit focuses on the concept of ratios and proportional relationships. Students will extend their understanding of ratios and develop understanding of proportionality to solve real-world and mathematical problems. They will engage in instructional tasks that provide them with opportunities to recognize and represent proportional relationships between quantities in different ways. Essential Question: How does ratio reasoning differ from proportional reasoning? Big Ideas/Enduring Understandings A ratio is a multiplicative comparison of two quantities, or it is a joining of two quantities in a composed unit. Forming a ratio as a measure if a realworld attribute involves isolating that attribute from other attributes and understanding the effect of changing each quantity on the attribute of interest. A number of mathematical connections link ratios and fractions: o Ratios are often expressed in fraction notation, although ratios and fractions do not have identical meaning. o Ratios are often used to make “part-part” comparisons, but fractions are not. o Ratios and fractions can be thought of as overlapping sets. o Ratios can often be meaningfully reinterpreted as fractions. Ratios can be meaningfully reinterpreted as quotients. A proportion is a relationship of equally between two ratios. In a proportion, the ratio of two quantities remains constant as the corresponding values of the quantities change. Proportional reasoning is complex and involves understanding that o Equivalent ratios can be created by iterating an/or Guiding Questions: 1. What is a ratio? 2. What is a ratio as a measure of an attribute in a real-world situation? 3. How are ratios related to fractions and division? 4. What is a proportion? 5. What are the key aspects of proportional reasoning? 6. What is a rate and how is it related to proportional reasoning? 7. What is the relationship between the crossmultiplication algorithm and proportional reasoning? 8. When is it appropriate to reason proportionally? partitioning a composed unit: o If one quantity in a ratio is multiplied or divided by a particular factor, then the other quantity must be multiplied or divided by the same factor to maintain the proportional relationship; and o The two types of ratioscomposed units and multiplicative comparisons are related. A rate is a set of infinitely many equivalent ratios. Several ways of reasoning, all grounded in sense making, can be generalized into algorithms for solving proportions problems. Superficial cues present in the context of a problem do not provide sufficient evidence of proportional relationships between quantities. NYS Common Core Learning Standards for Mathematics: 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. 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. Content Unit Rates Recognizing Proportional Relationships Representing Proportional Relationships Constant of proportionality (unit rate) Vocabulary/ Key Terms Ratio Rate Unit rate Proportion Proportional Relationship Constant of proportionality Complex fractions Skills Compute unit rates associated with ratios of fractions in like or different units Determine if two quantities are in a proportional relationship from a table, a graph, an equation and a verbal description Represent a proportional relationship using a table, a graph, and an equation Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, including the points (0, 0) and (1, r) where r is the unit rate ASSESSMENT EVIDENCE Performance Task(s): Please refer to the attached file named Performance Assessment. You should find the Grade 7 Performance-Based Assessment on Proportional Reasoning in the document. Alignment to NYS Common Core Standards for Mathematics: Content Standards: 7.RP.1, 7.RP.2 (a, b, c, d) Standards for Mathematical Practice: 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. Diagnostic and Pre/Post Assessments: Students will respond to the essential question at the start of the unit and at the end of the unit. (pre/post) Formative Assessments: 1. Discussions 2. Daily reflections at the end of class 3. Class work 4. Practice quizzes 5. Student mini-showcase Summative Assessments: 1. 2. 3. 4. Quizzes (graded) Interim assessments Unit test Portfolio assignments TEACHING PLAN Teaching and Learning Activities: 1. Administer the Pre-Test for Unit 2. 2. Pose this question: “How does ratio reasoning differ from proportional reasoning?” Have students brainstorm based on previous knowledge. 3. Review ratios. 4. Discuss rate and unit rate. Then have students work in pairs to complete instructional task/s. 5. Think-Pair-Share: “What is a proportion?” 6. Have students work on instructional tasks to help them learn the key aspects of proportional reasoning. 7. Check for understanding: How is rate related to proportional reasoning? 8. Conduct the formative assessment lesson from MARS: Developing a Sense of Scale. This lesson is intended to help assess whether students recognize relationships of direct proportion and how well they solve problems that involve proportional reasoning. (Please see attachments.) 9. Discuss how to represent a proportional relationship using a table, a graph, and an equation. 10. Have students work on instructional tasks to help them identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. 11. Explain what a point (x, y) on the graph of a proportional relationship means in terms of a given situation, including the points (0, 0) and (1, r) where r is the unit rate. 12. Have students complete the performance-based assessment task for the unit. 13. Use essential question as a post-assessment. (individual journal entry) 14. Administer the unit test. Resources Needed: IMPACT Curriculum GLENCOE Math Chart Paper http://www.projectpaced.com/index.html NYC Common Core Library CALENDAR Time Spent on Standard Standards 1 week Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. 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. 3 days Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. Unit Rate and the Constant of Proportionality Lesson 10.2 Proportions and Similarity Pp. 505-510 3 days 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. Representing proportional relationships by equations Lesson 10.4 Rates Pp. 540-550 1 week Topics To Cover Ratios of Fractions Unit Rates Equivalent ratios Proportional Relationships Graphing a proportional relationship, with special attention to the points (0, 0) and (1, r) where r is the unit rate Main Curriculum Lesson 10.1 Ratios Pp. 494-503 Lesson 10.4 Rates Pp. 540-546 Lesson 8.1 Rates Pp. 368-387 Lesson 10.1 Ratios P. 495 Recognizing proportional relationships in a table and graphs on a coordinate plane

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# 5: unit 2: ratios and proportional relationships