# Equations of Proportional Relationships

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1 Equations of Proportional Relationships Student Probe Tracy can ride her bike 30 miles in three hours. Write an equation that shows how far Tracy can ride her bike in h hours. Answer: y 10h Lesson Description This lesson builds on students understanding of ratios, proportions, and constants of proportionality (unit rates) to create equations of proportional relationships. will create ratio tables to assist them in writing equations to model proportional relationships. Rationale Representing proportional relationships as equations is an important bridge from arithmetic to algebraic thinking. These equations have a strong foundation in missing factor open sentences from earlier grades. By focusing on ratios and proportional relationships that they already understand, students can become adept at writing simple linear equations. At a Glance What: Represent proportional relationships with equations Common Core State Standard: CC.7.RP.c. Recognize and represent proportional relationships between quantities. (c) Represent proportional relationships by equations. Mathematical Practices: Model with mathematics. Who: who cannot write equations to represent proportional relationships Grade Level: 7 Prerequisite Vocabulary: ratio, proportion, ratio tables, variable, constant, equation, unit rate Prerequisite Skills: create ratio tables, write open sentences, find unit rates Delivery Format: Individual, small group Lesson Length: 30 minutes Materials, Resources, Technology: calculator (optional) Student Worksheets: None Preparation None

2 Lesson The Expect students to say or do If students do not, then the 1. Let s complete a ratio Prompt students as they table for this relationship: One out of three students Spanish All complete the table. Refer to Ratios and at Central Middle School 1 3 Proportional Thinking. take Spanish What is the unit rate of this relationship? 3. What patterns do you see in the table? Suppose there are 46 students taking Spanish and we want to know how many students are in the school. That would require a really big table. Let s use the patterns to make it easier. 4. I am going to re-write the All column in the table to show the pattern we have discovered. (Model for students.). What do you notice is staying the same in the All column? Because the 3 is staying the same, it is called a constant. 3 Refer to Unit Rates. Answers will vary, but listen for, The number of students is 3 times the number of students who take Spanish. 3 Spanish All Extend the table.

3 The Expect students to say or do If students do not, then the 6. What do you notice is changing in the All column? (Underline these numbers, if necessary.) That must be our variable. 1,, 3, 4 Variable means what is changing, or what varies. 7. Suppose we have x Spanish students. How many students would we have? 8. Now we can write an equation, or open sentence, to represent the relationship between the students who take Spanish and all students. What do you think the equation will be? 9. What is the relationship of the unit rate we found with the equation? 10. Now determine the number of students in the school if 46 of them take Spanish. Explain how you got your answer. 11. If there are 360 students in the school, how many can we expect to take Spanish? How do you know? 1. In one school district, out of students ride a bus to school. What is the unit rate of bus riders to all students? 3x Extend the table emphasizing 3 1,3,.... y 3x Model for students. Substitute values such as 1,, 3, etc., into the equation to verify that it is correct. They are the same. Prompt students. y Model for students x x Refer to Unit Rates.

4 The Expect students to say or do If students do not, then the 13. Predict what you think the Do not correct students at this y x. equation will be that gives time. the relationship between the students who ride a bus and all students. (Do not verify answers at this time.) 14. Build a ratio table showing this relationship. (You may need to assist students as they build this table.) 1. Let s look at what is staying the same (constant) and at what is changing (variable). What is the equation? 16. Repeat with additional problems as necessary. Bus Riders All y x. Refer to Multiplying Fractions. Teacher Notes 1. Emphasize to students that the multiplicative inverse of the unit rate (constant of proportionality) helps them determine the equation.. may require additional practice with ratios such as :3, 3:4, etc. Variations None

5 Formative Assessment Three out of every eight students at Eastern Middle School takes part in sports. Write an equation that gives the relationship between the numbers of students who take part in sports and all students. Answer: y 8 x 3 References Greenes, C. (000). Groundworks to Algebraic Thinking. Columbus, OH: Wright McGraw Hill. Russell Gersten, P. (n.d.). RTI and Mathematics IES Practice Guide - Response to Intervention in Mathematics. Retrieved December 7, 011, from rti4sucess: Paulsen, K., & the IRIS Center. (n.d.). Algebra (part ): Applying learning strategies to intermediate algebra. Retrieved on December 7, 011 from Van de Walle, J. A., & Lovin, L. H. (006). Teaching Student-Centered Mathematics Grades -8 Volume 3. Boston, MA: Pearson Education, Inc.

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