# Representing Proportional Relationships with Equations

Videos, examples, lessons, and solutions to help Grade 7 students learn how to represent proportional relationships with equations.

### New York State Common Core Math Grade 7, Module 1, Lesson 9

Lesson 8 and Lesson 9 Student Outcomes

Students use the constant of proportionality to represent proportional relationships by equations in real world contexts as they relate the equations to a corresponding ratio table and/or graphical representation.

Equations for Proportional Relationships

Points to remember:

• Proportional relationships have a constant ratio, or unit rate.
• The constant ratio, or unit rate, can also be called the constant of proportionality.
• m = y/x

Lesson 8 Summary:

If a proportional relationship is described by the set of ordered pairs that satisfies the equation y = kx, where k is a positive constant, then k is called the constant of proportionality. The constant of proportionality expresses the multiplicative relationship between each x-value and its corresponding y-value.

Lesson 8 Example 1: Do We have Enough Gas to Make it to the Gas Station?

Your mother has accelerated onto the interstate beginning a long road trip and you notice that the low fuel light is on, indicating that there is a half a gallon left in the gas tank. The nearest gas station is 26 miles away. Your mother keeps a log where she records the mileage and the number of gallons purchased each time she fills up the tank. Use the information in the table below to determine whether you will make it to the gas station before the gas runs out. You know that if you can determine the amount of gas that her car consumes in a particular number of miles, then you can determine whether or not you can make it to the next gas station.

a. Find the constant of proportionality and explain what it represents in this situation.
b. Write equation(s) that will relate the miles driven to the number of gallons of gas.
c. Knowing that there is a half gallon left in the gas tank when the light comes on, will she make it to the nearest gas station? Explain why or why not.
d. Using the equation found in part b, determine how far your mother can travel on 18 gallons of gas. Solve the problem in two ways.
e. Using the equation found in part b, determine how many gallons of gas would be needed to travel 750 miles.

Lesson 8 Example 2: Andrea’s Portraits

Andrea is a street artist in New Orleans. She draws caricatures (cartoon-like portraits of tourists). People have their portrait drawn and then come back later to pick it up from her. The graph below shows the relationship between the number of portraits she draws and the amount of time in hours needed to draw the portraits.

a. Write several ordered pairs from the graph and explain what each coordinate pair means in the context of this graph.
b. Write several equations that would relate the number of portraits drawn to the time spent drawing the portraits.
c. Determine the constant of proportionality and explain what it means in this situation.

Lesson 9 Summary:
How do you find the constant of proportionality? Divide to find the unit rate, y/x = k.
How do you write an equation for a proportional relationship? y = kx, substituting the value of the constant of proportionality in place of k.
What is the structure of proportional relationship equations, and how do we use them? x and y values are always left as variables, and when one of them is known, they are substituted into y = kx to find the unknown using algebra.

Lesson 9 Example 1: Jackson’s Birdhouses
Jackson and his grandfather constructed a model for a birdhouse. Many of their neighbors offered to buy the birdhouses. Jackson decided that building birdhouses could help him earn money for his summer camp, but he is not sure how long it will take him to fill all of the requests for birdhouses. If Jackson can build 7 birdhouses in 5 hours, write an equation that will allow Jackson to calculate the time it will take him to build any given number of birdhouses.
a. Write an equation that you could use to find out how long it will take him to build any number of birdhouses.
b. How many birdhouses can Jackson build in 40 hours?
c. How long will it take Jackson to build 35 birdhouses? Use the equation from part a to solve the problem.
d. How long will it take to build birdhouses? Use the equation from part a to solve the problem.

Lesson 9 Example 2: Al’s Produce Stand
Al’s Produce Stand sells 7 ears of corn for Barbara’s Produce stand sells 13 ears of corn for \$2.85. Write two equations, one for each produce stand that models the relationship between the number of ears of corn sold and the cost. Then use each equation to help complete the tables below.

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