Lesson 5: Two Equations for Each Relationship
Let’s investigate the equations that represent proportional relationships.
Illustrative Math Unit 7.2, Lesson 5 (printable worksheets)
Lesson 5 Summary
The following diagram shows how to find two constants of proportionality and write two equations for a proportional relationship.
Lesson 5.1 Missing Figures
Here are the second and fourth figures in a pattern.
- What do you think the first and third figures in the pattern look like?
- Describe the 10th figure in the pattern.
Lesson 5.2 Feeding a Crowd, Revisited
There are 100 centimeters (cm) in every meter (m).
- Complete each of the tables.
- For each table, find the constant of proportionality.
- What is the relationship between these constants of proportionality?
- For each table, write an equation for the proportional relationship. Let x represent a length measured in meters and y represent the same length measured in centimeters.
Are you ready for more?
- How many cubic centimeters are there in a cubic meter?
1 meter = 100 centimeters
1 cubic meter = 1003 centimeters = 1,000,000 centimeters
- How do you convert cubic centimeters to cubic meters?
To convert cubic centimeters to cubic meters we divide by 1,000,000
- How do you convert the other way?
To convert cubic meters to cubic centimeters we multiply by 1,000,000
Lesson 5.3 Filling a Water Cooler
It took Priya 5 minutes to fill a cooler with 8 gallons of water from a faucet that was flowing at a steady rate. Let w be the number of gallons of water in the cooler after t minutes.
- Which of the following equations represent the relationship between w and t? Select all that apply.
a. w = 1.6t
b. w = 0.625t
c. t = 1.6w
d. t = 0.625w
- What does 1.6 tell you about the situation?
- What does 0.625 tell you about the situation?
- Priya changed the rate at which water flowed through the faucet. Write an equation that represents the relationship of and w when t it takes 3 minutes to fill the cooler with 1 gallon of water.
- Was the cooler filling faster before or after Priya changed the rate of water flow? Explain how you know.
Lesson 5.4 Feeding Shrimp
At an aquarium, a shrimp is fed 1/5 gram of food each feeding and is fed 3 times each day.
- How much food does a shrimp get fed in one day?
- Complete the table to show how many grams of food the shrimp is fed over different numbers of days.
- What is the constant of proportionality? What does it tell us about the situation?
- If we switched the columns in the table, what would be the constant of proportionality? Explain your reasoning.
- Use d for number of days and f for amount of food in grams that a shrimp eats to write two equations that represent the relationship between d and f.
- If a tank has 10 shrimp in it, how much food is added to the tank each day?
- If the aquarium manager has 300 grams of shrimp food for this tank of 10 shrimp, how many days will it last? Explain or show your reasoning.
Lesson 5 Practice Problems
- The table represents the relationship between a length measured in meters and the same length measured in kilometers.
a. Complete the table.
b. Write an equation for converting the number of meters to kilometers. Use x for number of meters and y for number of kilometers.
- Concrete building blocks weigh 28 pounds each. Using b for the number of concrete blocks and w for the weight, write two equations that relate the two variables. One equation should begin with w = and the other should begin with b =.
- A store sells rope by the meter. The equation p = 0.8L represents the price p (in dollars) of a piece of nylon rope that is L meters long.
a. How much does the nylon rope cost per meter?
b. How long is a piece of nylon rope that costs $1.00?
- The table represents a proportional relationship. Find the constant of proportionality and write an equation to represent the relationship.
- On a map of Chicago, 1 cm represents 100 m. Select all statements that express the same scale.
A. 5 cm on the map represents 50 m in Chicago.
B. 1 mm on the map represents 10 m in Chicago.
C. 1 km in Chicago is represented by 10 cm the map.
D. 100 cm in Chicago is represented by 1 m on the map.
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