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By Michael Avidon, math editor
Tables and Verbal Descriptions: For Students
| More in this series |
|---|
| Lesson 2. |
| Lesson 3. |
| If you want to help your child, use our version for guardians. |
Performance Expectations (CCSS)
This lesson covers the following parts of 7.RP.A.2:
Recognize and represent proportional relationships between quantities.
- Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table …
- Identify the constant of proportionality (unit rate) in tables, … and verbal descriptions of proportional relationships.
MORE: Lesson 2 | Lesson 3 | Print Lesson 1 to work on | Guardians can catch up quickly through this overview
A bakery charges the following prices for cupcakes and bagels:
| Cupcakes | Bagels | |||
| Number | Cost | Number | Cost | |
| 1 | $3.50 | 1 | $1.50 | |
| 2 | $7.00 | 6 | $7.80 | |
| 3 | $10.50 | 12 | $13.20 | |
Let us compute the ratios of cost to number to see if there is a constant unit rate.
For the cupcakes: $3.50 ⁄ 1 = $3.50
$7.00 ⁄ 2 = $3.50
$10.50 ⁄ 3 = $_________
The ratios are all equal. There is a constant unit rate of $3.50 per cupcake.
For the bagels: $1.50 ⁄ 1 = $1.50
$7.80 ⁄ 6 = $1.30
$13.20 ⁄ 12 = $__________
The ratios are unequal. The rate per bagel is not constant.
The relationship between the cost and the number of cupcakes is proportional and the relationship between the cost and the number of bagels is not proportional.
A proportional relationship exists between a quantity y and a quantity x when the ratio y ⁄ x is constant. The ratio is known as the constant of proportionality. It is also known as the unit rate.
The constant of proportionality for the cupcakes was 3.50.
Example 1
| Distance (miles) |
Time (minutes) |
|---|---|
| 1 | 6 |
| 2 | 13 |
| 3 | 21 |
A middle-school track team member records some race times in the table. Is this a proportional relationship?
If so, what is the constant of proportionality?
Solution:
Calculate the ratios of time to distance to see if the minutes-per-mile rate is constant.
6 ⁄ 1 = 1
13 ⁄ 2 = 6.5
21 ⁄ ?? = ???
The rate is not constant. This is not a proportional relationship.
Example 2
| Brick machine | |
|---|---|
| Time (hours) |
Number of bricks |
| 2 | 1100 |
| 5 | 2750 |
| 8 | 4400 |
The table shows the output of a brick-making machine in a factory.
Is this a proportional relationship? If so, what is the constant of proportionality?
Solution:
Calculate the ratios of number to time to see if the bricks-per-hour rate is constant.
1100 ⁄ 2 = 550
2750 ⁄ 5 = ???
4400 ⁄ 8 = ???
The ratios are all equal. The number of bricks is proportional to the number of hours. There is a constant unit rate of 550 bricks per hour.
Note that if a quantity y is proportional to a quantity x, then it can also be said that the quantity x is proportional to the quantity y. Why is this true? If y ⁄ x is constant, then x ⁄ y is constant (but it is a different constant).
In example 2, the constant of proportionality is 550. But the number of hours is also proportional to the number of bricks and for that relationship the constant is 1 ⁄ 550 .
In many contexts, it is just more natural to state the relationship one way than the other.
Example 3
| Area (sq. ft.) |
Cost |
|---|---|
| 90 | $81 |
| 180 | $162 |
| 360 | $324 |
A gardener buys 2-foot-by-4.5-foot rolls of sod for $8.10 each.
Make a table for the cost of covering 90, 180, and 360 square feet.
Is there a proportional relationship between cost and area? If so, what is the constant of proportionality?
Solution:
Yes, cost is proportional to area. If you divide the cost by the number of square feet, each ratio comes out to $______/sq. ft. (or ____¢/sq. ft.). This is the constant of proportionality.
Example 4
| Length (feet) |
Area (sq. ft.) |
|---|---|
| 10 | 100 |
| 15 | 225 |
| 20 | 400 |
A gardener needs sod to create a square lawn. The area depends on the side length of the square. Make a table for the area when the side length is 10, 15, and 20 feet.
Is there a proportional relationship between area and side length? If so, what is the constant of proportionality?
Solution:
Compute the ratios of area to length:
100 ⁄ 10 = 10
225 ⁄ 15 = 15
400 ⁄ 20 = 20
The ratios [are/are not] constant. This [is/is not] a proportional relationship.
Example 5
| Ground Beef | |
|---|---|
| Weight (lbs.) |
Cost |
| 2.2 | 1$8.25 |
| 3.4 | |
| $14.25 | |
The table represents a proportional relationship between the cost of ground beef and its weight.
Determine the two missing numbers.
Solution:
First find the constant of proportionality: $8.25 ⁄ 2.2 $3.75.
So the unit rate is $3.75 per pound.
For the second row in the table:
cost ⁄ 3.4 = $3.75 → cost = ______ X $________ = $__________
For the third row in the table:
$14.25 ⁄ weight = $3.75 → $14.25 = $3.75 X weight
→ weight = $???? ⁄ $????/lb. = ???? lb.
Exercises for lesson 1
Does the table represent a proportional relationship?
If so, what is the constant of proportionality?
| x | y |
|---|---|
| 2 | 4 |
| 4 | 6 |
| 6 | 8 |
2.
| x | y |
|---|---|
| 3 | 9 |
| 5 | 15 |
| 8 | 24 |
3.
| x | y |
|---|---|
| 2 | 2.4 |
| 6 | 7.2 |
| 9 | 10.8 |
4.
| Production | |
|---|---|
| Time (hours) | Widgets |
| 3 | 1005 |
| 5 | 1675 |
| 7 | 2345 |
5.
| Apples | |
|---|---|
| Weight (lbs.) | Cost |
| 1 | $1.50 |
| 3 | $4.00 |
| 5 | $5.50 |
6.
| T-shirts | |
|---|---|
| Number | Cost |
| 2 | $23 |
| 4 | $46 |
| 5 | $57.50 |
7. An artist buys unprimed canvas in rolls that she will cut and use to make oil paintings. Each roll has 90 square feet of canvas and costs $54.
Is there a proportional relationship between cost and area? If so, what is the constant of proportionality or unit rate?
8. A plumber charges a set fee of $50 plus $60 per hour for labor. Is there a proportional relationship between cost and time? If so, what is the constant of proportionality?
9. There are 8 fluid ounces in a cup, 2 cups in a pint, 2 pints in a quart, and 4 quarts in a gallon. Is there a proportional relationship between ounces and gallons? If so, what is the constant of proportionality?
Each table represents a proportional relationship. Fill in the missing numbers.
10.
| x | y |
|---|---|
| 10 | 25 |
| 16 | 40 |
| 18 | 45 |
11.
| x | y |
|---|---|
| 5 | 2 |
| 8 | 3.2 |
| 12 | 4.8 |
12.
| x | y |
|---|---|
| 9 | 6 |
| 10 | 20/3 |
| 12 | 8 |
13. Challenge problem:
Suppose that there is a proportional relationship between quantity z and quantity y, and another proportional relationship between quantity y and quantity x. Must there exist a proportional relationship between quantity z and quantity x? Explain why or give a counterexample.
NEXT LESSON: Your next lesson awaits on proportional relationships | Jump ahead to Lesson 3