Courtesy of Wikimedia Commons
By Michael Avidon, math editor
Equations: For Students
| More in this series |
|---|
| Lesson 1. |
| 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.
- Identify the constant of proportionality (unit rate) in … equations … of proportional relationships.
- 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.
In the previous lesson, you looked at this proportional relationship:
| Cupcakes | |
| Number | Cost |
| 1 | $3.50 |
| 2 | $7.00 |
| 3 | $10.50 |
You saw that the ratio was a constant $3.50 per cupcake. If C is the cost of the cupcakes, and n is the number of cupcakes, then this ratio is represented by C ⁄ n . So
C ⁄ n = 3.50
Multiplying both sides by n yields the equation C = 3.5n. This represents the proportional relationship. For example, if n = 3, then C = 3.5(3) = 10.5. That is, the cost of ___ cupcakes is $_________, as it says in the third row of the table.
In general, if there is a proportional relationship between a quantity y and a quantity x with a constant of proportionality k, then
y ⁄ x = k, which implies y = kx
A proportional relationship between a quantity y and a quantity x that has a constant of proportionality k is represented by the equation y = kx.
If an equation in a different form can be rewritten as above, then it is a proportional relationship. If it cannot be rewritten as above, then it is not a proportional relationship.
We can take any table or verbal description of a proportional relationship and turn it into an equation. We just need to first determine the constant of proportionality or unit rate.
Example 1
| Brick machine | |
|---|---|
| Time (hours) |
Number of bricks |
| 2 | 1100 |
| 5 | 2750 |
| 8 | 4400 |
You saw in the previous lesson that the output of this brick-making machine follows a proportional relationship. Write an equation to represent the relationship. (To print this lesson out, click on the printable version.)
Solution:
Let N = the number of bricks and t = the time in hours. The unit rate is _____ bricks per hour. The equation is N = 550t.
Example 2
A 10-pound wheel of baby gouda cheese sells for $64. Write an equation relating the cost to the weight of the cheese.
Solution:
The cheese sells for $_________ per pound. Let C = the cost of the cheese in dollars and w = the weight of the cheese in pounds. Then the equation is C = 6.4w.
Remember that a proportional relationship works “both ways.” So in the above example, not only is C proportional to w, but w is also proportional to C. You can take the equation above and divide both sides by 6.4 to write the relationship the other way:
1 ⁄ 64C = w → w = 0.15625C
Example 3
The equation K = 0.4536P represents the relationship between pounds, P, and kilograms, K. Identify the constant of proportionality and give a verbal description of the relationship. Also, give a verbal description of the reverse relationship.
Solution:
The constant of proportionality is 0.4536. This means for every 1 pound there is 0.4536 kilogram. To reverse the relationship, you need the reciprocal of the constant. In this case, it is about 2.2. So for every 1 kilogram, there are about 2.2 pounds.
Example 4
| x | y |
| 3 | 18 |
| 5 | 30 |
| 7 | 42 |
Represent y = 6x as a table, with rows for x = 3, 5 and y = 42.
Solution:
When x = 3, y = 6(3) = _____.
When x = 5, y = 6(____) = ______.
When y = 42, 42 = 6x → x = 7.
Exercises for lesson 2
Represent each proportional relationship with an equation.
1.
| Area, A (sq. ft.) |
Cost, C |
|---|---|
| 90 | $81 |
| 180 | $162 |
| 360 | $324 |
2.
| Production | |
|---|---|
| Time, t (hours) | Widgets, w |
| 3 | 1005 |
| 5 | 1675 |
| 7 | 2345 |
3.
| T-shirts | |
|---|---|
| Number, n | Cost, C |
| 2 | $23 |
| 4 | $46 |
| 5 | $57.50 |
4. An artist buys unprimed canvas in rolls. Each roll has 90 square feet of canvas and costs $54. Let C = the cost of the canvas in dollars and A = the area of the canvas in square feet.
5. A 3-pound package of ground beef sells for $12. Let B = the cost of the ground beef in dollars and w = the weight of the ground beef in pounds.
6. A mason is building a brick wall. So far, he has laid 5 rows of brick (and cement) that reaches a height of 45 cm. Let n = the number of rows of brick and h = the height of the wall in centimeters.
For each equation, (a) represent the proportional relationship by a table with input values 1, 2, and 3, and (b) write the equation for the reverse relationship.
7. y = 4x
8. y = 0.2x
9. y = x ⁄ 3
10. Does each equation represent a proportional relationship?
(a) y = x + 5 (b) y = x2 (c) y = 2(x – 5) + 10
11. The equation y = 2.54x represents the relationship between inches, x, and centimeters, y. Give a verbal description of the relationship.
12. The equation y = 33.8x represents the relationship between liters, x, and fluid ounces, y. Give a verbal description of the relationship.
13. Challenge Problem:
While the constant of proportionality is usually positive in real-world problems, it could be negative. Let h = the number of hours after midnight, and T = the temperature in degrees Celsius. Explain what the relationship means in real-world terms at midnight and beyond.