By Michael Avidon, math editor
One theme that occurs often in mathematics is the idea of representing something in multiple ways. In these lessons, relationships between two quantities are studied (such as the cost of a product and the amount purchased). These relationships are represented by verbal descriptions, 2-column tables, equations, and graphs.
(Note: the use of the word decide in part a. of the standard is incorrect. The word determine should be in its place. You decide what clothes to wear. You determine the solution to a math problem.)
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Here is a bird’s-eye view of the material. The lessons have some fill-in-the-blanks for the student to complete. The correct fill-ins are included here.
These lessons are presented for guardians to help their child(ren) address the following Common Core State Standard (CCSS) for Grade 7:
- 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 or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.
- Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions 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.
- Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.
As with any math lessons, a certain amount of basic material and vocabulary (taught prior to grade 7) is assumed. Specifically, it is assumed that the student is familiar with the following material.
- The definition of ratio and unit rate
- Writing simple equations to represent relationships (e.g., y = 5x)
- Rewriting equations by multiplying or dividing both sides by the same number
- Graphing points on a coordinate grid
- Vocabulary from earlier grades: coordinate plane, ordered pair, origin, ray, x-axis, y-axis
Summary and vocabulary
Proportional relationship: exists between a quantity y and a quantity x when the ratio is constant.
Constant of proportionality: the ratio between two quantities in a proportional relationship; also known as the unit rate
Facts and Theorems: If a quantity y is proportional to a quantity x, then the quantity x is proportional to the quantity y.
The graph of a proportional relationship is a line through the origin or a ray whose endpoint is the origin.
Conversely, given a line through the origin or a ray whose endpoint is the origin, this represents a proportional relationship.
Formulas/Equations: A proportional relationship between a quantity y and a quantity x that has a constant of proportionality k is represented by the equation.
Other: You can take the equation for a proportional relationship and divide both sides by k to write the relationship the other way (switch the input and output).
Not all the points on the graph of a proportional relationship necessarily represent real values. For example, if the input is a number of people, then x-values must be whole numbers.
The point on the graph of a proportional relationship shows that the unit rate (or constant of proportionality) is r.
A line that does not go through the origin is not a proportional relationship.
Any curved graph, including one that goes through the origin, is not a proportional relationship.
In order to motivate the definition of proportional relationship, two examples of relationships between two quantities are given. One of these is a proportional relationship and one is not.
Page 1 fill-ins: $3.50; $1.10
Examples 1 and 2 present quantities in tables. It is a straightforward matter to compute the ratios of one quantity over the other to see if these ratios have the same value. Examples 3 and 4 require the student to create tables from verbal descriptions, and then check the ratios to see if there is a proportional relationship.
Example 5 is different because you are told that there is a proportional relationship at the start and the student must compute missing values in the table.
Page 2 fill-ins: 3, 7; 550, 550
Page 3 fill-ins: 0.90, 90; are not, is not; 3.4, 3.75, 12.75, 14.25, 3.75, 3.8
The first 6 exercises mimic the first two examples. Exercises 7 through 9 are similar to Examples 3 and 4. Exercises 10 through 12 are similar to Example 5.
The lesson starts with an example of a proportional relationship and explains how to write this as an equation. Then it shows that you can write a similar equation ( y =kx ) for any proportional relationship.
Page 1 fill-ins: 3, 10.50
The first two examples show proportional relationships represented as a table and a verbal description respectively, and ask for equations. Examples 3 and 4 give you equations and ask for a verbal description and a table. The point is to understand that you can represent the same relationship multiple ways, and to know how to go from one to another.
Page 2 fill-ins: 550; 6.40; 18, 5, 30
The first 6 exercises mimic the first two examples. Exercises 7 through 12 mostly match Examples 3 and 4. The information on the bottom of page 1 and the middle of page 2 about equations for proportional relationships and the reverse relationship are also needed to answer the latter exercises.
At this point, the student knows several ways to represent proportional relationships. In a straightforward procedure of plotting points, students discover that these points lie on a line through the origin (that is, the point (0, 0)).
Page 1 fill-ins: 3, 8
Page 2 starts with an explanation of why the points lie on a line or a ray (a line continues in both directions forever and a ray is “half” of a line). Example 1 starts with an equation, creates a table (as in Lesson 2), and then draws the graph of the relationship. It makes the important point that not all points on this graph correspond to real world values.
Page 2 fill-ins: 14, 17.50
Example 2 gives the student a graph and asks them to interpret it in terms of the context. It also asks the student to find the equation of the proportional relationship.
Page 3 fill-ins: 3, 90, 30
Example 3 exists to clarify which graphs do and do not represent proportional relationships. Examples 4 and 5 show relationships that are “linear” but are not “proportional.”
Page 4 fill-ins: 0, 0; origin; 0, 20
The first two exercises are similar to Example 1. The next two are similar to Example 4. Exercises 5 and 6 are similar to Example 5. Exercises 7 and 8 are most similar to Example 2 (look back to Lesson 2 for more examples of finding equations). Exercises 9 through 12 are like Example 3.
Exercises with answers
Does the table represent a proportional relationship?
If so, what is the constant of proportionality?
1. Answer: No
2. Answer: Yes; 3
3. Answer: Yes; 1.2
4. Answer: Yes; 335
5. Answer: No
6. Answer: Yes; $11.50
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?
Answer: yes; $0.60/sq. ft.
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?
Answer: yes; 128 oz./gallon
Each table represents a proportional relationship. Fill in the missing numbers.
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.
Answer: Yes. The z/y ratio is constant and the ratio y/x is constant. It follows that their product, z/x, is constant. This says that there is a proportional relationship between z and x.
Represent each proportional relationship with an equation.
1. Answer: C = 0.9A
2. Answer: w = 335t
|Time, t (hours)||Widgets, w|
3. Answer: C = 11.5n
|Number, n||Cost, C|
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.
Answer: C = 0.6A
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.
Answer: B = 4w
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.
Answer: h = 9n
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
Answer: x = ¼y
8. y = 0.2x
Answer: x = 5y
9. y = x ⁄ 3
Answer: x = 3y
10. Does each equation represent a proportional relationship?
(a) y = x + 5 Answer: no (b)y = x 2 Answer: no (c) y = 2(x – 5) + 10 Answer: yes
11. The equation y = 2.54x represents the relationship between inches, x, and centimeters, y. Give a verbal description of the relationship.
Answer: For every inch, there are 2.54 centimeters.
12. The equation y = 33.8x represents the relationship between liters, x, and fluid ounces, y. Give a verbal description of the relationship.
Answer: For every liter, there are 33.8 fluid ounces.
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.
Answer: At midnight, the temperature is 0°C. Thereafter, the temperature is dropping by 2°C per hour.
Draw the graphs of the given equations.
1. y = 3x, where x= the number of pounds of ground beef, and y = the price in dollars.
2. y = 0.5x, with no restrictions on the variables
Answer: Graphs below for exercises 1 through 6
For each table, draw the graph and determine if the relationship is proportional.
3. Answer: proportional
4. Answer: not proportional
For each description, draw the graph and determine if the relationship is proportional.
5. A tutor charges $50 for one hour, $90 for two hours, and $120 for three hours.
Answer: not proportional
6. Asparagus sells for $2.50 per pound.
7. For each relationship in exercises 3 through 6 that was proportional, find the equation.
Answer: For exercise 3: y = 1.25x ri. For exercise 6: y = 2.5x
8. The graph shows the relationship between the time, x, in minutes someone jogs on a treadmill and the distance, y, in miles they run. Explain the meaning of the points (4, 0.4), (1, .01) and (0,0) in this context.
Answer: The person jogs 0.4 mile in 4 minutes, a unit rate of 0.1 mile per minute, and no distance at the start.
Does each graph below represent a proportional relationship?
9. Answer: yes 10. Answer: no
11. Answer: no
12. Answer: no
13. Challenge Problem:
A line passes through the point (a, b), where both coordinates are positive. If the line represents a proportional relationship, what is its equation? Explain.
Answer: The ratio of output to input is the constant of proportionality. This is b ⁄ a, so the equation is y = b ⁄ ax.