Introduction
Finding the Constant of Proportionality is an important Grade 7 math skill because students are moving from simple answers toward explaining how the math works.
In this lesson, students use models, real questions, worked examples, practice problems, and two online quizzes to build confidence with finding the constant of proportionality.
What Is Finding the Constant of Proportionality?
Finding the Constant of Proportionality means choosing a model, naming what each number means, and explaining the strategy.
The goal is not only to get the answer. Students should be able to show the idea, explain the strategy, and check whether the answer makes sense.
Understanding Finding the Constant of Proportionality
Before solving, students should slow down and decide what each number, shape, unit, or label represents.
- Read the question carefully and identify what is being asked.
- Choose a model, equation, table, or diagram that matches the situation.
- Solve one step at a time and keep units or labels attached.
- Use the answer explanation to check that the result makes sense.
Visual Models
Visual Model 1
Question: The table shows a proportional relationship between pounds of sugar (\(s\)) and the number of cookies baked (\(c\)). In the equation \(c = ks\), what is the constant of proportionality (\(k\))?
| Pounds of Sugar | Cookies Baked |
|---|---|
| 2 | 24 |
| 3 | 36 |
| 5 | 60 |
- A. \(8\)
- B. \(10\)
- C. \(15\)
- D. \(12\)
Why it works: Using any pair in \(c = ks\): \(k = c/s = 24/2 = 12\) cookies per pound of sugar.
Answer: \(12\)
Visual Model 2
Question: The table shows the relationship between gallons of paint (\(g\)) and square feet painted (\(p\)). What is the value of \(k\) in the equation \(p = kg\)?
| Gallons | Square Feet |
|---|---|
| 1 | 400 |
| 2 | 800 |
- A. \(100\)
- B. \(800\)
- C. \(200\)
- D. \(400\)
Why it works: \(k = p/g = 400/1 = 400\) square feet per gallon.
Answer: \(400\)
Worked Examples
Example 1
Question: The graph shows a proportional relationship. One point on the graph is \((4,5)\). What is the constant of proportionality?
- A. \(0.75\)
- B. \(1.0\)
- C. \(1.5\)
- D. \(1.25\)
- The point \((4,5)\) is on the line; \(k = y/x = 5/4 = 1.25\).
Answer: \(1.25\)
Example 2
Question: A water pump fills a tank at a constant rate. The table shows the gallons of water (\(w\)) pumped over time (\(t\)) in minutes. What is the constant of proportionality?
| Time (minutes) | Water (gallons) |
|---|---|
| 2 | 16 |
| 4 | 32 |
| 6 | 48 |
- A. \(4\) gallons/min
- B. \(6\) gallons/min
- C. \(16\) gallons/min
- D. \(8\) gallons/min
- \(k = 16/2 = 8\) gallons per minute.
Answer: \(8\) gallons/min
Example 3
Question: The graph represents a proportional relationship, \(y=kx\). One point on the graph is \((2, 8)\). What is the constant of proportionality, \(k\)?
- A. \(2\)
- B. \(8\)
- C. \(6\)
- D. \(4\)
- From point \((2,8)\): \(k = y/x = 8/2 = 4\).
Answer: \(4\)
Real-World Word Problems
Problem 1
Question: The cost \(c\) of buying \(n\) identical books is proportional to \(n\). If \(4\) books cost \($18\), what is the constant of proportionality (unit price per book)?
- A. \($2.50\)
- B. \($4.00\)
- C. \($14.00\)
- D. \($4.50\)
Why it works: The constant of proportionality is \(k=c/n=18/4=$4.50\) per book.
Answer: \($4.50\)
Problem 2
Question: A bike rental costs \($2.50\) per hour. If \(h\) is the number of hours and \(c\) is the total cost, what is the constant of proportionality in the equation \(c = kh\)?
- A. \(1.0\)
- B. \(0.40\)
- C. \($5.00\)
- D. \(2.5\)
Why it works: In the equation \(c = 2.5h\), the coefficient \(k = 2.5\) is the constant of proportionality (unit rate per hour).
Answer: \(2.5\)
Common Mistakes
- Rushing before identifying what the numbers represent.
- Choosing an operation that does not match the situation.
- Dropping labels, units, or context from the answer.
- Skipping the estimate or reasonableness check.
Strategy Tips
- Underline the question being asked.
- Use a model before jumping to computation.
- Write an equation that matches the story or picture.
- Explain the final answer in a sentence.
Practice Questions
Question 1
Which equation represents a proportional relationship with a constant of proportionality of \(3\)?
- A. \(y = 3x + 2\)
- B. \(y = x + 3\)
- C. \(y = \frac{x}{3}\)
- D. \(y = 3x\)
Question 2
A car travels \(120\) miles in \(2\) hours at a constant speed. What is the constant of proportionality (miles per hour)?
- A. \(240\) mph
- B. \(90\) mph
- C. \(100\) mph
- D. \(60\) mph
Question 3
A recipe calls for \(3\) cups of flour to make \(12\) cookies. At this rate, what is the constant of proportionality (cups per cookie)?
- A. \(4.0\)
- B. \(0.75\)
- C. \(3.0\)
- D. \(0.25\)
Question 4
A streaming service charges a proportional rate. If \(8\) months of service costs \($96\), what is the constant of proportionality (cost per month)?
- A. \($8\)
- B. \($10\)
- C. \($16\)
- D. \($12\)
Question 5
In the proportional relationship \(y = \frac{2}{3}x\), what is the constant of proportionality?
- A. \(\frac{1}{3}\)
- B. \(2\)
- C. \(\frac{3}{2}\)
- D. \(\frac{2}{3}\)
Question 6
A gym membership costs \($45\) per month. If \(m\) is the number of months and \(c\) is the total cost, what is the constant of proportionality?
- A. \($22.50\)
- B. \($40\)
- C. \($90\)
- D. \($45\)
Full Answer Explanations Click to show all answers and explanations
Question 1
Answer: \(y = 3x\)
A proportional relationship has the form \(y = kx\) with no constant term. Here \(k = 3\).
Question 2
Answer: \(60\) mph
Speed is distance divided by time: \(k = 120/2 = 60\) miles per hour.
Question 3
Answer: \(0.25\)
\(k = \frac{\text{flour}}{\text{cookies}} = \frac{3}{12} = 0.25\) cups per cookie.
Question 4
Answer: \($12\)
\(k = 96/8 = $12\) per month.
Question 5
Answer: \(\frac{2}{3}\)
The coefficient of \(x\) is the constant of proportionality: \(k = \frac{2}{3}\).
Question 6
Answer: \($45\)
In \(c = 45m\), the constant of proportionality is \(k = 45\) dollars per month.
Connection to Standards
This lesson supports Grade 7 math expectations for reasoning, modeling, problem solving, and explaining answers clearly. It connects classroom skills to the kind of questions students see on state math assessments.
Summary
Finding the Constant of Proportionality becomes easier when students connect the question to a model, use clear steps, and explain why the answer fits.
GOLDEN RULE
Understand the model before choosing the operation.
