Introduction
Proportional vs. Non-Proportional Relationships is an important Grade 6 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 proportional vs. non-proportional relationships.
What Is Proportional vs. Non-Proportional Relationships?
Proportional vs. Non-Proportional Relationships 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 Proportional vs. Non-Proportional Relationships
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: Which table shows a proportional relationship between \(x\) and \(y\)?
| x | \(1\) | \(2\) | \(3\) |
|---|---|---|---|
| y (option A) | \(5\) | \(10\) | \(15\) |
| y (option B) | \(3\) | \(5\) | \(8\) |
- A. Option A
- B. Option B
- C. Both A and B
- D. Neither
Why it works: A relationship is proportional if \(y/x\) is constant. Option A has \(5/1=5\), \(10/2=5\), and \(15/3=5\), so it is proportional. Option B has \(3/1=3\) and \(5/2=2.5\), so the ratio is not constant.
Answer: Option A
Visual Model 2
Question: A recipe calls for 2 cups of flour for every 3 cups of sugar. Which statement correctly describes the table?
| Flour (\(f\)) | \(2\) | \(4\) | \(6\) | \(8\) |
|---|---|---|---|---|
| Sugar (\(s\)) | \(3\) | \(6\) | \(9\) | \(12\) |
- A. Proportional; \(s/f = 2/3\) throughout
- B. Not proportional; the ratio changes
- C. Proportional; \(f/s = 2/3\) throughout
- D. Cannot determine from the table
Why it works: The ratio of flour to sugar is \(f/s = 2/3\), \(4/6 = 2/3\), \(6/9 = 2/3\), \(8/12 = 2/3\). The ratio remains constant, so the relationship is proportional.
Answer: Proportional; \(f/s = 2/3\) throughout
Worked Examples
Example 1
Question: Which graph represents a proportional relationship between \(x\) and \(y\)?
- A. Graph A only
- B. Graph B only
- C. Both graphs
- D. Neither graph
- A proportional relationship must pass through the origin \((0, 0)\).
- Graph A passes through the origin; Graph B starts at \((0, 1)\), so it does not pass through the origin.
Answer: Graph A only
Example 2
Question: A table shows the number of hours worked (\(h\)) and the amount of pay earned (\(p\), in dollars). Which option gives both the correct unit rate and the correct equation for the relationship?
| Hours (\(h\)) | \(1\) | \(2\) | \(3\) | \(4\) |
|---|---|---|---|---|
| Pay (\(p\)) | \(12\) | \(24\) | \(36\) | \(48\) |
- A. Unit rate: $12 per hour; equation: \(p = 12h\)
- B. Unit rate: $24 per hour; equation: \(p = 24h\)
- C. Unit rate: $12 per hour; equation: \(h = 12p\)
- D. The relationship is not proportional
- The ratio \(p/h\) is constant: \(12/1 = 24/2 = 36/3 = 48/4 = 12\).
- That means the unit rate is $12 per hour, and the equation \(p = 12h\) represents the proportional relationship.
Answer: Unit rate: $12 per hour; equation: \(p = 12h\)
Example 3
Question: Which of the following does NOT represent a proportional relationship?
| x | \(2\) | \(4\) | \(6\) |
|---|---|---|---|
| y (option A) | \(6\) | \(12\) | \(18\) |
| y (option B) | \(4\) | \(7\) | \(10\) |
- A. Option A
- B. Option B
- C. Both A and B
- D. Both are proportional
- Option A has a constant ratio: \(6/2 = 3\), \(12/4 = 3\), and \(18/6 = 3\).
- Option B does not: \(4/2 = 2\), \(7/4 = 1.75\), and \(10/6 \approx 1.67\).
Answer: Option B
Real-World Word Problems
Problem 1
Question: A taxi charges $2 per mile traveled. If \(m\) is the number of miles and \(c\) is the total cost in dollars, which equation represents this proportional relationship?
- A. \(c = 2m\)
- B. \(c = 2m + 1\)
- C. \(m = 2c\)
- D. \(c = m + 2\)
Why it works: Since the taxi charges $2 per mile with no additional fee, the total cost is directly proportional to miles: \(c = 2m\). This passes through the origin when \(m = 0, c = 0\). Options B, C, and D either add a constant or reverse the relationship.
Answer: \(c = 2m\)
Problem 2
Question: The graph below shows distance traveled (\(d\), in miles) over time (\(t\), in hours). Does the relationship show a constant speed?
- A. No; the line passes through the origin
- B. No; the line is not straight
- C. Yes; the graph is a straight line through the origin
- D. Cannot determine from the graph
Why it works: A proportional distance-time graph is a straight line through the origin. Here, the line passes through \((0,0)\), \((1,3)\), and \((2,6)\), so the speed stays constant at \(3\) miles per hour.
Answer: Yes; the graph is a straight line through the origin
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
A store charges $5 per notebook. An employee has a discount and gets the first notebook free, then pays $5 for each additional one. For \(n\) notebooks, the total cost is \(c = 5(n - 1)\). Is this a proportional relationship?
- A. Yes; the equation is linear
- B. Cannot determine without the graph
- C. Yes; the cost depends on notebooks
- D. No; the ratio of cost to notebooks is not constant
Question 2
Which table represents a non-proportional relationship?
| x | \(1\) | \(3\) | \(5\) |
|---|---|---|---|
| y (option A) | \(7\) | \(21\) | \(35\) |
| y (option B) | \(2\) | \(4\) | \(6\) |
- A. Option A; the ratio is 7:1
- B. Option B; the ratio is not constant
- C. Option A; each \(y\) is \(x\) squared
- D. Neither; both are proportional
Question 3
A cyclist travels at 15 miles per hour. Which equation represents the distance \(d\) (in miles) after \(t\) hours?
- A. \(d = 15t\)
- B. \(d = 15 + t\)
- C. \(t = 15d\)
- D. \(d = 15t + 1\)
Question 4
Look at the data in the table. Is the relationship between hours worked and pay proportional?
| Hours | \(0\) | \(1\) | \(2\) | \(3\) |
|---|---|---|---|---|
| Pay ($) | \(5\) | \(10\) | \(15\) | \(20\) |
- A. No; the first data point is not at the origin
- B. Yes; the pay increases by $5 each hour
- C. Yes; the equation is \(p = 5h + 5\)
- D. Cannot determine without a graph
Question 5
A fruit stand sells apples for $2 each. Maria buys 1, 2, 3, and 4 apples. Which graph correctly shows the total cost?
- A. The graph correctly shows the relationship
- B. The line should not pass through the origin
- C. The line should have a different unit rate
- D. The graph should start at \((0,2)\)
Question 6
A car travels at a constant speed. After 2 hours, it has traveled 100 miles. After 4 hours, it has traveled 200 miles. Is this relationship proportional?
- A. Yes; the speed is constant at 50 mph
- B. No; the distance increases by 100 miles each hour
- C. Yes; the equation is \(d = 100t\)
- D. No; the speed is not constant
Full Answer Explanations Click to show all answers and explanations
Question 1
Answer: No; the ratio of cost to notebooks is not constant
This is not proportional because the cost per notebook is not constant. For example, \(1\) notebook costs \($0\), but \(2\) notebooks cost \($5\), so the ratio of cost to notebooks changes.
Question 2
Answer: Option B; the ratio is not constant
Option A has a constant ratio: \(7/1 = 7\), \(21/3 = 7\), and \(35/5 = 7\). Option B does not: \(2/1 = 2\), \(4/3 \approx 1.33\), and \(6/5 = 1.2\), so Option B is non-proportional.
Question 3
Answer: \(d = 15t\)
The proportional relationship between distance and time is \(d = 15t\). When \(t = 0\), \(d = 0\), indicating the line passes through the origin. Options B and D include constants (non-proportional), and option C reverses the variables.
Question 4
Answer: No; the first data point is not at the origin
For a proportional relationship, when hours \(= 0\), pay must \(= 0\). Here, when hours \(= 0\), pay \(= $5\) (a fixed fee), so the relationship is not proportional. The equation is \(p = 5h + 5\), which includes a non-zero constant term.
Question 5
Answer: The graph correctly shows the relationship
For apples at $2 each, the equation is \(c = 2a\). The graph passes through \((0,0)\), \((1,2)\), \((2,4)\), \((3,6)\), and \((4,8)\), representing a proportional relationship with a unit rate of $2 per apple.
Question 6
Answer: Yes; the speed is constant at 50 mph
The ratio of distance to time is \(100/2=50\) and \(200/4=50\) mph, confirming constant speed. The proportional equation is \(d=50t\), not \(d=100t\).
Connection to Standards
This lesson supports Grade 6 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
Proportional vs. Non-Proportional Relationships 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.
