Lesson 4: Identifying Proportional and Non-Proportional Relationships in Graphs
- A proportional relationship makes a straight line that passes through the origin (0, 0). The ratio stays the same.
- A non-proportional relationship may be straight or curved but does not go through the origin, or the ratio changes.
Part B
1. Hours worked and money earned
| Hours | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Money | 10 | 20 | 30 | 40 |
| Ratio | 10÷1=10 | 20÷2=10 | 30÷3=10 | 40÷4=10 |
Answer: Proportional, constant = $10 per hour
It is a straight line, which means money increases at a constant rate of $10 per hour.
2. Sandwiches and cost
| Sandwiches | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Cost ($) | 5 | 9 | 14 | 18 |
| Ratio | 5÷1=5 | 9÷2=4.5 | 14÷3≈4.67 | 18÷4=4.5 |
Answer: Not proportional
The relationship is not a straight line, which shows the cost does not increase at a constant rate for each additional sandwich.
3. Pencils and cost
| Pencils | 2 | 4 | 6 | 8 |
|---|---|---|---|---|
| Cost $ | 1 | 2 | 3 | 4 |
| Ratio | 1÷2=0.5 | 2÷4=0.5 | 3÷6=0.5 | 4÷8=0.5 |
Answer: Proportional, constant = $0.50 per pencil
It forms a straight line, showing the cost increases at a constant rate of $0.50 per pencil.
4. Tickets and cost
| Tickets | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Cost $ | 12 | 24 | 35 | 48 |
| Ratio | 12÷1=12 | 24÷2=12 | 35÷3≈11.67 | 48÷4=12 |
Answer: Not proportional
The line is not perfectly straight, which shows the cost per ticket is not constant. The first two tickets cost $12 each, but the 3rd and 4th tickets are slightly more expensive.
Part C. Extra Practice Problems
5. Hours and distance traveled
| Hours | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Distance | 40 | 80 | 120 | 160 |
| Ratio | 40÷1=40 | 80÷2=40 | 120÷3=40 | 160÷4=40 |
Answer: Proportional, constant = 40 miles/hour
It is a straight line through the origin, showing a proportional relationship: the object moves 40 km every hour at a constant speed.
6. Bags of apples and cost
| Bags | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Cost | 4 | 7 | 10 | 13 |
| Ratio | 4÷1=4 | 7÷2=3.5 | 10÷3≈3.33 | 13÷4=3.25 |
Answer: Not proportional
The cost goes up by $3 each time, but it does not start at zero. The first bag already costs $4. That means the relationship is linear but not proportional.
7. Minutes and texts sent
| Minutes | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Texts | 15 | 30 | 45 | 60 |
| Ratio | 15÷1=15 | 30÷2=15 | 45÷3=15 | 60÷4=15 |
Answer: Proportional, constant = 15 texts/min
It is a straight line through the origin, showing a proportional relationship: you send 15 texts every minute at a constant rate.
8. Notebooks and cost
| Notebooks | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Cost $ | 3 | 6 | 8 | 12 |
| Ratio | 3÷1=3 | 6÷2=3 | 8÷3≈2.67 | 12÷4=3 |
Answer: Not proportional
The points do not form a straight line, showing that the cost per notebook is not constant.
9. Hours and pay
| Hours | 2 | 4 | 6 | 8 |
|---|---|---|---|---|
| Pay | 30 | 60 | 90 | 120 |
| Ratio | 30÷2=15 | 60÷4=15 | 90÷6=15 | 120÷8=15 |
Answer: Proportional, constant = $15/hour
It forms a straight line through the origin, showing a proportional relationship: the pay increases at a constant rate of $15 per hour.
10. Pizzas and price
| Pizzas | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Price | 12 | 25 | 36 | 48 |
| Ratio | 12÷1=12 | 25÷2=12.5 | 36÷3=12 | 48÷4=12 |
Answer: Not proportional
The points do not lie on a straight line, showing the price per pizza is not constant. The first pizza costs $12, the second adds $13, the third adds $11, and the fourth adds $12.
11. Shirts and cost
| Shirts | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Cost $ | 8 | 16 | 24 | 32 |
| Ratio | 8÷1=8 | 16÷2=8 | 24÷3=8 | 32÷4=8 |
Answer: Proportional, constant = $8/shirt
It forms a straight line through the origin, showing a proportional relationship: each shirt costs $8, so the total cost increases at a constant rate.
12. Pens and cost
| Pens | 2 | 4 | 6 | 8 |
|---|---|---|---|---|
| Cost $ | 1 | 2 | 4 | 5 |
| Ratio | 1÷2=0.5 | 2÷4=0.5 | 4÷6≈0.67 | 5÷8=0.625 |
Answer: Not proportional
The points do not fall on a straight line. The cost per pen is not constant.
13. Days and pages read
| Days | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Pages | 12 | 24 | 36 | 48 |
| Ratio | 12÷1=12 | 24÷2=12 | 36÷3=12 | 48÷4=12 |
Answer: Proportional, constant = 12 pages/day
It is a straight line through the origin and shows a proportional relationship: you read 12 pages per day at a constant rate.
14. Video games and cost
| Games | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Cost $ | 20 | 40 | 55 | 80 |
| Ratio | 20÷1=20 | 40÷2=20 | 55÷3≈18.33 | 80÷4=20 |
Answer: Not proportional
The points do not lie on a straight line. The cost per game is not constant.
15. Minutes walked and calories burned
| Minutes | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Calories | 4 | 8 | 12 | 16 |
| Ratio | 4÷1=4 | 8÷2=4 | 12÷3=4 | 16÷4=4 |
Answer: Proportional, constant = 4 calories/min
It forms a straight line through the origin, so it is proportional: you burn 4 calories for every minute walked at a constant rate.
16. Books and cost
| Books | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Cost $ | 7 | 15 | 21 | 28 |
| Ratio | 7÷1=7 | 15÷2=7.5 | 21÷3=7 | 28÷4=7 |
Answer: Not proportional
The points do not form a straight line, so it is not proportional.
17. Jars of jam and cost
| Jars | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Cost $ | 6 | 12 | 18 | 24 |
| Ratio | 6÷1=6 | 12÷2=6 | 18÷3=6 | 24÷4=6 |
Answer: Proportional, constant = $6/jar
It is a straight line through the origin, showing a proportional relationship: each jar costs $6, so the total cost increases at a constant rate.
18. Days and total pay
| Days | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Pay | 50 | 100 | 160 | 200 |
| Ratio | 50÷1=50 | 100÷2=50 | 160÷3≈53.3 | 200÷4=50 |
Answer: Not proportional
The points do not fall on a straight line. The daily pay is not constant:
19. Hours cleaning and rooms cleaned
| Hours | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Rooms | 2 | 4 | 6 | 8 |
| Ratio | 2÷1=2 | 4÷2=2 | 6÷3=2 | 8÷4=2 |
Answer: Proportional, constant = 2 rooms/hour
It is a straight line through the origin, showing a proportional relationship: 2 rooms are cleaned per hour at a constant rate.
20. Soccer games and goals scored
| Games | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Goals | 1 | 3 | 4 | 7 |
| Ratio | 1÷1=1 | 3÷2=1.5 | 4÷3≈1.33 | 7÷4=1.75 |
Answer: Not proportional
The curve is not straight. The goals per game are not consistent.