(7) Lesson 1.9 - Direct Variation

1. Course 2, Lesson 1-9
1. Terrence’s phone bill one month was $27.60 for 345 minutes. The
following month his bill was $34.56 for 432 minutes. Draw a
graph of minutes versus cost. Find the numerical value of the
slope and interpret it in words.
2. The graph shows the amount of
money in an account over time.
What does the slope of the line
represent?

2. Course 2, Lesson 1-9
ANSWERS
1. slope = 0.08; The slope
represents the cost per
minute.
2. The amount of increase every month.

3. HOW can you show that
two objects are proportional?
Ratios and Proportional Relationships
Course 2, Lesson 1-9

4. Ratios and Proportional Relationships
Course 2, Lesson 1-9 Common Core State Standards © Copyright 2010. National Governors Association Center for Best Practices and Council
of Chief State School Officers. All rights reserved.
• 7.RP.2
Recognize and represent proportional relationships between quantities.
• 7.RP.2a
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.
• 7.RP.2b
Identify the constant of proportionality (unit rate) in tables, graphs,
equations, diagrams, and verbal descriptions of proportional relationships.

5. Ratios and Proportional Relationships
Course 2, Lesson 1-9 Common Core State Standards © Copyright 2010. National Governors Association Center for Best Practices and Council
of Chief State School Officers. All rights reserved.
Mathematical Practices
1 Make sense of problems and persevere in solving them.
2 Reason abstractly and quantitatively.
3 Construct viable arguments and critique the reasoning of others.
4 Model with mathematics.

6. • To find the constant of
proportionality of a direct
variation from a graph or an
equation
• To determine if a situation
represents a direct variation
Ratios and Proportional Relationships
Course 2, Lesson 1-9

7. Ratios and Proportional Relationships
Course 2, Lesson 1-9
• direct variation
• constant of variation
• constant of proportionality

8. Ratios and Proportional Relationships
Course 2, Lesson 1-9
Words A linear relationship is a Model
direct variation when the
ratio of y to x is a constant,
k. We say y varies directly
with x.
Symbols
Example y = 2x

= or = ,
where 0
y k y kx
x
k

9. 1
Need Another Example?
2
3
Step-by-Step Example
1. The height of the water as a pool is being filled is shown in the
graph. Determine the rate in inches per minute.
Since the graph of the data forms a line, the rate of change is
constant. Use the graph to find the constant of proportionality.
The pool fills at a rate of 0.4 inch every minute.

10. Answer
Need Another Example?
The amount of money
Serena earns at her job
is shown in the graph.
Determine the rate in
dollars per hour.
$10 per hour

11. 1
Need Another Example?
2
3
Compare the equation to y = kx, where k is the constant of
proportionality.
Step-by-Step Example
2. The equation y = 10x represents the amount of
money y Julio earns for x hours of work. Identify the
constant of proportionality. Explain what it represents
in this situation.
y = kx
y = 10x
The constant of proportionality is 10. So, Julio earns $10 for
every hour that he works.

12. Answer
Need Another Example?
Neil is practicing for his typing test. The
equation y = 45x represents the total number
of words y he can type in x minutes. Identify
the constant of proportionality. Then explain
what it represents in this situation.
45; Neil can type 45 words per minute

13. 1
Need Another Example?
2
3
Step-by-Step Example
3. Pizzas cost $8 each plus a $3 delivery charge.
Show the cost of 1, 2, 3, and 4 pizzas. Is there a
direct variation?
There is no constant ratio and the line does not go through
the origin. So, there is no direct variation.

14. Answer
Need Another Example?
A photographer charges a $30 sitting fee and
then $6 for each photograph ordered. Make a
table and a graph to show the cost of 1, 2, 3,
and 4 photographs. Is there a direct
variation? Explain.
no; Sample answer: ≠ ; Because there is
no constant ratio and the line does not go
through the origin, there is no direct variation.

15. 1
Need Another Example?
2
3
Step-by-Step Example
4. Determine whether the linear function is a direct
variation. If so, state the constant of proportionality.
Compare the ratios to check for a common ratio.
Since the ratios are the same, the function is a direct variation.
The constant of proportionality is .

16. Answer
Need Another Example?
Determine whether the linear function is a
direct variation. If so, state the constant of
proportionality.
no

17. How did what you learned
today help you answer the
HOW can you show that
two objects are proportional?
Course 2, Lesson 1-9
Ratios and Proportional Relationships

18. How did what you learned
today help you answer the
HOW can you show that
two objects are proportional?
Course 2, Lesson 1-9
Ratios and Proportional Relationships
Sample answers:
• By identifying a constant of variation and explaining
what it means in the context of a real-world situation

19. Explain what a
constant of proportionality
represents in a
direct variation.
Ratios and Proportional RelationshipsRatios and Proportional Relationships
Course 2, Lesson 1-9