Unit 5: Proportional Relationships

Unit 5: Proportional

Relationships

Learning Target

I know what the Constant of proportionality

I can find the Constant of proportionality of a table, graph, equation

I can determine when a graph is linear

I can determine when a graph is Proportional

Multiple Choice

a measure of the steepness of a line, or a section of a line, connecting two points

• Intersecting Lines

• Origin

• Proportional Relationships

• Slope

Vocabulary

Direct Variation: when two things change in the SAME way.
They are positive, proportional relationships to each other.
**NOTE that the change does not have to be positive. Because if they are both
changing negatively, then it will become positive.

Proportional Relationships: two quantities are in a proportional
relationship when they vary directly. That means, when one
thing increases, the other thing increases as well. When one
thing decreases, so does the other.

This pattern must be CONSISTENT!

Formula for Proportional Relationships:

All proportional relationships fit the same
formula:

y = kx

Total

(dependent variable)
Focus

(independent variable)

Constant of Proportionality!!!!

k =

y
x Does this look like anything you've seen

before?

RATES OF CHANGE

Slope

Unit Rate

Constant of Proportionality

Scale

Multiple Choice

Another word for slope is....

• rate of change

• fills the inside

• covers the outside

• change in size

Prior Knowledge

●

Who can tell me what the
fancy word is for Unit Rate
in proportional situations?

What letter do we use?

Constant of
Proportionality

“K”

Remember, on a graph…

The independent variable
appears on the x-axis.

The dependent variable
appears on the y-axis.

Linear Relationship: Shows a relationship between two
variables. When one variable changes, the other variable
also changes. The changes are consistent.

Proportional Relationship: Shows a relationship between
two variables. When one variable changes, the other
variable changes in the same way! The changes are
consistent.

Characteristics of a LINEAR Relationship:

Graph is a straight line.
Consistent.

Characteristics of PROPORTIONAL Relationship:

Graph is a straight line.
Goes through the Origin (0,0).
Consistent.
Slope is positive.

Multiple Choice

Which one does NOT belong to NON-PROPORTIONAL relationships?

• y=mx

• Does NOT cross through origin

• y/x is not the same

• equation will have plus (+) something

Multiple Choice

What makes a graph proportional?

• Straight Line

• The line goes through (0,0)

• Curvy line

• Straight line and passes through (0,0)

Proportional Graphs

The graph of a proportional relationship is a Straight-Line that
passes through the origin (0, 0).

Proportional

Not Proportional

Not Proportional

Is the graph proportional or non-proportional?

Why?

1)

YOU
TRY!

Is the graph proportional or non-proportional?

Why?

2)

YOU
TRY!

Is the graph proportional or non-proportional?

Why?

3)

YOU
TRY!

Multiple Choice

What makes a graph proportional?

• Straight Line

• The line goes through (0,0)

• Curvy line

• Straight line and passes through (0,0)

Multiple Choice

What best describes the data on this graph?

• This graph is linear and not proportional.

• This graph is proportional.

• This graph is non-linear.

▪ The rate for 1 of something is called the Unit Rate.

▪ On the previous slide, the cost for 1 concert ticket was $21.

▪ This value is also called the Constant of Proportionality.

Let’s look at the steps for finding the Constant of

Proportionality from a graph…

Multiple Choice

when rates are expressed as a quantity of 1, they are called

• Intersecting Lines

• Origin

• Proportional Relationships

• Unit Rate

Finding the Constant of Proportionality from a Graph

1)

YOU
TRY!

4
1

Y = 4x

2)

(R) Rand is the money for South
Africa.

YOU
TRY!

Notice that our previous technique would not work on
this graph because the graphed line does not pass
though a lattice point at the quantity of 1.
We need different steps for this one!

Finding the Constant of Proportionality from a Graph

This means that the
pool is filled with 5
gallons of water in 1
minute.

Multiple Choice

Which equation matches the graph?

• y = 7x

• y = 42x

• y = 6x

• y = 2x

Multiple Choice

What is the unit rate in this proportional relationship?

• 60 miles/1 hour

• 120 miles/ 2 hours

• 1 mile/ 1 hour

Multiple Choice

What is the unit rate/constant of proportionality?

• 80 miles in 1 hours

• 20 miles in 1 hours

• 40 miles in 1 hours

• 160 miles in 1 hours

Multiple Choice

Which statement matches the graph?

• Taylor can write 6 words per minute

• Taylor can write 25 words per minute

• Taylor can write 50 words per minute

• Taylor can write 150 words per minute

Multiple Choice

How much money does Miguel save per month?

• 15 dollars per month

• 30 dollars per month

• He does not save any money.

1

2

3

4

5

10

12

14

16

18

Determine if the table is linear or Proportional?
WHY

1

2

3

4

5

2

4

6

8

10

6

7

8

1

2

3

4

5

10

8

6

4

2

10

12

14

16

18

1

2

3

4

5

Linear or Proportional?

Multiple Choice

Which statement best describes the relationship between x and y in the table?

• Proportional

• Not Proportional

Bieber Fever

Total

Love Beb

200

60

250

75

400

120

550

165

• Is the relationship proportional?
• What is the “k”?
• What point (x, y) on a graph

represents “k”?

• What does the constant of

proportionality represent in this
relationship?

Yes, there is a

constant unit rate.

3 out of 10 people asked

love Justin Bieber.

Miles Walked

• Is the relationship proportional?
• What is the “k”?
• What point (x, y) on a graph

represents “k”?

• What does the constant of

proportionality represent in this
relationship?

miles

hours

The relationship is not proportional,
because there is not a constant rate.
There is no “k” because the
relationship is not proportional.

Is this a proportional relationship?

Is there a Constant of Proportionality?

Multiple Choice

Find the constant of proportionality on this table of values.

• k=3

• k=1

• k=0

• k=4

Multiple Choice

What is the constant of proportionality (in miles per hour) based on the table?

• 45

• 90

• 135

• 2

Multiple Choice

What is the equation y=kx for the table?

• y=1/3x

• y=3x

• y=5x

• y=1/5x

Multiple Choice

The constant of proportionality is 9/1. Interpret what that means in the context of this story.

• $9 per hour

• $9 for 3 hours

• 9 hours per $1

Linear Equations vs. Proportional Equations

y = mx + b

y = kx

Linear Equation:

Proportional Equation:

Multiple Choice

In the equation y = 14x, what is the constant of proportionality?

• y

• x

• 14

• y=14

Multiple Choice

What is the unit rate if the rate is 30 laps for 10 minutes.

• 3 laps per minute

• 10 laps per minute

• 15 laps per minute

• 10 laps per two minute

Multiple Choice

Samantha drives at a rate of 30 mph. Tracy drives at a rate of 60 miles every 2 hours. Who drives faster?

• Samantha

• Tracy

• They drive the same speed per hour

Multiple Choice

Does the equation show a proportional relationship?

• Yes

• No

Multiple Choice

Evan bought 5 pounds of bananas for $10. Christy bought 6 pounds of bananas for $9. Who has the better deal?

• Christy

• Evan