Proportional Relationships

Proportional Relationships

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Unit Rates

A rate is a comparison between two quantities with different units. 
            miles per hour; feet per minute; sq ft per day
A unit rate is a special rate with a denominator of 1. 

Poll

Which of these is a unit rate?

• 15 feet per 3 seconds

• 100 miles per hour

• $64 per 5 bags of kiwis

• $1.50 per 2 bottles

Proportions

Proportions are statements that two or more rates are equivalent.

Multiple Select

Which of these statements are proportional?

• $$\frac{2}{3}=\frac{4}{6}$$

• $$\frac{2}{1}=\frac{5}{10}$$

• $$\frac{7}{14}=\frac{50}{100}$$

• $$\frac{3}{6}=\frac{4}{3}$$

Rate of change

A rate of change shows how one quantity changes with respect to another. 
A constant rate of change is known as the constant of proportionality,

Characteristics of Proportional Graphs.

• They are in the shape of a straight line.

• They start or cross at the point of origin.

Multiple Choice

Which graph is non-proportional?

• A
• B
• C
• D

Multiple Choice

Does this graph represent a proportional relationship?

• Yes
• No

Multiple Choice

Is the graph proportional or non proportional?

• proportional
• non proportional

Multiple Choice

The origin is located where in the coordinate plane?

• (0, 0)

• (10,10)

• (100, 100)

• (50, 50)

Equations for proportional relationships

y = kx
k = constant of proportionality
x = a quantity (traditionally units of time)
y = a quantity 

Example: Steven earns extra money babysitting. He charges $31.25 for 5 hours and $50 for 8 hours.

 Is this a proportional relationship? Compare the two charges. 

$31.25/5hr = $6.25/hour

$50/8hr = $6.25/hour

Yes, this is proportional. 

Is this proportional?

To solve, check each ordered pair for a rate of change.
10/1 = 10
20/2 = 10
70/7 = 10
100/10 = 10
Yes, the rates are constant, so this is proportional.

Is this proportional

Check all the rates of change.

-10/70 = -1/7

-9/63 = -1/7

-5/35 = -1/7

-3/21 = -1/7

All of the rates of change are constant (-1/7), so this is proportional. 

Constant of Proportionality

Multiple Choice

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

• y
• x
• 14
• y=14

Multiple Choice

What is the constant of proportionality?

• $3
• $4
• $2
• $9

Multiple Choice

Proportional or Non-Proportional?

• Proportional

• Non-Proportional

Multiple Choice

Does this equation represent a proportional relationship?

• Yes

• No

Multiple Choice

Is this graph proportional or non-proportional?

• Proportional
• Non-Proportional

Multiple Choice

Does the graph show a proportional relationship? 

• No, it's a curved line
• No, it's a pretty line
• Yes, if you connect the line, it is kind of straight.
• Yes, the dots are pink

Multiple Choice

Does this table represent a proportional relationship?

• Yes
• No

Multiple Choice

Does the table represent a proportional relationship?

• Yes.
• No.

Multiple Choice

The constant of proportionality in the equation y = ½x is _______. 

• x
• y
• 1/2
• 2

Multiple Choice

Identify the constant of proportionality.

• k = 2x

• k = 2

• k = 4

• none (not proportional)

Multiple Choice

What is the constant of proportionality of the graph above?

• 16
• 1/16
• 1
• None of the above

Multiple Choice

What is the constant of proportionality for the following graph?

• 80
• 20
• 40
• 160

Multiple Choice

What is the constant of proportionality?

• 8
• 16
• 24
• 2

Multiple Choice

What is the constant of proportionality?

• 45

• 50

• 27

• 1