Proportional Reasoning to Solve Problems

Proportional Reasoning

by Kellany Bennett

​What is a Proportion?

• A proportion is an equation that states two ratios are equal

• If something is proportional, it means that all its ratios are EQUAL.

• ​If asked if two ratios form a proportion (or are they proportional?), check to see if they are EQUAL by...

  • ​simplifying the ratios

  • ​looking for multiplicative or division-based relationships across

  • ​using a calculator to divide the ratios and see their decimal forms

​Checking for Proportionality

• ​You may not like the methods from before

  • ​dividing the ratios into decimals

  • ​simplifying to see if both simplify to the same ratio

  • ​looking for relationships

• ​There's more! Remember cross products, or means and extremes? If two ratios are proportional, their cross products or means and extremes have to be equal.

Multiple Choice

What is a proportion?

• When two ratios can be simplified

• When two ratios are equivalent and show the same value

• When two ratios have the exact same numbers

• When two ratios can be multiplied by 2

Multiple Choice

Do the ratios form a proportion?

• Yes, they form a proportion.

• No, they do not form a proportion.

Multiple Choice

Determine if the ratios are proportional. 4/5 and 15/20

• Yes

• No

Multiple Choice

Are the two ratios proportional?

• YES

• NO

Multiple Choice

Are the two ratios proportional?

• YES

• NO

​Tables Can Represent Proportional Relationships

• ​Again, in order for something to be proportional, the ratios must all be equivalent

• ​You can physically write out each ratio and check them for equivalency

• ​It can take a long time, right?

• ​There is a shortcut!!

​Other Ways to Determine Table Proportionality

• An easier way to determine if ​a table is proportional is to look for a multiplicative rule across the rows.

• ​If every row uses the exact same multiplication-only rule (dividing works as well, since it's the inverse of multiplying) then it's proportional.

​You Try It!!

​Does this table represent a proportional relationship?

​

​Hint: Either write out all the ratios and check to see if they are equal, or find a single multiplicative (or division-based) rule going across the rows.

​156

Answer

​Does this table represent a proportional relationship? YES

​

Why:

​2/26, 3/39, 6/78, 10/130, 12/156 all are equal to 1/13

​

​Why:

​The constant rule is * 13.

* 13 =

​* 13 =

​* 13 =

​* 13 =

​* 13 =

​

​156

​You Try It!!

​Does this table represent a proportional relationship?

​

​Hint: Either write out all the ratios and check to see if they are equal, or find a single multiplicative (or division-based) rule going across the rows.

Answer

​Does this table represent a proportional relationship? YES

​

2/1, 4/2, 6/3, 8/4, and 10/5 all simplify to 2.

​

​The constant rule is divide by 2, or,

​

​This is why division rules work--we can MAKE them multiplication rules!

* 1/2 =

​* 1/2 =

​* 1/2 =

​* 1/2 =​

* 1/2 =

​

​You Try It!!

​Does this table represent a proportional relationship?

​

​Hint: Either write out all the ratios and check to see if they are equal, or find a single multiplicative (or division-based) rule going across the rows.

Answer

​Does this table represent a proportional relationship? NO

​

Why:​

1:12

​2:26 = 1:13

​3:30 = 1:10

​4:48 = 1:12

​

​There is no constant, multiplicative rule, so the relationship is not proportional.

* 12 =

​* 13 =

​* 10 =

​* 12 =

​

​A few special notes:

1. ​The point 0,0 is always proportional

   1. ​Any multiplicative rule times zero equals zero

   2. ​If you see a 0,0 point, ignore it but know it's proportional

2. ​Conversely, any point with a zero in it without another zero cannot be proportional!

   1. ​You cannot get from 0 to any non-zero number by multiplying!!

3. ​It only takes one ratio row being off the multiplicative rule to ruin the table for proportionality

​Not proportional

​Why? (0, 4), not (0,0)

​Also, no constant multiplicative rule

​+ 4 =

​* 5 =

​* 4 =

* 3 + 4 =

​

​Proportional

​Why? (0,0) is proportional

​Also, the constant rule is *3

​* 3 =

​* 3 =

​* 3 =

* 3 =

​

Multiple Choice

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

• Proportional

• Not Proportional

Multiple Choice

Does the table show a proportional relationship?

• Yes

• No

Multiple Choice

 Is this table proportional or non-proportional?

• Proportional

• Non-Proportional

Multiple Choice

Is this table proportional?

• Yes

• No

Multiple Choice

Is this table proportional?

• Yes

• No

Fill in the Blanks

This table is proportional.

What is the value of x?

(just type the number alone)

Fill in the Blanks

This table is proportional.

What is the missing value?

(Just type the number alone)

The Constant of Proportionality

• A number that shows the proportional relationship between two amounts (x and y)

• Also called: a constant ratio, unit rate, constant multiplier, the multiplicative rule

• ​The constant of proportionality for this table is 3, since the constant relationship is *3

• ​The constant of proportionality is always found where the x value is 1

• ​CONSTANT OF PROPORTIONALITY IS UNIT RATE!!!!

You can find the constant of proportionality in a table.

x values times the constant of proportionality is the y value

​

​use your super smart math brains to find the relationship

Sometimes the number being multiplied is not obvious, so we can kind of cheat and divide!

​Just take the y value and divide by the x value, for

​

y/x or

Multiple Choice

What is the constant of proportionality?

AKA: What is the unit rate?

• $3

• $4

• $2

• $9

Multiple Choice

The constant of proportionality is also the unit rate.

• True

• False

​IMPORTANT!

​If the values INCREASE from x to y, the constant is a positive number above 1

​

​If the values DECREASE from x to y, the constant is a fraction or decimal BELOW 1

​

​You have to make the rule go FROM x, TO y!!!

​

​In this table, the rule is divide by 5, or times

Fill in the Blanks

Based on the information in the table, what is the constant of proportionality?

Fill in the Blanks

Based on the information in the table, what is the constant of proportionality?

Multiple Choice

Is the given table a proportional relationship? If so, what is the constant?

$$\frac{y}{x}=$$  constant of proportionality

• Yes; 1/3

• Yes; 3

• Yes; 5

• No

Multiple Choice

What is the constant of proportionality?

Hint--left column is x, right column is y

$$\frac{y}{x}=$$  constant of proportionality

• 7

• 3.3

• 3/10

• 14

Multiple Choice

What is the constant of proportionality?

CAREFUL! It's the multiplicative relationship from x to y.

• 4

• 2

• $$\frac{1}{4}$$  

• -4

Multiple Choice

What is the constant of proportionality?

AKA: What is the unit rate?

Hint: Top row are x values, bottom row are y values.

• $18

• $18.50

• $22.50

• $22

Multiple Choice

What is the constant of proportionality?

• 1/30

• 30

• 1/10

• 10

Multiple Choice

What is the constant of proportionality?

• 5

• $$\frac{1}{5}$$  

• $$\frac{1}{4}$$  

• 0.2

Multiple Select

What is the constant of proportionality?

Hint--the left column is x, the right column is y

• 30

• $$0.\overline{3}$$  

• $$\frac{1}{3}$$  

• 3

We can write a COP (constant of proprotionality) in an equation, y = kx where k is the constant.

​Don't ask me why they didn't use c = constant. Just laugh at the bad spelling of "konstant" with me and move along.... 😆

• When the constant is 16, then the equation is y=16x

• When the constant is 57, then the equation is y=57x

Let's Put it All Together

What is the Constant of Proportionality for this table?

What would the equation of the line be?

Let's Put it All Together

What is the Constant of Proportionality for this table? 3

What would the equation of the line be?

​* 3 =

​* 3 =

​* 3 =

Let's Put it All Together

What is the Constant of Proportionality for this table? 3

​

What would the equation of the line be? y = 3x

​ * 3 =

​ * 3 =

​ * 3 =

​x * 3 = y

Multiple Choice

Which equation helps you solve for k?

• $$k\ =\ \frac{x}{y}$$

• $$k\ =\ \frac{y}{x}$$

• k = y + x

• k = 2x

​Notice that NONE of these equations have an addition or subtraction attached, ONLY multiplication!!! y = 3x - 2 would NOT be proportional!!!

Multiple Choice

Identify the constant of proportionality.

• k = $$\frac{1}{0.45}$$  

• k = 0.45

• k = -0.45

• k = $$-\frac{1}{0.45}$$  

• none (not proportional)

Multiple Choice

Identify the constant of proportionality.

• k = 2

• k = 4

• k = 1/2

• k = 1/4

• none (not proportional)

Multiple Choice

Write the equation for the table given.

• y = 16x

• y=8x

• y=12x

Multiple Choice

Write the equation for the table given

• y = $$\frac{1}{3}$$ x

• y = 3x

• y = $$\frac{1}{2}$$ x

Multiple Choice

Which equation would match this proportional relationship?

• y = 45x

• y = 30x

• y = 15x

• y = -45x

Multiple Choice

What is the equation y=kx for the table?

• y= $$\frac{1}{3}$$  x

• y=3x

• y=5x

• y= $$\frac{1}{5}$$  x

Multiple Choice

Write an equation for this relationship.

• y= $$\frac{1}{5}$$ x

• y= $$\frac{1}{4}$$ x

• y=2x

• y=5x