7th_IM_U2: Lessons 4-5

7th_IM_U2: Lessons 4-5

7.RP.A.2

Proportional Relationships UNIT 1

• Unit 1 - we learned there is a scale factor used to multiply the side lengths by to either enlarge or reduce a figure

• Unit 1: To scale back to the originial figure multiply the scale factor by the reciprocal of the scale factor

• Unit 1: In order for figures to be proportional their ratios must be equal.

• You can use the scale factor to find missing side lengths.

MOST IMPORTANT CONCEPT FROM UNIT 1

We can set up and solve proportions to find missing measures.

Proportional Relationships UNIT 2

• Scale Factor is also known as a constant of proportionality when we are working with tables

• To find the constant of proportionality (K) we divide y/x (dependent variable/independent variable)

• Equations to find the y value are Y = Kx (MULTIPLY K by X to find Y).

• We can also multiply by the reciprocal of the constant of proportionality to find the x value in a table just like when we scale back to the original from Unit 1.

Unit Rates and Constant of Proportinality

• Unit Rates are the same as constant of proportionality WHEN the simplified ratio has a denominator equal to ONE.

• 
  

   $\frac{30\ miles}{1\ hour}$  is a UNIT RATE

•  $\frac{60\ miles}{2\ hours}$  is not a unit rate because the denominator must be simplified to equal 1 to be a UNIT RATE but it could be a constant of proportionality. 

Proportional Relationships and Equations

Y = Kx

Dependent vs Independent Variables

y is the DEPENDENT variable

x is the INDEPENDENT variable

Multiple Choice

Red paint is the independent variable. What variable represents the RED paint?

• X

• Y

Write an equation.

Blue Paint is dependent upon the red paint.

Enter your answer on the next slide.

Multiple Choice

Which equation describes the proportional relationship?

• Y = 0.25X

• Y = 4x

• x = 4y

• cannot be determined

Write an equation to describe the relationship.

Cups of flour depends on the amount of honey.

Enter your answer on the next slide.

Multiple Choice

What is the equation for the table?

• Y = 0.8X

• Y = 1.25x

• X = 1.25y

• cannot be determined

Sometimes other variables can be used in place of Y and X.

What would be a better variable for flour and honey?

(Probably better to use F for flour and H for honey.)

Using variables other than Y and X.

It took Priya 5 minutes to fill a cooler with 8 gallons of water from a faucet that was flowing at a steady rate. Let (G) be the number of gallons of water in the cooler after (M) minutes.

Enter you answer(s) on the next slide. You will pick 2 equations.

Multiple Select

It took 5 minutes to fill a cooler with 8 gallons of water from a faucet that was flowing at a steady rate.

Which of the following equations represent the relationship between (G) gallons and (M) minutes? Select 2 that apply.

• G = 1.6M

• G = 0.625M

• M = 1.6G

• M = 0.625G

What do the numbers mean?

What does 1.6 tell you about the situation?

What does 0.625 tell you about the situation?

What does 1.6 tell you about the situation?

When you divide 8 gallons of water by 5 minutes, you get  $\frac{8\ gallons}{5\ \min}$  

 $\frac{1.6\ gallons}{1\ \min}$  which means 1.6 gallons of water is filled per minute. 

What does 0.625 tell you about the situation

 $\frac{5\ \min}{8\ gallons}$  = 0.625 which means  $\frac{0.625\ \min}{1\ gallon}$  

It takes 0.625 minutes to fill one gallon of water. 

What do you notice about 1.6 and 0.625?

They are RECIPROCALS!

1.6 = 8/5

0.625 = 5/8

Multiple Select

Priya changed the rate at which water flowed through the faucet. Write an equation that represents the relationship of G and M when it takes 3 minutes (M) to fill the cooler with 1 gallon (G) of water.

• M = 3G

• G = 1/3M

• G = 3M

• M= 1/3G

Which rate is faster? (HINT: Compare the UNIT RATES.)

8 gallons/5 minutes

or

1 gallon/ 3 minutes

A unit rate is the fraction simplified so that the denominator = 1.

Comparing the UNIT RATES.

• 8 gal/5 minutes = 1.6 gal per minute (FASTER)

• 1 gal/ 3 minutes = 1/3 gal per minutes

Feeding Shrimp

• At an aquarium, a shrimp is fed 1/5 gram of food each feeding.

• A shrimp is fed 3 times a day.

• Use the information for the next multiple choice questions.

Multiple Choice

A shrimp is fed 1/5 gram of food each feeding and is fed 3 times a day.

How much food does a shrimp get fed in one day?

• 1/5 grams

• 3 grams

• 3/5 grams

• 0.25 grams

Multiple Choice

How would you find how many grams of food a shrimp is fed after 1 day or 30 days?

• Multiply the days by by 1/5

• Multiply the days by 3

• Multiply the days by 3/5

• Multiply the days by 0.25

Multiple Select

What does 3/5 tell us about the situation? Select all that apply.

• 3/5 is the constant of proportionality

• The shrimp are fed 0.6 grams of a food per day.

• Nothing.

• The shrimp are fed 3/5 a gram of food everyday.

Multiple Select

Select the equations that apply to the a shrimp is fed 1/5 gram of food each feeding and is fed 3 times each day.

• Y = 3/5X

• F = 3/5D where F = Food and D = Days

• D = 5/3F where D = Days and F = Food

• X = 5/3y

• F = 0.25D where F = Food and D = Days

Multiple Select

If a tank has 10 shrimp in it, how much food is added to the tank each day? Select all that apply.

• 3/5(10)

• 300 grams

• 6 grams

• 30/5

Multiple Choice

When 2 quantities are in a proportional relationship, we write the equation y = kx and say, "y is proportional to x." What is the reciprocal of K?

• 1/x

• 1/y

• 1/k

• cannot be determined