1. Bell Ringers
Solve the following two step linear equations. Show your work. 1. 𝑥 = x = x = x + 8 = x – 5 = 135

2. 7.RP.1 Unit Rates

3. Ratios
A ratio is comparing a pair of non-negative numbers that have the same unit of measure. Neither of the numbers can be 0.  A ratio can be written in different ways.
Using words:  5 is to 7
Using a colon:  5 : 7
Using a fraction bar:  5 7

4. In a ratio problem, your answer will have one label
In a ratio problem, your answer will have one label. For example; ounces, miles, liters.  A ratio is never changed into a mixed number.  A ratio is NOT a fraction, it only looks like one.  It is a way of comparing two things using math.

5. For example
A shade of green paint is 2.5 quarts yellow paint and 1.5 quarts of blue paint for one gallon.  What is the ratio of the two types of paint?
To write the ratio we first need to write the ratio using a fraction bar. We will place the amount of yellow paint over the amount of blue paint.

6. Ratio
When getting Maddison ready for daycare I have to prepare bottles. A single bottle uses 2 scoops of powder and 4 ounces of water. What is the ratio of powder to water?
2 4 or 2:4

7. Equivalent Ratios
If you wanted to get a 5 gallon tub of paint in this same color, how can we use ratios to show the answer?
To find the ratio for a 5 gallon tub, we find an equivalent ratio. You can see that by multiplying by 2, we double the amount.  So to get 5 gallons, we multiply by 5.  We multiply both parts because we want the ratio to be equivalent. In other words we want the paint to be exactly the same color.
˟ =

8. How much milk and powder would I need to make Maddison 6 bottles?
Remember that when making equivalent ratios we need to multiply our ratio by the same amount on both the top and the bottom.
× =
What does mean?
To make 6 bottles I will need 12 scoops of powder and 24 ounces of water. Or I could state that for every 12 scoops of powder I use I will need 24 ounces of water.

9. Types of ratios There are two types of ratios
Part to part, is comparing the parts that make up the whole
For example: In my family there are two girls and three boys. The ratio of girls to boys would be 2:3 or
2. Part to whole, is comparing one part of the whole to the entire whole.
For example: In my family there are two girls and three boys. The ratio of girls to family members would be 2:5 or

10. Rates
A rate is a ratio that compares two unlike quantities. A rate can be written in the same forms as a ratio.   In a rate, the numbers (or quantities) have different labels.  In a rate problem, your answer will have a combination label.  For example 3 miles in 25 minutes, 9 ounces for $0.88.

11. Rate Example
Let's consider a runner who finished a 10 mile race in 56 minutes.  She would now like to predict her time to run a marathon (26.2 miles).  What is her predicted time for the marathon? This is a rate because there are two different units (miles and minutes).
First we set up this rate as a ratio. 
10 56
Then we set it equal to the rate of time for the marathon.
10 𝑚𝑖𝑙𝑒𝑠 56 𝑚𝑖𝑛𝑢𝑡𝑒𝑠 = 𝑚𝑖𝑙𝑒𝑠 𝑥 𝑚𝑖𝑛𝑢𝑡𝑒𝑠
Next we cross multiply or butterfly. We make a "mini" algebra problem.
10 𝑚𝑖𝑙𝑒𝑠 56 𝑚𝑖𝑛𝑢𝑡𝑒𝑠 = 𝑚𝑖𝑙𝑒𝑠 𝑥 𝑚𝑖𝑛𝑢𝑡𝑒𝑠
10 𝑥 = 56 ∙26.2
10 𝑥 =
Finally solve for 𝑥
÷10
𝑥 = , The runner will run 26.2 miles in minutes.

12. Maddison Rate Example
Maddison drinks 7 bottles in a 24 hour period. How many bottles will she drink in 168 hours (or 1 week).
Step 1: set up as a ratio
7 24
Set it equal for the rate of time for bottles
= 𝑥 168
Cross multiply or butterfly
24 𝑥 = 7∙168
24 𝑥 = 1176
Solve for 𝑥
𝑥 = 49. It takes 49 bottles per 168 hours.

13. Rate Problems
An ice cream factory makes 300 quarts of ice cream in 5 hours. How many quarts could be made in 48 hours? What was that rate per day?

14. Rate Problems
A jet travels 530 miles in 2 hours. At this rate, how far could the jet fly in 11 hours? What is the rate of speed of the jet?

15. Rate Problems
The bakers at Healthy Bakery can make 300 bagels in 10 hours. How many bagels can they bake in 22 hours? What was that rate per hour?

16. Journal- Linear Equations
Mike and Maddison were given the following linear equation: 3(-2x + 6) = 36. Mike argues that the only way to solve the equation is by using the distributive property. Maddison believes that you can solve the equation by first dividing both sides by 3 and then solving by isolating the variable. Who is correct and why? Try solving the problem using both strategies. Show your work.
InMike’s Strategy
Who is correct and why?
____________________________________
________________________________________________________________________
Maddison’s Strategy
Will these strategies be best for all equations?
________________________________________________________________________________________________________________________________________________

17. Unit Rate
A unit rate is a rate (which means it is also a ratio) with a denominator of 1. The word unit actually means 1. For example; miles per gallon, ounces per dollar. The word "per" means for one unit. Example: If apples are $.99 per pound. 𝑃𝑟𝑖𝑐𝑒 𝑜𝑓 𝐴𝑝𝑝𝑙𝑒𝑠 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑃𝑜𝑢𝑛𝑑𝑠 = .99 1

18. Maddison Unit Rate Example
It costs $24.99 for one container of Similac. What is the unit rate?
Write as a ratio with cost over units
$ 𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑒𝑟

19. Calculating Unit Rates
Unit rates are the most used and helpful of all types of rates.  Having the denominator of 1 makes it easy to find equivalent rates. It also makes it easy to compare rates.  Supermarkets use unit pricing, which is a form of unit rates, to help shoppers get the best value for their money. Unit rates can find the best value in a car's gas mileage.
If we are given a rate, finding the unit rate can give a baseline  or starting value when we are comparing two things

20. Calculating Unit Rate Example
Karen ran four miles in 20 minutes. Is this a ratio, rate, or unit rate.  Find the unit rate.
This is a rate.  We are comparing two different units, minutes and miles. First we write it as ratios
20 𝑚𝑖𝑛𝑢𝑡𝑒𝑠 4 𝑚𝑖𝑙𝑒𝑠 ? 𝑚𝑖𝑛𝑢𝑡𝑒𝑠 1 𝑀𝑖𝑙𝑒
To find the unit rate we simply cross multiply or make a butterfly.
× 𝑥 1
4 ∙𝑥=4𝑥
20 ×1 =20
4x = 20
X=5
5 𝑚𝑖𝑛𝑢𝑡𝑒𝑠 1 𝑚𝑖𝑙𝑒 or 5 minutes per mile

21. Calculating Unit Rates.
Example: Jeff bought ¾ lb. of candy for $1.80.  What is the unit price of the candy per pound?
Set up as ratios.
$ 𝑝𝑜𝑢𝑛𝑑𝑠 = $ 𝑥 1𝑝𝑜𝑢𝑛𝑑
If ratios are not using the same form of numbers convert to easiest form.
1.80 is a decimal and ¾ is a fraction. It will be easiest to convert to decimals to solve.
3 ÷4= .75
= 𝑥 1
Cross multiply or butterfly the equation.
= 𝑥 1
.75 ∙𝑥= .75𝑥
1.80 ∙1=1.80
.75x = 1.8
x = 2.4
$ 𝑝𝑜𝑢𝑛𝑑 $2.40 per pound

22. Calculating Unit Rates
Example: Mia walks her dog of a mile in   2 3 of an hour.  What is Mia's unit rate of speed in miles per hour?
= 𝑥 1
= 𝑥 1
2 3 𝑥= 5 6
2 3 ÷ 𝑥= ÷ 2 3
Keep, Change, Flip: x 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑦 𝑎𝑐𝑟𝑜𝑠𝑠
𝑚𝑖𝑙𝑒𝑠 1 ℎ𝑜𝑢𝑟
5×3 6×2 =
15 ÷12= =

23. Solving Word Problems Using Unit Rates
Sarah uses 5½ cups of flour for every 2 loaves of bread she bakes.  What is the unit rate of cups of flour per loaf of bread?  Using this unit rate, how many cups of flour will Sarah need to make 5 loaves of bread?
1. Set up ratios
= 𝑥 1
2. Change mixed fractions into improper fractions or into decimals.
= 𝑥 1
3. Cross multiply or butterfly
𝑐𝑢𝑝𝑠 𝑜𝑓 𝑓𝑙𝑜𝑢𝑟 𝑓𝑜𝑟 5 𝑙𝑜𝑎𝑣𝑒𝑠
= 𝑥 1
2x = 11 2
11 2 ÷2= ∙ = 11 4
4. Solve for x.
5. Rewrite ratio and multiply top and bottom
× = × ×5 =
6. Change improper fractions into proper fractions.

24. Comparing Unit Rates
If Tide sells 80 ounces for $11.99, and Gain sells 64 ounces for $8.99, which is the better value for the money? In order to compare them equally, we find the unit cost for each.
1. Set up ratios for both products and determine the unit rate.
= 𝑥 1
= 𝑥 1
= 𝑥 1 = 80x = 11.99
X= .1498
= 𝑥 1 = 64x = 8.99
X= .1404
2. Compare both solutions to see which is the better value.
.1498>.1404 𝑡ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒 𝐺𝑎𝑖𝑛 𝑖𝑠 𝑡ℎ𝑒 𝑏𝑒𝑡𝑡𝑒𝑟 𝑣𝑎𝑙𝑢𝑒.

25. Comparing Unit Rates
Solve the following problem in your notebook. You may work with the person sitting next to you.
John’s car gets 185 miles for every 5 gallons of gas. Charlie gets 192 miles for every 6 gallons of gas. Who’s car gets better gas mileage?

26. Comparing Unit Rates
Solve the following problem in your notebook. You may work with the person sitting next to you. Red Delicious apples cost $1.25 for 3 pounds. Granny Smith apples cost $2.20 for 5 pounds. Which is type of apple is the best value?

27. Comparing unit rates
You can buy 5 cans for green beans at the Village Market for $2.30. You can buy 10 of the same cans of beans at Sam's Club for $5.50. Which place is the better buy?

28. Comparing Unit Rates
You can buy 3 apples at the Quick Market for $1.20. You can buy 5 of the same apples at Stop and Save for $1.20. Which place is the better buy?