Unit 2 Introducing Ratios — Unit Plan

Title

Assessment

Lesson 1

Introducing Ratios and Ratio Language

A Collection of Animals

Here is a collection of dogs, mice, and cats:

Write two sentences that describe a ratio of types of animals in this collection.

Show Solution

Sample responses:

The ratio of dogs to cats is $6 : 4$ .
There are 3 dogs for every 2 cats.
There is 1 mouse for every 2 cats.
The ratio of cats to mice is $4 : 2$ .

Lesson 2

Representing Ratios with Diagrams

Stationery Sets

Lin has 3 sets of stationery. Each set has 2 erasers, 4 pencils, and 1 notepad.

Draw a diagram that shows an association between the numbers of erasers, pencils, and notepads that Lin has.
Complete each statement:
1. The ratio of $\underline{\hspace{.8in}}$ to $\underline{\hspace{.8in}}$ to $\underline{\hspace{.8in}}$ is $\underline{\hspace{.4in}} : \underline{\hspace{.4in}} : \underline{\hspace{.4in}}$ .
2. There are $\underline{\hspace{.5in}}$ pencils for every notepad.
3. There are $\underline{\hspace{.5in}}$ pencils for every eraser.

Show Solution

Sample response:
1. The ratio of erasers to pencils to notepads is 6 : 12 : 3.
2. There are 4 pencils for every notepad.
3. There are 2 pencils for every eraser.

Section A Check

Section A Checkpoint

Problem 1

In a set of coloring pencils, there are 10 red pencils, 4 yellow pencils, and 8 brown pencils.

Select all true statements about the coloring pencils:

A.The ratio of red pencils to yellow pencils is 4 to 10.

B.The ratio of red pencils to brown pencils is

10 : 8

C.The ratio of yellow pencils to red pencils is 2 to 5.

D.There are 5 red pencils for every 4 brown pencils.

E.For every yellow pencil, there are 2 brown pencils.

Show Solution

B, C, D, E

Problem 2

Noah and Jada are making hot cocoa using their own recipes.

Noah mixes 12 tablespoons of cocoa powder and 3 cups of milk.

Draw a diagram to represent the ratio of tablespoon of cocoa powder and cups of milk in his recipe.
This diagram represents Jada’s recipe:

Write two sentences to describe the ratio of tablespoons of chocolate syrup and cups of milk in her recipe.

Show Solution

Sample response:
Sample responses:
- The ratio of teaspoons of chocolate syrup to cups of milk is 6 to 2.
- For every 3 teaspoons of chocolate syrup, 1 cup of milk is needed.

Lesson 3

Recipes

A Smaller Batch of Lemonade

When Elena makes lemonade, she usually mixes 9 scoops of lemonade powder with 6 cups of water. Today, she doesn’t have enough ingredients.

Think of a recipe that would give a smaller batch of lemonade but still taste the same. Explain or show your reasoning.

Show Solution

Sample responses:

3 scoops of lemonade powder and 2 cups of water
6 scoops of lemonade powder and 4 cups of water

Sample reasoning:

$3:2$ represents the scoops of lemonade powder to the cups of water.
$3:2$ is equivalent to $9:6$ .

Lesson 4

Color Mixtures

Orange Water

A recipe for orange water says, “Mix 3 teaspoons of yellow water with 1 teaspoon red water.” For this recipe, we might say: “The ratio of teaspoons of yellow water to teaspoons of red water is $3:1$ .”

Write a ratio for 2 batches of this recipe.
Write a ratio for 4 batches of this recipe.
Explain why we can say that any two of these three ratios are equivalent.

Show Solution

The ratio of teaspoons of yellow to teaspoons of red is $6:2$ (or any sentence that accurately states this ratio). Note: A statement like “The ratio of yellow to red is $6:2$ ” describes the association between the colors but does not indicate the amount of stuff in the mixture.
The ratio of teaspoons of yellow to teaspoons of red is $12:4$ (or any sentence that accurately states this ratio).
Sample response: These are equivalent ratios because they describe different numbers of batches of the same recipe. To make 2 batches, multiply the amount of each color by 2. To make 4 batches, multiply the amount of each color by 4. As long as we multiply the amounts for both colors by the same number, we will get a ratio that is equivalent to the ratio in the recipe.

Section B Check

Section B Checkpoint

Problem 1

Lin is making pancakes using a pancake mix and water. In her recipe, the ratio of cups of pancake mix to cups of water is $4 : 3$ .

Lin mixes 20 cups of pancake mix and 15 cups of water. Is this ratio of pancake mix to water equivalent to that in the recipe? Explain your reasoning.
Find two ratios of cups of pancake mix to cups of water to make pancakes that would taste the same as those using the amounts stated in the original recipe. Show your reasoning.

Show Solution

Yes. Sample reasoning: Both the cups of pancake mix and the cups of water are 5 times the numbers in the original recipe ( $20 = 5 \boldcdot 4$ and $15 = 5 \boldcdot 3$ ). Lin is making 5 batches of the recipe.
Sample response: $8 : 6$ and $2 : 1\frac{1}{2}$ . The first ratio is for 2 batches of pancakes: $2 \boldcdot 4 = 8$ and $2 \boldcdot 3 = 6$ . The second ratio is for half a batch: 2 is half of 4, and $1\frac{1}{2}$ is half of 3.

Lesson 6

Introducing Double Number Line Diagrams

Batches of Cookies on a Double Number Line

A recipe for one batch of cookies uses 5 cups of flour and 2 teaspoons of vanilla.

Complete the double number line diagram to show the amount of flour and vanilla needed for 1, 2, 3, 4, and 5 batches of cookies.
If you use 20 cups of flour, how many teaspoons of vanilla should you use?
If you use 6 teaspoons of vanilla, how many cups of flour should you use?

Show Solution

8 teaspoons of vanilla
15 cups of flour

Lesson 7

Creating Double Number Line Diagrams

Representing Paws, Ears, and Tails

Each of these cats has 2 ears, 4 paws, and 1 tail.

Draw a double number line diagram that represents a ratio in the situation.
Write a sentence that describes this situation and that uses the word per.

Show Solution

Students may draw any 2 of the 3 number lines shown.
Samples responses:
- There are 2 ears per tail.
- There are 4 paws per tail.
- There are 2 paws per ear.
- There is $\frac12$ tail per ear.

Lesson 8

How Much for One?

Unit Price of Rice

Here is a double number line showing that it costs $3 to buy 2 bags of rice:

<p>Double number line. Cost, dollars. Rice, number of bags.</p>

At this rate, how many bags of rice can you buy for $12?
Find the cost per bag.
How much do 20 bags of rice cost?

Show Solution

8 bags cost $12.
The cost per bag is $1.50.
20 bags cost $30. Sample reasoning: Multiply 20 by the price for one bag, or find a ratio of 20 bags to cost in dollars that is equivalent to 3 bags to 2 dollars.

Lesson 9

Constant Speed

Train Speeds

Two trains are traveling at constant speeds on different tracks.

Train A:

<p>Double number line. Distance traveled, meters. Elapsed time, seconds.</p>

Train B:

Double number line. Distance traveled, meters. Elapsed time, seconds.

Which train is traveling faster? Explain your reasoning.

Show Solution

Train B travels faster. Sample reasoning:

It only took 4 seconds for Train B to travel 100 meters, while it took Train A 8 seconds to go the same distance.
Train B's speed is 25 meters per second. Train A’s speed is 12.5 meters per second.

Lesson 10

Comparing Situations by Examining Ratios

Comparing Runs

Andre ran 2 kilometers in 15 minutes, and Jada ran 3 kilometers in 20 minutes. Both ran at a constant speed.

Did they run at the same speed? Explain your reasoning.

Show Solution

They did not run at the same speed. Sample reasoning:

Andre would have run 6 kilometers in 45 minutes, and Jada would have run 6 kilometers in 40 minutes. Jada completes the 6 kilometers in less time, so she runs at a faster speed than Andre runs.
Andre would have run 8 kilometers in 60 minutes, and Jada would have run 9 kilometers in 60 minutes. Jada travels farther in the same amount of time, so she runs at a faster speed than Andre runs.

These examples explain why Jada runs faster and also explain why the two runners did not run at the same speed.

Section C Check

Section C Checkpoint

Problem 1

A painter mixes 3 cups of black paint and 20 cups of white paint to make a batch of gray paint.

Complete the double number line diagram to show the amount of each color needed to make 5 batches of the gray paint.
If the painter has 18 cups of black paint, how many cups of white paint are needed to make the same shade of gray?

Show Solution

15 cups of black paint and 100 cups of white paint for 5 batches
120 cups of white paint

Problem 2

At a carnival, it costs $5 for 4 people to play a game. What is the cost per person?

Show Solution

Sample responses:

It costs $1.25 per person to play a game.
The carnival charges $1.25 per person for a game.

Problem 3

Diego and Priya each rode their bicycle at a constant speed. Diego traveled 7 kilometers in 60 minutes. Priya traveled 2 kilometers in 15 minutes.

Did they travel at the same rate? Explain your reasoning.

Show Solution

No, they did not travel at the same rate. Priya traveled faster. Sample reasoning:

There are four 15 minutes in 60 minutes, so Priya would travel $4 \boldcdot 2$ , or 8, kilometers in 60 minutes, 1 kilometer more than Diego traveled in that amount of time.
Using a double number line, we can see that Priya would ride 8 kilometers in 60 minutes, while Diego rode 7 kilometers in 60 minutes.
Diego biked $\frac{7}{4}$ or 1.75 kilometers in 15 minutes, which is a shorter distance than Priya traveled in 15 minutes.

Lesson 11

Representing Ratios with Tables

Batches of Cookies in a Table

Here is a table that represents a cookie recipe we saw in earlier lessons.

Write a sentence that describes a ratio shown in the table.

flour (cups) vanilla (teaspoons)

5 2

15 6

$2\frac12$ 1
What does the second row of numbers represent?
Complete the last row for a different batch size that hasn’t been used so far in the table. Explain or show your reasoning.

Show Solution

Sample responses:
- The ratio of cups of flour to teaspoons of vanilla is $5:2$ .
- This recipe uses 5 cups of flour for every 2 teaspoons of vanilla.
- This recipe uses $2\frac12$ cups of flour per teaspoon of vanilla.
For 15 cups of flour, you need 6 teaspoons of vanilla.
Sample response: 10 cups of flour and 4 teaspoons of vanilla

Lesson 12

Navigating a Table of Equivalent Ratios

Price of Bagels

A shop sells bagels for $5 per dozen.

For each question, explain or show your reasoning. You can use the table if you find it helpful.

At this rate, how much would 6 bagels cost?
How many bagels can you buy for $50?

number of bagels	price in dollars
12	5

Show Solution

$2.50. Sample reasoning: Twelve bagels cost $5 and 6 is half of 12, so 6 bagels cost half of $5, which is $2.50.
120 bagels. Sample reasoning:

number of bagels price in dollars

12 5

6 2.5

120 50

Lesson 14

Solving Equivalent Ratio Problems

Sharpening Pencils

Jada is helping to sharpen colored pencils for an art class. She wants to know how much time it would take to sharpen all the pencils.

What information would she need to answer that question? How might she use that information?

Show Solution

Sample responses:

Jada would need to know how many pencils there are and how quickly she can sharpen pencils. She could measure the number of pencils sharpened in 1 minute and use this ratio to find the number of minutes needed to sharpen all the pencils.
Jada would need to know the number of pencils and her pencil-sharpening speed. She could measure the time needed to sharpen 1 pencil and multiply that by the number of pencils.

Section D Check

Section D Checkpoint

Problem 1

The table shows the time it took a swimmer to swim 4 lengths of a small pool at a constant rate. (One length of the pool is the distance from one end of the pool to the opposite end.)

distance in lengths of the pool	time in seconds
4	96
1

At that rate, how many seconds did it take her to swim 1 length of the pool?
What distance did she swim in 60 seconds?

Show Solution

24 seconds
2.5 lengths of the pool

Problem 2

At a store, 8 bars of soap cost $6. At that rate, how much would 60 bars of soap cost? Explain or show your reasoning. You can use a table if it is helpful.

Show Solution

$45. Sample reasoning:

At that rate, 40 bars would cost $30, so 20 bars would cost $15. Three times 20 is 60, and 3 times 15 is 45.
Using a table:

Lesson 15

Part-Part-Whole Ratios

Room Sizes

A house has a kitchen, a playroom, and a dining room on the first floor. The areas of the kitchen, playroom, and dining room in square feet are in the ratio $4 : 3 : 2$ . The combined area of these three rooms is 189 square feet. What is the area of each room?

Show Solution

The area of the kitchen is 84 square feet, the area of the playroom is 63 square feet, and the area of the dining room is 42 square feet. Sample reasoning: All three rooms amount to 9 parts. All three rooms amount to 9 units. All three rooms make 189 square feet. $189\div9=21$ , so each part of the tape diagram represents 21 square feet. $4 \boldcdot 21=84$ , $3 \boldcdot 21=63$ , and $2 \boldcdot 21=42$ .

Lesson 16

Solving More Ratio Problems

Pizza-making Party

You had a pizza-making party for a number of people. Each person needed 6 ounces of dough and 4 ounces of sauce for their pizza. A total of 130 ounces of dough and sauce were used at the party.

How many ounces of dough were used at the party?
How many ounces of sauce were used at the party?
How many people were at the party?

Show Solution

78 ounces of dough
52 ounces of sauce. Sample reasoning:

dough (ounces) sauce (ounces) total (ounces)

6 4 10

78 52 130
13 people. Sample reasoning: $6 \boldcdot 13= 78$ and $4 \boldcdot 13 = 52$ .

Section E Check

Section E Checkpoint

Problem 1

In a class, the ratio of students who wear glasses to those who don’t is 3 to 5. If there are 32 students in the class, how many students wear glasses? Show your reasoning.

Show Solution

12 students. Sample reasoning: The total number of parts in the ratio is 8. $32 \div 8 = 4$ , so each part represents 4 students, and $3 \boldcdot 4 = 12$ .

Problem 2

In a school, the ratio of teachers who wear glasses to those who don’t is 7 to 2.

If there are 10 teachers who don’t wear glasses, how many total teachers are at the school?
Explain why you chose the strategy you did to solve the problem.

Show Solution

45 teachers. Sample reasoning:
Sample response: With a tape diagram I can see the parts in the ratio (7 and 2) and the total number of parts (9). If the 2 parts for teachers who don’t wear glasses represent 10 teachers, each part represents 5 teachers and 9 parts represent $9 \boldcdot 5$ or 45 teachers.

Lesson 17

A Fermi Problem

No cool-down

Unit 2 Assessment

End-of-Unit Assessment

Problem 1

Select all the true statements.

<p>Three rows of shapes. First row contains six smiley faces. Second row contains two triangles. Third row contains four squares. </p>

The ratio of triangles to squares is 2 to 4.

The ratio of squares to smiley faces is $6:4$ .

The ratio of smiley faces to triangles is 6 to 4.

There are two squares for every triangle.

There are two triangles for every smiley face.

There are three smiley faces for every triangle.

Show Solution

A, D, F

Problem 2

Select all the ratios that are equivalent to $8:6$ .

$4:3$

$6:8$

$16:12$

$10:8$

$7:5$

Show Solution

A, C

Problem 3

A mixture of purple paint contains 6 teaspoons of red paint and 15 teaspoons of blue paint. To make the same shade of purple paint using 35 teaspoons of blue paint, how much red paint would you need? Use the double number line diagram to help if needed.

Double number line. Red paint, teaspoons. Blue paint, teaspoons.

12 teaspoons

14 teaspoons

18 teaspoons

26 teaspoons

Show Solution

14 teaspoons

Problem 4

Lin rode her bike 2 miles in 8 minutes. She rode at a constant speed. Complete the table to show the time it took her to travel different distances at this speed.

distance traveled (miles)	elapsed time (minutes)
2	8
1
	1

Show Solution

distance traveled (miles)	elapsed time (minutes)
2	8
1	4
$\frac14$	1

Problem 5

Three movie tickets cost $36. At this rate, what is the cost per ticket?
Three ice cream cones cost $8.25. At this rate, how much do 2 ice cream cones cost?
Three bananas cost $0.99. At this rate, how much do 5 bananas cost?

Show Solution

$12
$5.50
$1.65

Problem 6

A bag contains 120 marbles. Some are red and the rest are black. There are 19 red marbles for every black marble. How many red marbles are in the bag? Explain your reasoning.

Show Solution

114 red marbles. Sample reasoning: Make a table in which the first row has 19 red marbles and 1 black marble. To find the row where the total number of marbles is 120, multiply the first row by 6, giving 114 red marbles and 6 black marbles, which add to the correct total.

Minimal Tier 1 response:

Work is complete and correct.
Sample: $19 \boldcdot 6 = 114$ , so there are 114 red marbles and 6 black marbles.

Tier 2 response:

Work shows general conceptual understanding and mastery, with some errors.
Sample errors: Red and black marbles are in the correct ratio but do not add up to 120. Correct answer is given but with no work or explanation shown. Work involves some correct reasoning about equivalent ratios with some errors.

Tier 3 response:

Significant errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: Work does not show evidence of an understanding of equivalent ratios. Answer is incorrect and no work is shown.

Problem 7

To make an orange-flavored drink, Noah mixes 4 scoops of powder with 6 cups of water. Andre mixes 5 scoops of powder with 8 cups of water.

Create a double number line or a table that shows different amounts of powder and water that taste the same as Noah’s mixture.
Create a double number line or a table that shows different amounts of powder and water that taste the same as Andre’s mixture.
How do their two mixtures compare in taste? Explain your reasoning.

Show Solution

Sample response:

Noah’s recipe

scoops of powder	cups of water
4	6
2	3
1	1.5

Andre’s recipe

scoops of powder	cups of water
5	8
10	16
1	1.6

Noah’s recipe tastes stronger. Sample reasoning: As shown in the tables, Noah’s recipe uses less water for the same amount of powder.

Minimal Tier 1 response:

Work is complete and correct, with complete explanation or justification.
Acceptable errors: some mixing up of the terms “cup” and “scoop.”
Sample:

See table for Noah’s recipe.
See table for Andre’s recipe.
Noah’s tastes stronger because for him, 1 cup of water needs $\frac23$ of a scoop of powder, but for Andre, 1 cup of water needs only $\frac58$ of a scoop of powder.

Tier 2 response:

Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification.
Acceptable errors: A good explanation to Part C is based on incorrect powder-water combinations found in Parts A and B.
Sample errors: Work shows arithmetic errors in otherwise reasonable tables or double number lines. Explanation for Part C is on track but leaves the reader to connect too many dots, for instance, “Noah’s tastes stronger because $\frac23 >\frac58$ .”

Tier 3 response:

Work shows a developing but incomplete conceptual understanding, with significant errors.
Acceptable errors: A good explanation to Part C is based on incorrect powder-water combinations found in Parts A and B.
Sample errors: Representations in Parts A and B do not show evidence of an understanding of equivalent ratios. Correct representations are used in Parts A and B but explanation in Part C is missing or does not involve a comparison of unit rates.

Tier 4 response:

Work includes major errors or omissions that demonstrate a lack of conceptual understanding and mastery.
Sample errors: None of the work contains evidence of an understanding of equivalent ratios.