Lesson 2
Representing Ratios with Diagrams
Let’s use diagrams to represent ratios.
2.1: Number Talk: Dividing by 4 and Multiplying by $\frac14$
Find the value of each expression mentally.
\(24\div 4\)
\(\frac14\boldcdot 24\)
\(24\boldcdot \frac14\)
\(5\div 4\)
2.2: A Collection of Snap Cubes
Here is a collection of snap cubes.
![A collection of snap cubes arranged by color. The collection contains 2 green, 5 yellow, 5 red, 3 pink, 2 blue, and 1 black.](https://cms-im.s3.amazonaws.com/vQ7GBdM18JmcJUh4p9GkF5JL?response-content-disposition=inline%3B%20filename%3D%226-6.2.A2_Image_4.png%22%3B%20filename%2A%3DUTF-8%27%276-6.2.A2_Image_4.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240726%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240726T234719Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=034ad832edb56b34c570ca061f598c5bba5579f7f8fa7680673ff16b6cdcf359)
- Choose two of the colors in the image, and draw a diagram showing the number of snap cubes for these two colors.
- Trade papers with a partner. On their paper, write a sentence to describe a ratio shown in their diagram. Your partner will do the same for your diagram.
- Return your partner’s paper. Read the sentence written on your paper. If you disagree, explain your thinking.
2.3: Blue Paint and Art Paste
Elena mixed 2 cups of white paint with 6 tablespoons of blue paint.
Here is a diagram that represents this situation.
![Two large squares and six small squares.](https://cms-im.s3.amazonaws.com/U6L65gXWqX3S7LK5nmdnE1s4?response-content-disposition=inline%3B%20filename%3D%226-6.2.A2_Image_6.1.png%22%3B%20filename%2A%3DUTF-8%27%276-6.2.A2_Image_6.1.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240726%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240726T234719Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=a7672ae15f5927c6aa3490aacab74cf0f2ebcd5844667266319ad3c3f68bd314)
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Discuss each statement, and circle all those that correctly describe this situation. Make sure that both you and your partner agree with each circled answer.
- The ratio of cups of white paint to tablespoons of blue paint is \(2 : 6\).
- For every cup of white paint, there are 2 tablespoons of blue paint.
- There is 1 cup of white paint for every 3 tablespoons of blue paint.
- There are 3 tablespoons of blue paint for every cup of white paint.
- For each tablespoon of blue paint, there are 3 cups of white paint.
- For every 6 tablespoons of blue paint, there are 2 cups of white paint.
- The ratio of tablespoons of blue paint to cups of white paint is 6 to 2.
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Jada mixed 8 cups of flour with 2 pints of water to make paste for an art project.
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Draw a diagram that represents the situation.
- Write at least two sentences describing the ratio of flour and water.
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2.4: Card Sort: Spaghetti Sauce
Your teacher will give you cards describing different recipes for spaghetti sauce. In the diagrams:
- a circle represents a cup of tomato sauce
- a square represents a tablespoon of oil
- a triangle represents a teaspoon of oregano
![A photo of spaghetti sauce.](https://cms-im.s3.amazonaws.com/QAdG7kPuksYxuVaaqqYjUA13?response-content-disposition=inline%3B%20filename%3D%226-6.2.image_of_sauce.png%22%3B%20filename%2A%3DUTF-8%27%276-6.2.image_of_sauce.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240726%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240726T234719Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=8a87b8c3cab22d80668d04281b0b346af8431f42bab01299a54d64eff2bf1951)
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Take turns with your partner to match a sentence with a diagram.
- For each match that you find, explain to your partner how you know it’s a match.
- For each match that your partner finds, listen carefully to their explanation. If you disagree, discuss your thinking and work to reach an agreement.
- After you and your partner have agreed on all of the matches, check your answers with the answer key. If there are any errors, discuss why and revise your matches.
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There were two diagrams that each matched with two different sentences. Which were they?
- Diagram _______ matched with both sentences ______ and ______.
- Diagram _______ matched with both sentences ______ and ______.
- Select one of the other diagrams and invent another sentence that could describe the ratio shown in the diagram.
Create a diagram that represents any of the ratios in a recipe of your choice. Is it possible to include more than 2 ingredients in your diagram?
Summary
Ratios can be represented using diagrams. The diagrams do not need to include realistic details. For example, a recipe for lemonade says, “Mix 2 scoops of lemonade powder with 6 cups of water.”
Instead of this:
![A drawing of 2 scoops and 6 tea cups](https://cms-im.s3.amazonaws.com/dyXYLPKtdb9FiGENnmHLpubQ?response-content-disposition=inline%3B%20filename%3D%226-6.2.A2_Image_10.png%22%3B%20filename%2A%3DUTF-8%27%276-6.2.A2_Image_10.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240726%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240726T234719Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=b8c13af1aaf58df33945492cbec5f40066a243ece2a09cee0c91616b1e9eb08b)
We can draw something like this:
![A discrete diagram of small and large squares. The top row contains 2 small yellow squares and the bottom row contains 6 large blue squares.](https://cms-im.s3.amazonaws.com/fGPQGfvpD9cJLpKo3j36ehch?response-content-disposition=inline%3B%20filename%3D%226-6.2.A2_Image_11.1.png%22%3B%20filename%2A%3DUTF-8%27%276-6.2.A2_Image_11.1.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240726%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240726T234719Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=3904f8b8f5eb3b5719b097c030f19f7690b460f5f29ee5bf9c380bbf7bc8baa3)
This diagram shows that the ratio of cups of water to scoops of lemonade powder is 6 to 2. We can also see that for every scoop of lemonade powder, there are 3 cups of water.
Glossary Entries
- ratio
A ratio is an association between two or more quantities.
For example, the ratio \(3:2\) could describe a recipe that uses 3 cups of flour for every 2 eggs, or a boat that moves 3 meters every 2 seconds. One way to represent the ratio \(3:2\) is with a diagram that has 3 blue squares for every 2 green squares.