Lesson 1
Representing Ratios with Diagrams
Let’s use diagrams to represent ratios.
1.1: Number Talk: Dividing by 4 and Multiplying by $\frac14$
Find the value of each expression mentally.
\(24\div 4\)
\(\frac14\boldcdot 24\)
\(24\boldcdot \frac14\)
\(5\div 4\)
1.2: 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=20240726T233418Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=d7034fb61e1637b5c16dd39ed0ec57bddfe9e055dc8d31f7e0a8a37ddf9d0d8e)
- 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.
1.3: 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=20240726T233418Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=847fed3ae595f0c818aec35a022b507c0ba4aa57a8c70efcbedeb3e0e47cbcd6)
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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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1.4: 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=20240726T233418Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=4e0ac8e6e81732dd5265a71afb3e2173b2c5f4bfca1ef46b248c8b920cda8e21)
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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=20240726T233418Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=dadf5946211cb783b20771f7dda9a65e10bea059def509e60b620b9d6a7983dc)
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=20240726T233418Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=58cb8b791da9e59ec94663c6284d821d680cea51d41b438ac780efec8ed0e92b)
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.