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.
  1. Choose two of the colors in the image, and draw a diagram showing the number of snap cubes for these two colors.
  2. 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.
  3. 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.
  1. 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.

    1. The ratio of cups of white paint to tablespoons of blue paint is \(2 : 6\).
    2. For every cup of white paint, there are 2 tablespoons of blue paint.
    3. There is 1 cup of white paint for every 3 tablespoons of blue paint.
    4. There are 3 tablespoons of blue paint for every cup of white paint.
    5. For each tablespoon of blue paint, there are 3 cups of white paint.
    6. For every 6 tablespoons of blue paint, there are 2 cups of white paint.
    7. The ratio of tablespoons of blue paint to cups of white paint is 6 to 2.
  2. Jada mixed 8 cups of flour with 2 pints of water to make paste for an art project.

    1. Draw a diagram that represents the situation.

    2. Write at least two sentences describing the ratio of flour and water.

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.
  1. Take turns with your partner to match a sentence with a diagram.

    1. For each match that you find, explain to your partner how you know it’s a match.
    2. For each match that your partner finds, listen carefully to their explanation. If you disagree, discuss your thinking and work to reach an agreement.
  2. 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.
  3. 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 ______.
  4. 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?


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

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.

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.