Lesson 3

The purpose of this warm-up is to quickly remind students of different ways to write ratios. They also have an opportunity to multiply the number of each type of shape by 2 to make two copies of the flower, which previews the process introduced in this lesson for making a double batch of a recipe.

Arrange students in groups of 2. Ensure students understand there are 6 hexagons, 2 trapezoids, and 9 triangles in the picture, and that their job is to write ratios about the numbers of shapes. Give 2 minutes of quiet work time and then invite students to share their sentences with their partner, followed by whole-class discussion.

Students might get off track by attending to the area each shape covers. Clarify that this task is only concerned with the number of each shape and not the area covered.

Invite a student to share a sentence that describe the ratios of shapes in the picture. Ask if any students described the same relationship a different way. For example, three ways to describe the same ratio are: The ratio of hexagons to trapezoids is \(6:2\). The ratio of trapezoids to hexagons is 2 to 6. There are 3 hexagons for every trapezoid.

Ask a student to describe why two copies of the picture would have 12 hexagons, 4 trapezoids, and 18 triangles. If no student brings it up, be sure to point out that each number in one copy of the picture can be multiplied by 2 to find the number of each shape in two copies.

In this activity, three student volunteers participate in a taste test of two drink mixtures. Mixture A is made with 1 cup of water and 1 teaspoon of drink mix. Mixture B is made with 1 cup of water and 4 teaspoons of drink mix. The taste testers match diagrams with each mixture and explain their reasoning.

After the taste test, in front of students, recreate mixture A (1 cup water with 1 teaspoon of drink mix) and mixture B (1 cup water with 4 teaspoons of drink mix). Ask students to describe how the diagrams correspond with these mixtures. Then, conduct a demonstration in which 3 teaspoons of drink mix are added to Mixture A and a new diagram is drawn. Once Mixture A and Mixture B both contain one cup of water and 4 teaspoons of drink mix, both mixtures are combined in a third container labeled Mixture C.

As part of their work on the task, students reason that this combined mixture tastes the same as each individual batch. Students then conclude that a mixture containing two batches tastes the same as the mixture that contains just one batch, because mixing two things together that taste the same will produce a mixture that tastes the same. They should also note that each ingredient was doubled in the mixture.

Display the diagram for all to see:

Taste test: Recruit three volunteers for a taste test. Give each volunteer two unmarked cups—one each of a small amount of Mixture A and Mixture B. Explain that their job is to take a tiny sip of each sample, match the diagrams to the samples, and explain their matches.

Demonstration: Conduct a dramatic demonstration of mixing powdered drink mix and water. Start with two empty containers labeled A and B. To Container A, add 1 cup of water and 1 teaspoon of drink mix. To Container B, add 1 cup of water and 4 teaspoons of drink mix. Mix them both thoroughly. The first diagram should still be displayed.

Discuss:

• Which mixture has a stronger flavor? (B has more drink mix in the same quantity of water).
• How can we make Mixture A taste like Mixture B? (Put 3 more teaspoons of drink mix into Container A.)

Add 3 more teaspoons of drink mix to Container A. Display a new diagram to represent the situation:

Discuss:

• Describe the ratio of ingredients that is now in Container A. (The ratio of cups of water to teaspoons of drink mix is \(1:4\).)
• Describe the ratio of ingredients that is in Container B. (The ratio of cups of water to teaspoons of drink mix is also 1 to 4.)
• How do you think they compare in taste? (They taste the same. If desired, you can have volunteers verify that they taste the same, but this might not be necessary.)

Pour the contents of both A and B into a larger container labeled C and mix them thoroughly.

Discuss:

• How would the taste of Mixture C compare to the taste of Mixture A and Mixture B? (The new mixture would taste the same as each component mixture.)

Following this demonstration, students individually interpret the drink mixture diagrams. The work in the task will reiterate what happened in the demonstration.

Students may not initially realize that Mixtures C and B taste the same. You could ask them to imagine ordering a smoothie from a takeout window. Would a small size smoothie taste the same as a size that is double that amount? If you double the amount of each ingredient, the mixture tastes the same.

Mixing 1 cup of water with 4 teaspoons of powdered drink mix makes a mixture that tastes exactly the same as mixing 2 cups of water with 8 teaspoons of powdered drink mix. We say that \(1:4\) and \(2:8\) are equivalent ratios. Ask students to discuss what they think “equivalent” means. Some ways they might respond are:

• Mixtures that taste the same use equivalent ratios.
• A double batch of a recipe—doubling each ingredient—is an equivalent ratio to a single batch. 

Students continue to use diagrams to represent the ratio of ingredients in a recipe as well as mixtures that contain multiple batches. They come to understand that a change in the number of batches changes the quantities of the ingredients, but the end product tastes the same. They then use this observation to come up with a working definition for equivalent ratio.

Launch the task with a scenario and a question: “Let’s say you are planning to make cookies using your favorite recipe, and you’re going to ‘double the recipe.’ What does it mean to double a recipe?”

There are a few things you want to draw out in this conversation:

• If we double a recipe, we need to double the amount of every ingredient. If the recipe calls for 3 eggs, doubling it means using 6 eggs. If the recipe calls for \(\frac13\) teaspoon of baking soda, we use \(\frac23\) teaspoon of baking soda, etc.
• We expect to end up with twice as many cookies when we double the recipe as we would when making a single batch.
• However, we expect the cookies from 2 batches of a recipe to taste exactly the same as those from a single batch.

Tell students they will now think about making different numbers of batches of a cookie recipe.

For the fourth question, students may not multiply both the amount of flour and the amount of vanilla by the same number. If this happens, refer students to the previous questions in noting that the amount of each ingredient was changed in the same way.

Invite a few students to share their responses and diagrams with the class. A key point to emphasize during discussion is that when we double (or triple) a recipe, we also have to double (or triple) each ingredient. Record a working (but not final) definition for equivalent ratio that can be displayed for at least the next several lessons. Here is an example: “Cups of flour and teaspoons of vanilla in the ratio \(5:2\), \(10:4\), or \(15:6\) are equivalent ratios because they describe different numbers of batches of the same recipe.” Include a diagram in this display.