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.
Launch
Arrange students in groups of 2. Ensure that students understand that 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. Follow up with a whole-class discussion.
Activity
None
This flower is made up of yellow hexagons, red trapezoids, and green triangles.
Write sentences to describe the ratios of the shapes that make up this pattern.
How many of each shape would be in two copies of this flower pattern?
Student Response
Loading...
Building on Student Thinking
Students might get off track by attending to the area that each shape covers. Clarify that this task is concerned only with the number of each shape and not with the area covered.
Activity Synthesis
Invite a student to share a sentence that describes the ratio of two shapes in the picture. Ask if any students described the same relationship in a different way. For example, three ways to describe the same ratio are: The ratio of hexagons to trapezoids is .. 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.
3.2
Activity
Standards Alignment
Building On
Addressing
6.RP.A.1
Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was , because for every wings there was beak.” “For every vote candidate A received, candidate C received nearly three votes.”
This is the first of several activities that help students develop an intuition for equivalent ratios. First, a few volunteers taste two drink mixtures and consider diagrams that could represent the ratios of ingredients. Next, the class observes a demonstration of the creation of the two mixtures, a change to one mixture, and the creation of a third mixture. Students then consider how the three mixtures would taste and whether multiple batches of a recipe would taste the same as one batch.
Students see that combining two things that taste the same will produce a mixture that tastes the same. They should also note that each ingredient was doubled in the mixture.
Launch
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 to 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, invite volunteers to verify that they taste the same, but this might not be necessary.)
Pour the contents of both Container A and Container B into a larger container labeled ContainerC 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.
MLR8 Discussion Supports. To support a whole-class discussion, ask: “What if I add a half cup of water to Container C?” or “What if I add a teaspoon of drink mix to Container B?” To help students justify their reasoning, display a sentence frame such as: "Mixtures B and C will taste _____ because . . . ." Advances: Conversing, Speaking
Activity
None
Activity Synthesis
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 the ratios and are equivalent. Ask students to discuss what they think it means for the ratios to be “equivalent.” Students might say:
The ratios make the mixtures taste the same.
If there is twice as much of one ingredient, there is also twice as much of another ingredient, so the result still tastes the same.
3.3
Activity
Standards Alignment
Building On
Addressing
6.RP.A.1
Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was , because for every wings there was beak.” “For every vote candidate A received, candidate C received nearly three votes.”
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 ratios.
Launch
Ask students if they have ever cooked something by following a recipe. If so, ask them what they made and what some of the ingredients were.
Then, follow with these questions:
“How might we use ratios to describe the ingredients in your recipe?” (The ratios could associate the quantities of each ingredient being used.)
“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 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 teaspoon of baking soda, we use teaspoon of baking soda, and so on.
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.
Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer interactions. Invite students to talk about their ideas with a partner before writing them down. Display sentence frames to support students when they explain their strategy. For example: “I noticed _____, so I . . .” “In my diagram, _____ represents . . .” “To find a ratio that is equivalent, first I _____ because . . .” Supports accessibility for: Language, Social-Emotional Functioning
Activity
None
A recipe for one batch of cookies calls for 5 cups of flour and 2 teaspoons of vanilla.
Draw a diagram that shows the amount of flour and vanilla needed for two batches of cookies.
How many batches can you make with 15 cups of flour and 6 teaspoons of vanilla? Show the additional batches by adding more ingredients to your diagram.
How much flour and vanilla would you need for 5 batches of cookies?
Whether the ratio of cups of flour to teaspoons of vanilla is , , or , the recipes would make cookies that taste the same. We say that these ratios are equivalent.
Find another ratio of cups of flour to teaspoons of vanilla that is equivalent to these ratios.
How many batches can you make using this new ratio of ingredients?
Student Response
Loading...
Building on Student Thinking
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.
Activity Synthesis
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 , , or are equivalent ratios because they describe different numbers of batches of the same recipe.” Include a diagram in this display.
Lesson Synthesis
To highlight the key ideas from the lesson, consider asking students:
“When doubling a recipe, how does the amount of each individual ingredient change?“ (Each ingredient is doubled.)
“When tripling a recipe, how does the amount of each individual ingredient change?“ (Each ingredient is tripled.)
“How do different numbers of batches of the same recipe taste?“ (They taste exactly the same.)
"When doubling or tripling a recipe, the ratio of the ingredients is equivalent to the ratio in the original recipe. What do you think that means?" (The ratio creates a result that tastes the same.)
The main takeaways are:
To scale a recipe generally, we need to multiply each ingredient by the same number.
Scaling a recipe results in a substance that tastes the same as the original recipe.
We say that a ratio that represents a recipe is equivalent to a ratio that represents multiple batches of the same recipe.
Student Lesson Summary
A recipe for fizzy juice says, “Mix 5 cups of cranberry juice with 2 cups of soda water.”
To double this recipe, we would use 10 cups of cranberry juice with 4 cups of soda water. To triple this recipe, we would use 15 cups of cranberry juice with 6 cups of soda water.
This diagram shows a single batch of the recipe, a double batch, and a triple batch:
We say that the ratios , , and are equivalent. Even though the amounts of each ingredient within a single, double, or triple batch are not the same, they would make fizzy juice that tastes the same.
Standards Alignment
Building On
Addressing
6.RP.A.1
Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was , because for every wings there was beak.” “For every vote candidate A received, candidate C received nearly three votes.”
Here are diagrams representing three mixtures of powdered drink mix and water:
How would the taste of Mixture A compare to the taste of Mixture B?
Use the diagrams to complete each statement:
Mixture B uses cups of water and teaspoons of drink mix.
The ratio of cups of water to teaspoons of drink mix in Mixture B is .
Mixture C uses cups of water and teaspoons of drink mix.
The ratio of cups of water to teaspoons of drink mix in Mixture C is .
How would the taste of Mixture B compare to the taste of Mixture C?
Student Response
Loading...
Building on Student Thinking
Students may not initially realize that Mixtures C and B taste the same. Consider asking 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 we double the amount of each ingredient, the mixture tastes the same.
