Representing Ratios with Diagrams
2.1: Number Talk: Dividing by 4 and Multiplying by $\frac14$ (10 minutes)
This number talk helps students recall that dividing by a number is the same as multiplying by its reciprocal. Four problems are given, however, they do not all require the same amount of time. Consider spending 6 minutes on the first three questions and 4 on the fourth question.
In grade 4, students multiplied a fraction by a whole number, using their understanding of multiplication as groups of a number as the basis for their reasoning. In grade 5, students multiply fractions by whole numbers, reasoning in terms of taking a part of a part, either by using division or partitioning a whole. In both grade levels, the context of the problem played a significant role in how students reasoned and notated the problem and solution. Two important ideas that follow from this work and that will be relevant to future work should be emphasized during discussions:
- Dividing by a number is the same as multiplying by its reciprocal.
- We can multiply numbers in any order if it makes it easier to find the answer.
Display one problem at a time. Give students 1 minute of quiet think time per problem and ask them to give a signal when they have an answer and a strategy. Allow students to share their answers with a partner and note any discrepancies. Pause after the third question and ask, “What do you notice about the first three questions? Do you notice the same thing if we divide 5 by 4? Why?”
Supports accessibility for: Memory; Organization
Find the value of each expression mentally.
Ask students to share what they noticed about the first three problems. Record student explanations that connect dividing by a number with multiplying by its reciprocal. Revisit the meaning of “reciprocal” when the term comes up (or bring it up if it's not mentioned by students). Help students recall that the product of a number and its reciprocal is 1.
Discuss how students could use their observations on the first three questions to divide 5 by 4, and then any two whole numbers.
Design Principle(s): Optimize output (for explanation)
2.2: A Collection of Snap Cubes (10 minutes)
Here students read ratio information from a picture and represent it as a diagram. The activity serves two purposes: to reinforce ratio language introduced in the previous lesson, and to better understand the meaning of the term “diagram.”
As students work, check that they use ratios in their sentences and draw appropriate diagrams. Examples of sentences with ratios (from the previous lesson) should be posted in the room. Draw students’ attention to these existing examples as needed.
Look for students who write ratios involving the same values, e.g., “the ratio of blue to white is 1 to 1,” or make note if no one does so. If all examples of ratios students have come across so far involve pairs or sets with different numbers for their values, students may mistakenly conclude that quantities that have the same values cannot be expressed as a ratio.
For non-sighted or color-blind students, this activity can be adapted by giving them blocks of different shapes.
Orient students to the picture (if you have real cubes, use them). Review the meaning of “diagram.” For example, to represent two green snap cubes, you might draw two green squares on the board, or two squares labeled “G” if colors are not available.
Arrange students in groups of 2. Provide access to colored pencils.
Supports accessibility for: Conceptual processing
Here is a collection of snap cubes.
- 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.
Students might not draw discrete diagrams at first. They might be inclined to draw more detailed drawings. Emphasize that a diagram represents the number and type of objects, and does not need to represent details about the shapes of the snap cubes.
Invite one or two pairs of students to share their sentences. Press for details as they explain, asking them to clarify, elaborate, or give examples. Revoice student ideas to demonstrate mathematical language. Discuss whether or not students were able to interpret one another’s drawings accurately. If not, what may have led to confusion?
If no one wrote ratios in which all numbers are the same (e.g., 1 to 1, or \(3 : 3\)), ask if the following sentence is acceptable and why or why not: “The ratio of green cubes to blue cubes is 2 to 2.” If students suspect that ratios are only used to associate quantities with different values, clarify that this is not the case.
Design Principle: Maximize meta-awareness
2.3: Blue Paint and Art Paste (10 minutes)
In this activity, students continue to draw connections between a diagram and the ratios it represents. Students work in pairs to discuss different ways to use ratio language to describe discrete diagrams. They first identify statements that would correctly describe a given diagram. Then, they create both a diagram and corresponding statements to represent a new situation involving ratio.
As students work, monitor for different ways in which students draw and discuss diagrams of the paste recipe. Identify a few pairs who draw different diagrams and use ratio language differently to share later. A few things to anticipate:
- Some students may draw very literal drawings of cups and pints. Encourage them to use simpler representations.
- Students may choose to draw letters (X’s) or other symbols or marks instead of squares and rectangles.
- Students may use equivalent ratios to describe a situation, even though these have not been explicitly taught (e.g., they may say the ratio of cups of flour to pints of water is \(4:1\) instead of \(8:2\)). Though this is correct, be careful here. We have previously regrouped objects and might say, for example, that with a ratio \(8:2\), “for every 4 cups of flour there is 1 cup of water,” but we have not asserted that this ratio can be written as \(4:1\) yet. The idea of equivalent ratios is sophisticated and will be developed over the next several lessons.
- Correct descriptions may include fractions (e.g., for every tablespoon of blue paint, there is \(\frac13\) cup of white paint). Although students are not expected to work with fractions in this lesson, responses involving fractions are fine.
Arrange students in groups of 2. Provide them with the tools needed for creating a large visual display for the second part of the task. Ensure students understand they are supposed to select more than one statement for the first question. Consider having students take turns reading each statement and deciding whether they think it describes the situation or not.
Design Principle(s): Support sense-making
Elena mixed 2 cups of white paint with 6 tablespoons of blue paint.
Here is a diagram that represents this situation.
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.
Jada mixed 8 cups of flour with 2 pints of water to make paste for an art project.
Draw a diagram that represents the situation.
- Write at least two sentences describing the ratio of flour and water.
Some students may think all of the statements about the paint mixture are accurate descriptions. If so, suggest that there are two false statements. Have students discuss the statements again in determining which two are false.
Select students to share their paste diagrams and sentences with the class. Sequence the diagrams from most elaborate to most simple. Connect the many ways in which the paste can be represented and described. Compare more detailed pictures with a discrete diagram; point out how the discrete diagram is a more efficient way of showing the paste recipe.
2.4: Card Sort: Spaghetti Sauce (15 minutes)
Writing and using ratio language requires attention to detail. This task further develops students’ ability to describe ratio situations precisely by attending carefully to the quantities, their units, and their order in the ratio.
Students work in pairs to match ratios of sauce ingredients to discrete diagrams and to explain reasoning (MP3).
Arrange students in groups of 2. Place two copies of uncut blackline masters in envelopes to serve as answer keys.
Demonstrate how to set up and play the matching game. Choose a student to be your partner. Discuss what all the symbols mean. Mix up the cards and place them face-up. Point out that the cards contain either diagrams or sentences. Select one of each style of card and then explain to your partner why you think the cards do or do not match. Demonstrate productive ways to agree or disagree, e.g., by explaining your mathematical thinking, asking clarifying questions, etc.
Give each group cut-up cards for matching. Tell students to check their matches after they complete the activity using the answer keys.
Supports accessibility for: Language; Organization
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
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.
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.
Are you ready for more?
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?
If students disagree about a match, encourage them to figure out the correct answer through discussion and use of the answer key. Make sure that when students use the answer key, they discuss any errors rather than just make changes.
Students may think the shapes in the diagram need to be drawn in the same order the ingredients appear in the description. This is not the case. You could turn a diagram card upside down and it would still represent the same situation. The diagram just shows ingredients that get mixed together in a pot. It is the case, however, that within the description, the order of the words in the sentence must correspond with the terms within the ratio.
Once all groups have completed the matching, discuss the following:
- Which matches were tricky? Explain why.
- Did any pairs need to make adjustments in their matches? What might have caused an error? What adjustments were made?
- What if you were making this tasty sauce and got the ratios wrong? What would happen?
Design Principle(s): Optimize output (for explanation)
This lesson used diagrams to represent ratios. These diagrams omit details that are not necessary for understanding and solving the problem at hand. Discuss:
- What are some good things to remember when you draw a diagram of a ratio? (You only need necessary information. You could include shapes, color-coded boxes, or initials to represent each object within the set. It is helpful to organize the types of items in rows, and to arrange smaller groups so they are easier to see.)
- How can a diagram help you make sense of a situation involving a ratio? (It is easier to write correct statements about them. Also, you can see how the objects can be grouped.)
2.5: Cool-down - Paws, Ears, and Tails (5 minutes)
Student Lesson 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:
We can draw something like this:
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.