Unit 4, Lesson 3
Interpreting Division Situations (Optional)
Grade 6
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This Warm-up prompts students to carefully analyze and compare four situations involving equal-size groups. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology related to multiplication, division, and equal-size groups and how they talk about characteristics of the items in comparison to one another.
Arrange students in groups of 2–4. Display the four items for all to see. Give students 1 minute of quiet think time, and ask them to indicate when they have noticed three items that go together and can explain why. Next, tell students to share their response with their group and then together to find as many sets of three as they can.
Which three go together? Why do they go together?
Jada fills 4 jars with salsa. Each jar has 10 ounces of salsa. How many ounces of salsa are in all the jars?
Andre is filling 4-liter jugs with water. How many jugs can he fill if he has 10 liters of water?
Invite each group to share one reason why a particular set of three go together. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which three go together, attend to students’ explanations and ensure that the reasons given are correct.
During the discussion, prompt students to explain the meaning of any terminology that they use to describe equal-size groups, such as “4 times the same amount,“ “divide into 4 groups,” and “split into 4 equal parts.” Ask students to clarify their reasoning as needed. Consider asking:
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In this activity, students continue to investigate division problems in context, think of them in terms of equal-size groups, and represent them using diagrams and equations.
Both division problems result in quotients that are not whole numbers, but because they represent concrete quantities, students can make sense of the fractional values (number of groups and size of each group) in terms of each situation.
As students work, monitor for students who draw diagrams as they reason about the situations. Any diagram (concrete or abstract) is fine as long as it enables students to make sense of the relationship between the number of groups, the size of one group, and a total amount. Select students whose diagrams clearly show the meanings of division, and ask those students to share later. During class discussion, students will have an opportunity to look more closely at how tape diagrams can support thinking about division.
Arrange students in groups of 2. Tell students that they will look at two situations about equal-size groups of ingredients for a recipe for making bubble solution.
Ask students if they have had experience making homemade bubble mixture. Invite students to share what they know about the ingredients or the process.
Give students 5–7 minutes of quiet work time and then time to discuss their responses with a partner. During the partner discussion, ask them to compare their equations and diagrams in the first question, and their interpretations of division in the second question.
If needed, remind students that "fl oz" refers to "fluid ounce" and is a unit to measure the volume of a liquid.
Mai is making a bubble mixture. The recipe says to add sugar to help the bubbles last longer.
To make one batch of mixture, 14 grams of sugar is needed. To get that amount, Mai uses 4 packets of sugar.
Mai needs 26 fl oz of water to make a larger amount of the mixture. The only measuring tool she has is a 4-fl oz scoop. How many scoops will it take to measure 26 fl oz of water?
The goals of the discussion are to solidify students' understanding of the two interpretations of division (“How many groups?” and “How much in a group?”) and to clarify that division may not result in a whole-number quotient.
Invite students to share their responses and reasoning. Ask students who drew effective diagrams to display and explain them.
If no students drew tape diagrams, display the following diagrams as another way to reason about division.
Discuss what each part of the diagram and each label represent:
Ask students how they can tell if the results they get from dividing 14 by 4 and 26 by 4 are correct. If no students mention multiplying each result by 4 to see if the product is 14 and 26, respectively, discuss this idea.
When a situation involves equal-size groups, it is helpful to make sense of it in terms of the number of groups, the size of each group, and the total amount. Here are three examples:
Suppose we have 3 bottles with
We can think of this situation as a multiplication problem, “What is 3 groups of
The unknown value is the product. To find it, we can multiply 3 and
Next, suppose we have 20 ounces of water to fill 6 bottles equally, and we want to know the amount in each bottle.
We can also express this situation as a multiplication problem, “Six groups of what number make 20?” and with an equation,
The unknown value is a factor. To find it, we can divide:
Now, suppose we have 50 ounces of water to fill 12-ounce bottles, and we want to know how many bottles will be filled.
Again, we can see this as a multiplication problem, “How many groups of 12 make 50?” and write a multiplication equation,
The unknown value is a different factor. To find it, we can again use division:
In any situation that involves equal-size groups, we can use division to find the amount in each group or the number of groups.
Help us improve by sharing suggestions or reporting issues.
Some students may round their answers to the nearest whole number rather than including a fraction of a scoop or of a gram. Ask students to consider if it is possible to have a part of a scoop or a part of a unit of volume. Encourage them to think about how to show a part of a whole unit on a diagram.
Here are descriptions and diagrams of situations that involve equal-size groups.
Choose a situation.
If you choose one that is described in words, draw a diagram to represent the situation. Be sure to include labels.
If you choose an image or a diagram, write a story with a question that the image or diagram could represent.
Write a multiplication equation and a division equation to represent the relationships between the quantities.