The purpose of this Warm-up is to elicit the ways in which students understand and interpret a division expression with whole numbers, preparing them to explore the meanings of division in the lesson.
This activity uses the Collect and Display math language routine to advance conversing and reading as students clarify, build on, or make connections to mathematical language.
Student Lesson in Spanish
Launch
Arrange students in groups of 2. Ask students to write at least two different ways in which they think about . Explain that they can write what the expression means to them, how they think about it when finding the value of the expression, or a situation that the expression could represent.
Give students 1–2 minutes of quiet think time, followed by 1 minute of partner discussion. During the discussion, ask students to share their responses and notice what they have in common.
Use Collect and Display to direct attention to words collected and displayed from an earlier lesson. Collect the language that students use to describe their interpretations of . Display words and phrases, such as “dividend,” “divisor,” “equal groups,” “split equally into 4 groups (or parts),” and “put into groups of 4.”
Activity
None
Student Task Statement
Here is an expression: .
What are some ways to think about this expression?
Describe at least two meanings you think it could have or two questions it could be used to answer.
Student Response
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Building on Student Thinking
Some students may simply write the value of the expression because they are not sure how to describe how they think about it. Encourage them to think of a story with a question, in which the expression could be used to answer the question, or to draw a diagram that illustrates what the numbers could mean.
Activity Synthesis
Direct students’ attention to the reference created using Collect and Display. Ask partners to share the interpretations of that they had in common. Invite students to borrow language from the display as needed.
Record and display students’ responses for all to see. Ask students to notice any themes or trends in the range of responses.
Highlight the two ways students will be thinking about division in this unit:
Division means partitioning a number or a quantity into equal groups and finding out how many groups can be made.
Division means partitioning a number or a quantity into equal groups, and finding out how much is in each group.
2.2
Activity
20 mins
Bags of Clay
Standards Alignment
Building On
3.OA.A.2
Interpret whole-number quotients of whole numbers, e.g., interpret as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as .
This activity makes explicit the two ways of interpreting division in terms of equal-size groups: as a way to find the size of one group and a way to find the number of groups. Students explore these two ways of seeing division by connecting it to multiplication, making sense of it in the context of a situation, and drawing visual representations.
While tape diagrams are used to help students see what could mean, students may choose to draw other kinds of diagrams to represent other division expressions in the activity.
As students interpret equations and diagrams that represent the quantities in a situation and create their own representations, they practice reasoning abstractly and quantitatively (MP2).
Launch
Keep students in groups of 2. Ask students to keep their materials closed. Display the following question for all to see:
“An art teacher has 12 pounds of clay. She puts them in bags, so that each bag has the same weight.”
Ask students: “In terms of pounds and bags of clay, what could mean?”
Give students a minute of quiet think time and another minute to explain their thinking to their partner. Ask a few students to share their interpretations with the class. If students do not bring up one of the two ways to interpret the 6, ask them about it: “What else could the 6 mean in this situation?”
Once students see that the divisor could be interpreted in two ways, give them 5 minutes to complete the first question. Ask students to pause for a brief discussion before moving on.
Select students to explain Clare and Tyler's diagrams and equations. Highlight that, in this context, could mean 12 pounds of clay being divided equally into 6 bags, or 12 pounds of clay being divided so that each bag has 6 pounds.
Give students quiet time to complete the rest of the activity.
Action and Expression: Develop Expression and Communication. Develop fluency with tape diagrams. Provide access to blank or partially completed tape diagrams for students to use in answering the questions. Supports accessibility for: Visual-Spatial Processing, Attention
Activity Synthesis
The goal of this discussion is to make sure students recognize that a division expression such as can be interpreted as the answer to the question “4 times what number equals 12?” or the question, “What number times 4 equals 12?"
Select a few students to share their diagrams and equations for the problems in the last question. After each explanation, highlight the connections between the expression, the diagram, and the context.
Note that students may write either or for each interpretation as long as they understand what each factor represents. Because in these materials we tend to say “ groups of ,” we follow that order in writing the multiplication:
Point out that, in general, division can be interpreted as a way to find two values:
The size of each group when we know the number of groups and a total amount
How many groups are in a total amount given the size of one group
Lesson Synthesis
The goal of this discussion is to highlight that both multiplication and division can be reasoned in terms of equal-size groups. Just as we can interpret to mean “3 groups of 5 make 15,” we can think of division equations such as in terms of the number of groups and how much is in each group. Diagrams and equations can help us make sense of the numbers and what they mean.
Ask students questions such as:
"What are two ways to interpret ?" (“How many groups of 8 are in 20?" or "How much is in each group if there are 20 in 8 equal-size groups?")
“Suppose we interpret it as 'How much is in each group if there are 20 in 8 groups?' How might we draw a diagram to show this?” (A tape that represents 20 divided into 8 equal parts.) "What multiplication equation can we write?" (Either or can work, as long as we know what each factor represents.)
"If we think of it as 'How many groups of 8 are in 20?', how would the diagram be different?" (A tape that represents 20 divided into equal parts of 8.) "What multiplication equation can we write?" (Either or can work, as long as we are clear what each factor represents.)
A note about the term “group:”
Students may be most familiar with the idea of a group as a collection of people or objects. Clarify that the term is used more broadly here. A “group” can refer to a part, a batch, a bag, a section, or another quantity with a particular value. So “equal-size groups” can refer to collections with the same number of items or people in each, as well as parts with the same value, sections of equal length, bags of the same weight, and so on. As students reason about various multiplication and division situations, the meaning of “group” in each situation will become more intuitive.
Student Lesson Summary
Suppose 24 books are being distributed into boxes. The expression can be understood in two ways:
24 books are distributed equally into 3 boxes, as represented by this diagram:
In this case, the quotient, 8, is the number of books in each box.
24 books are distributed into boxes, 3 books in each box, as represented by this diagram:
In this case, the quotient, 8, is the number of boxes used.
These two ways of seeing division are connected to the way 3, 8, and 24 are related by multiplication.
can be read as “3 groups of 8 make 24.”
can be read as “8 groups of 3 make 24.”
If 3 and 24 are the only numbers given, the multiplication equations would be:
In both cases, the division can be used to find the unknown value, and the “?” can mean one of two things:
The size of a group (as in “3 groups of what number make 24?”)
The number of groups (as in “How many groups of 3 make 24?”)
Standards Alignment
Building On
3.OA.A.2
Interpret whole-number quotients of whole numbers, e.g., interpret as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as .
An art teacher has 12 pounds of clay. She puts them in bags, so that each bag has the same weight.
Clare and Tyler drew diagrams and wrote equations to show how they were thinking about in this situation.
Clare's tape diagram and equation. 2 equal parts labeled 6. Total, 12. Equation blank times 6 = 12.
Tyler's tape diagram and equation. 6 equal parts labeled 2. Total, 12. Equation 6 times blank = 12.
How do you think Clare and Tyler thought about ? Explain what the parts in each diagram and each equation (including the unknown number) could mean in the situation.
Pause here for a class discussion.
What could each division expression mean in the situation with the bags of clay? Draw a diagram and write a multiplication equation to show how you are thinking about each of these expressions.
Student Response
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Building on Student Thinking
Students may interpret to mean “divide 12 in half,” say that its value is 6, and draw a diagram showing 12 partitioned into 2 equal parts. Urge students to revisit their reasoning by comparing and . Discuss questions such as:
“Would and have the same value? Why or why not?” (No, because the 12 is divided by different numbers, so the quotients will be different.)
“If can mean “12 divided into groups of 2,” what can mean?” (12 divided into groups of s)
“To check if is 6, we can multiply to see if it is 12. To see if equals 6, we can find . Does it give 12?” (No, it gives 3, which means is not 6.)
5.NF.B.7
Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.Students able to multiply fractions in general can develop strategies to divide fractions in general, by reasoning about the relationship between multiplication and division. But division of a fraction by a fraction is not a requirement at this grade.
