Meanings of Division

10 min

Teacher Prep
Setup
Students in groups of 2. 1 minute of quiet think time, followed by partner and whole-class discussions.

Narrative

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.

Launch

Arrange students in groups of 2. Ask students to write at least two different ways in which they think about 20÷420 \div 4. 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 20÷420 \div 4. Display words and phrases, such as “dividend,” “divisor,” “equal groups,” “split equally into 4 groups (or parts),” and “put into groups of 4.”

Student Task

Here is an expression: 20÷420\div 4.

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.

Sample Response

Sample responses:

  • It could mean:
    • 20 things divided equally among 4 people or into 4 groups
    • 20 things divided equally so that each person or each group has 4 things
  • It could answer questions such as: 
    • How many groups of 4 are in 20?
    • How many are in each group if we split 20 into 4 groups?
    • How many 4s are in 20?
    • What times 4 equals 20?
    • What is the length of a rectangle with a width of 4 units and an area of 20 square units?
Activity Synthesis (Teacher Notes)

Direct students’ attention to the reference created using Collect and Display. Ask partners to share the interpretations of 20÷420 \div 4 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.
Anticipated Misconceptions
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.
Standards
Building On
  • 3.OA.2·Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 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. <em>For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8.</em>
  • 3.OA.A.2·Interpret whole-number quotients of whole numbers, e.g., interpret <span class="math">\(56 \div 8\)</span> 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. <span>For example, describe a context in which a number of shares or a number of groups can be expressed as <span class="math">\(56 \div 8\)</span>.</span>

20 min