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Domain Operations and Algebraic Thinking
Grade 3
Cluster Represent and solve problems involving multiplication and division.
Standard Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.<span class='footnote'>See Glossary, Table 2.</span>
Task Two Interpretations of Division

Two Interpretations of Division


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Alignments to Content Standards: 3.OA.A.3
Student View

Task

  1. Maria cuts 12 feet of ribbon into 3 equal pieces so she can share it with her two sisters. How long is each piece?
  2. Maria has 12 feet of ribbon and wants to wrap some gifts. Each gift needs 3 feet of ribbon. How many gifts can she wrap using the ribbon?

IM Commentary

Both of the questions are solved by the division problem $12 \div 3$ but what happens to the ribbon is different in each case. In the first case, the number of pieces of ribbon is fixed at 3 and the question is asking "how long is each piece?" (12 feet divided into 3 pieces gives 4 feet per piece.) In the second question, the size of each piece is fixed and the question is "how many pieces does one get?" (12 feet divided by 3 feet per gift gives 4 gifts.) The problem can be solved with a drawing of a tape diagram or number line. For problem 1, the line must be divided into 3 equal parts. The second problem can be solved by successive subtraction of 3 feet to see how many times it fits in 12.

In this case it is particularly helpful for the teacher to require students to justify their answers with a diagram. The way in which a student represents the problem can reveal whether or not he or she really understands the distinction between the two types of division problems shown here.

Solutions

Solution: Tape Diagram

  1. This question asks, "how long is each piece?" so is a "How many in each group?" division problem: $$3\times ?= 12$$

    1_b46f49f62498dbbf96088f6cbb628801

    2_45836aef16424869af17f63c9fdaa9d3

    $12 \div 3 = 4$, so each child gets a piece of ribbon that is 4 feet long.

  2. This question asks, "how many pieces does one get?" so is a "How many groups?" division problem: $$? \times 3= 12$$ 3_5e9a4bdb7d8ccd9796c16daf9cd1887c

    4_5fe9119ae5a2d77d2f21baf7928ea2e5

    $12 \div 3 = 4$, so Maria can wrap 4 gifts.

Solution: Number Line

  1. This question asks, "how long is each piece?" so is a "How many in each group?" division problem: $$3\times ?= 12$$ Sol5_8233d3a8eb4f7fe9ec2e705244c07b94
    Sol6_d10344896c2f82e821b038b37a7ae01d

    $12 \div 3 = 4$, so each child gets a piece of ribbon that is 4 feet long.

  2. This question asks, "how many pieces does one get?" so is a "How many groups?" division problem: $$?\times 3= 12$$

    7_a89f0ab611a5444c948beb05e976acc9

    8_61e04a2b22c73ce044e79a3ee2bec238

    $12 \div 3 = 4$, so Maria can wrap 4 gifts.

Two Interpretations of Division

  1. Maria cuts 12 feet of ribbon into 3 equal pieces so she can share it with her two sisters. How long is each piece?
  2. Maria has 12 feet of ribbon and wants to wrap some gifts. Each gift needs 3 feet of ribbon. How many gifts can she wrap using the ribbon?
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