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Analyzing Word Problems Involving Multiplication


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

Task

Many problems can be solved in different ways. Decide if the following word problems can be solved using multiplication. Explain your thinking. Then solve each problem.

  1. Liam is cooking potatoes. The recipe says you need 5 minutes for every pound of potatoes you are cooking. How many minutes will it take for Liam to cook 12 pounds of potatoes?
  2. Mel is designing cards. She has 4 different colors of paper and 7 different pictures she can glue on the paper. How many different card designs can she make using one color of paper and one picture?
  3. Nina can practice a song 6 times in an hour. If she wants to practice the song 30 times before the recital, how many hours does she need to practice?
  4. Owen is building a rectangular tile patio that is 4 tiles wide and 6 tiles long. How many tiles does he need?

IM Commentary

The purpose of this task is for students to analyze different contexts in which multiplication is appropriate. Research indicates that students often do not make sense of word problems. Instead, to come up with an answer, they just apply the most recent algorithm taught, chose operations based on the types of numbers involved, or apply other approaches that do not require sense making. In this task, the students are asked to analyze the problems and explain their thinking before they find an answer. In the process, they are thinking more carefully about different contexts that can be represented by multiplication.

Note that none of the contexts involve multiplicative comparison, which is intentionally left until 4th grade (see 4.OA.A.1).

Solutions

Solution: Analysis

  1. While you can solve this by repeated addition (5+5+5+….), this problem can also be solved by multiplication since adding 5 together 12 times is the same as $12\times5$.
  2. This can be solved by multiplication, because for each of the four colors, there are seven different pictures from which to choose, so 7 pictures on the first color + seven pictures on the second color + 7 pictures on the third color + 7 pictures on the fourth color gives four groups of seven or $4\times 7$.
  3. This is a division problem since the number of hours needed can be determined by counting the number of groups of six needed to make 30, or $$30 \div 6 = ?$$ On the other hand, we can think about this in terms of multiplication by asking what times six equals 30, which we can write symbolically as $$? \times 6 = 30$$
  4. It is a good idea to draw a picture:

    Tiles_1268f2e0b2ed53d06d5135d25cf1de37

    We can count the number of tiles one by one, or we can see that there are four rows, each with six tiles, so we can think of it as a multiplication problem $4 \times 6$.

Solution: Solving explicitly

  1. $12\times5$ is 12 groups of 5, which we can see in the picture below:

    1_d383c0f2e8161471cf07d20a78da31a2

    12 groups of 5 is the same as 6 groups of 10:

    2_b43d595bf410a3de977ebb72a4ed1844

    which is 60. So Liam needs 60 minutes to cook the potatoes.

  2. An array makes it easier to see:

    3_c10a3f9ac075f692a389120953c6df77

    There are 28 different possible cards.

  3. If we multiply the number of hours she practices by 6, we know the answer must be 30: $$?\times 6 = 30$$ Since 5 times 6 is 30, she has to play for 5 hours if she wants to practice the song 30 times.
  4. Looking at the picture above, we can see that Owen needs 24 tiles.

John says:

over 2 years

I would like to echo the comments about part b of this task. Combinations problems are an application of multiplication that the common core standards seem to have deliberately delayed until later work with permutations and combinations. It's a topic we still see in resources, but I agree that it doesn't really fit at this grade level or standard.

Mary Anne says:

about 4 years

I was also confused with task b. I liked the suggestion of using the shirts and shorts or ice cream and cones. We used to call them combination problems and show how to solve them using tree diagrams. Is a tree diagram still considered a useful or appropriate tool or should we always be guiding students to the array model?

Stephanie Biagetti says:

over 4 years

To avoid confusion, maybe the problem could be altered to be about clothing or ice cream choices or something like that. For example, Mel has 4 different shorts and 7 different shirts. How many outfits can he make from his shorts and shirts? OR Mel went to an ice cream shop where there are 4 different cones and 7 different ice cream flavors. How many different combinations of a cone and an ice cream flavor can Mel choose from?

Kristin says:

over 4 years

I think that is a great modification to this task.

Jason says:

almost 5 years

What if the prompt ended by saying "Then solve each problem."

Kristin says:

over 4 years

I added that part at the end. Is the solution adequately well developed?

k-8 Mathematics Coach says:

over 5 years

I was wondering if for "b" above, students won't think that there are only 7 possible cards because there are only 7 pictures. If there are only 7 pictures, then it wouldn't matter how many colors of paper - there are only 7 pictures to glue. Perhaps different wording is needed? Such as: How many different card designs could she make with 4 different colors of paper and 7 pictures?

Kristin says:

over 5 years

We struggled with the wording on this problem too. One problem is that there is no reason a student couldn't use more than one picture, for example. I think teachers may need to help clarify the intention of this part because writing it completely precisely and unambiguously is hard to do without making it too long. I think that is ok as this is best suited as an instructional task.

I did modify it a bit based on your suggestion.