Area Model

We will spend 15 minutes to complete the activity in this section.

What is it?

The area model is a way to visually depict two-digit multiplication. It stems from organizing groups used in repeated addition (such as 4 groups of 9 items) into an array with columns and rows.

Four groups of nine apples.

Four groups of nine apples organized into an area model, or closed array. The area of the large rectangle is 36 apples.

Once students are familiar with the concept of closed arrays representing equal groups of objects, they can begin working with open arrays (no grid lines). 4 x 9 = 36.

When working with open area models, we are not limited in how we can break up the array. We can use this strategy to demonstrate partial products. For instance, 8 x 24 can be broken into two pieces: 8 x 20 and 8 x 4. The addition of these two areas in the same as the original 8 x 24 area. This is modeled on the array below:

The areas of 8 x 20 (= 160) and 8 x 4 (= 32) can be added together to get the same area as 8 x 24 (= 192).

How does it support working memory needs?

Use of an area model to represent multiplication - particularly multiple-step multiplication - allows students to record their strategies in a very visual way. This can help them by not only "storing" intermediate steps/results on paper (instead of in their head), but also by helping them make sense of their solution.

Your Task: Partial Products with Area Models

Using either a piece of paper, a white board, or the Mathies Notepad tool/app (scroll down to Notepad), model the following statements using an area model.

Part 1:

Draw the array that corresponds to the following multiplication sentences. Once drawn, you can click on the underlined ones to check and see how you did.

HINT: Try to keep the scale roughly the same on each array.

4 x 20

4 x 6

4 x 26

4 x (20+6)

6 x 4

40 x 4

6 x 10

40 x 10

46 x 14

(40+6) x (10+4)

Reflection:

What connection can you see between the area model and the traditional algorithm? (If you were to calculate 46 x 14 by hand, are there any similarities with the area model?)

This demonstrates the use of partial products with area models. By breaking numbers into tens and ones, and multiplying each part separately, students have a friendlier way of multiplying two large numbers together. However, students with working memory needs might not be able to store each "piece" of the strategy in their heads. Drawing the area model allows them to track their progress as well as see the meaning of each step of the process directly in front of them.

Part 2:

Use the partial product strategy to draw the area model for the following multiplication statements. Then, solve each problem.

75 x 12

68 x 72

156 x 24

Extension: What might the area model of (x + 3)(x + 5) look like?