Lesson 6: Distributive Property of Multiplication


Activity 1: Arrays and the Distributive Property

Materials: counters (kit), drinking straw (kit), fine point dry-erase marker (kit), small sticky notes (kit), whiteboard (kit)
Give your child a large handful of counters, and ask her to create a 6 x 4 array using them (6 rows, 4 columns). Tell her to name her array on the whiteboard. She should write "6×4." Next, give her a drinking straw, and tell her to break her array into two pieces by placing the straw either vertically or horizontally across her array. The pieces don't have to be the same size. Say, "Now, you have two arrays. What are their names?" There are several possibilities, but she may now have a 2 by 4 array and a 4 by 4 array or a 1 by 4 array and a 5 by 4 array (there are other possibilities).

Ask her to name each of the arrays and write them as multiplication sentences next to each other on the whiteboard. Finish her work on the whiteboard by drawing parentheses around each of the multiplication sentences and adding an addition sign between them. For example, it may now read "(2×4) + (4×4)." Say, "You still have the same number of counters, but you've created different arrays using them. You've just proven that 6 times 4 is the same as adding 2 times 4 and 4 times 4."

Ask your child to repeat this process by following these steps:
  1. Create an array of your choosing.
  2. Write the name of this array on a sticky note.
  3. Break the large array down into two smaller arrays using the straw.
  4. Write the names of these arrays on separate sticky notes.
  5. Draw an addition sign on a sticky note, and place it between the last two sticky notes.
Now, help your child lay out the sticky notes to show that the first and larger array is equal to adding the smaller arrays together. Add an equal sign on a sticky note, and place it after the sentence for the larger array and before the two sentences for the smaller ones. For example, the sticky notes might show 5×3=2×3+3×3. Explain to your child that she's just proven an important property about multiplication and that she'll be working with it more during this lesson.

Activity 2: Exploring the Distributive Property of Multiplication

Materials: abacus (kit), fine point dry-erase marker (kit), whiteboard (kit)
Explain that, even though it's best to memorize facts for multiplication, sometimes we come across a fact that is more difficult than others to memorize. In this case, we can often use facts that we do know to find the product of the more difficult fact. Say, "When you were breaking a large array down into two smaller arrays, you were turning a more difficult multiplication problem into two easier multiplication problems that you then added together."

Write 8 × 9 on the whiteboard, and ask your child to picture it in her head as an array. Say, "That's a pretty big array." Now, say, "I'm going to show you how we can make this multiplication problem a little bit easier. I know that I can add 4 and 5 together to make 9. I'm going to use that to make this problem easier for me, and it will be the same as working with two smaller arrays." Below 8 × 9, write the following, making sure to line the numbers up in the columns as shown:

8×4 = 32
8×5 = 40
8×9 = ??
8×9 = 72

Ask the following questions, and discuss the answers with your child, leading her to them as needed by pointing to the numbers on the whiteboard:
  • What did I do with these products to get the product of 8×9? (added them together)
  • Does 8 times 9 equal 72? (yes)
  • How might you prove that? (allow your child to choose a way to prove it, for example, using the abacus or drawing a picture)
Explain that we call this approach the distributive property of multiplication.

Now, provide time for your child to watch the video at the following web link. Tell her to look for how the distributive property of multiplication can make math easier.
Web Link