Multiplication Clusters

The past few weeks we have been working on "building" multiplication clusters during our math workshop. The students learned how they could model clusters using array cards and then transfer their work onto graph paper. Be sure to take a look at the student work posted on our bulletin board and in the hallway so you can better understand how we use arrays to represent multiplication problems.

Multiplication clusters are a way of decomposing a multiplication problem into simpler facts which can be determined using mental math. The cluster is then recomposed to find the product of the initial problem. Multiplication clusters usually fall into three main categories (addition, subtraction, or doubling).

An example of an addition cluster for 9 x 8 might look like this:

...5 x 8 = 40
+ 4 x 8 = 32
= 9 x 8 = 72 (40 + 32)

In order to solve the more difficult problem of 9 x 8, the 9 can be decomposed into 5 + 4 and two simpler problems are created. In other words, 9 groups of 8 is equal to 5 groups of 8 plus 4 groups of 8. It is important to remember, at this stage, that we are only decomposing one of the two factors.

A subtraction cluster for 9 x 8 could be:

10 x 8 = 80
- 1 x 8 = 8
= 9 x 8 = 72 (80 - 8)

Similar to the addition cluster, 9 x 8 is looked at as a combination of two simpler problems. In this case, 10 groups of 8 minus 1 group of 8 equals 9 groups of 8.

Finally, a doubling cluster for 9 x 8 could be:

9 x 2 = 18
therefore 9 x 4 = 36 (18 doubled)
and 9 x 8 = 72 (36 doubled)

When you double one of the factors in a multiplication equation and keep the other factor constant, the product of the new equation is doubled. If you double the number in each group (9 groups of 2 doubles to 9 groups of 4), the total doubles from 18 to 36.

As we progress into bigger numbers, multiplication clusters will become an invaluable tool for our students.