This week in Math Workshop we have been using our understanding of 100 to build on larger numbers that are multiples of 100. In addition to relating these landmark numbers, we are also concentrating on how well we communicate our mathematical understanding to others. Below are responses from two of our students that were discussed during our Closing Meeting this past Monday with our morning class. The students in this class analyzed the work of these two students and many students either asked clarifying questions or connected this work in some way with their own work.

Ryan’s Work:

20 as a factor of 100:
In 100, there are 5 groups of 20.
In 200, there are 10 groups of 20.
In 300, there are 15 groups of 20.
In 400, there are 20 groups of 20.
and so on……….

After completing a table which reflects the information shown above, Ryan made the following statement regarding the number of groups of 20 in 300. He determined that there are 15 groups of 20 in 300.

“I know there are 15 groups of 20 in 300 because there are 5 groups of 20 in 100 so 300 ÷ 20 = 15. Because 100 ÷ 20 = 5 and 5 x 3 = 15. The 3 represents 300 and the five represents the groups of 20 in 100.”

We can make a connection to Ryan's written work by relating what he is saying back to our earlier work with multiplication clusters.

5 x 20 = 100
5 x 20 = 100
+ 5 x 20 = 100
15 x 20 = 300
(15 "groups of 20" is 300.)

Min’s Work:

10 as a factor of 100:
In 100, there are 10 groups of 10.
In 200, there are 20 groups of 10.
In 300, there are 30 groups of 10.
In 400, there are 40 groups of 10.
and so on……….

After completing a table which reflects the information shown above, he made the following statement regarding the number of groups of 10 in 300. He determined that there are 30 groups of 10 in 300.

“I know there are 30 groups of 10 in 300 because I know that 10 10’s are equal to 100 so you do 10 x 3 which equals 30 because 300 is the triple of 100 so to find how many 10’s are in 300 is to multiply 10 x 3.”

We can also make a connection to Min's written work by relating what he is saying back to our earlier work with multiplication clusters.

10 x 10 = 100
10 x 10 = 100
+10 x 10 = 100
30 x 10 = 300
(30 "groups of 10" is 300.)

Min and Ryan did a very nice job in communicating their mathematical understanding using words, numbers, and symbols. We will continue to focus on how well we communicate our mathematical understanding to others as the year progresses.

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. If you attend Open House tomorrow night, be sure to take a look at the student work posted on our bulletin board 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.