## Sunday, July 15, 2007

### M2N3 b - Multiplication Table

M2N3. Students will understand multiplication, multiply numbers, and verify results.
b. Use repeated addition, arrays, and counting by multiples (skip counting) to correctly multiply 1-digit numbers and construct the multiplication table.

In the previous post, I discussed that multiplication is not the same thing as repeated addition. Rather, repeated addition is a way to obtain the product (the answer to a multiplication problem). In the same way, skip counting is a method of getting the product. Therefore, repeated addition and skip counting are two strategies to obtain the product.

Arrays, on the other hand, is a slightly different object. An array may be used to model multiplication situation. An array with 3 rows and 4 columns possesses the equal group structure that can be described by multiplication. It is a very powerful model to help students construct their understanding of various properties of multiplication, such as commutative and distributive properties. However, if we are emphasizing the equal group meaning of multiplication, we need to be careful how we use arrays to model multiplication as students are first being introduced to the concept. That is because the distinction between the multiplier and the multiplicand is not clear in an array. There is not logical reason that a row or a column is to be considered as a group. Thus, an array with 3 rows and 4 columns may be 3x4 or 4x3. Thus, we should use the array model with caution.

Perhaps the most important part of this particular standard is that students are to “construct the multiplication table.” Thus, we should look at the strategies such as repeated addition and skip counting and the model like arrays as tools for students to construct the multiplication table. As students construct the multiplication table, one very powerful pattern that can be useful for them to explicitly think about is that when the multiplier increases by 1, the product increases by the multiplicand. In other words, if you add one more set or skip count one more time, the product will become greater by the size of the group. With an array model, you can see this that when you reveal one more column, the total number is increased by the number of rows.

So, how can we help students construct the multiplication table on their own? Here are a couple of suggestions. First, organize your lessons according to the size of multiplicand, that is the number in a group. So, when you are looking at the multiplication facts of 6’s, you are looking at 1x6, 2x6, … 9x6 (1 group of 6, 2 groups of 6, … 9 groups of 6). Second, start with the facts of 2’s and 5’s, then 3, 4, 6, 7, 8, and 9. Wait till the end to treat the multiplication facts of 1. You might find it strange to wait to discuss 1’s until the end, but if the focus of the initial treatment of multiplication is helping students understand the meaning of operation, not just get the answers, then you might notice that “group of 1” seems to be a rather strange idea – particularly for children. A “group” usually consists of more than 1 person or item. So, as far as multiplication is concerned, 1’s are special cases, and it is probably not a wise move to start with a special case.

One particular teaching strategy teachers may want to consider is asking students questions like the following:

• How much greater is the answer to 4 x 7 compared to 3 x 7?
• What do you need to add to the answer of 3 x 7 to get the answer to 4 x 7?
• How much greater will the answer be when the multiplier (i.e., the number of groups) increases by 1 in the 6’s facts (i.e., the multiplicand is 6)?

There are many strategies children can develop to master multiplication facts, but it is probably a good idea for all children to have at least one common strategy they can rely on. Don’t discourage other strategies (such as 4’s are doubles of 2’s), but make sure everyone understand this particular strategy, which is a special instance of the distributive property they will study later.