## Wednesday, September 30, 2009

### M3N3c - Developing multiplication algorithms (5)

M3N3. Students will further develop their understanding of multiplication of whole numbers and develop the ability to apply it in problem solving. c. Use arrays and area models to develop understanding of the distributive property and to determine partial products for multiplication of 2- or 3-digit numbers by a 1-digit number.
In the previous post, I discussed how students can develop a paper-and-pencil algorithm for multiplying 2-digit numbers by 1-digit numbers. Let's consider how we can help students extend the procedure to multiplication of 3-digit numbers by 1-digit number.

How can we multiply 312 x 3? How can students use what they have learned so far to calculate this? One possibility is to think of 312 as 300+12. Then, we can multiply 300x3 and 12x3. Both of these are already learned ideas. If students have already understood how to multiply a 2-digit number by a 1-digit number using a paper-and-pencil method, they can then combine their learning and record this multiplication something like this:

When extending the multiplicand from 2-digit to 3-digit, therefore, there isn't really any new concept involved. Even the idea of looking at 312x3 as 300x3+12x3 is really the same idea as looking at 12x3 as 10x3+2x3, i.e., the distributive property of multiplication, which will be formally studied in Grade 4.

One important thing to think about when we study multiplying 3-digit numbers by 1-digit numbers is different situations where re-grouping must take place, or when there is a 0 (or more) in either the multiplicand or the product. The example we just saw, 312x3, does not involve re-grouping and there is no 0 in the multiplicand nor the product. So, in a way, it is a "general" case of multiplying 3-digit numbers by 1-digit numbers. But, here are some of other cases:
Re-grouping is involved
• 227x3
• 227x5
• 162x3
• etc.

0 is involved
• 406x7
• 365x4
• 527x4
• etc.
I encourage you to think about other cases. As teachers, we must also think about how we want to deal with them. We can carefully sequence those cases and have students think about how they can adapt the written procedure they developed those situations. As you do, it will be helpful if you explicitly ask students what is different about each case compared to the most general one that we start with.

As we look at those special cases, it is important that students understand what is actually happening when we are multiplying 3-digit numbers by 1-digit numbers. For that, it might be useful to go back to the notation system that we used when we developed when we were multiplying 2-digit numbers by 1-digit numbers. For example, let's think about 427x4. Since we can think of 427x4 as 400x4+27x4, and we can use a pictorial notation like this:

Or, we can use more symbolic notation like this (with the previous agreement that we start recording with the partial product of the ones digits first):

We can combine some of the steps involved in this notation and develop a notation like this:

No matter how you approach this topic, what we cannot do is to start with the standard algorithm, which is the most sophisticated way of recording the processes. Help students extend what they have previously learned, which may be the standard algorithm for multiplying 2-digit numbers by 1-digit number by thinking about the structure of numbers and the meaning of operations. If necessary, go back to the intermediate notations that were used while developing the algorithm for multiplying 2-digit number by 1-digit numbers. By experiencing this extension, students can then think about how they can extend the algorithm for multiplying 3-digit numbers by 1-digit numbers to multiplying 4-digit (or even longer) numbers by 1-digit numbers. They have not only the experiences of multiplying two numbers but also the experience of "extending" their procedure from one case to another. So they can ask themselves not just "How did I multiply 2- or 3-digit numbers by 1-digit numbers?" but also "How did I extend the algorithm for multiplying 2-digit numbers to 3-digit numbers?" Therefore, when teaching multiplication of 3-digit numbers by 1-digit numbers, what is important is not the procedure but the idea of how to extend the previously learned procedure (2-digit multiplicands) to a new situation (3-digit multiplicands).