## Friday, October 9, 2009

### M3N3d - Developing multiplication algorithms (6)

M3N3. Students will further develop their understanding of multiplication of whole numbers and develop the ability to apply it in problem solving.
d.Understand the effect on the product when multiplying by multiples of 10.

This standard talks about multiplying by multiples of 10, for example 37x30. This situation is different from multiplying multiples of 10, 100, etc. (which we have discussed in a previous post) because we now have 30 groups of 37. Now, if we study this idea after students have already developed a paper-and-pencil algorithm, these problems can be considered as a special case where there will be a 0 in the product. So, procedurally, there are different ways to deal with these problems. Some will carry out the calculation exactly in the same manner as they do with other multipliers:

After students get used to this calculation, they might try to combine the steps to make it more efficient:

From this perspective, this multiplication isn't much different from something like 35x18. The important idea is that we have to write a 0 in the ones place as a place holder.

However, M3N3d states that students must understand "the effect on the product when multiplying by multiples of 10." Moreover, according to the GPS, students do not study how to multiply by 2-digit number until Grade 4 (next post). So, it seems rather odd to talk about multiplying by multiples of 10, which are 2-digit number, at this point. If students' don't know how to multiply by 2-digit number, then we can't focus on the procedural aspect discussed above. Rather, we want students to understand what is going on when we multiply by multiples of 10. Although we cannot use the idea of 10 as a unit in the same way as we did when we were multiplying multiples of 10, we can still use the idea of 10 as a unit when the multipliers are multiples of 10. For example, you can think of 37x30 as 37x3x10. Alternately, you can think of 37x30 as 37x10x3. Either way, multiplying a 2- or 3-digit number by 3 is something students have already learned. What students may not have studied is multiplying 2- (or 3-) digit number by 10. So, that seems to be the primary focus of this standard.

As we explore multiplying 2- and 3-digit numbers by 10, we may again want to go back to the area model of multiplication. For example, if students are to model 17x10 using base-10 blocks, they might at first construct something like this by simply extending what they have done previously:

At this point, some might notice that we can actually use a flat on the left side since there are 10 longs. Moreover, on the right side, since there are 10 rows of units, we can replace each column by a long, resulting in an arrangement like this:

Students can also record the process more abstractly like this, too:

They can also consider cases like 40x10 by extending their thinking of 40 as four 10's. If you have 10 groups of four 10's, you can think of that as 4 groups of ten 10's as well, or 100x4.

From these exploration, students may notice that when you multiply 2- and 3-digit numbers by 10, the product will contain the same set of numerals in the same order but every numeral is moved one place to the left - and there is a 0 in the ones place as a place holder.

Although we may be able to consider multiplying by multiples of 10 as a special case of multiplying by 2-digit numbers, students still need to learn the effect of multiplying by 10 before they can explore multiplying by 2-digit numbers. Moreover, once you study the effect of multiplying by 10, extending it to multiplication by multiples of 10 may be useful to help students deepen their understanding of multiplication operation. Although the formal study of properties of multiplication is done in Grade 4, Grade 3 students can use the associative property to reason about the effect of multiplying by multiples of 10. I believe that's the point behind this standard, not just the procedural fluency.