M3N3(b) - Developing multiplication algorithms (2)

M3N3. Students will further develop their understanding of multiplication of whole numbers and develop the ability to apply it in problem solving. b. Know the multiplication facts with understanding and fluency to 10 x 10.
The new idea here is multiplication with 10 as a factor, either the multiplicand or the multiplier. Let's first look at the cases where 10 is used as the multiplicand, i.e., 10x1, 10x2, 10x3, .... These can be interpreted as one 10, two 10's, three 10's, ... respectively. In Grade 1, when students studied the numbers up to 100, this is something they should have encountered. Thus, they can use that particular prior knowledge to figure out what these facts will be, i.e., 10x1=10, 10x2=20, 10x3=30, ....

What about 10 as the multiplier, i.e., 1x10, 2x10, 3x10, .... These means, respectively, ten 1's, ten 2's, ten 3's, .... For adults, these are obvious and we might think they should be obvious to children, too. However, although young students can answer the problem on the left very quickly, many of the same students have much more difficult time with the problem on the right.

So, how can students think about problems like 3x10? Hopefully, when they were constructing their multiplication table, they have used the idea that when the multiplier increases by 1, the product increases by the multiplicand. For example, the answer for 3x6 should be 3 (the multiplicand) more than 3x5. This idea, then, can be used to think about 3x10. The answer to 3x10 should be 3 more than 3x9, which is a part of the basic fact they have learned in Grade 2. This idea is really a particular case of the distributive property, which students will formally study in Grade 4. However, the distributive property plays an important role as students think about how to multiply by larger numbers. Therefore, it may be useful if this idea is discussed explicitly in classrooms.

Some of us grew up memorizing the multiplication table up to 12x12. Even today, some teachers/schools/districts still make their students consider the multiplication table up to 12x12. Although it may have some usefulness in everyday situations to know the multiplication facts of 11's and 12's, there is really no particular mathematical reason for expanding the multiplication table to 12x12. Once students develop an algorithm for multiplying by 2-digit numbers, they can calculate anything beyond 10x10 using the algorithm. On the other hand, students can also use the property of multiplication and think of 11's and 12's as simply 10's and 1's or 10's and 2's. Thus, 7x12 can be thought of as 7x10+7x2 and 12x8 can be thought of as 10x8+2x8. Perhaps it is much more important for students to develop that form of flexible thinking than simply memorizing those facts.