This is the third in a series of posts in which I discuss the development of multiplication algorithms in Grades 3 and 4. If you haven't read the first two, I encourage you to do so, either before or after reading this post.
One idea that is important as children continue to develop multiplication algorithms yet not explicitly mentioned in the GPS is the idea of multiplying multiples of 10 and 100, such as 40x7 and 600x3. Note, multiplying by multiples of 10 and 100 is a Grade 4 standard.
So, how can students make sense of multiplying multiples of 10 and 100? So, let's think about 40x7. Again, just a reminder that I am following the Japanese convention of writing the multiplicand (number in a group) first. Therefore, this problem is asking us to find the total amount when there are 40 groups in each and there are 4 groups.
An important idea here is the understanding of 10 and 100 (and 1000) as unit. When students first learned simple addition and subtraction of 2-digit numbers such as 30+40 and 70-20 in Grade 1, they used the idea of 10 as a unit. Since 30 and 40 are made up of 3 and 4 tens, putting those two numbers together meant there are seven 10's, or 70. In a similar way, we can think of 40x7 using 10 as a unit. Since there are 7 groups of 40, or 7 groups of four 10's, we see that there are 7x4=28 tens altogether. Therefore, the product is 280. Similarly, you can think of 600x3 as 3 groups of six 100's, or 3x6=18 hundreds, i.e., 1800.
The idea of 10 and 100 (and 1000) as units was the focus of M2N1(b):
Understand the relative magnitudes of numbers using 10 as a unit, 100 as a unit, or 1000 as a unit. Represent 2-digit numbers with drawings of tens and ones and 3-digit numbers with drawings of hundreds, tens, and ones.
Although the second part of the statement emphasizes looking at 3-digit numbers as composed of hundreds, tens, and ones, it is also important for children to understand numbers like 280 as twenty-eight 10's. This notion of relative magnitude ("relative size" in M3N1(b)) plays an important role in students' mathematics learning in the future. Therefore, it is important to help them deepen and consolidate their understanding as we teach multiplication of multiples of tens and hundreds.