Understanding Multiplication Part 1May 13, 2020
What does “solve 3 x 8” mean?
There are actually two crucial pieces of information to parse for this to mean anything.
“x” means “groups of”
First off, the multiplication sign x we generally read (in elementary school) as “groups of.” So we’re literally talking about “3 groups of 8.” The nice thing about this is that it now suggests a visual. I could actually build 3 x 8 by physically putting blocks into 3 groups of 8 each. Like this.
Or, if I want to lighten the cognitive load, like this.
There’s a huge advantage to organizing the blocks into an array or other structure. Patterns and structure allow our brain to be less busy blocking out the clutter and have more power to apply to the actual tasks at hand.
This is all fine, but why aren’t we done? The question asked for 3 groups of 8 and I’ve built it. What’s left to do?
2. “Solve” means “group by tens”
What’s funny about base ten is that we forget that it’s really just another way of expressing groups. Three groups of eight should be fine, but if we want to compare it to five groups of five, it’s hard to see which is larger. And if we want to add, subtract, multiply, or divide further, dealing with different group sizes can be a hassle. Better to choose a common group size and use that for everything.
And since we have ten fingers, ten became the de facto group size we use.
It’s easy to forget how many choices and structure went into finding the “answer” of 24. But it’s less of an answer and more of a translation. We’re taking our 3 groups of eight and rearranging them to form groups of ten. When we do, we find we’ve built 2 groups of ten with 4 blocks leftover. And we’ve got this great shorthand for groups of ten: 2 tens and 4 ones we just write as 24. Which we pronounce “twenty-four,” which is the old English way to say “2 tens and 4 ones.”
The Hundred Chart is a handy tool that suggests the same organization by tens. Even without the snap cubes, we can count by eights and see where we end up in terms of tens, which cap the rows.
Once you know that “x” means “groups of” and that “solve” means “group by tens,” you know what you need to know to really understand what multiplication means and begin digging into it. If kids get practice drawing or building expressions like 3 x 8, and forming them into groups of ten to find the “answer,” they’ll begin memorizing the multiplication facts from a position of strength and understanding, underscoring the meaning of what’s happening, the fact that you’re just translation from one group size to another.
And guess what? Division just does this in reverse, translating between grouping by ten and grouping by other numbers.
I’ll talk about division and multi-digit multiplication in future posts. But I hope this helps you talk about math at home. Slow down and start from understanding. You’ll find your kids have a much more solid understanding to build on going forward.
On that note, I’ll close with a quick plug for our Multiplication by Heart Visual Flash Cards, in their last 48 hours on Kickstarter at the moment of this writing. If you want flash cards that actually highlight the meaning and the most useful visuals, you can get them here.
But whatever you do, I encourage you to use meaningful visuals and physical models, and start from understanding. The “shortcut” of just memorizing steps to get a right answer will end up short-circuiting the thinking process in the long run. Lay the foundation right, and you can build a mighty structure indeed.