We all know the “trick” of multiplying by ten. Just “add a zero,” right? But let’s not have our students rely on a trick alone. Today, let’s examine why this works.
Take 10 x 23 = 230. Here is one group of 23 blocks (2 blocks-of-10 and 3 single blocks).
One group of 23 blocks
10 groups of 23 blocks
And here we have our 10 groups of 23, all lined up on the classroom floor.
There are several ways we could pack up the blocks to determine how many blocks there are all together. Today, I want to be strategic about how our class packs up the blocks so we can really examine what happens when we multiply by 10.
So… I help them see we have something convenient right before our eyes. There’s a special, efficient way we could pack them up:
Sometimes, all it takes is setting out some empty holders above our project. The students can envision each column of ten blocks sliding straight up to make a bigger block.
Each distinct column of 10 blocks is packed together…
Once packed, we can verify our answer: 230 (2 block-of-100 and 3 blocks-of-10).
But wait! Doesn’t this look awfully familiar?
I like to pause here so my students can really grapple with this. They realize that what they’re discovering is actually something they’ve known for a long time – that by definition 10 groups ARE 1 of the next sized unit! It’s almost too obvious. But, as with most things in mathematics, the elegance is in the simplicity.
Your students may not have the language to say this quite so specifically, but think of what incredible discoveries they’re making!
Now that’s powerful learning.
Sometimes, for a special twist, at the beginning of the lesson, I split a classroom into three groups. Each group models a slightly different problem, such as:
Group 1: 9 x 23 Group 2: 10 x 23 Group 3: 11 x 23
When each group has laid out the repeated rows of 23 blocks, we pause and briefly discuss the visual similarities to each group’s set-up.
Then, each group packs up their blocks and shows the class their answer:
Group 1: 207 Group 2: 230 Group 3: 253
This is a great way to prompt the observation that there’s something memorable to the look of Group 2’s blocks. In fact, by seeing the three problems side-by-side, there’s a distinct and fun “a-ha” moment: there really is something uniquely special about multiplying by 10.
Want to learn more? Check out Operations with Whole Numbers and Decimals.
A quick side note here. Perhaps your students will initially want to pack up the blocks like this:
Of course, this wouldn’t be wrong. In fact, it is vital that students be given plenty of opportunity to pack the blocks in whatever manner they like, to explore different possible packing patterns, and to reach the stage where they’re organically hungry to hear about why one method might be more strategic than another in certain scenarios. If they haven’t had enough free exploration time, forcing this method may feel overly rigid (or, even worse, it may convey the message that there is a single formulaic way to arrive at a correct answer).