The Commutative Property Cuts Your Child's Memorizing Almost in Half

The Table Is Lying About How Much There Is

Open any times table and it looks like a wall. Twelve rows, twelve columns, a hundred and forty four little boxes staring back at a seven year old. No wonder kids feel like they'll never get to the end of it.

Here's the good news I give parents every fall. More than half of those boxes are repeats. The commutative property of multiplication for kids is just a fancy name for something they can see with their own hands, that 6 times 8 and 8 times 6 land on the exact same answer. Learn one, you've got the other for free. Once a child truly believes that, the mountain of facts to memorize shrinks almost in half.

That's not a trick or a shortcut that skips understanding. It's the understanding.

Show It Before You Name It

Do not start with the word "commutative." Start with something they can count.

Grab a box of crayons, some coins, or dried beans, whatever's on the table. Lay out 3 rows of 4. Have your child count them. Twelve. Now turn the whole arrangement a quarter turn so it's 4 rows of 3. Have them count again. Still twelve.

Nothing was added. Nothing was taken away. You just turned it. That's the entire idea, and a six year old can feel it in their fingers. The arrangement looks different but the total can't change, because it's the same pile of beans.

I do this with a real class and you can watch the moment it clicks. Somebody always says, "Wait, so it's the same both ways?" Yes. Every time. That's the property, and now they own it before I've made it sound like homework.

If you want to drive it all the way home, do it with a couple of different arrangements so it isn't a fluke in their mind. Try 2 rows of 5, then 5 rows of 2. Ten both ways. Then 3 rows of 6 and 6 rows of 3, eighteen both ways. Two or three examples and a kid stops treating it as a coincidence and starts treating it as a rule they can count on. That shift, from "it happened to work" to "it always works," is exactly what you're after.

Then Give It Its Name

Once they've seen it a few times, tell them the word. Kids like knowing the grown up name for a thing they already understand. "That thing you just figured out, that flipping it doesn't change the answer, that's called the commutative property. Mathematicians named it. You already knew it."

Naming it after they get it does two things. It makes them feel smart, and it gives you both a shorthand. Later, when they're stuck on 8 times 7, you can just say "flip it" and they'll try 7 times 8, which they might know better.

Cross Off the Doubles

Here's where the savings become obvious. Print or draw the multiplication grid and do this together.

• Color in the diagonal, the squares like 3x3, 5x5, 7x7. Those only exist once. No twin.
• For everything else, find its pair. 6x9 lives in one spot, 9x6 lives in another. They're the same fact wearing two hats.
• Cross out one of each pair.

When you're done, the kid is looking at a grid that's been cut nearly in half. What felt like a hundred plus facts becomes a much shorter list, because they never needed to memorize both directions. This is honestly one of my favorite ten minute activities, because kids can literally see the workload drop in front of them.

Use the Easier Twin

The real payoff shows up in daily practice. When your child hits a fact they don't know, the commutative property gives them a second door.

Say they blank on 9 times 4. But they're solid on 4 times 9, because they've drilled the fours more. Same answer, thirty six. Or they freeze on 7 times 8 but they can get to 8 times 7 by counting up from 8 times 5. The property means there's always a spare route.

I tell kids to always reach for the easier twin. If one of the two feels friendlier, use that one. There's no rule that says you have to solve a fact in the order it's written. That small permission takes a lot of pressure off a kid who thinks they have to know every single box cold.

You can practice this at home without any worksheet. Call out a fact your kid finds hard, like 8 times 7, and ask them to say the flipped version out loud, 7 times 8, before they answer. Do it in the car, at the dinner table, wherever. After a couple of weeks the flip becomes automatic, and instead of freezing on the hard-looking ones they reach for the friendlier door on their own.

Fluency, Not Just Memorizing

I want to be careful here, because cutting the memorizing in half is not the goal by itself. The goal is real fluency, and fluency means a child understands why the facts work, not just that they can recite them under a timer.

A kid who memorizes 6x8 and 8x6 as two unrelated facts is doing twice the work and understanding half as much. A kid who knows they're the same thing seen from two sides is actually learning how multiplication behaves. That understanding carries forward. It shows up again in division, in fractions, in algebra years later when they meet the property by name and think, oh, I've known this since I was seven.

So when you work on times tables at home, keep pointing back to the beans and the turned arrangement. Let the property do the heavy lifting. Your kid has far fewer facts to learn than the table wants them to believe, and the ones they do learn will actually mean something.