The Sixes Trick

My friend sent me this video on Instagram last night. Please pause and go watch it, or else this post won’t make much sense to you.

I’ve seen the multiplication trick with nines, and I know one for elevens, but sixes? I’ve never seen a kid do that.

Most kids think of tricks like this as some kind of magic. The comments on the video suggest that most adults do too. The only thing magical about them is the math – yes, the number structure that makes these tricks work is really cool. So let’s delve into it.

Here’s a recap, in words, of what the girl in the video showed us. For 6 times any number – let’s call it the multiplier – start by holding up as many fingers as the multiplier tells you. For example, for 6 times 7, hold up 7 fingers.

Then skip count by 5 using those fingers to keep track: 5, 10, 15, 20, 25, 30, 35. Stop here – these are the tens of your product. Then hold up seven fingers again and count by ones: 36, 37, 38, 39, 40, 41, 42. There’s your answer: 42.

Seems like magic, right?

What we’re actually doing is breaking down the problem (6 x 7) into an equivalent problem: (5 + 1) x 7. All we did was decompose the 6 into 5 and 1. Why? Because skip-counting by fives is relatively easy for most of us.

Because of the distributive property, we can rewrite our expression into another equivalent one:

For the first term of our new equivalent expression (5 x 7), we can skip count by 5 seven times to figure it out. It might help to remember the commutative property here: 5 x 7 is equivalent to 7 x 5, or seven groups of five. Remember that multiplication is repeated addition. All the fingers are doing is keeping track of how many fives we’ve added.

Then we switch to counting by ones to solve the second term, 1 x 7.

Voila. What seems like magic is just numbers being decomposed.

This “trick” would also work if we decomposed the numbers in another way, or even just skip-counted by six seven times. The key to making it so simple a five-year-old can do it is that most kids learn to skip-count by fives in kindergarten or first grade. Skip-counting by sixes, or knowing their six times tables, comes later.

For the sake of argument, and to prove to you that there’s nothing magical going on here, let’s create a trick for the fours times tables. (Maybe this exists already, but I haven’t seen it — please let me know if it’s out there somewhere!).

Here’s our problem: 4 x 8

Again, because counting by fives is pretty easy, I’m going to rewrite this as (5 - 1) x 8. And then, again using the distributive property, rewrite that expression into an equivalent one:

Are you starting to think about how we would solve this using our fingers?

Just like before, hold up the same number of fingers as the multiplier, in this case 8. Then skip-count by 5 eight times: 5, 10, 15, 20, 25, 30, 35, 40.

Now we need to subtract the second term (1 x 8), so use those 8 fingers to count back from 40: 39, 38, 37, 36, 35, 34, 33, 32.

Was it magic that time? No, just a fun1 way to use your fingers to demonstrate that expressions can be rewritten an infinite number of ways. The only reason these “tricks” exist is because they call on skills that most students have down pretty securely (e.g. counting by fives).

We could probably invent some kind of “trick” for every digit’s times tables. Some of them would be so laborious, though, that they’d barely qualify as a trick. Six worked well because it’s so close to five, and we can just add the ones leftover after we skip-count by fives.

Next up: Why that nines trick works!

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