math tips

Dressed to the Nines

We have a wonderful base-10 number system. It makes a lot of things easy, and would make things even easier if the U.S. would get with it and switch to the metric system. Think about the poor Romans. Have you seen those years noted at the end of movies made in the twentieth century? Yuck.

The nature of the base-10 system makes for some interesting things with the number that’s just one shy of ten—nine. While learning your times tables, you may have noticed these properties of nine.

Up to 9 × 10, the digits of the products add to nine.

Again up to 10, there’s a cool bookend-reversing thing going on, as the first digits go up and the second digits go down:

09
18
27
36
45
54
63
72
81
90

A side-effect of this, along with the fact we have ten fingers, is a little trick I use with kids who still struggle to remember multiplication facts with nine. (I thought everyone knew this, but have found several adults who’ve never seen it, so I figured I’d share it here.)

Hold your hands in front of you, fingers spread. Whatever number you want to multiply nine by, count that many fingers from the left and put down the finger you land on. (So if you’re doing 3 × 9, count three fingers from the left, and put down your left middle finger.)

How many fingers are up to the left of the lowered finger? (In the example, two.) How many fingers are up to the right? (Seven.) Put those together, and you have the answer. (Two and seven … 3 × 9 = 27.)

And now, I have to go do some research on the title of this post, because I’ve always kind of wondered about that phrase.

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Impress Your Friends With Mental Math

You’ll have to take my word for it that I’m going to do this entire post without touching a calculator or scribbling calculations anywhere. I also don’t know how useful these tricks will be for you, but hey, it’s fun.

Divisibility by 3, 6 or 9

Have a large-ish number and need to know whether you can divide it by one of the above numbers? Easy. Just add up the digits. If the result is divisible by 3 (or 9), then so is the original number. If it’s divisible by 9, it’s automatically divisible by 3. If it’s divisible by 3 and is even, then it’s divisible by 6.

Example: 4,374

4 + 3 + 7 + 4 = 18

18 is divisible by both 3 and 9, so 4,374 is divisible by both. Since it’s even, it’s also divisible by 6. Go ahead and check it while I try a larger number.

Example: 5,660,193

5 + 6 + 6 + 0 + 1 + 9 + 3 = 30

30 is divisible by 3, but not 9. The original number is odd, so it’s only divisible by 3 (not 9 or 6).

Multiplying by 11

We all know that multiplying a single-digit number by 11 is easy—just repeat the number. 11 times 7 is 77, 11 times 3 is 33, etc. Multiplying by larger numbers is pretty easy, too.

The first and last digits stay the same. For the middle number(s), add adjacent numbers together.

Example: 11 × 35 = 3_5.
Since 3 + 5 = 8, that’s the middle digit.
So 11 × 35 = 385.

Bigger Example: 11 × 724 = 7_ _4.
7 + 2 = 9, and 2 + 4 = 6.
So 11 × 724 = 7,964.

What if one of those middle number sums results in a 2-digit number? Still works, you’ll just have to do a little carrying over to the next column to the left.

Example: 11 × 3852 = 3_ _ _2
3 + 8 = 11. Oops, carry that 1 over to the left, so the first digit is 4.
8 + 5 = 13. Oops again. Carry that 1 to the 1 in the 11 above. (Confusing, yeah.) Second digit is 2, third is 3.
5 + 2 = 7. Okay, nothing fancy here.
So 11 × 3,852 = 42,372.

Now go and impress your non-mathematical friends.

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Mental Math Tips for Your Kids (that Might Help You, Too)

Every once in a while, I’ll try to post some math tips that may or may not be helpful to people. We’ll start with a couple of simple ones this time, and you’ll have to trust me when I say I’m not using a calculator for any of the examples.

Multiplying By 12

I never memorized the multiplication facts for 12 when I was a kid. Why bother? It’s easy to multiply by 11, then just add one more of whatever number your multiplying by. For example, for 12 × 7:

11 × 7 = 77
77 + 7 = 84
so 12 × 7 = 84

In third grade, I was able to do that process quickly enough to still complete the timed multiplication tests, so my teachers never knew I hadn’t actually “memorized” those facts. (Do they still do those timed tests in elementary?) Of course, that assumes you can add a single-digit number to a larger number without counting it out.

What if you need to multiply something by 12 that’s beyond the typically memorized math facts? It’s a little trickier, but in a pinch, you can still do it in your head. Multiplying by 10 is even easier than 11. Then you just double whatever you’re multiplying by, and add the two together. For example, 12 × 15:

15 × 10 = 150
15 × 2 = 30
150 + 30 = 180

Maybe that one was too easy. Let’s try 12 × 43:

43 × 10 = 430
43 × 2 = 86
430 + 86 = 516

Again, this requires some mental addition ability. So let’s look at one of the simplest cases of this to start with.

Adding (or Subtracting) 9

As crazy as it makes me to see teenagers in advanced math classes add by counting on their fingers, it’s the worst when they do it for something as easy as adding or subtracting 9. Our base-10 number system is a marvelous thing. It makes adding/subtracting 10 super-easy. To add/subtract 9, just do 10, then move one space back the other direction.

For 27 + 9:

27 + 10 = 37
37 – 1 = 36

For 82 – 9:
82 – 10 = 72
72 + 1 = 73

Yes, pretty much all of us have cell phones with calculators these days. But seriously, for simple calculations, I bet you can do it in your head more quickly than you can take out your phone and punch it in. 😉

I’ll see if I can come up with more tips and tricks another week.

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