By Michael Hartley
Elsewhere on this site, I gave some general times tables tips. Here, we'll have a closer look at the nine times tables. You can use this page to show your kids the hidden patterns in the nine times tables, and make it easier for them to learn. Keep in mind that these patterns will mean infinitely more to your children if you can somehow coax them to realize the rule for themselves, rather than just pointing it out to them.
Let's have a look at the nine times table. By the way, you might like to download the printable nine times table chart on this site, and stick it to the wall of your classroom, or your kid's bedroom!
| 9 | x | 1 | = | 9 |
| 9 | x | 2 | = | 18 |
| 9 | x | 3 | = | 27 |
| 9 | x | 4 | = | 36 |
| 9 | x | 5 | = | 45 |
| 9 | x | 6 | = | 54 |
| 9 | x | 7 | = | 63 |
| 9 | x | 8 | = | 72 |
| 9 | x | 9 | = | 81 |
| 9 | x | 10 | = | 90 |
| 9 | x | 11 | = | 99 |
| 9 | x | 12 | = | 108 |
If you have a look at the tens digits here, you'll see a very interesting pattern, especially in the first ten rows :
| 9 | x | 1 | = | 09 |
| 9 | x | 2 | = | 18 |
| 9 | x | 3 | = | 27 |
| 9 | x | 4 | = | 36 |
| 9 | x | 5 | = | 45 |
| 9 | x | 6 | = | 54 |
| 9 | x | 7 | = | 63 |
| 9 | x | 8 | = | 72 |
| 9 | x | 9 | = | 81 |
| 9 | x | 10 | = | 90 |
| 9 | x | 11 | = | 99 |
| 9 | x | 12 | = | 108 |
That's right! At least up to nine times ten, you have this rule :
Now that's one rule that should make things easier to remember. Here's why it works.
Nine times something is ten-minus-one times something. According to the rule of arithmetic called the "distributive law", ten-minus-one times something is just ten-times-something minus one-times-something. For example, 9 x 7 equals (10-1) x 7, which is just 10 x 7 - 1 x 7. To do this subtraction, we need to borrow one from the tens digit of 10 x something, so we get the rule given above.
Well, that might help remember the tens digit, but what about the ones digit?
Ask your child to add up the digits in the results. If they get a number with more than one digit, do it again. They should end up with something like this table.
| 9 | x | 1 | = | 9, | |
| 9 | x | 2 | = | 18, | 1+8=9 |
| 9 | x | 3 | = | 27, | 2+7=9 |
| 9 | x | 4 | = | 36, | 3+6=9 |
| 9 | x | 5 | = | 45, | 4+5=9 |
| 9 | x | 6 | = | 54, | 5+4=9 |
| 9 | x | 7 | = | 63, | 6+3=9 |
| 9 | x | 8 | = | 72, | 7+2=9 |
| 9 | x | 9 | = | 81, | 8+1=9 | 9 | x | 10 | = | 90, | 9+0=9 |
| 9 | x | 11 | = | 99, | 9+9=18, 1+8=9 |
| 9 | x | 12 | = | 108, | 1+0+8=9 |
Note that every single result leads to nine!. In fact, this pattern continues...
- 9 x 13 = 117, 1+1+7=9
- 9 x 14 = 126, 1+2+6=9
- 9 x 15 = 135, 1+3+5=9
- 9 x 17 = 153, 1+5+3=9
- 9 x 21 = 189, 1+8+9=18, 1+8=9
- 9 x 111 = 999, 9+9+9=27, 2+7=9
- 9 x 42033 = 378297, 3+7+8+2+9+7=36, 3+6=9
- 9 x 7389359345798... Whoa there! Let's not go overboard here (although, the pattern would still work)
Your child can use these two patterns to help remember the nine times tables, at least up to 9x10. Just imagine a child called upon for an on-the-spot test... or playing one of the Times Tables Games elsewhere on this site...
- Teacher : Johnny? Stand up! What is nine times six?
- John, thinking : Nine times six, nine times six... ummm.... what is it?
- Teacher : Johnny? Nine times six! Do you know?
- John : Let's see, the tens digit... ummm... one less than six is five! It must be fifty something.... Ummm....
- Teacher : Johnny?!?
- John : Umm, fifty ... ahhh...
- Teacher : Yes, Johnny? Fifty-what?
- John : Now five plus what equals nine? Let's see... you have five... then six, seven, eight, ...
- Teacher : Johnny?
- John : Umm, fifty-four!
- Teacher : Well done! Nine times six equals fifty-four! See class, what a good memory Johnny has?
One useful trick I'd like to point out - because 9 is 10-1, 9 times any number can be worked out by subtracting the number from 10 times the number. For example, 9 times 8 is 80-8, or 72. Or 9 times 123 is 1230 minus 123, or 1107. Similar tricks work for the 99 times table, or the 999 times table. Since (for example), 99 = 100-1, therefore 99x365 = 36500-365 = 36135... Thanks to "Frostie" for reminding me to include this tip!
Want a "rule-of-thumb" for working out 9x1 to 9x10? The video below shows you how to work out these facts from the 9 times tables on your fingers...
One last amazing fact about the nine times table. If your child has started to learn multiplication of larger numbers, you might pose them this problem to work out. Trust me, it's not just another sum - the answer is surprisingly nice!