I posted a while ago about A, a 3rd grade student I have been tutoring.

We have been working really hard on multiplication, and memorizing times tables. *A* still has some gaps in her fundamental understanding of numbers, so we will be working with that a lot more in the future.

But in the past few weeks, we have been learning “tricks” for figuring out multiplication without having to count, or even having to think too hard about the numbers. We’re working really hard on recognizing patterns, using facts we know to find out things we don’t know, and of course, memorizing the times tables.

**The great 9’s**

After we learned fact families of 10, and practiced adding numbers to 10, we then started adding numbers to 9. This led to a discussion of 9 fact families, which I thought was a perfect way to start talking about the nine times tables, since the digits of every product of 9 adds up to 9. We started by listing the numbers 0-9 in a row down our paper:

Then we wrote the numbers 0-9 going back up the paper.

And then, we had the 9 multiplication table right in front of us. We had been talking about “subtracting to add” numbers, specifically nines. This means that instead of taking 9 + 5 and thinking 9, 10, 11, 12, 13, 14 we take one from the 5 and add it to the 9, making the problem 10 + 4 which we can much more easily see is 14. The same principle applies to multiplying by nine. The answer to 9 * 5 is found when we subtract one from 5, and we get 4, and then we just think about the “fact family” that includes 4. That gives us 45, and we’re done.

*A* got pretty good at thinking “minus one, then fact family” – meaning when I said “Nine times six” she would think “Five, four – fifty four.” She got really fast at figuring out the nine times tables, and now they are by far her best.

Now, the “minus one, then fact family” trick only works for 1-10. But 11 times tables are easy up to 9, because you just repeat the numbers – so 9 * 11 = 99. This is where we started learning to use something we already know to figure out a new problem. Instead of simply trying to memorize 9 * 12 = 108 arbitrarily, we think about “subtracting to add” again. We know that 9 * 11 = 99, and 9 * 12 is just adding one more group of nine, and when we add things to 9 (or 99) we can subtract one to add. So 9 * 12 = 9 * 11 + 9 = 99 + 9 = 100 + 8 = 108.

So the theme we are learning with the number 9 is “subtracting to add” – and we are learning that 9 and 1 are pretty much inseparable.

Tomorrow – Some Tricky 11’s

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