Friday, August 12, 2011

Three is a Magic Number

You might be wondering what our pattern was for the 3’s times tables. Well, instead of looking for a pattern (there isn’t a really obvious one) we just learned Three is a Magic Number from Schoolhouse Rock. I wasn’t really convinced that it would work for A, but it did! So, here the video of the song. Good luck with threes!

PS – you’ll notice that the song leaves out 3 * 11 and 3 * 12 – well, 3 * 11 is easy, right? We talked about that yesterday – it’s just “3 in the tens place, and 3 in the ones place.” 12 is easy, too - “3, and then 3 doubled” 3 * 12 = 36. We’ll talk more about the 12’s when we get to them, but the first few 12 times tables are pretty straightforward.

If you have questions, you can ask them in the comments, email me, ask on Facebook, or on Twitter. I am on Facebook and Twitter live from 3:00-3:30 pm Mon-Thurs MDT.

Thursday, August 11, 2011

Some Tricky 11’s – in each place

Remember that A and I have been talking about patterns in multiplication? Well, the easiest of the multiplication tables (besides 10s) are 11s – just double the number, right? Well, what happens when you get to 11 * 11? It isn’t as hard as you might think – you just have to figure out the right pattern.

Think about 11 * 9:
 image       We don’t even really have to think about it as “90 + 9” because we can just think of it as “write 9 twice”. But now when we move on to 11 * 10, looking at it this way will help when we do 11 * 11 and 11 * 12:

imageWe can think of the number 11 as “one in the tens place, and one in the ones place” – which when extended to multiplication (for example, 11 * 5) becomes “5 in the tens place and 5 in the ones place” (where you replace “5” with whatever number you are multiplying by.

For 9, that is easy, for 10, it isn’t quite as straightforward – because what does “10 in the tens place” mean? Well, in the illustration above, it means “100” – because a 10 in the tens place means we need to put another zero on the end, and a 10 in the ones place is really just 10. So that’s how we get 11 * 10 = 110.

Now we can move on to 11 * 11. We think the same thing; “11 in the tens place, and 11 in the ones place.” Let’s write it out like we did with 9 and 10:


So you can see that when we write “11 in the tens place” what we get is 110, and when we put “11 in the ones place” we get 11. So the answer is 11 * 11 = 121.

image We can do the exact same thing for 11 * 12, and we get 120 + 12 = 132.

Now you can probably figure out 11 * 13, 11 * 14, and 11 * 15! And after a while, you can probably see a new pattern as you go.

So when thinking about the 11’s times tables, you can just think “the number in the tens place, and the number in the ones place.” This rule works for all the times tables – from 1 to infinity!

If you have questions, you can ask them in the comments, email me, ask on Facebook, or on Twitter. I am on Facebook and Twitter live from 3:00-3:30 pm Mon-Thurs MDT.

Wednesday, August 10, 2011

An Update on A and The Great 9’s

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:

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

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

If you have questions, you can ask them in the comments, email me, ask on Facebook, or on Twitter. I am on Facebook and Twitter live from 3:00-3:30 pm Mon-Thurs MDT.