# Understanding algebraic rules can help arithmetic

Just recently I was trying to help a small group of primary students with their knowledge of tables. They were at the stage of having problems with things like 6, 7, 8 and 9 times tables. At one point, one student said “I know that 12 × 12 is 144” – and he seemed proud of his knowledge, so I congratulated him on being correct and I asked the group if I could tell them something interesting leading from what he just said.

I pointed out that knowing 12 × 12 was 144 led me to the knowledge that 13 × 11 was 143. I explained that when you know the value of something times itself, it is **always** true that, if you go up 1 and down 1, the answer is always 1 less than the original. We talked for a bit then I asked what 30 × 30 was – starting with 3 × 3 and moving on from there. When we agreed it came to 900, I then asked them what 31 × 29 would be – and got the immediate correct answer 899.

I don’t know if it was before or after that part of the lesson that one girl had called me the “algebra man” because of the “think of a number” trick I had done with her in a previous group. It certainly meant I had to repeat the trick with this group, so they started with a whole number – the better ones choosing larger numbers, the weaker ones choosing single digit numbers – and when we got to the stage where I said “cross the 8 off the end”, there was the usual combination of shock and amusement because, of course, they all had 8 on the end. When I then said that the number they had left was two more than the number they started with, there were, of course, cries of wonder when they realised it had worked for everyone. Then, as usual when I did this in primary school, I gave all the credit to this thing called algebra that meant it didn’t work just once like arithmetic – it worked all the time!

Going back to the 12 × 12 calculation, it is interesting that 14 × 10 is 4 less than 12 × 12. It is fairly easy to show that going up 2 and down 2 means the answer is always 4 less.

Going and up and down 3 would mean a change of 9 in the answer – so, if 30 × 30 is 900, then 33 × 27 is 891. Thinking about it, it could mean that if a student struggles with 7 × 7, you could work the algebra backwards. 7 × 7 would be 9 more than 10 × 4. 10 × 4 is 40, so 7 × 7 is 49.

Clearly, the algebra would extend to adding and subtracting any number – so we could prove it like this using a for “any number”:

Note that the significant thing here is that, whatever the value of n and a then na and –na will **always** cancel leaving n² - a².

What have we found? For example, given that 20 × 20 is 400, then

21 × 19 = 399 (400 – 1)

22 × 18 = 396 (400 – 4)

23 × 17 = 391 (400 – 9)

24 × 16 = 384 (400 – 16)

25 × 15 = 375 (400 – 25) and so on.

At this point I should highlight the fact that this algebra doesn’t work only with whole numbers. I know that 3½×3½ is 3×4+¼ = 12¼

meaning altogether we have 2 lots of (½ of 3) [that makes 1 lot of 3] plus 3 lots of 3 making 4 lots of 3 plus the extra quarter.

Generalising this for (n + ½) × (n + ½), we have

leading to (n + ½) × (n + ½) = n(n + 1) + ¼.

This means, for example, that 6½ × 6½ would be 6×7+¼ = 42¼.

We can now combine two things we’ve done and say that, because 6½×6½ comes to 42¼, then we can use the “one up, one down” rule to work out 7½ × 5½, since this will just be 1 less – so it’s 41¼. The “two up, two down” rule would give us 8½× 4½ as 38¼ and the process would continue indefinitely… and, yes, I have said indefinitely because it will even work for negatives! For example, going up and down 7 would give you 13½ × (- ½) with an answer of 42¼ - 49, which would give the correct answer of -6¾ !

The next thing I want to look at is whether knowing, for example, 30×30 will help me to find 31×31 ? Using a grid makes this easy to understand.

You can see that 31×31 is [30×30 + (30+30+1)] which is 30×30 + 30 + 31. Can we generalise this?

As you can see, we have n×n and to add to it we have n + (n+1), so all we have to do to move from one square to the next one is add the number we’re moving **from** to the number we’re moving **to**.

Hence, for example, 41×41 = 40×40 + 40 + 41 = 1681. Similarly, 7½ × 7½ would be 6½ × 6½ + 6½ + 7½, giving 42¼ + 6½ + 7½ = 42¼ + 14 = 56¼ (or 7×8 + ¼).

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