Poached from the_magician:-

Mental Arithmetic 002

Here's that table of single digits from the previous mathematics post.

Single Digits

5	1	3	2	7	5	8	1	6	2
0	9	2	5	1	3	4	1	9	6
7	1	0	4	8	4	7	2	9	3

You have previously tried doubling each digit before - reading 2 as 4, reading 9 as 18 and so on. Now it's time to learn a different technique.

Time to learn how to halve the digit, rounding down. Always rounding down.

Instead of seeing 5, read it as 2. Instead of 6, read it as 3. Odd numbers, when you halve them discard the fraction - the half. Just the integer portion: 0, 1, 2, 3, 4.

Here's a table - again from the previous post. It's to help you until you get the hang of thinking in halves as quickly and easily as you think of doubles.

"Halving" Digits

Digit	"Halved"
0, 1	0
2, 3	1
4, 5	2
6, 7	3
8, 9	4

Go to the above table and practice. Again, don't think "Half of six is three." Practice looking at the 6 and saying "3." Same thing if you're looking at a 7. Say "3."

Now let's try practicing something with this halving. Let's try multiplying by 6.

Most people who've come through the maths curriculum will probably be familiar with multiplication by 6. Probably it's second nature. That's not the point. These techniques - adding to the neighbour, halving down, doubling - are practice for doing things with individual digits that you can apply to long numbers in such a way that you won't be thinking of these daunting big numbers. All you'll see will be pairs and small groups of digits, patterns you'll be familiar with and which you can apply like running the slide along a zip, one pair of teeth at a time, until the whole thing is zipped up tight.

And with practice, you'll be able to do this sort of thing quickly, quietly and mostly in your head.

So, multiplying by 6. Here's what you do.

1. To each column, add half the neighbour.

If you see a number pair like 6,8 you start with the 6, look at the 8, say "4," then look at the first figure and say "10."

Try it with the following number (remember, assume that the "neighbour" of the right hand column is zero, so just write down the number:-

22648 x 6

Did you get 135888?

Work it through. 8 + half of 0 = 8;

0 2 2 6 4 8 x 6

           8    

4 + half of 8 is also 8;

0 2 2 6 4 8 x 6

         8 8    

6 plus half of 4 is, guess what, 8;

0 2 2 6 4 8 x 6

       8 8 8    

2 plus half of 6 is 5;

0 2 2 6 4 8 x 6

     5 8 8 8    

2 plus half of 2 is 3;

0 2 2 6 4 8 x 6

   3 5 8 8 8    

Finally, 0 plus half of 2 is 1.

0 2 2 6 4 8 x 6

 1 3 5 8 8 8    

There is, however, more to this. Remember the part about wondering where the 5 went when halving down an odd number? There's a second part to the technique of multiplication by 6, and it is this.

1. To each column, add half the neighbour.
2. Add 5 if the column is odd.

Try it with the following number exercise:-

0 7 3 5 1 9 x 6

                

Here, every column is odd, so you'll have to add 5 each time.

0 7 3 5 1 9 x 6

          ¹4    

0 7 3 5 1 9 x 6

        ¹1¹4    

0 7 3 5 1 9 x 6

      ¹1¹1¹4    

0 7 3 5 1 9 x 6

    ¹1¹1¹1¹4    

0 7 3 5 1 9 x 6

  ¹4¹1¹1¹1¹4    

0 7 3 5 1 9 x 6

 4¹4¹1¹1¹1¹4    

The result is 441,114.

Here's another table for you, this time to help you with the task of learning to add 5 to an odd digit.

Adding 5 To Odd Digits

Digit	"Add 5"
1	6
3	8
5	10
7	12
9	14

When you next encounter an odd digit, don't look at the digit and say "5 plus 5 is 10," just see the 5 and say "10."

Now try practicing with the digit pairs in the table below. Remember: add half the neighbour. Add 5 if the number is odd.

Digit Pairs

4 8	2 7	1 8	4 5	5 2	8 3	6 7	6 9	9 1	8 0
0 9	1 2	3 6	2 7	9 4	8 6	2 5	6 1	4 2	1 7
7 6	6 4	3 5	1 8	5 1	6 9	0 4	2 7	2 6	4 7

Now try this exercise on the following table of three digit numbers (not counting the leading zero, added for your convenience).

Three-Digit Numbers (+leading zero)

0709	0475	0154	0216	0352
0198	0847	0729	0117	0263
0229	0886	0212	0261	0811
0702	0260	0699	0370	0435
0735	0579	0545	0873	0567
0378	0456	0112	0396	0440

And now try the following on the five digit numbers below.

Five-Digit Numbers (+leading zero)

039447	039232	040222	034251	052497
070452	067924	003561	070302	051801
040681	038163	087386	058424	065359
033770	097925	058764	030034	091426
027759	062257	076715	083982	045638
038841	010937	088876	037494	080149

Lastly, practice something you've already done before: multiplying by 11 and 12. Give it a go with these six digit numbers below. Remember the techniques for multiplying by 11 and 12? Add each column to its neighbour for x11, Double each column then add its neighbour for x12.

Six-Digit Numbers (+leading zero)

0836325	0341267	0136900	0963637	0988439
0335986	0902530	0119677	0901084	0724990
0298444	0958696	0521690	0555354	0930256
0994412	0275789	0695617	0803280	0814464
0383324	0760391	0143876	0819288	0358038
0110864	0434222	0301688	0697005	0853435

More to follow next time, including further practicing of this halving down technique - next time, by multiplying by 5 and by 7.

And, just because I don't like leaving anybody in suspense, here are the methods you need to multiply by these two numbers:-

Multiplying by 5:-

1. Under each column, write down half the neighbour.
2. Add 5 if the column is odd.

Multiplying by 7:-

1. Double each column and add half the neighbour.
2. Add 5 if the column is odd.

I know, spoilers. But I'm trying to show you that there's no mystique to these techniques. You can, if you want, try these out on the above numbers between now and the next lesson if you wish. You won't get any extra points for enthusiasm. But it helps.