Activities/Turtle Art/Tutorials/Shortcuts to the Times Table

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The Multiplication Table

Here is a multiplication table for the numbers 0-10. With 11 rows and 11 columns, it has 121 entries. Schools routinely order children to memorize these entries as though they were random facts, with no relationships among them, and without any help on how to memorize facts. This is silly and for many children cruel.

Multiplication table
× 0 1 2 3 4 5 6 7 8 9 10
0 0 0 0 0 0 0 0 0 0 0 0
1 0 1 2 3 4 5 6 7 8 9 10
2 0 2 4 6 8 10 12 14 16 18 20
3 0 3 6 9 12 15 18 21 24 27 30
4 0 4 8 12 16 20 24 28 32 36 40
5 0 5 10 15 20 25 30 35 40 45 50
6 0 6 12 18 24 30 36 42 48 54 60
7 0 7 14 21 28 35 42 49 56 63 70
8 0 8 16 24 32 40 48 56 64 72 80
9 0 9 18 27 36 45 54 63 72 81 90
10 0 10 20 30 40 50 60 70 80 90 100

Memorization

First, let us get pure memorization out of the way. There is no other way to deal with, say, the names of the 50 US state capitals or the names of the 100+ chemical elements. They are just names chosen with almost no system, except for elements 113 and above, which have temporary numeric names, Ununtritium for 113 through ununoctium for 118. I don't know why children should know all of either collection of names, but if your child is assigned to learn them or anything else of the sort, get a flashcard program such as Anki. You may find a deck already put together for free downloading with just the information you need, but if not, making such a deck counts as practicing the material. But this is not how math works. (It is also not how chemistry works. If you want to memorize properties of the elements, not just names, if you want to understand how the number of protons in a nucleus and electrons bound around the nucleus determine those properties, start with a good periodic table and study the relationships it shows you.)

Mnemonics

Before the Gutenberg press and movable type launched a revolution that made books (and later magazines, newspapers and so on) cheap enough for any literate person to buy or borrow, the Art of Memory was as essential to a scholar or theologian as it still is to an actor. The Greeks had Mnemosyne, a Titan, the mother of the nine Muses, just for this function, and she has given her name to the mnemonics used to make memorizing easier. You may give thanks for all of the effort that has gone into creating mnemonics so that you can spend less effort remembering things, in this case multiplication facts.

First, let us note the commutative law, ab = ba, which makes nearly half of the table redundant. We can divide the table into three parts, the main diagonal of square numbers, and two triangles of 55 cells each. For multiplying counting numbers, the commutative law just says that turning a rectangle by 90 degrees leaves it the same size and shape, as in this Turtle Art program. It has a subroutine stack for generating a rectangle with given sides, and we can just place the turtle somewhere and turn it appropriately to get the two views. Try it with different inputs.

File:TACommutativeLawMultiplication.png

The square numbers form a sequence whose differences are the successive odd numbers, as discussed in Figurate Numbers. Here is the Turtle Art version of that fact again.

File:Activities/TurtleArt/Tutorials/Figurate Numbers/Square Numbers/TASquare Numbers.png

Some rows and columns of the multiplication table are really easy. Multiplying by 0 always gives 0 (0x = x0 = 0), and multiplying by 1 always gives the other number (1x = x1 = x). You can multiply by 10 just by putting a 0 on the end.

Multiples of 5 all end in 0 (for even multiples) or 5 (for odd multiples). Multiplying by five is the same as dividing by two and multiplying by ten,.or the other way around if you prefer.

Multiplying by 2 is easy if you know how to count by twos, 0 2 4 6 8 10...The result is always even,by definition. Similarly, any multiple of 4 or 8 is even, ending in an even digit. Not only that, but you can multiply by 4 by doubling twice. And you can multiply by 8 by doubling three times. So 4 × 7 = 2 × 14 = 28, and 8 × 7 = 4 × 14 = 2 × 28 = 56.

Next is 9. In decimal numbers, the sum of the digits of a multiple of 9 is also a multiple of 9. Thus, 0, (0)9, 18, 27, 36, 45, 54, 63, 72, 81, 90. Notice that the tens digits increase from 0 to 9 and the units decrease from 9 to 0 apart from the 0 at the beginning, keeping the sum the same. The reason for that is that adding 9 is the same as adding 10 and subtracting 1. If we want to, we can represent 0 as -1 in the tens column and 10 in the units column, giving 10-10 = 0, and continuing the pattern of digits. But that's silly. ^_^ Which doesn't mean that you can't do it.) The rule for multiplying any digit except 0 by 9 is to take one less than the digit for the 10s column, and find the digit for the units that makes the two digits of the answer add to 9. You don't need this rule for 0 × 9. We dealt with that earlier.

Right, we have done 0, 1, 2, 4, 5, 8, and 9. And 10, of course. That leaves 3, 6, and 7.

Now, you can check that multiples of 3 (and therefore of 6) have sums of digits that are divisible by 3, that is, 3, 6, 9, and so on. When you add 3 to 12, for example, the sum of the digits goes from 3 to 6. When you add 3 to 18, the tens digit goes up by 1, and the ones digit down by 7, which gives -6. That is, 1+8 = 9, and 2+1 = 3, 6 less. Carries that go to the hundreds column or beyond are only possible if those digits are 9, as in 99+3. First we add 3+9, adding 1 to the tens column and taking away 7 from the units. Then we add the carry, 1, to the 9 in the tens column, putting 1 in the hundreds column and 0 in the tens column. The answer is 102, and the sum of the digits is 3.

If you aren't sure what 3 × 8 is, you can look at it several ways.

  • 8 is even, so the result is even
  • The answer has digits adding to a multiple of 3.
  • The answer is rather more than 16, in fact it is 16+8.
  • The answer is less than 30, which is 3 × 10.

Which numbers fit that description?

20 No, digits sum to 2 21 No, not an even number 22 No, digits sum to 4 23 No, digits sum to 5 24 Aha! 25 No, digits sum to 7 26 No, digits sum to 8 27 No, odd 28 No, digits sum to 10 29 No, odd

So there you are. Or, alternatively, you can double 3 three times: 3, 6, 12, 24, or add three 8s: 0, 8, 16, 24.

Multiples of 6 are all even multiples of 3.

Now we have done everything except 7. In fact, there is only one multiplication fact left, which is 7 × 7 = 49. You might as well memorize that one as part of the sequence of squares.

Let us summarize. Here is a Turtle Art version of the multiplication table, colored according to the rules given above. Green entries are the easiest, multiplying by 0, 1, and 10. Blue is even numbers. Yellow for 5, orange for multiples of 3, 6, 9, and black for 7. The program is simplified by drawing in the opposite order, hardest to easiest, so that the colors for the easier rules are on top of the harder ones.

File:TAMultiplication Table.png

Now, the treatment above is not suitable for children. We have to think about making a sequence of lesson plans to address one idea at a time.