Teaching kids real math with computers/calculators

[Thomas Klemm]

(08-27-2018 05:01 PM)burkhard Wrote:  We want fluid numeracy in head, with requires some rote learning of things like the multiplication tables. There's really no way around it, I think.

Start with:

 1  2  3  4  5  6  7  8  9
 2  4  6  8 10 12 14 16 18
 3  6  9 12 15 18 21 24 27
 4  8 12 16 20 24 28 32 36
 5 10 15 20 25 30 35 40 45
 6 12 18 24 30 36 42 48 54
 7 14 21 28 35 42 49 56 63
 8 16 24 32 40 48 56 64 72
 9 18 27 36 45 54 63 72 81


Show them that multiplication is commutative: \(a\times b=b\times a\)
Thus we can forget about the lower left triangle:

 1  2  3  4  5  6  7  8  9
 2  4  6  8 10 12 14 16 18
 3  6  9 12 15 18 21 24 27
 4  8 12 16 20 24 28 32 36
 5 10 15 20 25 30 35 40 45
 6 12 18 24 30 36 42 48 54
 7 14 21 28 35 42 49 56 63
 8 16 24 32 40 48 56 64 72
 9 18 27 36 45 54 63 72 81


The first line is trivial. I assume they can count:

 1  2  3  4  5  6  7  8  9
 2  4  6  8 10 12 14 16 18
 3  6  9 12 15 18 21 24 27
 4  8 12 16 20 24 28 32 36
 5 10 15 20 25 30 35 40 45
 6 12 18 24 30 36 42 48 54
 7 14 21 28 35 42 49 56 63
 8 16 24 32 40 48 56 64 72
 9 18 27 36 45 54 63 72 81


Multiples of 5 are easy so off they go:

 1  2  3  4  5  6  7  8  9
 2  4  6  8 10 12 14 16 18
 3  6  9 12 15 18 21 24 27
 4  8 12 16 20 24 28 32 36
 5 10 15 20 25 30 35 40 45
 6 12 18 24 30 36 42 48 54
 7 14 21 28 35 42 49 56 63
 8 16 24 32 40 48 56 64 72
 9 18 27 36 45 54 63 72 81


Multiples of 9 are easy as well: first comes the factor decremented by 1 and then the difference to 10.
Or if you prefer: the digits add up to 9.

 1  2  3  4  5  6  7  8  9
 2  4  6  8 10 12 14 16 18
 3  6  9 12 15 18 21 24 27
 4  8 12 16 20 24 28 32 36
 5 10 15 20 25 30 35 40 45
 6 12 18 24 30 36 42 48 54
 7 14 21 28 35 42 49 56 63
 8 16 24 32 40 48 56 64 72
 9 18 27 36 45 54 63 72 81


Thus we end up with 6 + 9 + 6 = 21 essential multiplications.
However the multiples of 2, 4 and 8 have a lot in common.

So you might show them that 2 × 6 = 2 × 2 × 3 = 4 × 3 = 12.
And that 4 × 6 = 4 × 2 × 3 = 8 × 3 = 24.
So they learn the associative law. And see some of the results appear multiple times.

Thus we end up with a few multiples of 3 and 7 that they still have to learn by heart.
Show them the diagonal: these are the squares.
And teach them the powers of 2 until 2¹⁰.

Best regards
Thomas