I call my approach to memorising multiplication tables Stitching together because it relates the entries in the multiplication table to each other in three different ways. I present the third way Number Runs first, for the benefit of those glancing at these pages, in search of something new.
About the same time one is learning tables, one learns about Pythagorean Triples, 3²+4²=5²,6²+8²=10², 5²+12²=13², etc. Squares of numbers are of great interest, but have you ever looked at the numbers nearby?
Compare 8×8=64 with 7×9=63 or compare 5²=25 with 4×6=24;. Whenever you have three consecutive numbers, the square of the one in the middle is one more than product of the two on either side. This is a fact of algebra and does not depend on working in base ten or restricting oneself to single digit numbers. For example, the board for the Chinese game of Go is nineteen by nineteen, so how many playing points are there? 19² obviously. Well 9²=81, which immediately makes working out 19² by long multiplication look unattractive. Consider the sequence 18,19,20. Is it obvious to you that 18×2=36?. If not, then this is a case for method 2 Factor Bashing.18×2=(9×2)×2=9×(2×2)=9×4=36And if 18×2=36 then 18×20=360. So the square of the number in the middle is one more, i.e. 19²=360+1=361.
Sometimes this is directly helpful, as when 9²=8×10+1=80+1=81. With other numbers it is tying together two equally obscure facts such as 49=7²=6×8+1=48+1. You might think that if you can remember neither 7² nor 6×8, then knowing that 7²=6×8+1 is of no value, but I disagree. The challenge in memorising tables is to discover the inner meanings and hidden links that make an otherwise indigestible heap of random facts acquirable by the human memory.
Pushing this further, one can ask about five consecutive numbers. The square in the middle is four more than the product of the outer pair. The sequence 6,7,8,9,10 tells us that 8² is four more that 6×10, thus 8²=60+4=64. The sequence 10,11,12,13,14 tells us that 12² is four more than 10×14; not 140 but 144. Which perhaps reminds you of the pythagorean triple earlier: 5²+12²=13². Had I got it right? Just what is 13²? The sequence 11,12,13,14,15 says it is four more that 11×15. If it is clear to you that 11×15=165, then 13²=165+4=169. Alternatively, one could press on to sequences of seven consecutive numbers. The square in the middle is 9 more that the product of the outer pair, (1 more, 4 more, 9 more, there is a pattern here) so 10,11,12,13,14,15,16, tells you that 13²=10×16+9=169. But perhaps one should stick with sequences of five numbers, because one can run the principle backwards. If the square in the middle is four more that the outer pair, then equally the outer pair is four less than the square in the middle. What is 8×12? The sequence of five consecutive numbers is 8,9,10,11,12, so the square in the middle is 10²=100, thus 8×12=100-4=96. And finally, the product of the inner pair is three more than the product of the outer pair. So in the sequence 9,10,11,12,13, since we know that 10×12=120, we immediately have 9×13=120-3=117. Bet you didn't know that!I've leapt from sequences of three consecutive numbers to sequences of five. What happened to four? For example 2,3,4,5. 3×4=12 and 2×5=10. Or 3,4,5,6. 4×5=20 and 3×6=18. The product of the inner pair is two more than the product of the outer pair. This is sometimes very useful. As a child I struggled with 6×7=42. For some reason, knowing that 3×7=21 didn't help, even thought factor bashing 6×7=(2×3)×7=2×(3×7)=2×21=42 was easy in this case. I wish I had been taught that counting 5,6,7,8 tells you that 6×7 is two more that 5×8. I learned very easily that 5×8 was 40. Adding two to get 6×7=42 would have come quickly. 8×9=72 is another very hard one to learn, and yet is is really very easy because it is just two more than 7×10=70.
Pushing on to sequences of 6 consecutive numbers might seem unreasonable, but bear with me. You can see the pattern with 1,2,3,4,5,6. 3×4 is 12, 2×5 is 10, two less, and 1×6=6, six less than twelve, four less than ten. Actually the pattern is even clearer if one can overcome ones fear of zero and say 0,1,2,3,4,5. Working from outside to inside the multiplications are 0×5=0, 1×4=4, 2×3=6. I call this the 0,4,6 pattern. Becoming dependent on base ten for a moment, this exact pattern of final digits recurrs starting on every multiple of ten. 10,11,12,13,14,15 gives 10×15=150, 11×14=154, 12×13=156. 20,21,22,23,24,25 gives 20×25=500, 21×24=504, 22×23=506. I know that is not helpful, but I wanted to be sure you saw the pattern, because it also starts on multiples of 5. 5,6,7,8,9,10 gives you 5×10=50, 6×9=54, 7×8=56. That is quite a revelation. Far from being part of a devious plot to make multiplication tables impossible to memorise, 6×9 and 7×8, have to come to 54 and 56, to take their places in the 0,4,6 pattern.
Elsewhere I suggest, as an advantage of linear text over hypertext, that the author can shape the earlier material to anticipate the later. This principle applies especially to education as a whole. A child who has learnt their tables by methods including number runs, is well set to start algebra.
n²=(n+1)(n-1)+1summarises facts they are already familiar with, and its derivation explains them. Better yet
(a+b)(a-b)=a²-b²is now about something; the mysterious pattern alluded to earlier
n²=(n+1)(n-1)+1
n²=(n+2)(n-2)+4
n²=(n+3)(n-3)+9
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Lots more work needed to finish this and split it over pages, but a friend said he was interested so up goes work in progress
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