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A Study of Patterns

August 19, 2012

One of my schoolteachers defined a mathematician as someone who spends three hours finding a way to save five minutes. (His point was that this easier way can then be used hundreds of times, saving much more than the initial three hours).

There is almost always more than one way to solve a maths problem or calculation. It may look difficult or long at first, but often there is a pattern lying underneath. If you find the essential pattern, it becomes much easier to solve that problem and others similar to it.

Here are a few examples:

1. Adding numbers easily

Karl Friedrich Gauss (1777-1855) is generally acknowledged as one of the finest mathematicians who ever lived.

The story goes that his grade two teacher wanted a quiet afternoon and asked his class of 7 year olds to find the sum of the numbers from 1 to 100. All the other students started with 1+2=3, 3+3=6, 6+4=10, 10+5=15 and so on. Gauss simply wrote 5050 on his slate and gave it to his teacher.

How did he do it? (He was correct.)

Gauss had noticed that the numbers could be paired up:

1          +          100
2          +          99
3          +          98
:                       :
48        +          53
49        +          52
50        +          51

Then, each number pair adds up to 101.
And there are 50 number pairs.
So the sum of the pairs is 50 x 101 = 5050.
Voilà!

We use exactly the same argument when introducing arithmetic progressions in Year 11, and may also do so in earlier years as an introduction to quadratic expressions (x2 etc).

2. Multiplying numbers easily

Finding products of two digit numbers, such as 47 x 67, would have most people reaching for their calculator. Maybe you would still do so after reading this, or…

The patterns below can be proved using middle secondary algebra.

a.    (50 + X)2 = 2500 + 100X + X2 

With the pattern in the equation above, we get:

502  =   50 x 50  =  2500
512  =   5x 51  =  2601, which is 2500 + 100 + 12
522  =   52 x 52  =  2704, which is 2500 + 200 + 22
532  =   53 x 53  =  2809, which is 2500 + 300 + 32

and so 572 = 5x 57 would be 2500 + 700 + 72 = 3249
and so 582 = 5x 58 would be…?
This even tells us that 622 = (50 + 12)2 = 2500 + 1200 + 122 = 3844.

b.    (50 – X)2  = 2500 – 100X + X2

We also have the same thing for numbers below 50 (almost – note the minus signs):

502  =   50 x 50  =  2500
492  =   (50 – 1)2 =  2401, which is 2500 – 100 + 12
482  =   (50 – 2)2 =  2304, which is 2500 – 200 + 22
472  =   (50 – 3)2 =  2209, which is 2500 – 300 + 32

and so 432 = 2500 – 700 + 72 = 1849
and so 422 = …?

(Note: The two patterns a. and b. above work only for numbers not too far from 50. How far from 50 depends on your memory of times tables.)

c.    (10X + 5)2  = 100X(X + 1) + 25 

This is similar in style (but different in details) to a. and b. above:

152   =     225, which is (1 x 2) x 100 + 25
252  =     625, which is (2 x 3) x 100 + 25
352  =   1225, which is (3 x 4) x 100 + 25
452  =   2025, which is (4 x 5) x 100 + 25

and so 652 = (6 x 7) x 100 + 25 = 4225
and so 852 = …?

d.    Difference of Two Squares: X2 – Y2 = (X-Y)(X+Y)

The difference of two squares is one of the most important patterns in algebra, appearing in all sorts of places, such as:

60×60                               =  3600
59×61  = (60 – 1) x (60 + 1) =  3599, which is 60212
58×62  = (60 – 2) x (60 + 2) =  3596, which is 60222
57×63  = (60 – 3) x (60 + 3) =  3591, which is 60232

and so 83 x 97 = 902 – 72 = 8051
and so 72 x 88 = …?

   

In fact, combining the above and associated patterns, between a quarter and a third of all products of two numbers less than 100 can be quickly multiplied (again depending on your times tables memory).

And so to our example above product:

47 x 67 = (57 – 10) x (57 + 10) = 572 – 102 = 3249 – 100 = 3149 !

Who needs calculators all the time?

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