Question
Introduction
Numerical
patterns are a challenge to learn and to teach.
Those of us who are teachers, usually the better students of our time,
tend to think that the pattern is “obvious”, and hence do not bother to explain
and/or facilitate classroom discussions regarding patterns.
This article is
about a pattern that involves the
sum of the first few natural numbers (or wholenumbers). This pattern, common in our primary school mathematics patterns, is a special case of the
sum of an arithmetic progression, but its
formula is not usually taught explicitly.
The story is told of the great mathematician
Karl Friedrich Gauss, who figuredout a short-cut for adding up 1 + 2 +... + 100. Lesser mortals in primary
schools are left to struggle with frustration, or to copy “model answers” from
their tutors or teachers without understanding
how the solutions were obtained.
As in a
previous article, I attempt to illustrate the pattern visually. I shall highlight the linkages to facilitate discovery
of the general method, as well as show how the last part (part(c)) can be
solved.
Solution
Many pupils are able to deduce the answer to part (a) by analysing the differences between successive answers, which is equivalent to asking what must you add to get the next number. For example, from the 1st number 1, you add 2 to get the 2nd number 3, you add 3 to get the third number 6. To get the 4th number, you add 4 to get 10. This approach works, but will not help you much for parts (b) and (c) of the question. The better way is to look for a method that does not require you to keep on adding numbers. That would allow one to kill all the birds with one stone. How to do this?
Imagine the given
dots being doubled, rotated and then put together into parellelogram-like
matrices. If you want, you can imagine
them as rectangular arrays. I use the
colour orange for the originals, and blue for the copies.
As you can see, for figure 1, the total
number of blue and orange dots is 1 ´ 2. For figure 2, the total is 2 ´ 3. For figure 3,
the total is 3 ´ 4. For figure 4, the total is 4 ´ 5. The pattern is: for whatever number representing the position of the figure, the total is this ordinal number multiplied by another factor that is one more than this number. Hence the number of original dots (shown in orange) is this product divided by 2. With
this insight, we can fill up the table to answer part (a). The answer for figure 5
is 30.
We can also figure out that the answer to part (b) is 55, with
the above-mentioned pattern.
For part (c), we can use trial and error or “guess and check”. Trying 21
gives 21´22 = 462 which does not work. We try 22:
22´23
= 506. Yes! Bingo!
You may also use a calculator
to help you. Since the two unknown factors
are close together (they differ only by one), it is almost like multiplying a
number with itself, or squaring. So to guess our number, we may use the square root (the opposite of squaring) to estimate it. The square root of 506 is
about 22.49. We guess
22 and verify that 22´23 =
506.
Summary
The pattern involving
the sum of the first few whole numbers may be deduced by making a copy of the original
figure, rotating it and joining it to form an array. Just multiply accordingly and then divide
by 2
to get the sum for the original figure.
To solve for a figure’s ordinal number (which figure has a certain given
number of dots), one may use guess and check, or use square roots. A primary school pupil should be able to all
these without the knowledge of advanced techniques like the sum of arithmetic
progressions or quadratic equations.
H02. Use a diagram / model
H03. Make a systematic list
H04. Look for pattern(s)
H05. Work backwards
H07. Use guess and check
H09. Restate the problem in
another way
Suitable Levels
* Primary School Mathematics (algebra unnecessary)
* GCE ‘O’ Level “Elementary” Mathematics (Number patterns, with algebra)
* GCE ‘A’ Levels H2 Mathematics (Number patterns, with algebra)
* anyone who loves a challenge to unravel a pattern
