MERCY COLLEGE - THE DEAN'S COLUMN
Discovering Mathematical Relationships
By Alfred Posamentier
In today’s world of mathematics instruction, we must take notice of the Common Core State Standards (CCSS). These require students to verbalize what they are doing mathematically. In other words, they should typically be able to speak about mathematics. What better way to accomplish this than to “discover” some mathematical relationships or patterns. We provide such a situation here. Consider the following question: Which numbers can be expressed as the sum of consecutive integers? You may want to have your students experiment a bit before trying to generate a rule. They might try to express the first batch of natural numbers as the sum of consecutive integers. We will provide some in the following list.
2 = not possible 21 = 1+2+3+4+5+6
3 = 1+2 22 = 4+5+6+7
4 = not possible 23 = 11+12
5 = 2+3 24 = 7+8+9
6 = 1+2+3 25 = 12+13
7 = 3+4 26 = 5+6+7+8
8 = not possible 27 = 8+9+10
9 = 4+5 28 = 1+2+3+4+5+6+7
10 = 1+2+3+4 29 = 14+15
11 = 5+6 30 = 4+5+6+7+8
12 = 3+4+5 31 = 15+16
13 = 6+7 32 = not possible
14 = 2+3+4+5 35 = 17+18
15 = 4+5+6 36 = 1+2+3+4+5+6+7+8
16 = not possible 37 = 18+19
17 = 8+9 38 = 8+9+10+11
18 = 5+6+7 39 = 19+20
19 = 9+10 40 = 6+7+8+9+10
20 = 2+3+4+5+6
These consecutive number sum representations are clearly not unique. For example, 30 can be expressed in more than one way: 30 = 9+10+11, or 30 = 6+7+8+9.
An inspection of the table shows that those numbers, where a consecutive number sum was not possible were those that are powers of 2.
This is an interesting fact. It is not something that one would expect. By making a list of these consecutive number sums, your students ought to begin to see patterns. Clearly the triangular numbers are equal to the sum of the first n natural numbers**. A multiple of 3, say 3n, can always be represented by the sum: (n-1) + n + (n+1). Your students may discover other patterns. That’s part of the fun of it; not to mention its instructional value – seeing number patterns and relationships, and verbalizing the findings.
For the purists and the more ambitious students or those that are properly math-prepared, we now prove this (until-now) conjecture. First we will establish when a number can be expressed as a sum of at least two positive integers.
Let us analyze what values can be taken by the sum (S) of two or more consecutive positive integers from a to b. (b>a). We get S = a + (a+1) + (a+2) + K + (b-1)+ b = ((a+b)/2)(b-a+1) by applying the formula for the sum of an arithmetic series*. Then doubling both sides we get: 2S = (a+b)(b-a+1). Letting (a+b)=x and (b-a+1)=y, we can note that x and y are both integers and that since their sum, x+y=2b+1, is odd, one of x, y is odd and the other is even. Note that: 2S= xy. There are two cases to consider.
Case 1: S is a power of 2.
Let S=2n. We have 2(2n)=xy, or 2n+1=xy. The only way we can express 2n+1 as a product of an even and an odd number is if the odd number is 1. If x=a+b=1, then a and b cannot be positive integers. If y=b-a+1+1 then we have a=b, which also cannot occur. Therefore, S cannot be a power of 2.
Case 2: S is not a power of 2.
Let S=m2n, where m is an odd number greater than 1. We have 2(m2n)=xy, or m2n+1=xy. We will now find positive integers a and b such that b>a, and S= a+ (a+1)+K+b.
The two numbers 2n+1 and m are not equal, since one is odd and the other is even. Therefore one is bigger than the other. Assign x to be the bigger one, and y to be the smaller one. This assignment gives us a solution for a and b, as x+y=2b+1, giving a positive integer value for b, and x-y=2a-1, giving a positive integer value for a. Also, y= b-a+1>1, so b>a, as required. We have obtained a and b. Therefore, for any S that is not a power of 2 we can find positive integers a and b, b>a, such that S= a+ (a+1)+K+b.
Just to recap, a number can be expressed as a sum of (at least two) positive integers if, and only if, the number is not a power of 2. This may appear harmless, but you should alert your students to the notion that they just accomplished some nice mathematics, while admiring the beauty of the subject. Now according to the CCSS students should be able to explain what they found and how they established its truth.#
** Remember that the natural numbers are the counting numbers: 1, 2, 3, 4, 5, 6, … .
*The sum of an arithmetic series is S= (n/2)(a+l), where n is the number of terms, and a is the first term and l is the last term.
Alfred S. Posamentier is Dean of the School of Education, and Professor of Mathematics Education, Mercy College, New York. He was Distinguished Lecturer at New York City College of Technology of CUNY and is Professor Emeritus of Mathematics Education at The City College of CUNY.