One after the other
Consecutive sums is a topic that has appeared from time to time on work cards or in textbooks, but usually in a dull and restrictive way, with some firm guidance about what the pupils are supposed to do. This is not in what I understand to be the spirit of investigational work, and I prefer to leave it to the children to ask the questions.
Furthermore, most of the introductions to the idea are very prosaic, laying out the rules in a necessarily routine way that hardly captures the imagination. I prefer to present the situation in a more challenging mode, so that some sort of problem is posed right from the outset.
I write on the board: 4 + 5 + 6 = 15 and ask what else makes 15.
There are various other suggestions. I write 7 + 8 under the above, but other offers I put on the other side of the board. I invite the class to guess why I am separating their answers into these two groups. What is special about the two I have chosen?
Before long there are suggestions that "the numbers you are adding come after each other". I get them to clarify this, and introduce the word "consecutive". I reinforce their understanding of the word by talking about consecutive letters of the alphabet, and the days of the week. I ask if there are any other ways of getting 15 by adding consecutive numbers. Someone suggests: 1 + 2 + 3 + 4 + 5 and I write this under the two we already have. They cannot think of any more, and have some ideas about why there are no more.
I ask if they can make up other sums of consecutive numbers. They suggest: 1 + 2 + 3 = 6 2 + 3 + 4 = 9 3 + 4 + 5 = 12 I invite them to investigate these and other possibilities over the next few lessons, and see what they can find out.
Some children followed up this last sequence. They noticed that the numbers "go up in threes", and explained that this was because on each successive line you added one to each of the three numbers. The answers are also multiples of three, something fresh in their minds from their previous activity, in which a stick of Unifix cubes was arranged red, yellow, blue, red, yellow, blue. . . and numbered in sequence, so that all the multiples of 3 were blue.
After some discussion they decided that each triple of consecutive numbers contained a multiple of 3, and if this was a blue number then the other two were always a red and a yellow. They knew already that red + yellow = blue, and blue + blue = blue, so this was all consistent with their previous experience, and they had already why this was so. (A red number was 1 more than a multiple of 3 and a yellow was 1 less, so when you added a red and a yellow the differences cancelled out so you got a multiple of 3.) If the sum of three consecutive numbers was a multiple of 3 then it was reasonable to assume that the sum of four consecutive numbers was a multiple of 4. They investigated.
1 + 2 + 3 + 4 = 10 2 + 3 + 4 + 5 = 14 3 + 4 + 5 + 6 = 18 These were certainly not multiples of 4, although they were even numbers. It was not easy to describe them, other than to say that they were "even numbers which were not multiples of 4", which of course was perfectly in order. However, after some discussion we decided that even numbers were "2 times something". The above numbers were: 10 = 2 x 5 14 = 2 x 7 18 = 2 x 9 so we were able to describe them as "odd multiples of 2", as opposed to multiples of 4, which were "even multiples of 2".
Consecutive sums of five numbers gave multiples of 5, and after some further investigation it seemed that odd sums gave multiples of the number of numbers, but even sums did not. We looked at a representation of the triples with Unifix cubes: and decided that we could move a cube from the tallest column to the shortest to make each column the same, and obviously that meant that we had a multiple of 3. Furthermore, the middle column told us which multiple of 3 it was.
Someone now suggested that this all applied to any odd number of columns, so for instance the sum of five consecutive numbers gave a multiple of 5, which was 5 times the middle number.
The situation with four numbers was different.
Now it was impossible to even up the columns: as one boy put it, you would need "half-cubes". And this applied to any even number of consecutive numbers.
One special case of this was the sum of two consecutive numbers, 1 + 2 = 3 2 + 3 = 5 3 + 4 = 7 and this obviously gave the odd numbers. This was because two consecutive numbers had to be one odd and one even, and the sum of odd plus even was odd. (Again, this tied up with the previous activity, which had begun with colouring the numbers red, yellow, red, yellow. . . and observing for example that red + yellow = red!) It was natural to ask whether the sum of an odd number of numbers giving a multiple of that odd number happened the other way round, that is, whether, say, a multiple of 5 could always be written as the sum of five consecutive numbers. It appeared that this was true. One could, for instance, write 25 as 5 + 5 + 5 + 5 + 5 and then, viewing this as five equal stacks of Unifix cubes, move cubes from the two left stacks to the two right stacks to give 3 + 4 + 5 + 6 + 7, the reverse of the process we used before.
Now we were ready to discuss another problem with which some of the class had been grappling. What other numbers, like 15, could be written as the sum of consecutive numbers in more than one way?
Well, why did it happen with 15? 15 was a multiple of 3, so it could be written as the sum of three consecutive numbers; it was also a multiple of 5, so it could be written as the sum of five consecutive numbers; and it was also odd, so it could be written as the sum of two consecutive numbers.
Now we could look for other candidates for this multiplicity. For instance, if we wanted a number which could be written as the sum of 5 or 7 consecutive numbers then we required a number which was odd, and was a multiple of 5 or 7. This we could easily construct by multiplying the numbers together: 5 x 7 = 35.
It was not so easy for some of the children to see how they could find other numbers which fulfilled this condition, that is, that any number which had both 5 and 7 as divisors would work: 70, or 105, or 140, etc, but they managed eventually, though quite how they did this was difficult to see; word gets around the classroom, children help each other, and one never knows as a teacher exactly how much is fully understood, and how much is just following someone else's rule.
This seemed a very worthwhile set of activities with which to start a new year with a strange class in a newly opened school. It was a way of entering the familiar field of number without getting involved in "revision of four rules", although there was plenty of calculation to do. It was a way of looking at properties of numbers: odds and evens, multiples and divisors, number patterns. It was also a way of accustoming children to an unfamiliar investigative approach, in which discussion, both in class and on paper, was something new.
One of the most difficult things was to get them to use words when they produced written work; they were obviously used to mathematics as a lot of numbers to be marked right or wrong, and found it strange that one could write sentences in English.
One girl gave the perfect assessment at the end of one lesson. "It's difficult," she said, "but it's fun!"
David Fielker took early retirement a few years ago and now teaches part-time