Consecutive sums/Consecutive sums activity
This page is available to download as a .doc (Word, etc.) file File:Consecutive sums Activity.doc The Steps activity referred to in the document is also available as a separate download File:Steps activity.doc
|Pedagogic rationale||Tackling an extended problem is difficult. By definition, a problem is something that you do not immediately know how to solve, so learning how to solve something unfamiliar is not straightforward.
This lesson gives pupils an opportunity to work on developing their problem-solving skills using a problem that is accessible but which has a number of interesting features (such as the link between diagrammatical and numerical ways of presenting the problem).
This plan gives several ways of presenting what is the same underlying task. It should be useful for teachers to compare these different presentations and either to select the one that they feel will be most useful for their pupils or to explore ways of allowing the pupils to make links between them. The assessment ideas, using other pupils solutions from the nrich website (nrich.maths.org) are applicable to other problems and situations too.
e.g. Duration, resources, learning objectives, differentiation, links, type of lesson, follow up/extension, recommendations/class set up, other methods of teaching same topic.
|Type of Lesson/length: Iisten and/or follow. Work in pairs. The lesson might last from between 45 minutes and several hours (perhaps spread over several timetabled periods).
Age/Level/Key Stage: Ages 11-13 / Years 7-9 / KS3
By the end of this lesson students should be able to:
Understand more about how to investigate and what makes a good piece of investigative work.
They may do some of the following things to help them:
I first came across this as part of a collection of investigations that colleagues had assembled at school. It has since appeared in several different formulations on NRICH: http://nrich.maths.org/507
The NRICH website includes problems, teacher notes, hints, solutions and printable pages.
The first NRICH version was called ‘Consecutive Sums’. A poster of this version is at: http://nrich.maths.org/7999
This is an investigation that can be used in a large number of ways with pupils of all different levels of attainment and prior experience. It can be used in group work, as a whole-class task or individual work.
I like moving between episodes of individual work, group work and whole-class discussion, because this allows pupils to have their own ideas, but also to benefit from sharing these, both explaining them to others and listening to other people’s explanations. If the teacher wants the class to produce individual work, pupils could then either decide whether to continue with their own original course and ideas or ‘borrow’ someone else’s approach. An alternative is for the groups to produce a combined piece of work linking all of their ideas together.
Introduction to the task/One teacher's experience of this lesson
Questions that may be asked:
“If you add up consecutive integers what answers can you make?”
With some groups I put this on the board and ask the class whether there are any words they think someone else might not understand. This allows us to discuss the meaning of ‘consecutive’ and ‘integer’. I then ask if there are things they want to ask me. This might involve finding out whether negatives are allowed, or requesting an example.
With other groups I might start directly with some examples such as:
9 = 2+3+4, 11 = 5+6, 12 = 3+4+5, 20 = 2+3+4+5+6 “What is special about the numbers I am adding up?”
Having established the ‘rules’, we then look at formulating a question.
With some classes I ask a question, with others they create it for themselves, either as a class or individually. Examples might be: “Can every number be made by adding consecutive numbers?”, “What is special about the numbers that can be made by adding 5 consecutive numbers?”.
I often find it useful for pupils to be able to try a few sets of numbers for themselves (what the pupils describe as “random numbers”). Having tried a few out we can then discuss how to work more systematically.
Some pupils want to make a list and try to find ways to make each number, starting from 1 …
Others have the idea of narrowing the task to adding up two consecutive numbers. This one is often more fruitful and the idea of simplifying a task to gain more insights is an important one.
I like to encourage pupils to explain not only what is going on but also the reasons.
Two consecutive numbers always give an odd answer. This can be explaining using ‘odd + even is odd’, or using some algebra (n + n + 1 = 2n + 1. n is an integer, so 2n is even, which makes 2n + 1 odd). Turning this round, if I tell you an odd number you can tell me the two numbers to add by halving it and taking the two integers either side of it. Eg: 27 ÷ 2 = 13.5, so we use 13+14.
Three consecutive numbers throw up some interesting things. Some pupils miss out some of the possibilities. They start out with 1+2+3 and then try 3+4+5 or 4+5+6, missing out 2+3+4. Some, when they include every possibility, will remark that the answers alternate between being even and odd. Of more use is that the answers are always multiples of 3. Every multiple of three (above 3) can be made in this way. If you divide a multiple of 3 by 3 you can then take the numbers either side of the answer and the three numbers you have add up to the original number. Eg: 33÷3 = 11, so we use 10+11+12. This can be explained in lots of ways, including referring to working out the mean of a set of numbers. Algebra can also help: the three numbers can either be treated as m, (m+1) and (m+2), which have a total of 3m+3, or as (n-1), n and (n+1), which have a total of 3n. An alternative, geometric way of exploring and explaining this is shown later.
Four consecutive numbers have an even total, but are not multiple of four, whereas five consecutive numbers give a multiple of five. This can lead to hypotheses about odd and even numbers.
Further work/other ways of teaching this topic
The original NRICH version of this problem stated:
Many numbers can be expressed as the sum of two or more consecutive integers.
What can you say about numbers which can be expressed in this way?
Try to prove your statements.
The most recent version is now called Summing Consecutive Numbers and is at http://nrich.maths.org/507
I have also presented this in a geometrical way as a problem I call ‘Steps’. A task-sheet is available File:Steps activity.doc
This sort of diagram can suggest ways of explaining particular sums. For example, three steps (which can be thought of as summing three consecutive numbers) can then be split up in several different ways.
The first square in the bottom row can be cut off and added to the top row to make a rectangle with height three, showing that this is multiple of 3.
A similar thing happens with multiples of other odd numbers and the middle number is always the important one.
Exploring even numbers of steps is interesting and gives an insight into why making even numbers in this way is more difficult.
Another way of splitting up the diagram is shown here:
This means that every consecutive sum answer can be considered to be a triangular number plus a rectangle that has the same height as the triangle.
Depending on how the task is presented, some pupils think they have found a way of ‘breaking’ the problem. If you are allowed to include negative numbers you can make every single answer by summing consecutives. For example, 4 can be made as: 4 = -3 + -2 + -1 + 0 + 1 + 2 + 3 + 4. Either this can be excluded explicitly in the wording of the question (the Steps version clearly does not permit this!) or it can be left for the pupils to find, and the statement of the question altered if/when they do.
It is often difficult for pupils to assess their own investigative work. One way to encourage and support this is to use the answers that are available on the NRICH website. New problems on the NRICH website are provided with Teachers’ Notes and an opportunity to submit solutions. Pupils (or their teachers) can send in solutions either as attached documents or typed into a form. These solutions are aggregated at NRICH and a summary is posted the following month, referencing the first name of the pupil who submitted each idea, with the ideas arranged roughly in ascending order of complexity.
After pupils have completed their own work I sometimes use the NRICH solutions with them.
One way to do this is to give them copies of the full solutions and to ask them to read them through. Then they need to position their own work with the solutions. Is it the same as someone else’s work? Is it more complex than some? What could they have done next to make their work even better? Are there any parts of the solutions they don’t understand? I print these off, often on A3 paper, and give them to pupils to work on in pairs. The older solutions from a previous version of the NRICH problem are available here as a Word document. The most recent solutions are on the NRICH site at: http://nrich.maths.org/507/solution
An alternative is to cut out the different parts of the NRICH solution and to provide them on individual cards. The pupils need to assemble them in order of complexity themselves.