Consecutive Sums

Task 7 ... Years 4 - 8

Summary

Problem: Given number discs from 0 to 25, how many ways can you build addition equations with consecutive numbers as addends and the total among the discs?

How many with two addends?

How many with three addends?

How many with four addends?

Materials

Discs numbered 0 - 25

Content

basic arithmetic skills in addition, multiplication and division with whole numbers
number patterns
algebraic notation
generalisation and the concept of proof

Iceberg

A task is the tip of a learning iceberg. There is always more to a task than is recorded on the card.

There are two critical elements in this task:

The addends in any equation and its answer must be among the discs numbered 0 to 25.
The addends must be consecutive each time.

The first question is designed to ensure students are clear about these parameters. For example, this is not a task about finding all the two addend equations that equal 25. Only those that can be made with consecutive addends, which can only be 12 and 13. But what happens if the solution is 24, or 23, or 22, or ...?

The task only asks students how many solutions they can find; it doesn't ask to find them all. However encouraging them with questions like:

Great, you've found three ways so far. Do you think there might me more? ... Look at the mathematicians' questions on your Working Mathematically page. Which ones do you think a mathematician would ask now?

can lead to collecting more data and listing or tabulating it. For example for two addends:

25:(12, 13)
24:None
23:(11, 12)
22:(None)
21:(10, 11)
and so on...
Patterns may become apparent.
Hypotheses might be constructed and tested.

Digging Deeper

For each part of the task ask the mathematician's question:
How do you know when you have found all the solutions?
Look more deeply at the situation with three addends. In each case the total can be divided by 3 and one third of the total is the middle number. What can the students find out about the total and the list of addends in the other cases?
What happens if there are more than 4 consecutive addends?
Suppose n is the smaller of the two consecutive numbers in the first question. How will you write its partner?
Jackie says if you know the total in the two disc case you find the two numbers by taking off 1 and then dividing by 2.
Use n and its partner to explain Jackie's idea.
What happens if, instead of numbers that are 1 apart, we investigate sums of numbers which are 2 apart, 3 apart, 4 apart, ... p apart?

Note: There is a strong link between Consecutive Sums and Task 4, Window Frames.

Whole Class Investigation

Tasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works.

The whole class lesson for this task is detailed in Maths300 Lesson 139, Consecutive Sums. To convert the task to a whole class lesson each student cuts or tears a piece of paper into 16 sections. This is easy folding and gives enough pieces per pair for the numbers 0 - 25 with some left over for + and = signs. You will find that with most classes these are soon abandoned in favour of scrapbook work with written numbers, but without the pieces some students may not become engaged in the problem in the first place.

For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 139, Consecutive Sums, which also includes a Poster Puzzle to stimulate the investigation.

Is it in Maths With Attitude?
Maths With Attitude is a set of hands-on learning kits available from Years 3-10 which structure the use of tasks and whole class investigations into a week by week planner.
The Consecutive Sums task is an integral part of:

MWA Number & Computation Years 3 & 4

MWA Number & Computation Years 7 & 8

The Consecutive Numbers lesson is an integral part of:

MWA Number & Computation Years 9 & 10

Follow this link to Task Centre Home page.