Task 8 ... Years 4 - 8

Summary

Given a particular whole number, find whole number pairs which sum to that number.
• How many such pairs are there for the number?
• How do you know when you have found them all?
• If I give you any whole number, can you predict how many whole number pairs will sum to that number?

Materials

• Discs numbered 0 - 16 plus one extra of discs 5 & 6

Content

• basic arithmetic skills, especially counting and addition
• 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.

1. To generalise this task to any whole number introduces a rule which needs two parts. It depends whether the number is odd or even. Expressing this generalisation in words (or symbols) is an extension in itself.
2. What happens if we investigate whole number triples?
Task 216, Square Pairs, is related in that the starting point is numbers 1 - 16 being added in pairs to make square numbers.

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 18, Addition Totals. To convert the task to a whole class lesson teachers need A4 size pieces of paper with the same numbers as the counters above. Selected students hold these in front of them and demonstrate pairing. Explore 11, then 12, then predict first and check 13. Students may wish to tear up paper to make the equivalent of counters, but most seem happy to use paper and pen once the task is introduced in this physical manner.

Maths300 lessons help teachers build whole class investigations to encourage students to work like a mathematician. In this case the essence of the Maths300 lesson plan is presented here so that, if you are not a Maths300 member, you can get a sense of how these lessons are structured. However, no matter how good the lesson, children sometimes add joyful surprises, as this email shows:

Hi Doug,
I teach year 5/6 kids at Narrawong District Primary School. My kids have absolutely loved all the Maths300 lessons we have done - I believe it is the open entry/open ended nature of the tasks as well as the physical movement involved that gets the kids keen ... anyway.

Today we began the Addition Totals lesson: we digressed beautifully, as is often the case. During the introduction I asked, Are we sure we have all the pairs to make 11? One of my Year 5 boys suggested adding another level of complexity by including negative numbers. After a bit of think time he showed the class how this isn't really that much more complex and got some of my level 3 kids on board.

Then we got back on track...they loved that too! Just thought I'd share this with you.

Cheers
Danita Smith

For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 18, Addition Totals, which includes an Investigation Guide.

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.