# Bob's Buttons

### Task 123 ... Years 2 - 8

#### Summary

At first sight the task is closed. It could in fact be a text book question and some students will solve it by application of 'automatic' number knowledge. Others will need to arrange and rearrange the buttons. Both methods are valid. However, there are two hints on the card that there may be more than one answer. Seeking multiple answers, and the patterns that develop from them, is the iceberg of this challenge.

#### Content

• making, counting & arranging groups
• division with remainders - sharing
• multiples, factors & primes
• seeking & seeing pattern

#### Iceberg

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

The answer to the question on the card is 26.

Jamie and Jack, Year 5, Ashburton Primary School, show how this number satisfies both conditions in this Cube Tube Video.

When we look further, into the card we notice:

• it restricts the number of available buttons to 'more than ten buttons', and
• the question asks for the smallest number of buttons Bob could have.
These are the clues that there are multiple solutions.

Encourage the students to first look at numbers less than 10 which satisfy the conditions of the problem. They will find 6 is also satisfies the conditions. Exploring beyond 26 eventually reveals that 46, 66, 86 ... also satisfy the conditions and a pattern of counting by 20s has appeared.

But the groups involved are 4 and 5. Perhaps the skip counting pattern of 20 could have been predicted by multiplying the 4 and 5? Further investigation is required. Perhaps the counters are shared between:

• 3 friends with 2 left over and 5 friends with 1 left over
• 3 friends with 2 left over and 6 friends with 1 left over
or in any other way the students choose. They will find that if the group sizes are numbers that do not have a common factor, then the hypothesis of multiplying the group sizes to predict the skip counting number will work.

Extensions

1. What happens if three group sizes involved? For example, Bob shares his buttons between 4 friends with 2 buttons left over, 5 friends with 1 button left over and 3 friends with 1 button left over.
Note: This investigation has been included in Maths At Home. In this form it has fresh context and purpose and, in some cases, additional resources. Maths At Home activity plans encourage independent investigation through guided 'homework', or, for the teacher, can be an outline of a class investigation.
• For this specific activity click the Learners link and on that page use Ctrl F (Cmd F on Mac) to search the task name.

#### 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.

Using the students as 'buttons' is a great way to begin this lesson outdoors. Ask them to walk around in an enclosed space (such as the netball court) and on your signal make groups of a certain number. Record the number of groups of this size and the left overs. No-one is out, so, with the same number of students, repeat the activity five or six times. Return to the classroom where students use Poly Plug, or an alternative, on their desktop to make a 'table top model' of the outdoor experience and gather data for other possible groups 'called by the teacher'.

When the students are comfortable with making groups and left overs in this way and recording them, introduce the challenge something like this:

When I played this game with my class last year, I called groups of 4 and there were 2 left over. Then the kids walked around again and I called groups of 5 and there was 1 left over. I can remember that ... but I can't remember the number of students in the class that day. Can you help me work it out?
Follow on from here guided by the iceberg information above.

For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 10, Bob's Buttons, which includes an Investigation Guide and companion software.

Also, Calculating Changes Members will find that for young children the activity Buttons is related.

Visit Bob's Buttons in Menu Maths Pack D.

#### 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 Bob's Buttons task is an integral part of:

• MWA Number & Computation Years 5 & 6

The Bob's Buttons lesson is an integral part of:

• MWA Number & Computation Years 5 & 6
• MWA Number & Computation Years 9 & 10