# Bob's Buttons Years 4 - 8

### Preparation

• About forty (40) objects such as buttons, plastic screw caps, pasta, tiny toy people, beads or pebbles.
• Write the title of this challenge and today's date on a fresh page in your maths journal.

### Getting Started

• Click the photo on the right to play a video that will be Introducing Bob's Buttons.
You can play the video again whenever you want to.

The photo below is the Recorder's book from the video.

• Copy the picture into your journal.
• Play the clumps game with 17 pretend people five (5) more times.
Draw a new arrow on your page each time and write its number story.

Have fun exploring Bob's Buttons.

### The Teacher's Questions

1. If I called out sixteen (16) what number of players would be left out?
2. If I called out three (3) what number of players would be left out?
3. If I called out seventeen (17), what number of players would be left out?
4. If I called out one (1), what number of players would be left out?
5. Five (5) players were left out, what numbers might I have called?
6. Four (4) players were left out, what numbers might I have called?
In your journal, draw a table like the one below for all the numbers I could call in the 17 clumps game. Fill in all the boxes.

 No. Called No. of Groups No. Left Out Equation 1 17 0 17 x 1= 17 2 8 1 (8 x 2) + 1 = 17 ... ... ... (5 x 3) + 2 = 17 ... ... ... ...

When you have recorded all the data write or draw about anything interesting in the table.

### Special Questions to Think About

(You don't even have to look at these if you don't want to.)

Mathematicians like to ask unusual questions just for the fun of thinking about them. Questions like these:

• What would happen if the teacher called eighteen (18)?
The players can't stand apart like they are social distancing, because that would be groups of 1.
What could they do?
What would the equation be?
• What if the teacher called any number greater than seventeen in the 17 game?
• What would happen if the teacher called zero (0)? Is there an answer?? Could we write an equation??

### More of the Teacher's Questions

Seventeen is a special number because it has exactly two ways of having zero left overs. If the teacher calls 1 or the teacher calls 17.
A number like this is called a Prime Number.
A Prime Number has exactly two ways it can make groups with zero left overs.
One way is in groups of 1.
The other way is the one group made by the number itself.
1. Make a list of all the Prime Numbers up to twenty-five (25).
(Mathematicians say the number 1 is not a prime number. If you want to, you can try to explain why.)
2. Choose one number that is a prime number up to 25 and one that isn't.
For each of them draw an arrow picture or a table using a few 'call out' numbers.
If you want to, you can play the game as you do it, like on the video, but remember to check your number of players two ways.
3. Make up your own clumping game question and try it out.

### Bob's Buttons Question

This is the question Bob made up when he was working with buttons as his pretend people.
Find the answer to Bob's challenge and explain it in your journal.

### Bob's Extra Challenges

When you have worked out the answer to Bob's question and explained it, watch this video and see what you think of Jamie and Jack's explanation.
They were in Year 5 when they made the video.

 Hint: They start off with groups of 5 and 1 left over. One of the groups of 5 is on the left page. But what do they end with? It works, but is it Bob's question? Could you explain to them how they could change groups of 5 with 1 left over to make the other part of Bob's challenge?

Bob knew something extra and he has given us clues.

• Why does the challenge say 'the smallest number of buttons' instead of just 'the number of buttons'?
• Why does it say that he had '...more than 10 buttons'?
Explore ... and record your experiments.

If you aren't sure about Bob's clues, there is a photo in the Answers & Discussion section that explains what happens if he had less than 10 buttons. But don't look ... unless your really, really can't figure it out for yourself.

If you don't have to just find the smallest number of players (buttons), then Bob's challenge has many answers.
One answer for the number of players is even smaller than the one in the boys' video and all the other answers are bigger.
• Make a list of the first ten (10) answers to Bob's question.
• Describe the pattern.
• What happens if Bob's question was groups of 3 with 2 left over and groups of 5 with 1 left over.
• If there is a pattern of answers, describe the pattern.
Now we have two pieces of data:
• If the 'call out numbers' are 4 & 5 with left overs of 2 & 1 there are many answers to the number of players and there is a counting pattern of ... between the answers.
• If the 'call out numbers' are 3 & 5 with left overs of 2 & 1 there are many answers to the number of players and there is a counting pattern of ... between them.
There is a way to predict the counting pattern number from the 'call out' numbers.
What is it?
Mathematicians love Bob's Buttons question because it has more than one answer. They love it more because the answers make a pattern. They love it even more because they can ask 'what if' questions and find even more patterns.

### More Challenges

A problem is never finished; it is only ever finished for now.
If you have had enough of Bob's Buttons then you can go to the next section, then stop.
Mathematicians do that all the time
... BUT if another good question about the problem comes along, they might jump right back in again.

Perhaps one of these questions might encourage you to return to Bob's Buttons one day.

• Is it true that the counting pattern number can always be predicted from the 'call out' numbers?
• Is it true that any two group numbers with their remainders produces solutions?
If not, what can you learn about the numbers that do give solutions?

A mathematician must be able to explain their discoveries to other mathematicians, so when you have finished finding out stuff, prepare a report of your discoveries. You might write a report, or make a slide show, or a video, or a poster or explain in some other way.

### Just Before You Finish

Read your Working Like A Mathematician page again and write three or more sentences explaining how you worked like a mathematician.