# Monkeys & Bananas Years 5 - 10

A mathematician's work begins with an interesting problem.
The problem in this activity begins with a story which many young mathematicians have found interesting.

### Preparation

• Find thirty (30) objects that you can pretend are bananas.
It doesn't matter what they are - screw caps or pebbles or ... - as long as you can move them around on your table.
• Blank paper and markers for drawing and doodling as you think through the problem.
• Print the Working Mathematically page.
• Write the title of this challenge and today's date on a fresh page in your maths journal.

### Understanding The Story

• In a moment you are going to click a link to see a story about monkeys and bananas.
• It will open in a new tab.
• Read it through once or twice - no more - then close the tab so you can't see the writing.
• Use a marker on blank paper to write or draw or doodle the important things you remember from the story.
• Then open the link again and check and change what you remembered.
Here's The Story

When you think you understand what happened in the story, move on to...

• Write them on your doodle page.

### Getting Started With Questions

• Open this Monkeys & Bananas Starter.
It has two questions to get you started.
If they are the same as your questions that's great.
• Now you have the story and the questions on the same page.
To discover answers, you will need to dig into the strategy toolbox at the bottom of the Working Like A Mathematician page.
Which strategies might work?
It's good to have more than one strategy in mind.
You can start a different way if your first strategy doesn't work or it gives you a different way to check your work,
Have fun exploring Monkeys & Bananas

These teachers seem to be acting out the story with their 'bananas'.

Hint
You might have acted out the story by choosing a number for the original pile (P) and trying it out.
- Whichever number you choose you have to take away one for the banana eaten by the first monkey?
- So what has to be true for one less than the pile number (P - 1)?
- If you know that you won't have to test every pile number.

### When You've Found The Answers

When you know the number for the original pile of bananas (P) and its morning share number (S) for each monkey:
• Write a recipe, or draw a diagram, or create a poster, or pen a comic strip, or make a slide show or ... to explain to someone else how to do it.
If you know how to draw a Flow Diagram it might help you explain.

### Tracking & Backtracking

In this problem there are only two things we don't know.
• We don't know the original pile number (P).
• We don't know the morning share number (S).
If we knew just one of those we could work out everything else, because we know everything else that happened.
• A tracking diagram is a way of showing all the steps in order
This is part of a Enrico's tracking diagram.

Copy it into your journal and finish it.

• Check that it works to give you the answer you already know.
• Why does the diagram have ÷3 in it for each monkey?
• Why does the diagram have x2 in it for each monkey?
• At which points do the hidden bananas leave the track?
Julia thought Enrico's tracking diagram was great, but she wanted to guess the morning share number (S) first and work backwards.
So she changed Enrico's tracking diagram into Julia's backtracking diagram.

Copy it into your journal and finish it. (If you want to, you can just copy the hops onto Enrico's diagram.)
• Check that it works to give you the answer you already know.
• Why does the diagram have x3 as the first hop at the share end?
• Why does the diagram have ÷2 in it for each monkey?
• Why does the diagram have x3 in it for each monkey?

### Digging Deeper

1. Before looking at what comes next, look back to the questions you wrote at the beginning and see if there are any you want to try.

2. Guess what. There is another starting pile of bananas that works. And another. And another. And...
• Try to find at least one more solution - one more (P, S) pair.

You might want to use a tracking or a backtracking diagram.

3. Mathematicians love a problem with more than one answer. It means there might be a pattern in the answers and looking for that pattern could be their next interesting problem. But there's a lot of calculations to do in Monkeys & Bananas to find enough solutions for a pattern to start showing up. One tool in their toolbox that can speed up the hunt for solutions is a Spreadsheet.

It's really only Julia's backtracking diagram in a different way.

• Click on Cell B5 and look up in the formula line. What is it telling you?
• Repeat for C5, D5, E5,... until you are convinced the spreadsheet has the same operations as Julia's diagram.

You already know the result (2, 25) which is there when the spreadsheet opens.

• Change 2 to the second S number you found. Does the spreadsheet give the answer you found for the P number?
• Experiment with the S number to find more that work.
• Is there a pattern?

If you want to change it to try something out, go ahead.
If you mess it all up, you can click the link and save the original version again.

### More Questions

A mathematician is never finished with a problem. They are only 'finished for now' until someone asks the next interesting question about it. Even then, they don't have to tackle it straight away. They can put it on their 'to do' list and come back later.
• There is a pattern in the successful S numbers, but can we explain why?
• The spreadsheet is just an equation linking S and P, but it has been broken into parts.
Can you create a 'normal looking' equation from the formula information in the spreadsheet (or Julia's backtracking).
• The answer is yes and it is below, but don't look until you have tried for yourself - it's really too beautiful.
• What happens if we investigate the hidden number of bananas?
• What happens if the story is about two monkeys who each take a half, or four monkeys who each take a quarter, or...?

### Just Before You Finish

• How did you feel as you were exploring Monkeys & Bananas? Draw and/or write.
• What did you learn about working like a mathematician?
• What mathematics 'stuff' did you bring out of your skill toolbox?
• What mathematics 'stuff' did you learn?
• What mathematics 'stuff' do you think you should now learn better.