# Monkeys & Bananas

### Task 212 ... Years 2 - 10

#### Summary

Three monkeys work all day to collect a big pile of bananas and then flop into bed too tired to eat them. The card then goes on to explain how each monkey got up in the night, ate a banana and then hid one third of the bananas remaining at that time. Despite these night hungries - and displays of selfishness - the monkeys were still able to have equal shares of bananas at breakfast. How many bananas might have been in the original pile.

This cameo includes an Investigation Guide which offers a similar problem as an extension. It also has a From The Classroom section which takes up an idea in the whole class lesson notes below and converts the lesson into a Poster Problem Clinic. This is adapted further by Beth Bright into Minions & Bananas complete with mobile phone apps.

• 30 'bananas'

#### Content

• basic arithmetic skills
• fractions of a set of objects
• problem solving strategies
• multiplicative thinking
• spreadsheets as a problem solving tool
• linear algebra

#### Iceberg

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

Some students will begin this problem by guessing a pile number; others by guessing a number the monkeys might have shared at breakfast. Whichever way is chosen the guess will either lead to a fraction of a banana - which doesn't exist in the problem - or will lead to a solution. Along the way, students might discover some, as yet, unrealised information in the problem. For example:

• Suppose the original pile is 23. Then Monkey 1 eats one and divides the remainder into three equal shares. Hey that won't work because 22 doesn't divide equally by 3. Okay then, what about 24. No that won't work either because then there would be 23 to make into thirds. 25? Yep, that's okay!

Monkey 1 eats one, which leaves 24, hides one third of that, which leaves 16 for Monkey 2 to find.

Then Monkey 2 eats one...

• Whatever is left at the end must be a multiple of three. It also has to come from a multiple of three because Monkey 3 had to hide one third and leave two thirds. Suppose there were 3 bananas for breakfast. What multiple of three number could that come from by making thirds? It can't come from 3 because two thirds of that is 2. And it can't come from 6 because two thirds of that is 4. So 3 bananas to share at breakfast can't work. What about 6 bananas to share at breakfast? What multiple of three could that come from by making thirds? If it's there, it has to be bigger than 6. Try 9. Hey that works!

Dividing 9 into three parts and keeping two of them gives 6 bananas for breakfast, so Monkey 3 must have found 10 bananas when he woke up.

But this had to come from a multiple of three number...

Following either of these directions will lead to a conclusion of 25 bananas as the smallest possible original pile, which leads to a breakfast share of 2 bananas. For most students (and teachers?) it will be essential to at least begin their reasoning by using the materials. It is also essential that when the problem is solved students are challenged to explain how to do it. This could be as a diagram or slide show or text. The reason for requesting this is that ...we are learning to work like mathematicians and mathematicians have to publish their work for others to understand.

#### Extensions

• The use of the word 'might' on the card suggests that there also might be another solution. Encourage the students to re-apply their reasoning to look for at least one more.
• This is a great problem for introducing, or reinforcing, how a spreadsheet works. Support students to design a spreadsheet which starts with the breakfast share and works backwards, column by column, to the original pile. Starting from the left, the question used each time to find the formula for the next cell, is What had to happen to the number in the next cell to get the number in this cell? Now reverse that.

It seems to help students crystallise their thinking to break the problem into parts like this. The approach is essentially the same as that in Maths300 Lesson 19, Backtracking.

Setting up formulas for each column also creates a tool that can investigate what happens for any given morning share and thus help uncover other solutions. The introduction of the spreadsheet into this problem often revitalises student interest in it because the burden of the calculations is taken away. Once set up, it only takes milliseconds for the spreadsheet to find several answers and the mathematical focus shifts to looking for patterns in the answers and why they might be there.

A further extension would be to ask questions like What happens if there are four monkeys in the team and during the night they each eat one and hide one fourth?.

Another approach to extension is to offer a related problem as Kerry Wode, St. Thomas the Apostle School, Kambah, has done with this Tale of Two Pyramids Investigation Guide.

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.

Prepare the monkey story as a poster or a slide. Ask students to sit in groups of 3 and hand out 'bananas'. Poly Plug works well, especially if you convert the story to Kids & Cup Cakes (see below). Challenge the groups to find a solution. Perhaps there could be a real banana for each member of the team that finds a solution first. Take time for several groups to explain their reasoning and then ask each group to prepare a poster (or other form of report) to explain how to work out the problem.

Raise the possibility of another solution and encourage searching for one. (It may be that groups have already found different solutions.) Encourage students to express their insights into the small step calculations in the problem. First you have to... then you have to... then next you... etc. Introduce the idea that a computer is designed to do perform repetitive calculations like this and suggest exploring how a mathematician uses a spreadsheet to take advantage of the computer's mindless calculation power. This is a great opportunity to justify the school's laptop or tablet computers.

Use the spreadsheet to explore every possible morning share number up to say 50 will emphasise that by removing the drudgery of calculation, mathematicians can seek and see patterns, for in fact, a pattern does begin to emerge showing that solutions for the morning share number are separated by 8, ie: 2, 10, 18, ... Other patterns can also be found.

In senior classes, pushing the investigation further into one algebraic expression linking the final share number and the original pile helps to show why the difference of 8 occurs.

For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 113, Monkeys & Bananas, which also includes a prepared spreadsheet.

Visit Monkeys & Bananas on Poly Plug & Tasks.

#### 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 Monkeys & Bananas task is an integral part of:

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

The Monkeys & Bananas lesson is an integral part of:

• MWA Number & Computation Years 9 & 10

## From The Classroom

#### Högskolan Malmö

Teacher In-Service Course
I was asked to run a workshop that included Monkeys & Bananas because these teachers were doing a course in algebra and functions. There weren't enough 'pretend bananas' in the storeroom but there were plenty of Poly Plug, so I decided to use the Kids & Cup Cakes variation. In addition, so these teachers would experience a new teaching craft approach, I prepared this poster and presented the lesson as a Poster Problem Clinic. Show the poster in full screen view using Ctrl L.

The purpose of these clinics is to help students realise the importance of:

• Read & understand the problem.
• Plan a strategy to start the problem.
• Carry out the plan.
• Check the result.
which is part of the Working Mathematically process.

The teachers appreciated the application of literacy teaching techniques (story context and read & retell for example) that makes the Poster Problem Clinic much more than just teaching strategies for problem solving and very much needed to use the concrete materials to start their thinking.
Doug. Williams

#### St. Mary MacKillop College, Canberra

Beth Bright
Hi Doug,

I attended your Professional Learning session entitled 'Numeracy Essentials' held at the Rheinberger Center (ACT) in September. I found the Poster Problem Clinic a very engaging task which clearly taught students to slow down their problem solving so as to maximise understanding of the question prior to solving. I took this strategy back to my school, St Mary MacKillop College, by slightly altering your 'Monkeys and Bananas' activity to one involving 'Minions and Bananas'. I also used my 'Problem Solving Apps' to assist in the process.

(Note: The links are to PDF files. Use Ctrl L to toggle between full screen and page views. Use full screen for display).

This activity was used in a peer observation lesson to great success. The students were incredibly engaged and realised the value of reading for understanding, rather than just rushing in. I have subsequently passed this activity onto my colleagues at school and thought you also might like a copy of my alteration and Problem Solving Apps.

The 'Apps' are just a poster (I wish I was able to develop actual apps...maybe that can be a project for me next year?)...it's just another way of presenting the 'mathematical toolbox'... even my year 9s enjoy playing with the big 'mobile phone', pretending to push the screen when they select an 'app' to use (and inevitably place the poster up to their ears pretending to talk on their 'phone').

Kind regards,
Beth Bright