Monkeys & Bananas

Summary

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.

Materials

• 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

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...

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.

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.

Whole Class Investigation

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.

Högskolan Malmö

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