# Jumping Kangaroos

### Task 182 ... Years 4 - 10

#### Summary

This is a very widely known puzzle that exists in many variations in many countries. It is a great puzzle, but it becomes mathematically richer when the underlying algebra behind the moves can be explicitly recognised and explored. To solve it requires a systematic approach to the movement of the 'kangaroos', and the patterns of moves yields an unexpected quadratic relationship. Students are encouraged to work out the number of moves required for different numbers of discs and to look for patterns as the number of discs increases. This allows them to make predictions for even higher numbers of 'kangaroos' on either side.

This cameo includes an Investigation Guide and two From The Classroom entries:

• Teachers' notes for a 1972 version of the task discovered in the A.A.M.T. office by Matt Skoss. Matt invites us to compare and contrast the support offered by those notes with the support by this Cameo and its Maths300 companion lesson.
• Two student reports on their investigation of this problem - one primary and one secondary. These are also available in the Recording & Publishing link.

#### Materials

• Ten 'kangaroos' in two sets of 5

#### Content

• seeking & seeing patterns
• What happens if...?
• quadratic expressions
• problem solving
• examining all possibilities
• trying a related simpler problem
• graphing

#### Iceberg

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

One way teachers extend learning using tasks is to write an Investigation Guide. The usual procedure is to discuss with students what they have done from the card, then ask a question that opens the door to the deeper investigation. For this task we have included an Investigation Guide (and discussion of answers) to model this approach. It has been extracted from the Pattern & Algebra Replacement Unit, which is now out of print. However, there is a Replacement Unit built into the Maths With Attitude Pattern & Algebra kit for Year 7/8.

#### 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.

Tasks have three lives:

• an invitation to two students to work like a mathematician
• a deeper investigation led by an Investigation Guide
• a whole class investigation to model how to work like a mathematician
In this case the whole class lesson is easy to begin by marking seven spaces on the floor (a good use for text books?) and using the students as kangaroos - sleeves up and sleeves down, or boys and girls will identify the two groups. As this physically involving approach proceeds, students can be encouraged to take counters and try the puzzle on their table. Once the students are confident they can do the problem the attention turns to 'in how many moves'? Indeed what is the minimum number of moves? Then, what happens if we change the number of kangaroos on each side? Is there a pattern in the table of minimum moves? If so, how do we explain the pattern? What does the pattern look like if we graph it? What happens if we start with different numbers of kangaroos on each side?

For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 42, Jumping Kangaroos, which also includes one teacher's assessment approach in its Classroom Contributions.

Visit Jumping Kangaroos in Menu Maths Pack A.

#### 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 Jumping Kangaroos task is an integral part of:

• MWA Pattern & Algebra Years 5 & 6
• MWA Pattern & Algebra Years 9 & 10

The Jumping Kangaroos lesson is an integral part of:

• MWA Pattern & Algebra Years 5 & 6
• MWA Pattern & Algebra Years 9 & 10

## From The Classroom

#### Historic Find

Matt Skoss, Consultant
When based in the A.A.M.T. office in Adelaide Matt Skoss made a discovery.
While having a fossick through some historical stuff at the AAMT office ... came across an early version of Jumping Kangaroos ... Bedford Park Teachers' College 1972. I was just a pup back then, but it is heartening to know that rich mathematical tasks were recognised way back then! I've attached a scan of the relevant pages.
(This is a PDF file. Use Ctrl L to toggle between full screen and page views).
And then from follow up emails:
...going to use these old notes with contemporary M300 lesson notes to compare/contrast the journey in how a good mathematical idea was elaborated upon.
Please put them on the Mathematics Centre site. My interest was piqued when I saw the sheer contrast of what was offered to teachers and the current richness of the task compared to yesteryear.

#### Student Reports

This document includes two student reports about Jumping Kangaroos - one Year 6 and one Year 7. They were gathered during school trialling in the mid-1990s of the Replacement Unit model for integrating tasks into the curriculum.

Follow this link to Task Centre Home page.