# The rabbit problem

Purpose

This is an activity based on the picture book The rabbit problem

Achievement Objectives
NA4-9: Use graphs, tables, and rules to describe linear relationships found in number and spatial patterns.
Specific Learning Outcomes
1. Students will be able to represent the Fibonacci sequence in a diagram which models how the sequence is generated.
2. Students will be able to describe how their diagrams relate to the mathematics that generated the sequence.
Description of Mathematics
1. Numeric patterns can be represented schematically to illustrate the relationships between members within the sequential pattern.
2. Rules are used to describe the relationship that produces a pattern and to calculate other members of the pattern.
Required Resource Materials
The rabbit problem by Emily Gravett

Copymaster: Representing Rabbits

Large sheets of paper (A3 or A2)

Activity

Representing Rabbits
This activity is based on the picture book The rabbit problem

Author: Emily Gravett
Paper Engineering: Ania Mochlinska
Publisher: MacMillan (2009)
ISBN: 978-0-330-50397-6

Summary:
This is a representation of the classic Fibonacci problem of reproducing rabbits. The problem of how many pairs of rabbits will you have after 1 year if you start with 1 pair and they each take 1 month to mature and produce 1 other pair each month afterwards is illustrated through a calendar. The paper engineering is creative and the narrative is told through a series of “problems” the every increasing population experiences each month. A small sign keeps the reader updated as to how many pairs are now living in Fibonacci’s Field. The extras, like the Ration Book and the Newspaper, contain great launch items for statistics discussions.

Lesson Sequence:

1. Prior to reading, present the rabbit problem. (It is on the inside cover)
If a pair of baby rabbits are put into a field, how many pairs will there be: a) at the end of each month, and b) at the end of one year? Criteria: Rabbits are fully-grown at 1 month and have another pair of bunnies at 2 months. Each pair is comprised of 1 male and 1 female and no rabbits die or leave the field.
This is the classic rabbit problem Fibonacci used to generate the sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144…
2. Ask students to work together in pairs and estimate the number of pairs they would have at the end of one year.
3. You could further set the context for the book by having students explore the sequence and see if they can discover a rule for generating the next number (each number is the sum of the previous two numbers) or you could discuss some biographical or mathematical ideas related to Fibonacci the mathematician.
4. After a first reading, quickly flip through a second time demonstrating how the pairs of rabbits are illustrated as unorganized sets. Ask students to work together to create an illustration of how the population has expanded over a year. Their diagram needs to be organized so that the set for each month is easily found. Try not to give too many directions about this assignment, as it will be a valuable assessment opportunity to see how students think about organizing a pattern.
a. What do they know about using a tree diagram or a flow chart?
b. Do they see it as a “branching” scheme or a layering scheme or a more linear scheme?
c. There are many ways to create a schematic representation and encouraging creative responses to this may provide you with surprises. The copy-master presents common organisations should you need to guide some students or give them a “launch”.
5. Ask students to present their diagrams to each other and locate the common elements and the differences between them. Generate some agreed criteria for representing sequential patterns: what makes for a clear and easily understood diagram?
6. As a follow up you may want to explore the sequence as it is found in nature or as the spiral generated by a series of squares working out from a centre. There are also units on nzmaths that explore Fibonacci’s ideas further. For example: http://nzmaths.co.nz/resource/fibonacci-i
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