Crazy Animals

Task 102 ... Years 2 - 10

Summary

This investigation began when a child in the first year of school brought along a book that allowed the reader to connect the head of one animal with the body of another with the legs of another. Talk about Crazy Animals! Over the years it has become a favourite at all levels. Everyone seems to want to make and name the animals and work out how many there are? But there is much more to the task than that.

Materials

9 crazy animal pieces made from 3 parts of 3 animals
1 dice

Content

combination theory
problem solving strategies
Venn diagrams
tree diagrams
powers
number patterns
expansion of (n + 1)³
more or less likely events & informal probability experiences
probability - long run frequency
collecting & recording data
probability theory
short-term variability versus long term average

Iceberg

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

Using only two animals, there are 8 crazy possibilities, including the 'normal' animal. For three animals there are 27 possibilities. But how do you know?

There are several strategies children could use to work this out.

Guess, Check & Improve:
Keep trying and recording until they can't make any more.
Break the Problem into Smaller Parts:
Fix one part, say the head, then fix a body to it. There will now be three possible legs. Now do it again with the same head and a different body. That's three more. Now again with the same head and the last possible body to make three more. That's nine so far. Now repeat the process with each of the two other heads.
Draw a Diagram 1:
This tree diagram is for two animals each with three parts. It maps the choices at each stage, beginning with the choices for head. Encourage students to sketch a similar diagram for 3 animals with 3 parts.
Draw a Diagram 2:
A Venn Diagram shows the ways the three animals can 'intersect'. Putting your finger into a region shows which 'normal' animal(s) contribute to the crazy animals in that region. You can then explore all the animals that can be made with those pieces, eg:
Write an Equation:
In this case it is about choices. There are 3 choices for a head and for each of these there are 3 choices for a body to match it (9 pairs so far) and for each of these pairs there are 3 choices for legs.
So, No. of Crazy Animals = 3 x 3 x 3 = 27

At this stage the investigation can be extended with these questions:

What happens if we change the number of animals?
What happens if we change the number of parts?
If I tell you any number of animals with any number of parts, can you tell me how many crazy animals can be made?

A further investigation develops by encouraging a focus on a particular animal, say giraffe. How many 3-part, 2-part, 1-part and 0-part giraffes can be made from 2 or 3 or 4 or ... starting animals? This table suggests developing number patterns which can be generalised.

No. of Animals	3-Part	2-Part	1-Part	0-Part	Total
2	1	3	3	1	8
3	1	6	12	8	27
4	1	9	27	27	64
5
6
...	...	...	...	...	...
10

The second part of the card explores probabilities related to making a crazy animal at random. In this case the animal identified as the student's favourite. Each animal has an equal chance of being made, ie: in the two animal case - 1 in 8, and in the three animal case - 1 in 27. So, in the long run we would expect, respectively, an average of 8 or 27 rolls to make the chosen favourite.

Here is an opportunity to experience the concept of short term variability and compare it with a long term average. It may be that in a given trial a student makes their favourite the first roll. However, it may be that it takes 200 rolls for the favourite to appear.

How much data is enough to be reasonably sure of the number of rolls it is likely to take to make the favourite?
How much data is enough for the experimental result to be close to the theoretical result?

Question 4 asks only that the students experiment and keep a tally. This is an invitation to design and carry out an experiment that attempts to answer either or both of these questions. In doing so students have to make decisions about what data to collect, how to display it and what statistics to use describe it.

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.
Question 4 on the card can be an introduction to a whole class lesson. It actually takes many experiments to be able to get a clear idea of the expected number of rolls to make a favourite. More than any pair is likely to do. So the instruction to 'keep a tally' can be re-interpreted in the whole class environment. Over time each pair explores the task and records their 'trials to make the favourite' results on a class chart (or spreadsheet). Discussion develops on a regular basis as the data is viewed and reviewed.
Alternatively, if you make your own Crazy Animal books or cards, the investigation can be compressed into a couple of lessons with each pair (or person) carrying out as many 'trials to favourite' experiments in, say, 10 minutes as possible. This approach, and several other directions for the investigation, are explored in the companion Maths300 lesson, which includes software.
Visit Crazy Animals on Maths300:
Guest ... Member

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 Crazy Animals task is an integral part of:

MWA Pattern & Algebra Years 3 & 4

The Crazy Animals lesson is an integral part of:

MWA Pattern & Algebra Years 3 & 4

MWA Pattern & Algebra Years 9 & 10

Clearly, this task or its whole class investigation could also be used to supplement Chance & Measurement kits in both the primary and secondary areas.
Crazy Animals task is also included in the Task Centre Kit for Aboriginal Students and the Primary Library Kit. Solutions for tasks in the latter kit can be found here.

Follow this link to Task Centre Home page.