Return to Free Tour Crazy Animals

 

Summary

 

Lesson 57

  • Years: K - 12
  • Time: 2 - 6 lessons
  • Strands: Probability, Statistics, Number, Algebra

One day a young child brought a new book into class for her 'show and tell'. It was a book which mixed up the heads, middle and legs of a set of animals to make a whole collection of 'crazy animals'. The teacher had an idea about investigating how many different animals could be made. Another teacher thought of including a dice to explore the possibility of making these animals. Another thought of naming the animals to add interest and to make recording and publication easier. Yet another realised the algebra possibilities. This lesson is truly the composite of creative teachers sharing and learning from each other.

 

Resources

1. Animal Books

If possible show a children's crazy animal book which mixes up heads, bodies and legs. The library is a good place to look. You will find many children have one at home and will volunteer to bring it in.
How do the people who make the books make sure that the parts join up?
Sample Horse & Giraffe
Would you like to make one of these books yourself?
This question usually produces instant excitement. A good time to introduce the Animal Book. For older children you may prefer to begin with the children designing their own with the help of the Draw Your Own Animal Book.

NB: The dark band on both sheets represents the spine of a crazy animal book that the children could staple together, so you might like to emphasise that the students leave it on.

I decided to use only the giraffe and horse to begin with. The children cut these into three segments.

Because the students use their own animals to produce their own data, many teachers have reported an added interest in the problem.

Introduce the students to the problem of deciding how many different animals can be made, and allow time to explore. Display new combinations as they are found. Give the students time to colour and name their creations.

If you allow students to design their own animals it is advisable that you counsel against combining animals such as a fox and duck. These choices lead to problems at the naming stage!

At this level it is likely that students will approach the question of deciding that there are no more by reasoning that, "We can't find any more." More sophisticated reasoning would explain that:

  • there are two choices for the head position
  • and for each of these there are two choices for the body position
  • and for each of these there are two choices for the legs position
resulting in 2 x 2 x 2 = 8 possibilities.

The informal discussion about whether we had found them all showed me a lot about their emerging problem solving strategies.

Accept reasoning appropriate to the level and note that the total includes two 'normal' animals.

Eight Possibilities

Lesson Stages

  1. Animal Books
  2. Dice games
  3. Computer simulation
  4. Predicting whole and part giraffes
  5. Adding a third animal
  6. Analysing data
  7. Publishing

Learning Outcomes

  • using everyday language to describe chance events
  • recognising and describing more and less likely events
  • making predictions based on informal probability assumptions
  • systematically collecting and organising data
  • physical graphing
  • number
    • powers
    • clever counting
      (also known as selections & arrangements or permutations & combinations)
    • pattern
  • algebra
    • symbolic representation
    • generalising
    • expanding brackets and collecting like terms
    • expansion of (n + 1)3

Software Contribution

The five options of this software extend and enhance the problem solving. The first option simulates making the animals and many students find making Crazy Animals becomes even more enjoyable on the computer. In the other options a very large number of trials happen very quickly which demonstrates the power of the computer as a tool. As a result students begin to use the computer in the same way as a mathematician does - as an analysis tool.

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References

Visit the Task Cameo at Mathematics Centre.
Crazy Animals Task

Materials are important in the challenge to find all possible crazy animals. They provide the gateway through which all may enter the problem. If presented in the context of traditional mathematics education the problem would likely be reduced to a pencil and paper exercise with animals represented by letters - an algebraic presentation. Certainly, this is a valid objective in the learning program, but as a starting point, it can become the finishing point for many students. It seems fairer, and more likely to allow more students to exhibit their preferred intelligence, if the animal books are used.

2. Dice Games

For this group of activities it is useful to have the animal parts assembled into books. The Animal Book master page has a darker band along each animal. Stack the three animal pages and staple in the darker band to make the 'spine' of the book. Then, and only then, cut the animals into their three parts, making sure that the book stays together. This reduces the chance of losing a piece of an animal.
Game 1: Random Animals & Human Graphs
If playing with two animals, assign 1, 2, 3 on a dice to mean using a horse part, and 4, 5, 6 to mean using a giraffe part. (Similarly if using three animals assign each pair of numbers on a dice to represent a different animal.) The first roll tells the which animal's head to use, the second which animal's body and the third which animal's legs.

I took a large dice and put stickers marked H over three numbers and stickers marked G over the other three. Then the dice always rolled either a giraffe or a horse. I demonstrated that the first roll was for the head, the second for the body and the third for the legs. Then I gave the children stickers so they could make their own dice and roll to make an animal.

When you have made your animal, the letters on the parts will tell you how to name it. For example if you roll Giraffe head, Horse body and Horse legs, you have made a Gir-or-se.
When everyone has made an animal, put up signs around the room to name the eight possible animals (assuming you are using parts from only two animals).
Now move over to the sign for the animal you just made.
Check that all eight possibilities are represented and the number of each. In theory each group should be equal in number, but for small amounts of data chance doesn't ever do what is expected. It is this short term variability which aggregates to create the long term averages which are the numbers assigned by probability theory. Demonstrate this by recording the totals for each group, playing another game, making a new human graph and adding to the running total for each type of animal.

We played the game several times and kept a tally of how many of each animal were made. The tallies moved closer to each other and in discussion the children agreed that this meant that each animal had an equal chance of being made.

Game 2: How Many of each Animal will Our Class Make?
This game focuses on the chance of a single event occuring. One student chooses an animal as the class target, eg: a complete giraffe. Everyone then uses the dice to create an animal and stands up if they make the target. What does the class expect? Discuss the results.

On average we had about three successes. It was fascinating to listen to the students' explanations of why we should expect about three successes in our class of 24.

Game 3: Making Your Favourite Animal
This game focuses on the expected number of trials to make a particular animal. The children choose their favourite animal, write down its name, and roll the dice until they make it. (It is a good idea to write down the animal made in each trial, rather than just keeping a tally of the number of trials.) The number of trials is then counted.

Some were lucky and made their animal quite quickly. For others it took as many as twenty rolls. When we averaged our results though it was about 8 trials, as expected.

Game 4: Three Dice Together

We sped up the rolling by using three different coloured dice - one for the head, one for the body and one for the legs. In pairs I asked one person to roll the three dice and the other to make the animal indicated by the dice. There was lots of small group discussion related to recognition and ordering.

3. Computer Simulation

Now that the students have first hand experience with the game, it is appropriate to move to the computer and try the first option. Allow them to enjoy this first option using both two and three animals to create crazy animals. Check that the computer gives similar results to those above. For example, if you have access to a lab with one computer per student, ask each student to play one game and stand if they make a giraffe, say. Is the number of giraffes about what is expected?

The second option allows up to 10,000 trials to explore the chances of each crazy animal being made. Students can choose to explore using either 2 or 3 animals.

10,000 trials with three animals

Note that even 10,000 trials is insufficient to show that the chance of making each animal is exactly 1 in 27. How many sets of 10,000 trials have to be combined before the theoretical probability results?

This natural variation is confirmed when students try the third option which records the average number of trials to make their favourite. Although the theoretical average is reached on occasion,

Crazy Horse

most experiments of 10,000 trials only come close.

4. Predicting Whole & Part Giraffes

Note that an alternative approach to this section, which leads into algebra usually tackled in upper secondary school, is developed in the Whole & Parts Appendix.
In Game 3 (above) students are likely to notice and comment on the fact that they get close to the animal they are after, but don't quite make it. An example is getting the head and legs, but not the body. This section investigates the chances of getting all part, two part, one part and no part animals. The giraffe is used as an example.

I drew four columns on the white board and headed them Complete Giraffe, Two-Part Giraffe, One-Part Giraffe, No-Part Giraffe. Then we wrote the eight crazy animal names in their correct columns.

In a moment we will roll. Before we do I want you to count how many children there are in the class and decide how many animals will be made in each group.
Do the dice rolling and ask the students to stand in front of the correct section of the board.

They had quite a time explaining why the groups were no longer equal until they realised that the reclassification allowed the two-part animal (for example) to be made in three different ways.

The expected proportions are:

  • 3-part: 1 in 8
  • 2-part: 3 in 8
  • 1-part: 3 in 8
  • 0-part: 1 in 8

When children work together to solve problems there are benefits which derive from the mathematical conversation. Students challenge and learn from each other (something mathematicians have done forever) as they refine and develop the problem. Mathematics becomes something you do, and succeed at, in community. For many, to the degree that this adds to their self-esteem, the lesson is uplifting.

Option 4 of the software can be used to explore the number of trials needed before these theoretical results show up in the experimental data.

5. Adding A Third Animal

If appropriate to your class, use all the same activities again with three animals.

I added the duck and we began to explore the possibilities. My children were quite surprised that the number went up to 27. We arranged them into sets like this:

Venn Diagram

After 18 rolls we had six crazy animals and the students said there was definitely more chance of getting a horse! They agreed to continue experimenting to see if their hypothesis was correct.

Again Option 4 allows an exploration of the number of trials needed before the experimental data comes closer to the theoretical average (or probability).

6. Analysing Data

This section is an opportunity to use data in context as the basis of a toolbox lesson. Use the Investigation Sheet (Years 1 - 3). Some find it surprising that young children are able to use and understand the symbolic representations on the sheet. In so doing, they are being prepared for using symbolic manipulations skills in later years.

(A toolbox lesson is one which practises a skill the students have already applied as part of an investigation. It relates to phase of the Working Mathematically process when a mathematician reaches into their toolbox of mathematical skills to find one that will assist with the current problem. Becoming a better mathematician includes becoming better at choosing and using mathematical skills. So, those skills need to be practised.)

Analysing the worksheet was an excellent link between the activity and its symbolic representation. Students liked looking through the list and it taught them to look systematically.

Changing Parts

Let's imagine that we had three animals in the book but each one was only cut into TWO parts - top half and bottom half. How many different crazy animals could be made.
Give groups time to sort this out then discuss the reasoning with the class.
This is a problem involving choices. In our toolbox we have a tool called a tree diagram that helps us to sort out choice situations.
Illustrate with a drawing that shows there are three choices for the top part of the body and for each of these there are also three more choices for the bottom part. So, in total:
3 x 3 = 9

Try another example such as two animals cut into four parts. (16 possibilities).

When students have the idea, introduce Option 5 of the software and allow them to search for a rule.

3 Animals with 2 Parts
If I give you any number of animals and any number of parts, I want you to be able to tell me how to work out the number of possible crazy animals.

Publishing

One of the key aspects of this lesson is to discover how many crazy animals can be made. This challenge is introduced through two key questions which are central to the Working Mathematically process:
  • How many solutions (animals) are there?
  • How can you convince me that you have found them all?
Publishing is an integral part of the process of Working Mathematically. This lesson usually creates a good deal of interest among students, so it is a good one to push into the publishing area. This is where the written language and the mathematics curricula can be integrated to help address outcomes in both. However students have to be taught this form of writing, so if this is your first attempt at mathematical publishing with your students, laying the groundwork as a class, perhaps through the development of a wall story, is essential.

Publishing could be in the form of a letter to a friend. In preparing the letter these questions would guide the writing:

  • What do they need to know to understand what we did?
  • What did we do?
  • How did we do it?
  • What did we find out?
  • What did it lead us to investigate next?

Or build your publishing lesson around the examples in the Recording & Publishing section of Mathematics Centre.

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Answers: Investigation Guide Years 1 - 3

  3-PART 2-PART 1-PART 0-PART
Giraffes 2 13 18 7
Horses 0 10 14 16
Ducks 1 8 21 10

 

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