## 1. Animal Books
## 2. Dice Games
Game 1: Random Animals & Human GraphsIf 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.
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). 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.
## 3. Computer SimulationNow 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.
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,
most experiments of 10,000 trials only come close. ## 4. Predicting Whole & Part GiraffesIn 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.
Do the dice rolling and ask the students to stand in front of the correct section of the board.
The expected proportions are: - 3-part: 1 in 8
- 2-part: 3 in 8
- 1-part: 3 in 8
- 0-part: 1 in 8
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 AnimalIf appropriate to your class, use all the same activities again with three animals.
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 DataThis 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.)
## Changing PartsGive groups time to sort this out then discuss the reasoning with the class. 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: 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.
## Publishing
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.
## Answers: Investigation Guide Years 1 - 3
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