Monster Munch

This activity introduces students to the concepts of randomness and mathematical modelling.
In statistics, “random” means that there is no pattern or reason behind a selection. In particular, it means that a person running an experiment or activity has no control over whom or what is selected. For example, Lotto numbers are selected randomly, using a machine built for the purpose and designed so that its operation is transparent. The balls can come out in any order, and participants usually have confidence that the process is fair and impartial. That is, they believe that every combination of numbers has the same chance of being selected.
Some students may think that making a random selection means choosing “a few from here and a few from there for no particular reason”. If they use this approach, they can’t be sure that their selection does not have an unintended bias. Randomness requires a system or method that guarantees that the selection is free from bias, including hidden bias.
Once the concept of randomness has been explored, the idea of random numbers follows naturally. Random numbers are numbers that occur in no particular order and with no pattern. No matter how carefully you study a sequence of random numbers, you cannot be sure what the next number in the sequence will be. Each number has the same chance of coming next.
Convenient sources of random numbers include:
• A calculator. Press the RAN# key. (Sometimes the shift key needs to be pressed first.) On a calculator, random numbers are normally displayed as decimals. If the students need a random number between 0 and 9, they should use the last digit; if they need a number between 0 and 99, they should use the last two digits.
• Car number plates. Use the last digit only for a random number between 0 and 9. Ignore personalised number plates.
• The phone book. Again, use the last digit for a random number between 0 and 9. The first three digits of a phone number will not be random because they usually denote an area.
• A computer spreadsheet program. Select Insert/Function/RAND, then use the cursor to drag the bottom right-hand corner of the active cell down the column to give a sequence of decimal numbers. Use the last digit in each cell as your random number.
You should discuss the concept of a mathematical model before the students attempt the activity.
The term “model” describes a piece of mathematics we use to imitate or replicate something that happens in the real world. A model can be a statistical experiment (like the one in the activity) or, for example, a table, an equation, or a graph.
In general, models simplify a real-life process by focusing on the important features only. For example, what happens when we drop a rock off a bridge? We can model its acceleration due to gravity using a formula that ignores the impact of wind resistance. In this activity, the students can explore how peanuts are distributed in a batch of cookies, without the ingredients, the facilities, or the mess.
Some students may find the instructions in question 2b hard to follow, but the task itself is straightforward. Check that everyone understands that the random numbers relate to the cookie numbers, not to the number of peanuts.
It could be useful to get the students to record the results of their experiments on the board. They should notice very quickly that very few (if any) of the sets of results are identical. This can lead into a discussion of the fact that when sampling, each sample is likely to give a different result.
It is important that the students come to see that, when they are dependent on something that is random, they can never guarantee the outcome. Even if 1 000 peanuts were added to a batch of 10 monster brownies, this would not absolutely guarantee that there would be at least 15 peanuts in each!

Answers to Activity

1. It will not be enough because they can’t be sure of an even distribution of nuts. For example, if one cookie has 16 nuts, another will be 1 short.
2. a.–e. Answers will vary.
f. Not necessarily. A random distribution will never guarantee anything.
3. Answers will vary. Options include recommending that the company replaces the guarantee of 15 nuts with a statement that there is an average of 15 nuts in each cookie or changes the production method so that the nuts are machine-counted into the individual cookies.