Year 7 Common Task: Dice Games

A first lesson:

Drawn on the board is the following:

“Start at the black circle; if you roll a 1,2,3 move up, if you roll 4,5,6 move down. We are going to roll the dice until we get to the end and I want you to predict on which letter we will finish. (Possible addition: You win 5 points if you guess right, you lose 1 point if you guess wrong.)”

Now get a student to roll and draw the route on the board.

“Okay, it ended up on ___. (Addition: Is the game fair? How many points should you get for winning?) We are going to play again and you can change the letter you are betting on if you want … Has anyone changed their letter? Why? (Hear from several students) … Has anyone not changed their letter? Why? (Again, hear from several students)”

After several games students should come up with some conjectures eg ‘you can’t end on B, D, F, H’ or ‘the middle letters are more likely to win’. Push them to give reasons for these conjectures eg ‘the middle letters are more likely to win because there are more ways of getting to them’.

At the point where there are conjectures to test, get students to play the game in pairs (have grids copied on to card and, say counters - they can choose more than one letter when playing in pairs) – they should keep a tally of which letter wins, and class totals can be collected after, say, 5 minutes.

“Okay, we had several conjectures, in total the class have played the game (eg) 100 times, is there evidence for or against the conjectures?” (Discuss results)

It could be at this point, or for homework, students write up their interpretation of the class results. Or, the class may be talking about the different ways of getting to each letter, in which case, in pairs, they should work on finding all the different routes. (The class must do this at some point.) After 5 or so minutes this will need to become a class discussion with the aim of agreeing on all the routes there are. The teacher will then need to show how these can be converted into probability fractions.

Where this can go
It is important with the first grid, as a class, to agree on the number of routes to each letter and to derive the associated probabilities. These probabilities should then be compared to the class’ results from playing the game (or, after getting the probabilities, the class can play again and collect results to test against the theory – try to get a multiple of 22 games in total – eg get each pair to play 22 times). This can be a good time to introduce (remind) about zero probability implying something is impossible and 1 or 100% meaning certain. There can be rich discussions about how individual results may be far from what is expected but that, as a class, the results are pretty close – why is this so?

Individuals can then choose to look at different size grids, play the game, and try to work out the probability. The probability must then be tested against actual results (they can decide how many times to play).

Probabilities can be collected on a common board going as far as you like. (You may decide not to collect results on B, D, etc):

This board should provide rich material for conjecture. All probabilities should be tested against experiment and comparisons commented upon (this is important – getting students in to the habit of commenting on results for Ma4 coursework in year 10).

Guidelines for notes at the end of the topic

(Get sentences in the students’ own words.)

Probabilities are measured on a scale between 0 and 1 or 0% and 100%, where 0 or 0% means impossible and 1 or 100% means certain.

If you find all the ways something can happen, then the (theoretical) probability is that number over the total number of ways for everything. (This will be put much better by students themselves, and could be illustrated with an example.)

The more times you do an experiment the more likely it is that your results (relative frequency) will fit with what is expected the theoretical probability.

Vocabulary
(To be defined as a class and copied by each student into the back of their note books.)

theoretical probability -

relative frequency -

denominator -

numerator -

Typical mathematical content
Definition of probability
Calculation of probabilities by finding all possible outcomes
Comparison of theoretical and experimental results
Link between fractions, decimals and percentages
Index notation (2n is the denominator of the fractions)
Pascal’s Triangle and the Binomial Theorem

A homework to be done by everyone
To be given at the end of the dice games project.

“Throw two dice 36 times and record your results. Comment on what you notice. Can you work out the probability of each total? Comment on how your experiment compares