Probability

1. Probability Aim of session: • Intoduce activities that explore and develop ideas of probability • Links with NZC and Progressions • Links with National Standards and NZ illustrations

2. Probability How might we adapt this activity for our Level 1 students? (refer to Progressions) What is “everyday language”? Will won’t might always no Yes perhaps no way maybe never Impossible certain

3. Probability How might we adapt this activity for our Level 2 students? Simpler / less ambiguous statements? Measurable outcomes?

4. Some Content Knowledge! • The Probability Scale: • The probability of an event that is certain to happen is 1. • The probability of an event that will never happen is 0. • The probability for all other events is between 0 and 1. The more likely it is to happen, the closer the probability is to 1.

5. Some Content Knowledge! • Simple probability terminology: • - impossible, certain, likely, unlikely, even chance…

6. Some Content Knowledge! • Probability of an event: • Number of favourable outcomes • Number of possible outcomes • But… count/list outcomes first. • Expressing probabilites as fractions not expected until Y8.

7. Some Content Knowledge! • Variation: despite what is expected (from models of of possible outcomes), actual results may differ from this.

8. Some Content Knowledge! • Independence: what happens in one event, has no • affect on another.

9. Some Content Knowledge! Sample size: the larger the sample, the closer the actual results will be to the expected results.

10. Some Content Knowledge! Be careful when interpreting data!

11. Good, rich tasks needed! • Noticing distributions • Describing shape, centre and spread • Understanding variation • Acknowledging sample size

12. Fancy a Flutter? A Horse Race game to develop ideas of probability Is it fair? Give your reasons.

13. Complete a chart showing how may times each horse won a race. Which horse never moved? Why?

14. Complete a chart showing how may times each horse won a race. Which horse won most often? Why?

15. Use your data to investigate outcomes that are possible and decide whether the game is fair.

16. What are the possible outcomes?

17. What are the possible outcomes?

18. Using a two-way table to find all possible outcomes Dice 1 Dice 2

19. Using a two-way table to find all possible outcomes Dice 1 Dice 2

20. Models of All Possible Outcomes • What is the total number of combinations (outcomes)? • Order the outcomes • Probability of an event = Number of favourable outcomes Number of possible outcomes • Work out the probability for each horse to move (express it as a fraction)

21. Expected Outcomes After 36 Races

22. Experimental Results • Play the game 36 times, 72 times… (divide task between class) • Graph results • Compare to models of all possible outcomes The more we play the game, the closer these two become? (Level 4)

23. Making the Task even Richer…How could we make the Game Fair? • Only use 6 horses • Don’t include Horse 0 • Don’t include Horses that don’t move • very easily (0, 1, 12, 2, 11) • Use a 12-sided dice • Pull names out of a hat • Adjust the race-course so that Horse 7 • has to move 6 spaces to win, • Horses 6 and 8 have to move 5 spaces • etc.

24. What might we expect students to do in order to meet each of the standards? In groups, develop criteria to help make judgments in relation to the standards for the Horse Race investigation. Use the illustration “Dicey Differences”, NZC Second Tier support, and your copy of Mathematics Standards to help.

25. What might we expect students to do in order to meet each of the standards? After 2 years: Identify all of the horses that might win. After 3 years: Identify which horses are more likely and less likely to win.

26. By the End of Year 4 Identify that horse 7 has the best chance of winning and that it’s impossible for horse 1 to win. Identify that horses 2 and 11 could still possibly win, even though the other horses are more likely.

27. By the End of Year 5 List the possibilities and order the probabilities for horses to win correctly, noting that e.g., horse 7 is “most likely” to win, horses 6 and 8 have an “equal likelihood”, horses 2 and 11 are equally “most unlikely” to win and that it’s “impossible” for horse 1 to win.

28. By the End of Year 6 Develop a model (e.g. 2-way table) to show all possible outcomes. From the model, explain that e.g. “there is only one way for horse 2 to move”, etc.

29. By the End of Year 7 • Create a model of all possible outcomes and identify, e.g. that horse 7 can move as a result of 6 of the 36 possible outcomes. • Predict that this outcome should occur about once every 6 rolls of the dice, but recognise that the actual experimental results are unlikely to be identical to this. • Recognise that their results may well differ from their neighbour’s due to the variability and independence of samples.

30. By the End of Year 8 • Organises results systematically. • Creates a model (eg 2-way table) for all possible outcomes. • Expresses likelihoods as fractions, concluding that e.g. the chance of Horse 7 moving is 1/6 but accepts that their results may not exactly reflect this.

31. Cross the River Place your 12 counters along the river bank. You can place more than one counter on different numbers. Roll two 1-6 dice and add the numbers. Move a counter across if you have on on that space You are aiming to move your counters across the river before your opponent does. 9 10 1 8 2 11 3 4 7 5 6 12

32. Probability Aim of session: • Intoduce activities that explore and develop ideas of probability • Links with NZC and Progressions • Links with National Standards and NZ illustrations