1. ENRICHING STUDENTS MATHEMATICAL INTUITIONS WITH PROBABILITY GAMES AND TREE DIAGRAMS NCTM PRESENTATION BY: ADOLFO CANON 2005-2006 Rice University Summer Mathematics Program

2. TWO DICE SUM GAME

3. TWO DICE SUM Materials needed: 2 Dice 2 Game Sheets 30 Counters

4. DIRECTIONS 1.PLACE THE 15 MARKERS NEXT TO THE NUMBERS ON THE GAME SHEET. YOU MAY PUT AS MANY OR FEW AS YOU WANT NEXT TO EACH OTHER.

5. DIRECTIONS 2.ROLL THE TWO DICE

6. DIRECTIONS 2.ROLL THE TWO DICE AND ADD THE RESULTS 3 + 4 = 7 2.ROLL THE TWO DICE

7. DIRECTIONS 3.REMOVE ONE MARKER FROM THE NUMBER EQUIVALENT TO THE SUM. 3 + 4 = 7

8. DIRECTIONS 4.TAKE TURNS WITH THE OTHER PLAYERS IN YOUR GROUP.

9. DIRECTIONS 5.THE WINNER IS THE FIRST PERSON TO REMOVE ALL OF THEIR MARKERS.

10. DIRECTIONS 6.REPEAT THE GAME 3 TO 5 TIMES MORE.

11. 1-2 2-1 1-11-4 2-3 3-2 4-1 1-3 2-2 3-1 1-6 2-5 3-4 4-3 5-2 6-1 1-5 2-4 3-3 4-2 5-1 4-6 5-5 6-4 2-6 3-5 4-4 5-3 6-2 5-6 6-5 3-6 4-5 5-4 6-3 6-6

12. SUMMARY: 1.DISCUSS WITH YOUR STUDENTS WHAT STRATEGIES THEY HAVE FOR WINNING THE GAME. 2.DISCUSS WHAT NUMBER FREQUENTLY COMES OUT 3.LIST ALL THE POSSIBLE OUTCOMES AND DISCUSS THE RELATIONSHIP BETWEEN THEIR OBSERVATIONS AND THE LIST

13. An instinctive feeling that something has happened or is about to happen Webster Standard Dictionary

15. COIN TAIL HEAD GIF Animation freeware from www.gifanimations.com

17. STUDENT 1 STUDENT 2 RESULT ODD EVEN ODD EVEN ODD

19. Materials Needed Paper Bag or Vase 3 Blue Cube 2 Red Cube

20. Rules: 1.Two students form a red team and a blue team 2.Blue and red cubes put in the bag 3.Team take turns without looking, removing cubes and recording the color, and replacing it. 4.Team record their score by tallying the color they picked. 5.Play game up to 25 rounds 6.The winner is the team that has drawn the most cubes matching the team color.

21. FIRST GAME 1 BLUE AND 1 RED BLUE RED

22. SECOND GAME 2 BLUE AND 2 RED BLUE RED

23. THIRD GAME 2 BLUE AND 1 RED BLUE RED

24. FOURTH GAME 3 BLUE AND 1 RED BLUE RED

27. In essence, this means that in any chance event, when the event happen repeatedly, the actual result will tend to be the calculated or planned result. Picture from http://www.mathematik.ch/mathematiker/jakob_bernoulli.phphttp://www.mathematik.ch/mathematiker/jakob_bernoulli.php IF AN EXPERIMENT IS REPEATED A LARGE NUMBER OF TIMES, THEN THE RELATIVE FREQUENCY WITH WHICH AN EVENT OCCURS EQUALS THE PROBABILITY OF THE EVENT JACOB BERNOULLI SWISS MATHEMATICIAN THE LAW OF LARGE NUMBER IN PROBABILITY THEORY

28. START BLUE 2 BLUE 3 BLUE 1 RED 1 START BLUE RED START BLUE 2 RED 1 BLUE 1 RED 2 START BLUE 2 BLUE 1 RED 1

30. Rules: 1.One team is the one-color team, and the other is the two-color team. 2.Blue and red cubes are placed in a bag 3.Team take turns reaching in without looking, removing two cubes; examining and recording whether the cubes are one color or two colors; and replacing the cubes. 4.Team record their score by tallying the color they picked. 5.Play up to 25 rounds 6.The winner is the team that has drawn the most score.

31. Materials Needed Paper Bag or Vase 3 Blue Cube 2 Red Cube

33. FIFTH GAME 2 BLUE AND 1 RED

34. FIFTH GAME 2 BLUE AND 1 RED 20 One-Color 30 Two-Colors

35. FIFTH GAME 2 BLUE AND 1 RED START BLUE 2 BLUE 1 RED 1 BLUE 1 BLUE 2 RED 1 One color Two colors One color

36. SIXTH GAME 2 BLUE AND 2 RED

37. SIXTH GAME 2 BLUE AND 2 RED START BLUE 2 BLUE 1 RED 1 One color Two colors BLUE 2 RED 2 RED1 BLUE 1 RED 2 RED 1 BLUE 1 RED 2 BLUE 2 BLUE 1 RED 1 BLUE 2 RED 2 One color Two colors

38. SEVENTH GAME 3 BLUE AND 1 RED

39. SEVENTH GAME 3 BLUE AND 1 RED START BLUE 2 BLUE 1 BLUE 3 One color Two colors BLUE 2 RED 1 BLUE 3 BLUE 1 RED 1 BLUE 3 BLUE 1 RED 1 BLUE 2 BLUE 1 BLUE 3 BLUE 2 RED 1 One color Two colors One color Two colors One color

40. SUMMARY OF THE STAGES IN DEVELOPING MATHEMATICAL INTUITION 1.CHECK FOR STUDENTS’ INTUITION IN DETERMINING FAIR AND NOT FAIR GAMES BY DISCUSSING RESULT WITH FLIPPING OF COINS AND “ODD IT OUT” GAME. SHOW THE TREE DIAGRAM TO DESCRIBE THE PROBABILITY OF BOTH CASES. 2.ACTIVITIES 1-4 INCREASE THE CONFIDENCE LEVEL OF STUDENTS IN DECIDING IF A GAME IS FAIR OR NOT USING ONLY THEIR INTUITION AS BASIS FOR DECISION. CONFIRM THEIR INTUITION WITH THE TREE DIAGRAM AND THE RESULT OF THE COLLECTIVE DATA OF THEIR EXPERIMENT. 3.ACTIVITIES 5-6 CHALLENGES THE “ONCE” RELIABLE INTUITION OF THE STUDENTS. THE EXPERIMENTAL RESULTS IS NOT ENOUGH TO CONFIRM OR REFUTE THEIR INTUITIONS. SHOWING THE TREE DIAGRAM IS A WAY TO COME UP WITH A CONCLUSIVE ANSWER. 4.ACTIVITIES 6-7 ENCOURAGES STUDENTS TO DEVELOP MATHEMATICAL INTUITION BY REFLECTING THEIR DECISION ON COLLECTIVE EXPERIMENTAL RESULTS AND VISUAL REPRESENTATION SUCH AS THE TREE DIAGRAM.