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2010-2011 Workshop Series for YISD Teachers of 6 th Grade Math

2010-2011 Workshop Series for YISD Teachers of 6 th Grade Math Conceptual Understanding & Mathematical Thinking Workshop 4: Sample Space and Probability (Feb 28, 2011) Kien Lim Dept. of Mathematical Sciences, UTEP. Item 0. What is the fundamental counting principle ?.

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2010-2011 Workshop Series for YISD Teachers of 6 th Grade Math

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  1. 2010-2011 Workshop Series for YISD Teachers of 6th Grade Math • Conceptual Understanding & Mathematical ThinkingWorkshop 4: Sample Space and Probability(Feb 28, 2011) • Kien Lim • Dept. of Mathematical Sciences, UTEP

  2. Item 0

  3. What is the fundamental counting principle? What key ideas should student learn in order to truly understand fundamental counting principle? • Multiplication as systematic exhaustion without duplication (counting all without double counting) • Representational tools to generate all cases systematically • A tree diagram • An exhaustive list of outcomes

  4. How can we create learning opportunity for students to experience the need for the fundamental counting principle? Talk To Your Partners

  5. How can we create learning opportunity for students to experience the need for the fundamental counting principle? Consider this problem:Bobbie Bear is planning a vacation. With 3 colored shirts and 2 colored pants, how many outfits can he make?

  6. Use the Manipulatives to Create as Many Outfits as You Possibly Can Consider this problem:Bobbie Bear is planning a vacation. With 3 colored shirts and 2 colored pants, how many outfits can he make?

  7. http://illuminations.nctm.org/ActivityDetail.aspx?ID=3 Consider this problem:Bobbie Bear is planning a vacation. With 3 colored shirts and 2 colored pants, how many outfits can he make?

  8. How can we create learning opportunity for students to experience the need for the fundamental counting principle? • Concrete manipulatives to pictorial representation to formal rule • Ample time to explore • Opportunity to experience inefficiency (students appreciate the efficiency of a conceptual tool only when they experience the “hardship” without it)

  9. Logging In Procedure • 1. Turn-on your clicker • 2. Wait until it says “Enter Student ID”(Enter your 5-digit ID) • 3. The screen should display “ANS”

  10. Item 1

  11. What is the difference among • An outcome • An event • A sample space is the result of an action. is an outcome or a group of outcomes. is the set of all possible outcomes.

  12. The sample space has 52 simple outcomes. {Black} is an event with 26 simple outcomes.

  13. Item 2

  14. The sample space has 52 simple outcomes. The event space {Black, Red} has 2 outcomes. {Black} is an event with 1 simple outcome.

  15. Item 3

  16. The sample space has 52 simple outcomes. The event space {Heart, Diamond} has 2 outcomes. {Black} is an event with 0 simple outcome.

  17. Create probability question involving a bear who has 3 shirts and 2 pants for students to understand the difference among • An outcome • An event • A sample space Talk To Your Partners

  18. Bobbie Bear has 4 colored shirts and 2 colored pants. He randomly chooses a shirt and a pair of pants. If the shirt and pants have the same color, he will put them back and randomly choose again. What is the probability that his outfit will have a green shirt? 1/2 1/3 1/4 1/6 1/8 Item 4

  19. Bobbie Bear has 4 colored shirts and 2 colored pants. He randomly chooses a shirt and a pair of pants. If the shirt and pants have the same color, he will put them back and randomly choose again. 6 Sample space Sample space consists of ___ possible simple outcomes. 2 Favorable event (green shirt) consists of ___ simple outcomes. Prob(green shirt) = Number of favorable outcomes 2 1 = = Number of possible outcomes 6 3

  20. Bobbie Bear has 4 colored shirts and 2 colored pants. He randomly chooses a shirt and a pair of pants. If the shirt and pants have the same color, he will keep the shirt and choose the other pair of pants. What is the probability that his outfit will have a green shirt? 1/2 1/3 1/4 1/6 1/8 Item 5

  21. Bobbie Bear has 4 colored shirts and 2 colored pants. He randomly chooses a shirt and a pair of pants. If the shirt and pants have the same color, he will keep the shirt and choose the other pair of pants. Is this correct? Prob(green shirt) = Number of favorable outcomes 2 1 = = Number of possible outcomes 6 3

  22. Bobbie Bear has 4 colored shirts and 2 colored pants. He randomly chooses a shirt and a pair of pants. If the shirt and pants have the same color, he will keep the shirt and choose the other pair of pants. Prob(green shirt) = 2 Is this correct? 6

  23. Bobbie Bear has 4 colored shirts and 2 colored pants. He randomly chooses a shirt and a pair of pants. If the shirt and pants have the same color, he will keep the shirt and choose the other pair of pants. 1 4 1 4 1 4 1 4 Prob(green shirt) = 1 So is 1/4 or is it 1/3? 4

  24. Bobbie Bear has 4 colored shirts and 2 colored pants. He randomly chooses a shirt and a pair of pants. If the shirt and pants have the same color, he will keep the shirt and choose the other pair of pants. 1 2 1 2 1 4 1 4 1 4 1 4 1 4 1 8 1 4 1 8 1 8 1 8 1 2 1 2 0 1 0 Prob(green shirt) = + 1 = + 0 = 0 Prob(green shirt) = or 1 1 1 1 1 1 1 1 1   4 8 8 4 2 3 2 4 4

  25. Bobbie Bear has 4 colored shirts and 2 colored pants. He randomly chooses a shirt and a pair of pants. If the shirt and pants have the same color, he will keep the pants and randomly choose another shirt. What is the probability that his outfit will have a red shirt? 1/2 1/3 1/4 1/6 1/8 Item 6

  26. Bobbie Bear has 4 colored shirts and 2 colored pants. He randomly chooses a shirt and a pair of pants. If the shirt and pants have the same color, he will keep the pants and randomly choose another shirt. Is this correct? Prob(red shirt) = Number of favorable outcomes 1 = 6 Number of possible outcomes

  27. Bobbie Bear has 4 colored shirts and 2 colored pants. He randomly chooses a shirt and a pair of pants. If the shirt and pants have the same color, he will keep the pants and randomly choose another shirt. 1 4 1 4 1 2 1 4 1 2 1 4 1 4 1 4 1 4 1 4 Is this correct? Prob(red shirt) = Number of favorable outcomes 1 = 6 Number of possible outcomes

  28. Bobbie Bear has 4 colored shirts and 2 colored pants. He randomly chooses a shirt and a pair of pants. If the shirt and pants have the same color, he will keep the pants and randomly choose another shirt. 1 8 1 8 1 4 1 4 1 4 1 4 1 4 1 8 1 4 1 4 1 2 1 2 1 4 1 8 1 8 1 8 But the sum of six 1/8s is only 6/8, not 1! Prob(red shirt) = 1 1 1 Is the answer 1/8 or 1/6? =  2 8 4

  29. Bobbie Bear has 4 colored shirts and 2 colored pants. He randomly chooses a shirt and a pair of pants. If the shirt and pants have the same color, he will keep the pants and randomly choose another shirt. 1 4 1 4 1 2 1 4 1 4 1 4 1 4 1 4 1 2 1 4 1 3 1 8 1 8 1 8 1 8 1 8 1 3 1 3 1 3 1 3 1 3 1 8 1 24 What have we learned? The usefulness of a tree diagram. Prob(red shirt) = + 1 3 + 1 1 1 1 1 1 1 1 = + =    = 24 2 4 2 8 3 4 6 24

  30. A fair die is rolled three times and the results are recorded in the order that they appear. For each roll, the die lands with the number 1, 2, 3, 4, 5, or 6 facing up. The following outcome is the LEAST LIKELY to occur: (A) 2 4 2 (B) 3 4 5 (C) 5 5 5 (D) 6 4 1 (E) All of the above sequences are equally likely Item 7

  31. Two fair dice are rolled and the resulting numbers facing upwards on the dice is summed. The following sum is the MOST LIKELY to occur: (A) Sum = 3 (B) Sum = 5 (C) Sum = 6 (D) Sum = 9 (E) All of the above sums are equally likely Item 8

  32. Source: Tami Dashley 2 = 36 1 18 P( Sum = 3 ) = P( Sum = 5 ) = P( Sum = 6 ) = P( Sum = 9 ) = 4 = 36 1 9 5 36 4 = 36 1 9 (A) Sum = 3 (B) Sum = 5 (C) Sum = 6 (D) Sum = 9 (E) All of the above sums are equally likely

  33. Item 9

  34. Why Clickers and Voting? • Requires students to participate actively • Provides immediate feedback • Facilitates class discussion/debate • Creates a fun atmosphere • (Cline, Zullo, & Parker, 2006)

  35. Cline, K. S. (2006). Classroom voting in mathematics. Mathematics Teacher, 100 (2). pp. 100-104.

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