Bansho for Probability



Bansho for Probability (grade 4 & 5)

Overall expectations (grade 5)

• Represent as a fraction the probability that a specific outcome will occur in a simple probability experiment, using systemic lists and area models

Overall expectations (grade 4)

• Predict the results of a simple probability experiments, then conduct the experiment and compare the prediction to the results

Specific expectations (Grade 5)

• Determine and represent all possible outcomes in a simple probability experiment (tossing a coin, rolling a die), using area model (rectangle divided into 2 represents outcome of coin toss experiment, or spinners)

• Represent the probability of an event using a simple fraction (heads is a ½ probability)

• Pose and solve probability problems, conducting experiments (using appropriate method of recording results; tally chart, line plot, bar graph)

Specific expectations (Grade 4)

• Predict the frequency of an outcome in a simple probability experiment, explaining their reasoning, conduct the experiment, compare the result with prediction

• Determine, through investigation, how the number of repetitions of a probability experiment can affect the conclusion drawn

Thinking

• Predict, conduct experiment

• Consider affect of number of trials (explore misconception that previous trials affect future trials)

• Consider ALL possible outcomes (2 coins; HH, HT, TH, TT; therefore 1 H and 1 T has a probability of ½, but the probability of 2 heads is ¼)

• Extend the number of coins; looking for theoretical probability

1 coins 1H 1T

2 coins 1HH 2HT 1TT

3 coins 1HHH 3HHT 3HTT 1TTT

4 coins 1HHHH 4HHHT 6HHTT 4TTTH 1TTTT

5 coins 1HHHHH 5HHHHT 10HHHTT 10HHTTT 5HTTTT 1TTTTT

1. 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

Developing the “Big Idea”

Question: If you toss a pair of dice 20 times (adding the sum), how many times would you expect to get 12? 7? 1? Explain your thinking. Conduct an experiment. Compare your results with your predictions.

(Dispel idea that if a 6 is rolled, then it is unlikely it will occur again (dice don’t remember what they rolled)

Method 1 – Experimental Probability

Game 1: On a 12X12 grid, place 12 chips. Roll 12 times. If there is a chip in the column about the sum, they remove it.

Play more than once. What strategies do they come up with

Game 2: On a grid, fill in space each time students roll the dice. The ideas is to race to the top, creating a bar graph, or roll 36 times. What numbers are rolled most often?

Place grids about class.

Calculate mean, median, mode

Notice graphs are different, but pattern emerges.

Method 2 – Theoretical Probability

Theoretical probability is determined by calculating all possible outcomes. 7 shows up more often than 3 because there are more combinations that make 7.

Using standard chart for probability of dice, fill in possible combinations to arrive at sums. (Use different coloured dice)

6+1

5+1 5+2 6+2

4+1 4+2 4+3 5+3 6+3

3+1 3+2 3+3 3+4 4+4 5+4 6+4

2+1 2+2 2+3 2+4 2+5 3+5 4+5 5+5 6+5

1+1 1+2 1+3 1+4 1+5 1+6 2+6 3+6 4+6 5+6 6+6

1 2 3 4 5 6 7 8 9 10 11 12

Extend by: using 3 dice, changing the numbers on the dice, use 8 sided dice.

Problems:

Predicting and experimenting; fractions; all possibilities; specific possibilities

1. Bag problem – focus: number of repetitions affect conclusion

2. Dice game 1 – focus: experimental probability

3. Dice game 2 – focus: experimental probability

4. Chart introducing theoretical probability - fractions

5. Coin toss 1 – Race (experimental and theoretical - fractions)

6. Coin toss 2 – (introduce tree diagram - fractions)

7. Fun question 1 – sandwiches

8. Fun question 2 – clothes

9. Fun question 3 – marbles

Question 1 - Bag problem (grade 5: 14 papers, grade 4: 10 papers)

Activation:

Discuss: meaning of probability, real life situations, vocabulary

Problem:

In a paper bag there will be a number of slips of paper with colours written on them.

“There are 14 pieces of paper in your bag. By reaching into the bag (without looking) and pulling one piece of paper out at a time (and them returning that piece of paper), can you determine what is in the bag?”

Consolidation:

Put all the results on a table in front of the class.

Looking at the table, can you draw any conclusions?

The more trials you have, the more accurate your results are.

Grade 5

|Orange |Red |Red |Red |

|Blue |Blue |Blue |Green |

|Green |Green |Green |Green |

|Green |Green | | |

Grade 4

|Orange |Red |Red |Blue |

|Blue |Green |Green |Green |

|Green |Green | | |

Question 2 (grade 5: dice; grade 4: coin)

Activation:

Read page 57 (Data Management and Probability – Guide to Effective Instruction) about the “Lucky Loonie”. Can a coin bring luck? Lucky objects? (Make predictions)

Problem:

Prove or disprove:

If you roll a number on a die, it is unlikely you will roll that number again.

If you flip heads on a coin, it is unlikely you will flip heads again.

Consolidation:

Dispel idea that if a 6 is rolled, then it is unlikely it will occur again (dice don’t remember what they rolled).

Question 3 – Dice experiment:

Activation:

Review yesterday’s ideas. Make predictions for today’s game.

Problem:

If you toss a pair of dice 20 times (adding the sum), how many times would you expect to get 12? 7? 1? Explain your thinking. Conduct an experiment. Compare your results with your predictions.

Consolidation:

How accurate were you? Why?

Question 4 – Dice game 1 ‘Chips Away’:

Activation:

Problem:

Game 1: On a 12X12 grid, place 12 chips. Roll 12 times. If there is a chip in the column about the sum, they remove it.

Play more than once.

Consolidation:

What strategies do they come up with

[pic]

Chips Away

| |

| |

D1 = the roll on die # 1

D2 = the roll on die # 2

How many possible rolls are there? _____________________

What is the probability a 7 will be rolled? ____ out of _____

What is the probability a 4 will be rolled? ____ out of _____

What is the probability a 12 will be rolled? ____ out of _____

What is the probability a 1 will be rolled? ____ out of _____

[pic]

Question 8: Coin Toss

Activation:

What does the numerator and denominator in yesterday’s chart represent?

What is the probability a coin will land on heads or tails?

Problem:

(Use 2 different coloured coins – tokens)

If you use 2 coins, what is the probability that you will flip 2 heads?

1. Prediction

2. Figure out theoretical probability

3. Do you want to change your prediction?

4. How many “Heads-Heads” do you think you would get if you had 100 trials.

5. Try it out using 100 trials.

Consolidation:

Introduce tree diagram

HW: Try figuring out the theoretical probabilities using 3 coins.

Question 9:

Here are the options you can get at a sandwich shop.

You have to choose 1 from each column.

|Bread choices |Topping |Dressing |

|White bread |Turkey |Lettuce |

|Brown bread |Tuna |Pickles |

| |Cheese |Tomatoes |

Calculate the theoretical probabilities.

What is the probability that a random order will be a ‘Brown bread, Turkey and Tomato sandwich?

Question 10:

You have a bag with 5 red marbles and 5 blue marbles. You add 2 yellow marbles and 2 green marbles. Have you changed the probability of pulling out a red marble?

More questions:

You have 5 red marbles and 5 blue marbles in a bag. You add 5 more red marbles, 3 yellow marbles and 2 green marbles. Have you changed the probability of pulling out a red marble?

The probability of drawing out a red marble is likely. The probability of drawing out a pink marble is unlikely. The probability of drawing out a blue or green marble is equally likely. What marbles might be in the bag?

You have 3 pairs of socks lying loose in a drawer, one red pair, one blue pair and one green pair. What is the probability, if you reach in without looking, that you will pull out a matching pair?

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