The lesson today will involve independent events, a fairly sophisticated topic in probability. For students, the emphasis will be on understanding that there may be more outcomes than first appear.
“Today we are going to play a game and decide whether it is fair or unfair.”
Tell students that they will play a game where each player has four ways to win, so it seems like a fair game. Give groups of three students a copy of the Number Cube Game tally sheet (M-3-5-3_Number Cube Game.doc) and a pair of regular six-sided number cubes. The sheet asks students to think about whether the game is fair and to predict who will win. They will reflect on this after they play also. Do not discuss the possible outcomes and reasoning until after students have had time to think and play the game.
“Think about the game as you get ready. Do you think it will it be fair or unfair?”
To play, two regular six-sided number cubes are rolled. Player 1 gets a point if 1, 2, 3, or 4 is rolled. Player 2 gets a point if 5, 6, 7, or 8 is rolled. Player 3 gets a point if 9, 10, 11, or 12 is rolled. The game ends when one player fills his/her row. [IS.4 - Struggling Learners]
Clearly the game is unfair, because we know that there are many more outcomes for 5, 6, 7, and 8 than for the other numbers. Students will know, or at least begin to suspect, that the game is not fair based on their results. Students will quickly realize that a 1 is never rolled because it is not possible. Discuss whether the game is fair or unfair, but do not spend too much time, as the next activity will give students another experience with two six-sided number cubes. Discuss the following:
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“Was the game fair? Why?” [IS.5 - Struggling Learners]
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“Which player won?” (Most likely, all groups will report that Player 2 won.)
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“Did each player have four outcomes?” (No, Player 1 had only three.)
“Let’s play another game with two six-sided number cubes. Your experience in the last game may help.”
Give student pairs a copy of the Number Cube Cover Up worksheet (M-3-5-3_Number Cube Cover Up.doc). To play the game students will use 11 counters to place in any of the columns, and in any combination they choose. You will roll two six-sided number cubes and call out the number. If students have a counter in that number’s column, they remove a counter. The first person to remove all of their counters wins. Obviously, students should think about the likelihood of each number as they decide where to place the counters. They do not know the likelihood of each number yet, but they have some clues to go on, based on the first game.
Once somebody wins, discuss the game. Ask students how they decided where to place their counters. Now that they have played, how would they change their strategy?
“Now that you have played the game once, play again with your partner. Think about where you would place your counters, based on the game we just finished.”
As students play, walk around and observe where they place their counters and ask students how they made their choices: “Why did you place four counters on 6?” At this point, students should be able to justify their choices.
When students finish, discuss the results. Did you do better this time? Some students may feel that they had a better idea where to put their counters based on the frequency of the numbers that they are seeing, but still they may not have been able to remove their counters quickly. They are probably also realizing that randomness is involved. Students have had enough experience with rolling two six-sided number cubes to see that rolling a 2 is unlikely. They have noticed that 6, 7, and 8 are rolled more often, but that they still do not have quantitative evidence about the frequency of each number. Lead the discussion toward thinking about this.
“How did you decide where to put your counters? Which numbers do you think are most likely?”
Be sure to include the following questions in the discussion:
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Which numbers do you think are more likely and less likely to be rolled?
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Why do you think this is happening?
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How can you roll a 2, or a 12? How can you roll a 7?
Students will begin to realize (if not already) that the reason the numbers are rolled with different frequency is because there are more ways to roll some numbers. Talk about each number; ask students to state the possible ways to roll that number.
There will be some discussion about whether a roll such as 2 – 1 is different from 1 – 2. Ask students what they think. Does the order matter? Explain that 2 – 1 is 2 on the first number cube and 1 on the second number cube, while 1 – 2 is a 1 on the first number cube and a 2 on the second number cube, so they are different. Remind them of the Heads and Tails combinations in Lesson 1.
As students tell you the combinations for each number, write them on the board, or on a large sheet of 1-inch square chart paper, stacking the combinations for each number. This will give a sort of informal graph of the ideal frequency of each number. The list should look something like this (the combinations do not need to be in any particular order):
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6 + 1
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5 + 1
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1 + 6
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6 + 2
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4 + 1
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1 + 5
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5 + 2
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2 + 6
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6 + 3
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3 + 1
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1 + 4
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4 + 2
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2 + 5
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5 + 3
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3 + 6
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6 + 4
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2 + 1
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1 + 3
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3 + 2
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2 + 4
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4 + 3
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3 + 5
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5 + 4
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4 + 6
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6 + 5
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1 + 1
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1 + 2
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2 + 2
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2 + 3
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3 + 3
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3 + 4
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4 + 4
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4 + 5
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5 + 5
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5 + 6
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6 + 6
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2
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3
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4
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5
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6
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7
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8
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9
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10
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11
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12
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“Now that we have a better idea why some numbers are more likely to be rolled than others, let’s make a line plot of rolls and see what results we get.”
Give student pairs copies of the Number Cube Line Plot worksheet (M-3-5-3_Number Cube Line Plot.doc). Each pair will roll two six-sided number cubes 50 times, filling in a box for each roll and creating a line plot. If there is time, have students do up to 100 rolls, or until one column is filled. The more results there are, the more likely the graph will resemble the expected results.
When students have completed their graphs, discuss the results. Compare the line plot to the chart of all the combinations. Have students compare their line plots to each others’ (it would good to display the graphs later). Students’ line plots will not look exactly like the chart of combinations, but students will see the connection between the two. They will also not expect the two to look the same, and they will understand why. Students will begin to see informally, that the more data there are, the closer they will get to the expected results. [IS.6 - All Students] Ask the following questions in the discussion:
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“Does your line plot look like our chart of combinations? Why do you think so?”
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“Does your line plot look like those of your neighbors?”
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“How would the line plot change if we made more rolls?”
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“If we did the Number Cube Cover Up game with 50 counters, how do you think you would do?”
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“How likely is it that we would roll a 7?”
As you conclude the discussion, make sure students understand the context of rolling each number. The likelihood of rolling a 7 is far greater than rolling a 2, but the likelihood of rolling a 7 when rolling two six-sided number cubes is still fairly low.
Extension:
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Routine:To help students see how more data affects the overall results, keep an ongoing line plot of number cube rolls. Prepare a large sheet of 1-inch chart paper as a line plot with 2 through 12 numbered along the bottom. Have one student a day roll 10 times and fill in squares on the line plot. Over time, students will see the shape change, and eventually settle in to a consistent shape.
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In Lesson 1, the idea of using the Unlikely Likely scale as a daily way to quickly discuss likelihood was introduced. A statement is written on the board for students to respond how likely or unlikely they think the event is. As the year goes on, you can progress from, “It will rain,” to, “If I roll two six-sided number cubes, I will roll a 6, 7, or 8.”
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Students can also repeat the Number Cube Cover Up game (M-3-5-3_Number Cube Cover Up.doc), perhaps with more counters.
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Students can first explore how many different outcomes there are. This can be done informally and students can list them on paper. Students will conclude that heads can go with each number on the number cube, and so can tails:
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H 1, H 2, H 3, H 4, H 5, H 6 and T 1, T 2, T 3, T 4, T 5, T6
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Once they know that there are 12 outcomes, they can decide how to collect data. They could tally tosses and see if the 12 outcomes occur equally, or choose groups of outcomes and make a fair game. They could predict how many outcomes they would get before they do 100 tosses, or play a version of a cover-up game. All of this can be done informally on blank paper as students are experimenting.
In this lesson, students explored independent events in probability, although this was not formally discussed. Students gain more experience and evidence in building their understanding of probability. Students learned that the number of outcomes, or ways to have a certain result, is what matters in probability. Students also reinforced the idea that events are random and don’t come out as expected, but do come out fairly close to expected with more results.
