We started looking at probability by asking:
We started looking at probability by asking:
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The students came up with answers that looked like this:
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Consolidation: Most groups pulled out pieces of paper from the bag (with colours written on them) only 10 times. But some groups continued as many times as they could. Their reasoning was:
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So we had a discussion emphasising the idea that in 'Experimental Probability' the more trials you have, the more accurate the results.
Problem: Prove or Disprove
Grade 4's - If you flip heads on a coin, it is more likely that you will flip tails the next time.
Grade 5's - If you roll a 6 on a die, 6 is less likely to be rolled the next time than a 4.
The students created some experiments (unfortunately no pictures)
Consolidation: The probability experiments the students came up with showed that both of the statement were false. A coin does not remember that it landed on heads, so the following flip has an equal chance of landing on heads or tails.
Problem:
We played a game. The students tried to guess, if they rolled 2 dice, what number would be rolled the most. The students would roll the dice and then mark down what was rolled (colour in the square). The first number to the top would win. Here are the results:
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Consolidation:
The students noticed that the numbers in the middle were more likely to win, and the numbers on the end were less likely to win. They also noticed there was a general shape of a pyramid or triangle happening.
Theoretical Probability:
So then we looked at all the possible rolls 2 dice could make and we created a chart. The students discovered that 'theoretically' there were 36 possible results that could happen when you roll 2 dice (die one could roll a 2 and die two could roll a 1 to make a total of 3). When they put it all on a chart they noticed that the triangle was very similar to the one the students found in their experiments. In fact the 7 was the most likely number to show up because th theoretical probability was 6 out of 36, higher than all the other numbers:
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We discussed how we display theoretical probability (as a fraction). The numerator being the possible outcomes of what we want (the number of ways to roll a 7) and the denominator being all the possible results (36 possible rolls). Therefore the probability of rolling a 7 is 6/36, and the probability of rolling a 2 is 1/36 (there is only one way to roll a two, both dice have to roll ones).
Today's problem (June 9):
Activation: We reviewed that theoretical probability means finding out ALL the possible outcomes first.
Then we asked:
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Some students explained what they thought, without showing all the possibilities. There was an AHA moment because some groups focused on the number of FACES on two coins, and how many of those faces are heads (2/4), as opposed to thinking of two faces (2 flips) being one possibility.
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Some groups showed all the possibilities they could think of using a table (flipping a coin three times, what are all the possibilities (8) - and then out of all those possibilities, how many are heads-heads-heads (1)) resulting in a theoretical probability of 1/8:
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Some groups used lists:
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Consolidation: We had a discussion about using tables and lists when figuring out ALL the possibilities of an event. Students noted that it is easy to make silly mistakes, like writing the same possibility twice, or not being sure if you actually have all the possibilities. We looked for another way to show our results. The idea of a tree diagram came up (like the one we are using for our speeches), and we drew them:
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The tree diagram is more efficient and will result in fewer mistakes.
Today's problem (June 14):
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Find:
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Some students used lists:
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Some students used tables, in this case in a very efficient pattern:
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Some students used tree diagrams:
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Some students were able to start a tree diagram and then predict from one outcome, what the rest would be. In this case they figured out how many sandwiches had cheese and white bread (4), and then extrapolated that ham and white will also have 4, etc... therefore all possibilities for white bread = 16. And then they realized that brown bread must also have 16 (total = 32):
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One group was able to extrapolate a formula, the number of choices for bread (2), times the number of choices for toppings (4), times the number of choices for garnishes (4) = 32 (2 x 4 x 4)
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Consolidation: We focused our discussion around the use of a formula and how to apply it to probability questions.
Today's problem (June 15):
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Before we began, the students realized there were two parts to this question, a probability to find at the beginning, and a probability to find at the end:
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Some students concluded that the probability changed from 5/10 to 10/20:
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Some groups realized that 5/20 and 10/20 both equal 1/2 and therefore the probability did not change:
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Some students used visuals effectively to display their thinking:
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Some students used excellent vocabulary in their explanations (in this case realizing we were talking about 'theoretical probability'):
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Consolidation: The discussion focused on equivalent fractions and what that means. The students realized that if the fractions are different, but equivalent, then the probability has not changed:
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