Probability

Today we started our unit on probability.

Today’s problem (May 18th):

To activate our brains and start to thing about probability, we tried to come up with games of chance:

Then we read today’s problem:

Most students answered the question by pulling out slips of paper exactly 10 (or 14) times.  This was then their answer as to what was in their bag (i.e.: we pulled a green slip 4 times so there must be 4 green slips):

Some groups tried pulling out slips many more times than 10 (or 14), yet could not come up with a solution:

 

Consolidation:  The students were asked if they thought it made a difference how many times they reached into the bag and pulled out a slip (how many tries).  There was not a concencus, some thought it was better to have more tries, some disagreed.  The question was then asked, looking at the following results, if they could not answer they question exactly, could they guess?

They agreed that most of the 10 slips would be green, the least amount of the slips would be orange, and that red and blue would be about the same.  We will continue these thoughts over to our next problem.

Today’s problem (May 22nd):

To activate our brains, we reviewed the last classes experiment in probability.

Today’s problem:

Some students organized their work by writing down the colour they pulled out of the bag each time:

Some students used a tally system:

Some students made a table:

Consolidation:  We recorded all the results of attempt #1 (10 trials) and attempt #2 (30 trials), and compared them to the actual items in the bag (5 green, 2 red, 2 blue, 1 orange):

What the students noticed was that the attempts that used more trials were more accurate.  We were able to look at this as a “rule of experimental probability“, the more trials you have, the more accurate the results.

Today’s (yesterday’s actually) problem (May 23-24th):

To activate our brains wer reviewed the work done the last class, and we talked about why/how we came up with our rule about experimental probability. 

We also had a discussion about luck.  Many students remarked that they had lucky objects (coins, key chains, rabbits feet):

The discussion then focused on coins.  Students were asked, if you flip a coin do you call heads or tails.  Most students said they chose tails.  They were then asked, “If heads was flipped on a coin, what will the next flip be, heads or tails?”  There were two camps of thought here, some students thought it was more likely to land on tails, because heads was just flipped, other students thought it was equally likely to land on heads or tails.

And so today’s problem became:

Some students simply tallied all their coin flips to see which came up more often, heads or tails:

Some students only recorded the coin tosses AFTER they flipped a “heads”.  But some groups used only a few trials (less than 10 trials).  The results of these experiments tended to favour either heads or tails:

Some students only recorded the coin tosses after they flipped a “heads” and they had more than 10 trials.  These experiments tended to demonstrate an equal chance of heads or tails being flipped:

Consolidation:  Our discussion focused on the results that had more trials, because they would be more accurate.  The students remarked that coins have no “memory” and therefore cannot remember what they flipped the last time.  The chance of heads or tails being flipped is therefore equal (every time!)

Today we worked on a similar question with similar results:

Most students used tallies like this:

Today we tried adding up the total number of trials done in the class to see what kind of results we would get.  There were about 442 trials.  A die has 6 sides, to the chance of any number coming up is 1/6 (or 73 out of 442).  And the results show that most number had about a 1 in 6 chance of being rolled after you roll a 6 (the first role does not influence the next).  Results:  a 1 was rolled 75 times, a 2 was rolled 54 times, a 3 was rolled 63 times, a 4 was rolled 73 times, a 5 was rolled 95 times, and a 6 was rolled 82 times.

Today’s problem (May 25th):

To activate our brains we reviewed some of the language and rules of probability:

Then we talked about today’s problem.  It was a game of dice (grade 4’s using 2 dice, grade 5’s using 3 dice):

Grade 4 charts looked like this:

The grade 5 charts looked like this:

We will consolidate next class.

Consolidation (May 28th):

Today we looked at last classes game.  The students notices a pattern, that most of the results appeared (to a greater or lesser degree) in the shape of a triangle. 

In order to understand why that might be we talked about “theoretical probability”.  From our experiment (game) we realized that the numbers in the middle showed up more often than the numbers at the end.  So we looked at what all the possible ways there are to roll each number.  Here is what we came up with:

There was a total of 36 possible outcomes when rolling 2 dice (dice #1 having 6 sides and dice #2 having 6 sides)

Some students noticed 6 x 6 = 36

The grade 5’s had a bigger job, and it started to look like this:

In the end all the possibilities look like this:

There are 216 possible outcomes.  Some students predicted (correctly) that 216 = 6x6x6 (three 6 sided dice).

Today’s problem (May 29th):

To activate our brains we reviewed the concept of “theoretical probability”:

Today’s problem (2 problems actually):

Some students tried to solve the problem using an experiment (experimental probability):

Some students focused on one of our previous problems.  Each coin having a 1/2 chance to land on heads (2 coins = 2 heads and 2 tails) therefore the probability is 2/4.

Some students correctly focused on the theoretical probability, realizing that in order to answer the question they needed to list ALL the possible outcomes:

One group also managed to focus on the formula to calculate all possible outcomes (all possible of first event x all possible outcomes of second event, etc… ):

We also looked at how you can show the same answer using a tree diagram:

Consolidation:  We reviewed the idea that theorectical probability means you have to show ALL POSSIBLE OUTCOMES.  We also reviewed the idea that a formula can be used to find out all the possible outcomes (the denominator to the answer), but the numerator has to be figured out.

Today’s problem (May 30):

To activate our brains we took up the homework question (What is the theoretical probability of flipping heads on a coin 4 times in a row):

We explored the use of a list and a tree diagram to answer the question (the chance is 1 out of 18 possibilities).  We also noticed that we could predict how many possibilities there were by using our formula (2 x 2 x 2 x 2 = 18; because there are 4 coins and each coin has 2 possible outcomes).

Then the students were given today’s problem:

Some students answered the question using lists:

Some students answered the question using a tree diagram:

We will consolidate next class.

Consolidation:  We focused on how the students solved the problem, noticing that it is easier to find all possible outcomes using a tree diagram.

Today’s problem (May 31):

To activate our brains we looked at the probability of pulling out a red marble from this bag (2/4), and how the students knew that (there are 2 red marbles out of a total of 4 marbles):

Today’s problem:

Some students answered simply using words:

Some students used drawings to show their work:

Some students included an equation to help explain their thinking:

Some students also included all the other possible outcomes, as well as simplifying their answers to the lowest fraction (i.e.: 5/10 = 1/2)

Consolidation: We focused our discussion on the difference between finding all possible outcomes (used when there is more than one choice), versus the theoretical probability when there is only one choice.

Today’s problem (June 4):

To activate our brains, we reviewed the definitions of experimental and theoretical probability.

Today’s problem: 

Some students answered the question by simply drawing a picture:

 

Some students solved the problem by giving a large number to red, a small number to pink and an equal number to blue and green (almost as if it was an experiment):

Some students focused on giving the theoretical probability for each marble colour:

Consolidation:  This allowed us to focus on the meaning of the numerator and denominator when writing theoretical probability.  The bottom number always meaning ALL POSSIBLE OUTCOMES, and the top number meaning THE OUTCOME DESIRED.

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