More Fun with Grade Six Probability Concepts!

I had so much fun teaching Math today!  We are still working on Probability concepts, and on Friday, I had the students play a game I call "Odds and Evens".  Students work with a partner and they each have a number cube.  They roll their dice and add the sum.  One student chooses to be "odd", the other "even".  If the sum is even, then the "even" person scores a point, and if it is odd, the "odd" person scores a point.

Each pair of students conducted fifty trials.  Then we tallied all of the trials for the class and determined the Experimental Probability.  Next, I showed them how to create a tree diagram for the two dice, and we determined all of the possible outcomes.  Students could see that if you found the sum of the two dice, exactly half of the total number of outcomes were even, (18/36) and half were odd.  So we called this a "fair game" because each student had a 50% chance of winning. 

Today, they played the "odds and evens" game again, but we changed one of the rules.  Once again, they each had a number cube and they played with a partner.  This time, when they rolled the dice, they had to find the product of the two numbers rolled.  Again, one of them had to choose "even" and the other "odd".  I asked them first to predict whether or not they thought this would be a "fair game".  They all believed that, like with the sum game, this would be a fair game.  

It was so fun walking around the room and listening to their conversations as they played.  They weren't long into it when I started hearing:  "This game can't be fair!"  So I walked around the room asking them them questions:

• Do you still think this is a fair game?  Why or why not? How can you tell?
• Why do you think "even" is winning?
• What is your experimental probability? Is it what you expected?

It was so much fun to walk around the room and hear the learning as it was happening.  They were so excited when they discovered that both numbers rolled had to be odd in order to end up with an odd product!

I asked them to conduct 40 trials.  (I like to use different numbers of trials so that they get used to finding percentages with different denominators.)  When they had completed all of their trials, they calculated the Experimental Probability.

I then asked them to find the Theoretical Probability for the experiment.  It was very interesting.  They all made their tree diagrams correctly.  They all knew that there were 36 possible outcomes.  And yet, out of 10 groups, only 2 correctly found the Theoretical Probability.  They all determined that there were 27 even outcomes and only 9  odd outcomes.  But they used 40 as their denominator again.  

The last step in the question asked them to find the Theoretical Probability for 500 trials.  They understood they needed to use equivalent fractions, but as they had been using a denominator of 40, their answers were not quite right.

During the consolidation, I asked one of the groups with the wrong answer to step forward and explain their theoretical probability.  The two groups who did it properly, at first, began to doubt themselves.  Why did most of the class get a different answer than they did?  A lively conversation ensued.  But soon enough, they realized what the majority of the class did wrong - that they had used 40 (the number of trials in our experiment) vs 36 (the total number of possible outcomes) for their denominator. 

I'm hoping that because they constructed their own understanding and clarified their own misconceptions, it will be understood and remembered! 

Learning Goals in Math

I had a very interesting conversation with my friends Janice and Celiza the other night.  Janice and Celiza are currently Curriculum Consultants in two other school boards in Ontario.

We were discussing the use of Learning Goals in Math.  Janice was of the mind that we shouldn't be naming the Learning Goals up front at the beginning of the Math lesson if we are truly teaching through problem solving.  I understand her point.  If you are teaching Math through problem-solving - which I hope you are - then you are using a constructivist approach, and you want the children to construct their own learning.  You don't want to give away the punch line.  Janice was adamant that the Learning Goal should come out during the Consolidation (the third part of the three part Math lesson).

<For a brief review of the Three-Part Math Lesson click HERE.>

I was very fortunate to go and hear Dr. Marian Small speak last May (I reference her often, I am quite a fan of her work).  She addressed this very question.  She said that we have to tell the students something about what we want them to do.  She asked (and I quote) "Can we say: 'We are going to develop strategies to compare two fractions'?"

What I liked about Marian's talk was that she never told us what we should be doing.  She only ASKED us what we might want to consider.  The fact is, there is no right answer here.  It is trial and error.

I have tried it both ways, not telling my students anything, and also giving them a simple statement about what we are hoping to learn about.  Not what we are going to learn specifically, but what we are learning about.  I have found that giving them a Learning Goal up front helps give them more direction.  It sets a purpose up front for why I am giving them the task that I am giving them.  And, I admit, it speeds up the problem-solving and consolidation.  The fact is, I don't have unlimited amount of time to teach them everything I want to teach them.  I only have 9 months to teach them 10 months of curriculum (because I have EQAO testing in the beginning of June).

What I don't do is give them Success Criteria for the content expectations up front.  That is what I want them to come up with in the consolidation.  I DO however, give them Success Criteria for the process expectations up front. (The process expectations include communication, representing, selecting tools and computational strategies, problem-solving, reflecting, reasoning and proving). Confused yet?

For example, I might write:

• We can reflect on the reasonableness of a solution 
• If we determine our solution doesn't make sense we can look for a different solution
• We can share our solution with others in a way that makes sense to them
• We can explain our thinking so that someone else "gets it"
• We can connect our solution to someone else's solution
• We can restate the problem in our own words

Here was our Math Learning Goal from Thursday and Friday:

For our Minds On, I gave the students fractions to add to a number line.  This allowed me to differentiate by giving different students different fractions.  As the students placed their fractions on the line, I asked them how they knew where to put it.  "Tell me what you are thinking in your head as you are deciding where to put that fraction."

Here are some strategies we came up with for comparing fractions in our Consolidation:

We are still using the "ShowMe" app and "Explain Everything" app to show how we solve a Math problem.
Here is a video from one of the problems my students solved on Friday.  I was only going to give them one problem, but some of the students solved it so quickly and easily, we went to the text book to find a second problem.


I think it is important at this point to mention the obvious.  This lesson could not have happened if it weren't for the previous lessons.  We spent Monday, Tuesday and Wednesday reviewing what fractions ARE.  Before students could begin comparing fractions, they had to fully understand what a numerator and denominator are, and they had to understand what mixed and improper fractions are.  For example, many students knew that an improper fraction was a fraction with a numerator larger than the denominator, but very few students knew that this meant we were talking about a number greater than one.

I also had one student who could compare fractions by using a common denominator.  He had learned how to do this in a previous school in grade five.  But he did not know how to represent those two fractions, and he did not know how to compare those two fractions without using a common denominator.  For example:  students should know that 5/6 is less than 7/8 because sixths are larger pieces than eighths. Just because they can convert them into 20/24 and 21/24 doesn't mean that they understand that concept.

I guess my point here is this:  You have to be extremely deliberate in everything you do when you are teaching.  Before you post a Learning Goal, you have to think, "How should I word this?" "How will knowing this help my students?" "What do my students already know and think?" "What skills do they need?" We have to be reflective practitioners; I am very fortunate to have friends that are interested in having these conversations with me!