I had a brain wave mid lesson today.
I introduced my year 8 class to tree diagrams today. I have developed an approach that I find works really well for this, with one minor issue (which I will come back to).
I start with just basic tree diagrams with two options at each stage and two stages. I keep the structure the same each time to start with to help them build confidence around where things go on the diagram. For example, we start with this question
"Fill in the probability tree below to display the outcomes of flipping a fair coin twice"
I introduced my year 8 class to tree diagrams today. I have developed an approach that I find works really well for this, with one minor issue (which I will come back to).
I start with just basic tree diagrams with two options at each stage and two stages. I keep the structure the same each time to start with to help them build confidence around where things go on the diagram. For example, we start with this question
"Fill in the probability tree below to display the outcomes of flipping a fair coin twice"
We compare this to the sample space method, but I discuss the limitations of sample spaces (e.g. how they don't work for more than 2 trials, and how large they can get for lots of outcomes)
Then they have a go at this one.
"Draw the probability tree to display the outcomes of spinning a fair five sided spinner twice, for getting odd or even."
Some of them start drawing 5 branches and need reining in, but they all get there pretty quickly.
We do a couple of counters from bags examples WITH REPLACEMENT, and then we look at this one:
"The probability that it will rain on Monday is 0.2. The probability that it will rain on Tuesday is 0.3. Draw a tree diagram to show this."
I have chosen the examples carefully to showcase different types of questions that lead to the same underlying structure. But this one always throws them. Most of the class put Monday and Tuesday at the ends of the branches. And some students have always struggled with what to choose to put at the ends of the branches in cases like these. Many students just 'see' it, but I had always struggled to break this down for those that didn't. Until today!
Back to my brain wave today.
It is about the language of the question. A tree diagram is broken down into the vertical strips representing the trials, and the outcomes at the end of the branches. But this language just trips some kids up. So today I tried this.
WHEN it is Monday (trial - title at top) WHAT could happen (outcomes - end of branch)
Before, the 'Monday/Tuesday' question always stumped them. But with 'WHEN it is Monday, WHAT could happen (rain or not rain)? WHEN it is Tuesday, WHAT could happen (rain or not rain)?' it clicked.
Then I realised this always works
WHEN I first flip the coin, WHAT could happen (heads or tails)?
WHEN I draw the first counter, WHAT could happen (red or blue)?
WHEN they play the tennis match, WHAT could happen (win or lose)?
WHEN I eat the first chocolate, WHAT could happen (milk, dark or white chocolate)?
WHEN the bus comes, WHAT could happen (late or not)? WHEN I get to work, WHAT could happen (late or not)?
This structure of thinking about it really helped students to see what needed to go where in the diagram.
As an aside, the rest of the sequence of lessons then introduces WITHOUT REPLACEMENT, but still in the same structure of 2 outcomes and 2 trials. I follow this by looking at different structures (3 outcomes, 3 trials, terminating trials). And finally bring it together to look at finding probabilities of events that combine different outcomes (e.g. exactly two counters the same colour).
Then they have a go at this one.
"Draw the probability tree to display the outcomes of spinning a fair five sided spinner twice, for getting odd or even."
Some of them start drawing 5 branches and need reining in, but they all get there pretty quickly.
We do a couple of counters from bags examples WITH REPLACEMENT, and then we look at this one:
"The probability that it will rain on Monday is 0.2. The probability that it will rain on Tuesday is 0.3. Draw a tree diagram to show this."
I have chosen the examples carefully to showcase different types of questions that lead to the same underlying structure. But this one always throws them. Most of the class put Monday and Tuesday at the ends of the branches. And some students have always struggled with what to choose to put at the ends of the branches in cases like these. Many students just 'see' it, but I had always struggled to break this down for those that didn't. Until today!
Back to my brain wave today.
It is about the language of the question. A tree diagram is broken down into the vertical strips representing the trials, and the outcomes at the end of the branches. But this language just trips some kids up. So today I tried this.
WHEN it is Monday (trial - title at top) WHAT could happen (outcomes - end of branch)
Before, the 'Monday/Tuesday' question always stumped them. But with 'WHEN it is Monday, WHAT could happen (rain or not rain)? WHEN it is Tuesday, WHAT could happen (rain or not rain)?' it clicked.
Then I realised this always works
WHEN I first flip the coin, WHAT could happen (heads or tails)?
WHEN I draw the first counter, WHAT could happen (red or blue)?
WHEN they play the tennis match, WHAT could happen (win or lose)?
WHEN I eat the first chocolate, WHAT could happen (milk, dark or white chocolate)?
WHEN the bus comes, WHAT could happen (late or not)? WHEN I get to work, WHAT could happen (late or not)?
This structure of thinking about it really helped students to see what needed to go where in the diagram.
As an aside, the rest of the sequence of lessons then introduces WITHOUT REPLACEMENT, but still in the same structure of 2 outcomes and 2 trials. I follow this by looking at different structures (3 outcomes, 3 trials, terminating trials). And finally bring it together to look at finding probabilities of events that combine different outcomes (e.g. exactly two counters the same colour).
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