Wednesday, 17 June 2015

Probability Enquiries

Lead in:

We began by sharing strategies we have been using with our TV game show probability games. Those that we have personally created that we find successful and those that don't seem so effective in clearly understanding what we are doing.

From this discussion, I introduced our line of inquiry we would be exploring:

FUNCTION: How we can work out probability visually

We watched this Khan Academy youtube for some ideas: 

 Khan Academy Tree Diagram for Flipping Coins

I didn't want to show them these strategies earlier on in our unit so they could have the opportunity to create their own problem solving methods.  I felt that now was a good time to show them some different strategies to add to their bank.

After, we discussed the visual strategies shown in the video.

We felt that the first 'Probability Distribution Table' wasn't very helpful since the fractions were all the same. We wondered what was the point in doing that. Questioning & evaluating minds. I love it. :)

We thought it WAS important to list and organise all the possible outcomes, but we didn't think this was an effective way so we decided we wouldn't use it for our enquiries.

I then presented 4 possible ideas they may want to investigate and practise using the new visual strategies we just learnt about.

Or they could think of their own idea.  

One boy decided he wanted to explore the probability of a 20-sided die.
Sure, why not?

Another group wanted to find out the probability of winning the jackpot in the Euromillions lottery. Nice idea- they see this advertised in their daily lives. Why not find out what the maths is behind it? 

The rest wanted to investigate some of the suggestions given.  We had looked at "lucky" 7 earlier on in the year and we had already enquired into its higher probability so no one latched on to that. I threw it in as a suggestion in case some hadn't really got it the first time.

It was fascinating listening into discussions and seeing different ways to approach the enquiry. So much rich, mathematical language being used and loads of interesting and deep questions were being raised within their groups. A room full of chattering and passionate mathematicians.  Many busily recording ideas and experimenting with those new visual startegies introduced. Others preferring to stick to their original startegies and others wanting to test out their theories by doing the chance event and seeing if the probabilities they had worked out would actually occur that way.

How some of the enquiries began to look like:

The 20-sided die investigation so far:

This group began by investigating the mathematics involved with a pack of cards.

- Wow!  A pack of cards are symmetrical!

- A pack of cards is balanced!

- If we add the jokers, it changes the probability because the pack isn't symmetrical anymore!

- That must be why we usually take the jokers out when we play games!

- Cards are real life maths. It's not just random when we win games!

They then decided to start working out the probability of pulling out certain cards from a pack and grouped them into columns of LOW, EVEN or HIGH probability of occurring.  An interesting approach and idea.

This group tried out the new ways we had learnt for visually working out probability.

They had also started testing out their probability theories they were creating which raised more questions when the results didn't always much their predictions.

"This is the best maths activity I have ever done!" explained the human calculator in our class. 


They had started measuring the probability of flipping 3 coins and the odds of getting at least 1 head and 1 tail.

They ended up calculating that when we flip 25 coins (they had started at flipping 1 coin and continued)and need to get at least 1 head and at least 1 tail, the chances of losing (ie getting 25 heads OR getting 25 tails) is:


Wow!  Listening to how they worked this out even started my own mind spinning trying to catch up with what they had clearly and confidently managed to measure.   I have no qualms telling kids in my class that they have a stronger mathematical mind than I do because its true. This was another occasion where it needed to be said. 

We then began creating posters to show our discoveries into probability.  We will finish and share these tomorrow.  I came up with an idea that we will have a silent sharing.  The kids will simply hold their poster up and we should be able to fully understand the mathematical thinking they did just by reading it.

That's the challenge.  To visually communicate effectively so we don't have to ask any questions.  

Such enthusiastic, enquiring minds.  Kids know how to ask the best mathematical questions and it reinforced in me how important it is to give children the option to explore their own ideas.  I wouldn't have thought about including the 20 sided die and I certainly didn't expect a group would start working out the probability of flipping up to 25 coins, but these are the great things that happen when we give kids the freedom to enquire into maths.  They aren't being fed strategies. Instead they are creating their own and evaluating the success or lack thereof in using them.  That's what great mathematical minds should be always encouraged to do.

Can't wait to see what they share tomorrow! :)


Day 2:

We kicked off with a boy explaining how he was chatting with his dad about how to define luck and he shared with us his dad's definition:


I think a barometer of whether you are doing something great in maths is when you find out your students are talking about it outside the classroom.

This was also a really interesting definition so we placed it on the wall with our others.

We continued with our investigations into probability and continued making posters that could visually communicate our findings effectively.

To make our sharing a bit more interesting, we did a silent gallery walk.

The posters were placed apart in the hallway.  Groups had a few minutes to read a poster and if they had questions about the mathematics involved, they attached them with post its.  The idea is that tomorrow when we share our findings, we can use those mathematical questions as a base knowing what the class really wants to know.  After a few minutes, we rotated around to the other posters.


We discussed the different ways we visually showed the probability and thought about which were more effective than others and why we thought so.

We then looked back at our central idea:

How have our enquiries helped us to understand our central idea more?

- We are able to measure chance events so that is why it is mathematical thinking.

- When we look at things that have more than one chance of happening, we can measure if it has a high or low probability.

- I used to think when we play games with dice that it is just luck, but now I know it is really maths.

- When we play cards, we can look at it from two different perspectives: either we are lucky to get the cards we want to win OR we can look at the cards as having a higher / lower probability.

- It's not just games we can measure, mathematicians probably look at everything that happens in life and see if they can measure the likelihood of it happening or not.  Will it rain this weekend?  We might think it is unlucky if it rains this weekend and we have plans to play outside.  But a mathematician might measure if it is a high /low chance of raining based on evidence.

I loved their creativity throughout their enquiry and could see real passion for what they were investigating.  They often excitedly shared with each other discoveries they were making and the rich mathematical language being practised and used was remarkable.  I wondered how different their passion would have been if I hadn't given the option to explore different investigations. I think giving kids the freedom to explore their own questions is crucial for successful and meaningful enquiry to take place. 

Tomorrow, we will use the maths questions generated from the other groups to help create a more effective sharing of our new discoveries. When they were leaving for lunch, I overheard some students still discussing our central idea. One child was talking about whether the probability of being born a boy or girl could be measured mathematically or not.  The bell had gone, but the curiosity and passion for enquirinf certainly hadn't stopped......

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