What's in the box?

A lot of the activities I have talked about on this blog, or the ideas I talk about, are generally ones I have put significant time into.  The activity I am talking about in this post has been a very successful one for me over the last few days.  I would like to say that it was because of the careful and deliberate planning I did on the task prior to the lesson, but that would be a lie.  Sometimes as the lesson is unfolding you see an opportunity present itself, and by following it through, sometimes planning the next step on the run, you can have a really good lesson.  Planning the next step on the run was not a result of being disorganised, but a result of identifying an emerging need and recognising the need to follow up on it before moving any further.

• 10 blocks put into a box
• I took the box around to students, 5 took out a block and showed it to the class, this was recorded on board.  Students had to guess at what the last 5 blocks were
• Discussed guesses, first 5 blocks were all red and blue so all the guesses were also red and blue
• drew out another 2 blocks (green and another blue) and asked them to guess again, told them that two of the blocks are colours that they have already seen
• Did the reveal on the other blocks

We had begun our preliminary work in looking at probability and it became clear that students did not have a solid idea of the concept of uncertainty.  So to demonstrate the concept I placed 10 coloured blocks into a box and told students that they were going to predict what was in the box. I then proceeded to have five of them draw out a block each and I recorded the colours.  The blocks the drew out are shown opposite.  

They then had a conversation with their table groups about what they thought would be the colour of the other 5 blocks. Not surprisingly all the responses were some combination of red and blue blocks, however despite knowing that there were lots of other colours of blocks in the storage tub, they had not considered they could be part of the final five.  

.Following this two further blocks were revealed by the students and the green one came in.  When it now came to guessing the final three blocks, there was much more variation in the, there was much more uncertainty, that new colour had led them to believe that there could be more colours they had not yet seen,  So in this case i showed them the final colour they had not seen which was the yellow one, however they were told, that the two remaining blocks are colours they have already seen in the set of blocks already revealed. 

As I listened to their conversations about what the two remaining blocks may be I could tell the opinions now were much more divided, but also much more reasoned, they became to come to the unerstanding that there was no way to tell. they didn't know if having half blue meant that there were more blues in it or whether it meant that we had now picked them all out, there was nothing on which to base their opinion but guessed it would two of the same colour.

​Following this I did the final reveal of the two blocks remaining

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Next I wanted to introduce the idea of using probabilities to describe exactly what is in the box so I gave them the information listed below and then gave them the time in groups to have the discussions required to figure it out

I have filled the box now with blocks according to the criteria below, can you tell me how many blocks of each colour are in the box?

Criteria:          Pr (red) = 2/3         Pr ( blue) = 1/12          Pr (green) = 1/4          # blocks < 20

What was exciting about seeing them work on this was that I finally saw the classroom culture that I have been trying to build all year with them. In working together on the problem they were talking about the problem, they were critiquing each other's reasoning, they were asking questions of each other, they were willing to tell the group when the explanation still didn't make sense to them, forcing the person giving the explanation to justify their thinking more strongly.  I think one of the most important aspects of their work though was their confidence with their answer.  With these sorts of questions when they tell me they have the answer I try to ask a few questions to head their thinking down a line that creates some doubt that they have found the answer. This isn't done to trip them up, but is more designed to see if they have got to a point where they feel the have considered everything and have come to the only answer that works, to gauge their confidence in their own thinking.  Normally when I ask students a question about their answer they take this as an indication that their answer is wrong, but  this time, no matter what question I asked them about their answer they had confidence with it as they had determined that 12 was the only possible number of blocks as you could not have parts of blocks.

The next day the aim was to move towards students being able to determine how to calculate the probability of pulling a block of a certain colour out of the box, to help facilitate this i made up a simulation of the box compostion from the previous day using excel.  The excel file and a screen shot is shown below.

whats_in_the_box_simulator.xlsx
File Size:	11 kb
File Type:	xlsx

Download File

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Like the day before i got students drawing the blocks from the box and this time we tracked the probabilities.  when the first green block was drawing I asked them to predict what would happen to the probabilities of each colour.

They were able to notice the trend of whenever you pull out a block of a certain colour the probability of drawing that colour goes down and the probability of drawing the other two goes up, but there were a few who were puzzled by how the fractions were changing.  For example with the green block it started at 1/4, then went to 2/11 and then to 1/5.

With further time to discuss it they were able to talk about how we started with 12 blocks so 1/4 could also be shown as 3/12. It goes to 2/11 next because the total number of blocks is now 11 as we drew out 1, and that block was green so the number of greens went from 3 to 2.  The next one would be 10 blocks so 1/5 is really 2/10, so the block chosen was not green.  They were able to clearly articulate reasons for what they were seeing.

Overall what started as a sidetrack, developed into some really great thinking, on some really important concepts and I couldn't be prouder of that lot today.