A few weeks ago my twitter sphere contained a discussion about how to make those simultaneous equation problems involving coins a bit more engaging for kids. I read a few tweets and connected to something Dan Meyer was doing. First up I saw his picture of coins, American coins, and I thought, well I can’t use this with Australian kids, I’ll have to do something myself. I never actually read the rest, (I must go back to that) ,so I don’t know how Dan’s lesson unfolded, possibly very different from mine. I did, however, want to include some aspects that I have seen in a lot of Dan’s work – making the question accessible to everyone and establishing a need to know.

This year, for the first time, we are beginning a formal process of peer observations, so I thought this would be a good lesson to have my colleague come along to give feedback on the lesson and how it could be improved.

So I started by projecting the picture below and told the (true) story, that my husband puts all his 5c and 10c coins in this tin as he doesn’t like carrying around change, and then when it’s full he takes it to the bank.

I then asked them what they were wondering and proceeded to record these on the whiteboard.

I should have taken a photo of this, as I can’t remember them all, but they included the following:

How much money is there?

Is it real money?  (Yes)

How much does it weigh?

How many coins are there?

How many 10c and how many 5c?

Was the tin full? (No)

How much money would there be in the whole tin?

My colleague suggested that I should have had them discuss their questions for a couple of minutes in groups before they shared them, so that everyone had a chance to contribute. A good suggestion and I’ll do that next time.

I then asked them to write down two estimates of how much money was there: one that they thought was too small, and one they thought was definitely too big. We shared some of these and then I asked them to write down their actual estimate.

Then I told them, that for now, I’d like to know how many 10c and 5c pieces there were, but that if we had time, we’d get to some of the other questions.

My next question was what would you like to know to help you work out the answer?

To the best of my memory I some of the questions above repeated:

‘How much money is there?’

‘How much does it weigh?’

‘How many coins are there?’

Along with ‘What is the ratio of the number of 10c to 5c pieces?’

We estimated again and came up with our range of estimates for how many 5c and 10c pieces there were.

I gave the answers to 2 of those questions: There were 136 coins totalling $9.70.

Most then launched into finding the solution, though 3 or 4 were hesitant to start. One student commented ‘I’m not smart enough for this’. At this point I probably should mention that this is an advanced class, however, many of these students are uncomfortable with being asked to do something without being given a method first – it’s something I have been working on all year with them, along with trying to instil a growth mindset and the value to learning of making mistakes. So, while obviously there is still room for improvement in the lesson or my approach, the fact that it was only 3 or 4 who resisted rather than more, was not too bad. My colleague commented on the wide range of mathematical conversations taking place; some more complex than others. One group went straight to simultaneous equations, the other 4 groups used trial and error. This was a one off ‘problem- solving’ lesson, we hadn’t been studying equations, so I was happy with that and it gave me a chance to reinforce that trial and error is a perfectly valid mathematical tool.

Having arrived at the answer the first question, one student said – but there could be more than one answer couldn’t there? This generated another short discussion. Having decided it was in fact the only possible answer, we moved on. This time I said I wanted to work out how much money I would have if the tin was full. To make it a little easier, I asked, what if it was all 10c pieces, or all 5c pieces – and a student chimed in ‘and which would be better?

Again I asked them what they would like to know to solve this problem:

Questions included:

What is the circumference of the tin?

What is the diameter of the 10c coin?

How high is the tin?

What is the diameter of the tin?

What proportion of the tin did the coins in the picture take up?

What is the volume of the tin?

What is the ratio of the weight of 5c coins to 10c coins?

My colleague wondered if a non-extension class would come up with these same questions. I guess she’ll have to decide if she wants to try it and see.  I’d like to think they would, particularly if they had been doing other work on measurement.

I gave the first four pictures below, along with the information that the tim was 13 cm high, with a diameter of 11 cm. As they made progress, and asked for the information, I added the ones showing how many coins would fit in one layer. I had thought about providing coins  for them to do this themselves, but I didn’t have enough and the counters I had thought of giving them to use instead weren’t close enough in size.

 

 

 

 

 

Armed with this information, they attacked the task with vigour, arriving at answers just as the class ended.

 

 

 

 

 

If I there had been more time, I would have liked to have students share their strategies and to have them consider the following:

Why there were differences arrived at by different students?

How accurate they think their solutions are?

Whether they think their solutions under or over- predict the real value and why they might be different.

Overall, I’m pretty happy with how the lesson went. Students were definitely more engaged than if I had just given them the information and and asked them to work out an answer using a prescribed method. As far as peer observation, it was great to be able to discuss the pros and cons with a colleague and to have another set of eyes and ears observing the student reactions, their questions and their discussions.