Chocolate Frogs and Car Tyres

Sometimes ideas for lessons get away from me and take me to some unexpected places, and sometimes the reaction of classes to those lessons are also unexpected. I was thinking about a lesson idea for circumference and I thought about a lesson I saw once from  Andrew Stadel (lesson link here). Where he wanted students to determine how many rolls of tyre it would be for a tyre to hit a target, I liked to lesson but was not sure how my students would "buy in" to the task.  I wanted to redesign the task a little to give students a greater reason for getting into the maths.

Like with many of my ideas I ran it past a teacher at the school that I often collaborate with on tasks such as this and we came up with what I think is an interesting alternative to that original task.

The Task
Students had to determine where on the ground they were going to put a chocolate frog so that it met two criteria:

1. It had to be more than ten rolls of the tyre away from the starting position, otherwise it would be squashed by the tyre.
2. It had to be less than 1 metre from the final position of the tyre.

The twist with this task was that they were only allowed to use 1 item from their bag to measure both the tyre and the distance on the ground.  

In relation to this task I think there are three incredibly important aspects to it.  Firstly is that they really have to think about the positives and negatives of anything that might be in there bag in relation to it's use as a measuring device, there was nothing in anyone's bag that could accurately measure the circumference of the circle in one measurement so they had to think about aspects such length, flexibility and having actual measurements on the device.  Secondly was the idea that the criteria and the restrictions I provided made the task of achieving it quite difficult,  Using something that was not ideal as a measuring device meant that there was going to be inherent errors in the way they could possibly measure the tyre.  They had to be very accurate in what they used as the measuring errors could compound quickly as they measured both the tyre and the ground.Finally there was a high degree of ownership over the task as they seemed to really want to be able to not have a squashed frog.  That little element of competition was very valuable in this task, That element of competition I have found to be very useful in a range of tasks I have done previously.

The Approaches and Challenges

The range of measuring devices people used to measure the tyre was quite interesting.  Most chose their ruler from their bag to measure it.  However since the ruler was only 30 cm and the tyre was significantly bigger than that both in circumference and diameter they needed to make lots of measurements and therefore compounding any measurement errors.  Most were also using a rigid wooden ruler to measure a curved surface which made the measuring more difficult again, some had a more flexible ruler that they could wrap around, but some of the compounding errors remained.  Many of these students decided to use a ruler as they could then stick within their comfort area of standard units of measurement.

Others chose to use headphone cables as measuring devices, they found them to be much more flexible and longer than the rulers, therefore they could accurately follow the contours of the tyre and it was long enough that the measurement errors were not as significant.  However they seemed to encounter difficulty when dealing with lengths that were not quite a full headphone cable length.  They were not sure how to figure out what fraction or proportion of the headphone cable was left over.  When Introduced to the idea of folding the headphone cable to figure out how many parts were used they found this area.  For example in the image below if they measured the tyre and got a total diameter of the blue part of the line below, they could fold it as shown to give an approximate length of just over 3/4.

The other part of this that students seemed to find challenging is using headphone cables as a unit of measurement in a mathematical formula.  The seemed to have this impression that the formula for circumference of a circle would only work if you were working in standard units of measurement.  They didn't realise that  If you said that the diameter was 3/4 headphone cables then the diameter would be 3.14 x 3/4 = 2.36 headphone cables.  This however was not unexpected, in mathematics we get so used to dealing with measuring devices with standard units of measurement that we do not often realise that sometimes the non-standard units of measurement can be just as valid and just as accurate.

Some of the students who engaged in the task a little more strategically came up with multiple ways to measure the tyre using the same device.  The measured the circumference directly, but then they also measured the diameter and used to formula to determine the circumference, where there was significant differences between the two values they took measurements again to verify their results and if they were accurate then they had some discussion about which of their measurements they felt was more accurate.  They realised that a discrepancy of 3 cm was not much but when you multiply it over 10 rolls you have just potentially lost a third of your target area.

The final challenge most of them faced was where to place the frog because that question is not as simple as it seems.  If you put it too close to where you think the tyre will finish and your measurements are off then your frog will get squashed. Conversely if you put it close to 1 m away from where you think the tyre will finish and your measurements are off then you could be outside the target range.  Most people put it closer to the tyre and a few put it closer to the end of the 1 m, very few considered putting it where they thought of the middle of the range may be which in my humble opinion is where I would be aiming.

The Results

Overall when we rolled the tyre the level of excitement was high, they were all keen to see how their calculations faired.  Of the 8 groups that placed frogs down 1 group was more than 10 rolls away but outside the 1 m target range, four groups had their frog squashed (but by less than 15 cm) and 3 groups met both criteria.  They seemed to really enjoy the activity and everyone got to eat the frogs at the end.