Thursday, January 28, 2016

Measurement to the Fore

I am sometimes concerned that our students' experience of Measurement consists primarily of memorising formulas and conversions at the expense of the the actual skills of estimating and measuring objects. It is all very well if a student can tell me how many millimetres are in three kilometres, but if they cannot measure the length of a floor accurately, then they have not learned about Measurement. 
My experience has been that as students use metre sticks, measuring jugs, and sets of scales, they develop such good measurement sense that they can see that since one metre is 100cm, then they will also understand that 5m is 500cm or that 725cm is 7.25m. In other words, they will use their measurement sense to make conversions. Those who rely solely on the staircase method for conversions will not develop their measurement sense so easily. 
As such, I like giving students activities which actually gets them measuring. It allows me to see if the are using the tools available to accurately measure objects. In two classes this week, I began the following task:
Design a container to hold twelve golf balls.
We gave each pair of students a single golf ball and told them that they could choose to use any further tools they might need from around the classroom. Right from the get go, all students were engaged in the task. They quickly began thinking about how they should arrange the golf balls. Some initially decided to design a bucket (but quickly abandoned this idea!). Others looked at a 12 by 1 arrangement; or a 2 by 6 arrangement; or a 3 by 4 arrangement. A couple of groups even thought about a 2 by 2 by 3 box. As they began to measuring the golf ball we noticed a few misconceptions:

1) Measuring from the edge of the ruler and not the zero mark.
I reckon most teachers have seen some of their students make this mistake or a variant of it (e.g. measuring from the 'one' mark). One way to correct this is explicit teaching: demonstrate to the student that measuring this way leads to inaccuracies and have them move the object (or the ruler) so that they measure from the zero mark.

2) Measuring the circumference and not the diameter
Maybe some students 'see' this attribute (circumference) more readily than the diameter. Having measured this, one group multiplied by 12 to get 158 cm. We asked them how they would use this value to design their container. When they seemed unsure, we showed them a metre stick to help them realise that this measurement was greater than a metre. At this point, they began to realise that they were in a bunker so we prompted them with 'Is there a better measurement to use?'

3) Spheres are difficult to measure!
We could have given the students a similar task with objects that are much easier to measure (e.g. design a box that will hold 12 juice boxes) but that being said, I think it was great that they were challenged; it brought out some creative approaches. For students who really struggled, I put a ruler either side of the golf ball and they measured the distance between these two rulers.

4) Only thinking two-dimensionally
This was fairly common and we dealt with it using this prompt: 'OK, you are phoning me, the manufacturer of these containers. Tell me all the information I need to know to make your container accurately.' This got the students to think about the height of the container.

The next step for both of these classes is to take their plans and actually make the boxes out of card. However, before this happens, the students need to know the old adage, 'Measure twice, cut once'. Most of them measured the diameter as 4cm when really it is about 43mm. They need to factor this exact measurement into their design (and consider if they need to add a little extra) before they actually begin construction.
Once all these boxes are constructed (again, this will involve a lot of hands-on measurement), they can argue which of the boxes is best for the job. I already know one student who is convinced that the 1 by 12 box will be best as it will fit in nicely alongside the clubs in the golf bag: it will be interesting to hear the other students' thoughts on this!
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I once gave students the task of designing a pasta box that would hold 1000 cubic centimetres. The choice boiled down to a 10 by 10 by 10cm cube or a 10 by 5 by 20cm rectangular prism. Many students made the argument that the cube was the better box as it had less surface area and therefore would be cheaper to manufacture. One student though argued against the cube as it was difficult to hold and therefore pouring the past into a pot of boiling water would be an issue. Another student argued that the cube had a smaller 'front' than the rectangular prism (100 compared to 200 square centimetres) so it would not be as visible on the shelf of a grocery store which would impact sales. 
It goes to show that sometimes the mathematically best solution is not the best solution practically. 

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