Q: I teach secondary school maths in Scotland. In lessons about measurements with Secondary 1 classes, I find pupils are unwilling to estimate. In "estimate first then measure" exercises they tend to try and just do the measuring, and either miss the estimating part of the question or write in the answer after they have measured. They don't appreciate that the estimating part is the main purpose of the exercise. How can I make sure they understand this?
A: I have an activity that I do with pieces of string. I have three pre-measured pieces, with labels A, B, and C attached to them. One piece is less than 10 centimetres, one between 10cm and 1 metre and one longer than 1m.
I also hand out envelopes that each contain a piece of string and three sticky labels, with "do not open until asked to do so" written on the outside of the packet. No rulers are handed out.
Pupils are asked to work in pairs and estimate the length of each piece of string and to write their answers on a piece of paper, along with their names. I then get them to swap their answers with another pair of pupils. I tell them the length of each piece of string in turn, demonstrating the measuring. The closest correct answer wins a reward of some kind.
Now hand out some scissors and rulers, including metre rules and tape measures, between each pair of pupils. Get them to measure part of their body for 1cm, 2cm, 10cm, 50cm, and 1m lengths, and any other measurement they think would be useful for using as a guideline for estimating.
Next, tell them to put away their measuring implements so they're not visible, and ask them to open their envelopes. Write on the board D = 125cm, E = 40cm and F = 8cm. They have to estimate these lengths with their string, using their body estimates, and cut the string, labelling each piece as D, E or F. Then they measure the pieces of string to check their answers. Again, a reward can be given for accurate estimates.
I also do this lesson with adults and it's hilarious. They have great fun and it really breaks the ice. It's particularly useful for helping them compare imperial and metric measures.
Introducing the history of measurements to complement this activity is really helpful, especially as measurement of length is originally related to body parts.
* For an international history visit www-groups.dcs.st-and.ac.ukhistoryHistTopicsMeasurement.html
A history of measurement in UK is at www.npl.co.ukabouthistory_length
For a history of metric measures visit www.nrdc.org.ukcontent.asp?CategoryID=520
Q: I'm having a "thick moment" - I want to round - 1.5 to one significant figure. Would the answer be - 2 or - 1? I don't know why I don't know this, but I've looked through all my textbooks and can't find any rules, or even examples.
A: This question appeared on The TES maths forum under the topic title "Rounding negative numbers", and it certainly created an interesting debate. My initial instinct was to round the - 1.5 to - 2. I tried this on my calculator and got - 2 and this is how Microsoft Excel treats the rounding of - 1.5. Then I read the messages on the maths forum.
When we round positive integers we talk of rounding up or rounding down, so 1.5 rounds up to 2 and 1.4 rounds down to 1. For negative integers - 1.5 would surely have to round down to - 2 and up to - 1.
I consulted Professor Alan Davies, who is head of the physics, astronomy and maths department at the University of Hertfordshire. He says: "The consensus is that it is - 1. However, rounding a 5 is always a little problematical. If 50 has been rounded already - maybe up from 48 or down from 53 - then we know what to do. If we don't know then consistently rounding up will lead to bias in the result. I use 'if the digit before 5 is odd then round down, otherwise round up'. So 1.5 becomes 1 and 2.5 becomes 3."
The most important part of working with approximations is to state the approximations that you are using. For exam purposes, check with your exam board to see what convention they are using. Are you any less confused?
* To read the "Rounding negative numbers" maths forum topic, visit www.tes.co.uksectionstaffroomthread.aspx?story_id=2091772path=Mathemati cs
Wendy Fortescue-Hubbard is a teacher and game inventor. She has been awarded a three-year fellowship by the National Endowment for Science, Technology and the Arts (NESTA) to spread maths to the masses.
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