Lib Dems best at Maths?
Answer these questions, without using a calculator.
1. What is the angle between the hands of Big Ben at 9.15?
2. Nick and Gordon each receive presents shaped like cuboids (or "boxes"). Each is tied with three loops of string - one in each of the three possible directions. Nick's package has loops of lengths 40cm, 60cm, 60cm, while Gordon's package has loops of lengths 40cm, 60cm, 80cm. Decide whose package has the larger volume, and find the volumes of the two packages.
They were posed by the think-tank Reform as part of challenge to delegates at the party conferences back in September.
And the interesting news is that the Lib Dems came out best, with an average score of 83%, with the Conservatives on 71% and Labour on a measly 65%.
I wonder if they included Vince Cable, the darling of The Guardian today, in the sample?
Reform wanted to draw attention to the problems that occur when the teaching of mathematical skills in schools is reduced to numeracy alone. They quote Dr Tony Gardiner, founder of the UK maths challenge, who set the questions:
"What is needed is better teaching of elementary mathematics, rather than intensive, highly focused efforts directed towards a narrow interpretation of ‘numeracy'. English school mathematics now systematically trains its best pupils to assume that all problems are mindlessly trivial."
You can download the full set of questions and answers here, but I'll reproduce the answers, with Tony Gardiner's comments, to the two questions above.
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1. This is an excellent example of a simple, everyday problem, which requires one to coordinate two thoughts at the same time. One's first thought may be to picture the minute hand pointing at the "3" and the hour hand pointing at the "9" and so to answer "180°".
However, one should immediately realise that between 9 and 9.15 the hour hand has moved one quarter of the way from "9" to "10". Since each "hour" corresponds to one twelfth of a full turn (namely 30°), in one quarter of an hour, the hour hand moves 7½°, so the angle between the two hands may be given either as 172½° or as 187½°.
Working recently with a very select group of 200 Year 10 pupils - from the top 1-5% of the ability range - almost three quarters failed to see beyond the knee-jerk response "180°", which indicates that English school mathematics now systematically trains its best pupils to assume that all problems are mindlessly trivial.
2. This is a classic instance in which naïve "intuition" is totally misleading: Nick's package is in fact the larger! In mathematics, one always has to calculate - not guess.
Let Nick's package be xcm by ycm by zcm.
The first loop has length 2x + 2y = 40, second 2y+ 2z = 60, third 2z + 2x = 60.
Adding gives 4x + 4y + 4z = 160; so x= (x + y + z) - (y + z) = 40 - 30 = 10.
Similarly y = (x + y + z) - (z + x) = 40 - 30 = 10; z = (x + y+ z) - (x + y) = 20.
So Nick's cuboid has volume 10´10´20 = 2000 cm3.
If we do the same with Gordon's pcm by qcm by rcm, we get
2p + 2q = 40, 2q + 2r = 60, 2r + 2p = 80, so p + q + r = 45; so p = (p+ q + r) - (q + r) = 45 - 30 = 15; q= (p + q + r) - (r + p) = 45 - 40 = 5; r = (p + q + r) - (p +q) = 25.
So Gordon's cuboid has volume 15´5´25 = 1875 cm3
Comments: 4
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(For the benefit of people who don't know him, he's one of the clever Lib Dems)
Comment modified)
In those days all detergents were sold in cuboid boxes. A package designer had asked him to come up with a box design which minimised the surface area (and hence cost of cardboard) for a given volume of detergent powder. He suggested a sphere.
Well, the package designers decided that was too difficult to make, pack and stack, so he was asked to design a box with flat faces. Naturally he came up with a cube.
"Ah! But we want the largest area on the front of the box so we can brand and promote the product on the shelves". Hence the standard shape box, even though it costs more to make....until they switched over to liquids, of course, and that no doubt raised other design issues.
In fact, there is a flaw in your argument. The circle formed by a loop of string does not have the same area as the rectangle shape that it was wrapped around. In fact, as you jiggle the loop into different shapes so you alter the area that it encloses. The maximum area is given when the loop is made into a circle. The area approaches zero as the string is pulled into a long thin rectangle.
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