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What fractions can you find between the square roots of 65 and 67?

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At the corner of the cube circular arcs are drawn and the area enclosed shaded. What fraction of the surface area of the cube is shaded? Try working out the answer without recourse to pencil and paper.

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Canny Fraction

Weekly Problem 4 - 2012
What fraction of the volume of this can is filled with lemonade?

Counting Fish

Age 14 to 16 Challenge Level:

Why do this problem :

Beyond the reasoning challenge to sort out how the fraction re-caught suggests an estimate for the size of the whole population, this problem introduces thoughtful students to the concept of 'confidence' in hypothesis testing ' consideration of how likely an estimate is to be wrong by some specified amount.

Possible approach :

Present the group with this problem on paper and ask them to read it and discuss in pairs what the situation is and what is asked for. This may lead some pairs to successfully solve the problem, in which case the main activity now becomes the task of explaining not just the calculation, but also the justification, to the other students in the group. If however this is a problem that isn't quickly solved a simulation with counters or coloured cubes is an excellent aid to visualisation.

Key questions :

  • Describe the procedure used.What is this procedure trying to do?
  • What do we know?
  • What do we need to find?
  • Is it the actual population or an estimate?
  • How close do you think it is?

Possible extension :

Conduct the same simulation as below for 'Possible Support' but draw attention to the variation that occurs as the simulation is repeated, and invite students to investigate how much their calculated estimates vary and in general use of the term how confident they might think it safe to be with their estimate. For example what 'plus or minus' amount might they attach to their answer. This situation is then gradually generalised to different size populations and different relative size of sample.

Possible support :

Simulation with counters or coloured cubes is the most useful aid to visualisation. For example put 20 counters into a bag and explain that the bag is the pond and the whole fish population in this instance is 20. Remove 5 counters and replace them with counters of a different colour, explaining that these five are the first sample, and the different colour allows the counters 'caught' for a second time to be identified. Now make the second sample of five.

This establishes the context or procedure being discussed so that attention can now rest on solving the problem.

A population of 100, with a sample size of 20, might give estimates closer to the actual population, and this may perhaps help students to see how to use the fraction re-caught.