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Archimedes and Numerical Roots

The problem is how did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?

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Weekly Problem 22 - 2012
What number could replace * so that */5 is between 3 and 4?

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Time to Evolve

How many generations would link an evolutionist to a very distant ancestor?

More or Less?

Stage: 4 Challenge Level: Challenge Level:1

Why do this problem ?

This problem gives practice with the use of estimating numbers and making choices about the information to use in a question. These are crucial mathematical skills in the sciences. These interesting questions will allow students to practise these skills whilst developing awareness of orders of magnitude in scientific contexts. As with any problems involving approximation, they offer opportunity for classroom discussion and justification.

Possible approach

There are several parts to this question, arranged in approximate order of difficulty. The individual pieces could be used as starters or filler activities for students who finish classwork early. Enthusiastic students might work through them in their own time. Since there is no absolutely 'correct' answer to many of these questions, they might productively be used for discussion: students create their own answers and then explain them to the rest of the class. Does the class agree? Disagree? Is there an obvious best 'collective' answer?

The questions are arranged for printing out into cards in this Word 2003 document .

Key questions

  • What assumptions will you need to make in this question?
  • What formulae will you need to use?
  • How accurate do you think the answer is?
  • What 'order of magnitude' checks could you make to test that your answer is sensible?

Possible extension

Can students make up similar questions? Can they put any upper or lower bounds on the actual numbers that would arise from a detailed calculation?

Possible support

Students might struggle with the 'open' nature of the questions. To begin, they might like to read the Student Guide to Getting Started.