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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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More or Less?

Are these estimates of physical quantities accurate?

Big and Small Numbers in Physics

Stage: 4 Challenge Level: Challenge Level:1

Why do this problem?

The ability to work with big and small numbers is a very important scientific skill. These questions provide a variety of contexts where handling such numbers is necessary, while offering practice at identifying key information and making decisions about estimates and assumptions.

Possible approach

These questions could be used individually as starters or fillers within lessons. Alternatively, a class could work on different questions in small groups and then present their answers with reasoning to the rest of the class to stimulate discussion. Some questions can be tackled with little specialist knowledge; others may require research and additional information, as well as some assumptions, in order to come up with an answer.
It is important to stress that there is no "correct" answer for some of these questions - the value of this task comes from having a go at estimating and calculating with big and small numbers, and discussing assumptions that need to be made and extra data that need to be sought in order to find a solution.

Key questions

What assumptions have you made?
What other information do you need?
Are there any questions which give an exact answer? Can you say anything about the accuracy of those answers which aren't exact?

Possible extension

This problem could be done in conjunction with Big and Small Numbers in Biology. Similar questions could be created and shared within the class.

Possible support

Start with those questions which have familiar content, and encourage whole class discussion of the ideas in them, before tackling any of the questions which require extra information. It is well worth suggesting that students read the article Student Guide to Getting Started with Rich Tasks before tackling a very open problem like this one.