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Whole Numbers Only

Can you work out how many of each kind of pencil this student bought?

Squaring the Circle

Bluey-green, white and transparent squares with a few odd bits of shapes around the perimeter. But, how many squares are there of each type in the complete circle? Study the picture and make an estimate.


What number could replace * so that */5 is between 3 and 4?

Does This Sound about Right?

Age 11 to 14
Challenge Level

Why do this problem ?

This problem gives practice with the use of estimating numbers and deciding whether an estimated answer is reasonable. 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. If students disagree with each other, or with the answers provided, this could lead to productive discussion.

The questions are available as a worksheet.

Key questions

Do you have all the information you need to check the calculation? If not, where can you find out what you need?
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

Start with questions which seem most accessible and encourage whole class discussion of the values given.  The article Getting Started with Solving Rich Tasks might be helpful.