Doesn't Add Up

In this problem we are faced with an apparently easy area problem, but it has gone horribly wrong! What happened?

Mediant

If you take two tests and get a marks out of a maximum b in the first and c marks out of d in the second, does the mediant (a+c)/(b+d)lie between the results for the two tests separately.

This problem in geometry has been solved in no less than EIGHT ways by a pair of students. How would you solve it? How many of their solutions can you follow? How are they the same or different? Which do you like best?

Perfect Eclipse

Stage: 4 Challenge Level:

Why do this problem?

This problem gives a good opportunity to practise some trigonometry in context. Students have the opportunity to make decisions about how to model the situation and what diagrams to draw in order to address the questions raised. Students will need to take into account considerations of lower and upper bounds, and the effects of changing measurements.

Possible approach

Perhaps start with some discussion about the apparent sizes of the sun and the moon, as viewed from Earth, and how the very similar apparent sizes makes solar eclipses possible.

Explain that the task is to work out the apparent sizes of the sun and the moon. In order to display this, students could cut out circles to represent the sun and the moon.

Ask students to consider in pairs what information they think they would need in order to work this out, and then provide any information they ask for from the data in the problem or using these information cards.

It may not be obvious to students at first that they can work out angles to the edge of the sun or moon, or how to use this to compare the apparent sizes of the sun and moon. The diagram in the Hint may help.

Key questions

What assumptions do you need to make to model the situation?

Can you draw a diagram to represent the appropriate lengths and angles?

What difference does it make that the orbits involved are elliptical?

Possible extension

Once students have had a chance to work out the necessary data for our moon, and drawn a diagram to show their findings, the problem provides data for other moons so that students can investigate how rare a perfect eclipse is within our solar system. This task could be divided up between groups with each group looking at the moons of a different planet, with everyone presenting their findings at the end of the lesson.

Possible support

The problem can be scaffolded by discussing modelling assumptions together as a class and working out what sort of diagrams will be helpful - the image in the Hint could be used as a starting point for discussion.

Tangent. physics. Probability. STEM - physical world. astronomy. Radius (radii) & diameters. Investigations. Plans & elevations. Maths Supporting SET. engineering.

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