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### Number and algebra

### Geometry and measure

### Probability and statistics

### Working mathematically

### For younger learners

### Advanced mathematics

# Three by One

### Why do this
problem ?

In this one problem you meet many important aspects of mathematics.
It illustrates how much mathematics is inter-related. It shows the
value of not being content to find one solution, but of asking
yourself "could I solve this another way?" or "have I found the
best method?" It is very satisfying to feel you have somehow got to
the very essence of a mathematical idea by looking at it in the
right way.

Also this problem provides a good example of how you can generalise from a result that is really a simple case of a much more general result. Are you content just to solve a problem or do you ask yourself "what if ...." and try to find more general results?

### Possible approach

You could challenge your class to find as many different methods of
solving this problem asposible. You could tell them that one pair
of school students found 8 different methods. At some stage of this
work you might mention that these two students used respectively
sines, cosines, tangents, vectors, matrices, coordinate geometry,
complex numbers and pure geometry.

Perhaps your students could work in pairs. They could come to the board and present their solutions to the rest of the class and/or make posters for the classroom wall. You could see collectively how many different methods your class can find.

### Key questions

What lengths in the diagram can we find?

What do we know about sines, cosines, tangents, vectors, matrices, coordinate geometry, complex numbers and pure geometry that we can use to prove this result?

How might we generalise this result?

### Possible extension

The linked article
Why Stop at Three by One? beautifully generalises this
result.

Or search by topic

Age 16 to 18

Challenge Level

- Problem
- Getting Started
- Student Solutions
- Teachers' Resources

Also this problem provides a good example of how you can generalise from a result that is really a simple case of a much more general result. Are you content just to solve a problem or do you ask yourself "what if ...." and try to find more general results?

Perhaps your students could work in pairs. They could come to the board and present their solutions to the rest of the class and/or make posters for the classroom wall. You could see collectively how many different methods your class can find.

What do we know about sines, cosines, tangents, vectors, matrices, coordinate geometry, complex numbers and pure geometry that we can use to prove this result?

How might we generalise this result?