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Darts and Kites

A rhombus PQRS has an angle of 72 degrees. OQ = OR = OS = 1 unit. Find all the angles, show that POR is a straight line and that the side of the rhombus is equal to the Golden Ratio.

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Pent

The diagram shows a regular pentagon with sides of unit length. Find all the angles in the diagram. Prove that the quadrilateral shown in red is a rhombus.

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Pentakite

ABCDE is a regular pentagon of side length one unit. BC produced meets ED produced at F. Show that triangle CDF is congruent to triangle EDB. Find the length of BE.

Golden Thoughts

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Why do this problem?

This problem offers opportunities to experience several key problem-solving processes including making a useful transformation into a simpler task and making use of algebra to represent the situation before interpreting results as meaningful solutions within the problem context.

Possible approach

Any teaching approach always involves a choice about the level of problem-solving process students are to experience. For example this problem could be used as a guided example of algebraic modelling, but the problem might equally successfully be presented as a simple diagram and students left to make their own progress with the question over time. Both approaches are valid, but the outcomes for the student will be very different with each.

Taking a route between those two approaches :

Ask a student to explain what the problem requires, then ask the group for any thoughts about where to start or what might work. Students may choose to work on specified (and simple) rectangle dimensions, hoping to generalise later once they have some results. This is a good and valid strategy.

Before long, try to catch someone questioning whether the rectangle dimensions actually matter. This is a key moment and it is worth stopping the group to allow discussion. We can work with the unit square (which simplifies the task) because areas are only required to match, and any stretch (to create a rectangle) scales all areas by the same factor. This transformation of the problem into a simpler problem is a key mathematical moment. Although students who can see how to proceed without this reduction should be allowed to continue unhindered.

It is possible to use a spreadsheet to explore the change in area distribution.

Through questions and discussion guide students into nominating an independent variable (for example the distance from R to X), and use this to go on to expressing the areas of interest in terms of that variable. If this is pursued algebraically a quadratic equation emerges with two distinct solutions only one of which fits the context being modelled.

Key questions :

  • What does this problem require ?
  • Where might you start - what might work ?
  • Would it help to fix on one specific rectangle, or is there another way to start ?
If students have settled on using the Unit Square :
  • Why can this be generalised?

Possible extension :

For more on algebraic modelling try Lap Times

For more reasoning about triangle areas try Triangle in a Triangle

More extension work on many other problems relating to the golden ratio, and articles on the subject, are collected in the package Golden Trail.

Possible support :

Isosceles is still a challenging task but might make a good 'warm up'