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Bendy Quad

Age 14 to 16
Challenge Level Yellow starYellow starYellow star
Secondary curriculum
  • Problem
  • Getting Started
  • Student Solutions
  • Teachers' Resources

Why do this problem?
This problem involves the interpretation of a very simple concrete structure, a linkage of 4 rods, and the angles that the quadrilateral formed by the rods could make if the joints between the rods at the vertices are totally flexible. Experimental evidence will offer ideas which then need justification and proof by forming convincing arguments.

The solution uses the cosine and sine rules. To find the constraints on the angles in the general case requires an argument using inequalities.

Possible approach
You might allow time for learners to explore the quadrilateral using strips of card or plastic and split pins, or a dynamic geometry package. This will help them to identify what can be varied and what not.

Discuss the freedoms and constraints within the problem, the impact these might have and how they could influence the structure of any investigation (what can be changed and what cannot).

Encourage groups to identify ideas that they would like to investigate. Spend time planning what they might do and sharing ideas before developing them.

Share findings and approaches.

Key questions
  • What are your variables?
  • If you flex the quadrilateral can the angles be any size?
  • Can you find a relation between the cosines of opposite angles?
  • What constraints would you like to impose? For example, that the quadrilateral is cyclic.

Possible support
Try the problem Diagonals for Area, also about bendy quads but only using the area of a triangle.

Possible extension
Try Biggest Bendy , Flexi Quads , Flexi Quad Tan

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Dividing the Field

A farmer has a field which is the shape of a trapezium as illustrated below. To increase his profits he wishes to grow two different crops. To do this he would like to divide the field into two trapeziums each of equal area. How could he do this?

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The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice.

NRICH is part of the family of activities in the Millennium Mathematics Project.

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