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Four rods, two of length a and two of length b, are linked to form a kite. The linkage is moveable so that the angles change. What is the maximum area of the kite?

### Making Rectangles, Making Squares

How many differently shaped rectangles can you build using these equilateral and isosceles triangles? Can you make a square?

This gives a short summary of the properties and theorems of cyclic quadrilaterals and links to some practical examples to be found elsewhere on the site.

##### Age 11 to 14Challenge Level

Why do this problem?

This problem gives students the opportunity to derive the various properties of quadrilaterals whilst reinforcing the importance of using appropriate language and terminology. These may include angle rules, identifying parallel lines and symmetry which could lead to solving more complex questions. Along the way, the challenges will provoke some insightful discussion linking together different geometrical ideas.

Possible approach

The task could be used when students are not yet familiar with these quadrilaterals (they can be labelled a-h). They should be familiar with right angles, parallel lines and symmetry. Pupils could play the game in pairs or begin as a whole class against the teacher. They should hone their strategies in pairs or small groups, which will involve defining each shape based upon its characteristics. Encourage the use of mathematical language.

You might want to move on to the questions to consider at the bottom of the problem.

The task could also be used as a whole-class plenary exercise. The teacher can think of a quadrilateral and the teacher can select students to ask questions. Praise should be given to pupils who use the correct mathematical language.

It may be useful to print and cut out the shapes so that students can label and group them more easily. Alternatively, the quadrilaterals could be displayed on the board. The learners thinking may be brought together by groups of students presenting their findings, for instance posters of their flow charts. There is also room for a whole-class discussion using the key questions for prompts to extend their ideas further.

Encourage pupils to play the game without using the handout. Can they remember the shapes and their properties without looking at them?

Key questions

• Who has found the shape after the smallest number of questions?
• What are ‘good’ and ‘bad’ questions?
• How many questions do you need to ask to guarantee you know your friend's shape?
• If your friend says "Yes", what would you ask next? If your friend says "No", would you still ask the same question?
• Charlie says he can always identify Alison's shape after asking just three questions. Which question might Charlie ask first? Which follow-up questions might he then ask?
• Can you string the eight special quadrilaterals in a line with just one difference between them?

Possible support

Initially, the students could be prompted to note the properties of each quadrilateral under given headings after being taught the relevant concepts. An activity whereby students fill in a table could be useful (see below for example). This could be followed by a grouping exercise to sort the shapes providing a starting point as to how they can be distinguished from one another.

 Shape # of right angles # of pairs of parallel lines # lines of symmetry # sides of equal length Isosceles Trapezium Rectangle Kite Rhombus Parallelogram Square Trapezium Arrowhead

Additionally, the students could each be given a single shape to work with and tasked with labelling various features. They could compare this with their peers to initiate collaboration.

Possible extension

Can you invent a game with more shapes, where you can always identify your friend's shape in four questions? What is the maximum number of shapes you could have in such a game?