/learn/s23/t15253 /learn/node/42478 Learners are used to being given definitions, accepting them and memorising them. This lesson aims to help them realise that polygons can be defined in more than one way, and that definitions can be contested and argued. Learners need to learn to write their own definitions which are mathematically accurate and logical. Before showing the video, you could have a class debate to give learners a chance to think about the issues for themselves. The topic for the debate could be something like: ‘A square is a rectangle – true or false?’ /learn/node/42479 We look at two ways to define quadrilaterals, using inclusive and exclusive defintions. /learn/node/42480 This lesson develops from the previous lesson, and it helps learners to write their own definitions. It is important for learners not to memorise alternative definitions, but to understand why definitions are important. A good definition is one that is correct, sufficient and necessary. A useful and efficient way to define a shape is to add some properties to a shape that is already defined. So, for example, a rectangle can be defined as a parallelogram with extra properties. Discuss the different definitions of the isosceles trapezium that are shown in the video, and identify which of these is inclusive and which is exclusive. /learn/node/42481 The focus of this lesson is on using transformation geometry to prove a conjecture. We have chosen a particular definition of a kite to work from. When talking about kites, it is useful to refer to their vertex angles (the angles between each pair of equal adjacent sides) and their non-vertex angles (the angle between two sides that are not equal) separately. Your learners may choose a different definition as their starting point, so their investigations may take a different path. If your learners have not seen our series, Investigating Geometry 1, you could show them lesson 5 of that series, which demonstrates how to write a plan for a proof and how to write a paragraph proof. Paragraph proofs help learners to develop confidence in writing proofs. They allow learners to focus on making sense of the proof without worrying about the layout of the proof. So don’t skip this important step. /learn/node/42482 Show learners some examples of flowchart proofs to help them understand what a flowchart proof is, and then let them practise making their own flowchart proofs. Remember that the value of this lies in the process of working it out, not in copying down a ‘model’ answer from the board! Some learners will prefer this visual way of writing down a proof as it makes the steps of the process very clear. /learn/node/42483 We use co-ordinate geometry to identify quadrilaterals by proving the properties they have. We make use of the five co-ordinate geometry ‘tools’ or formulae that are summarised at the beginning of the lesson. These tools are also explained in the video series Co-ordinate and Transformation Geometry. In this lesson, we show that a particular triangle is isosceles by placing it on the Cartesian plane and using the distance formula to calculate the length of its sides. We also use co-ordinate geometry to prove in two different ways that a given quadrilateral is a rectangle. You may want to show your learners further examples that use the tools of co-ordinate geometry in different ways to work with shapes. /learn/node/42484 In this lesson, learners make an investigation and arrive at their own conjecture. The conjecture is about the area of a rectangle and the area of a parallelogram between the same parallel lines. We have used an area theorem because it is fairly simple and allows learners to concentrate on the processes being taught rather than the knowledge of the content of the theorem. Your learners could use square dotted paper to do the investigation themselves before they watch the video. /learn/node/42485 We investigate the areas of a parallelogram and a rectangle that are on the same base and between the same parallel lines.