Although the words peripheral and subtend are not essential, instead of avoiding them why don't you encourage students to deduce the meanings? Subtend is from Latin 'to stretch underneath'. In class you may refer to theorems using catchy names such as opera house (or bow-tie or crown), arrowhead, ice-cream (or hat), but in a GCSE exam your students will need to use more formal language. We need to look at proofs along the way too. Once each theorem is established, students should do a lot of practice, starting with simple cases then moving onto more complex questions involving more than one theorem. Most teachers then get their students doing some exploration into angles in circles. So let's look at a few teaching ideas.Ī good place to start when teaching circle theorems is a recap of angle facts relating to triangles and parallel lines. I'm pleased that circle theorems are here to stay. The trickier problems can really test our logical thinking skills and are particularly satisfying. Mathematicians enjoy applying circle theorems to solve interesting geometrical problems. Circle theorems may have little direct application in 'real life', but that doesn't matter. Although there's been a small reduction in the range of theorems covered over the last 60 years, the majority have survived. So it's long been accepted that circle theorems are worth including in secondary mathematics education. The new OCR specification contains a similar level of detail. The new Edexcel GCSE maths specification (effective from 2015) is far less specific, simply stating: The extract below is from the 2012 Edexcel GCSE specification.Įxtract from Edexcel GCSE 2012 Mathematics A specification The current GCSE specifications are very specific about which theorems are included ( most of circle theorems on the 1957 Syllabus have stood the test of time). The 1984 O level (an excellent exam - similar to today's higher GCSE but more challenging) contained only two circle theorem questions. Over the following 40 years, circle theorems content was reduced (in line with a gradual move to include more topics in secondary mathematics, but less depth). In the question below, our current GCSE students would be able to do part a, but not part b. Again, circle theorems feature heavily in the Geometry section of the 1974 O level exam. Interestingly the wording (but not the substance) of the circle theorems content on the 1974 O level Syllabus differs notably from the 1957 Syllabus. There was also a question in the 1957 paper relating to the Power of a Point Theorem, which now features in iGCSE (specifically the Intersecting Chords Theorem) but not GCSE. Our students might enjoy having a go at questions like this - the maths that their grandparents did at school. The question below is perhaps the easiest, and is similar in style to the questions seen on current GCSE papers. In the 1957 O level exam there were many questions relating to circle theorems, including a number of proofs. Note that the term 'cyclic quadrilateral' isn't used, but instead 'angles in opposite segments are supplementary'. Most of this looks very familiar, although the theorem 'if two circles touch, the point of contact is on the line of the centres' is not covered anymore. Looking back at the 1957 O level Syllabus, we have the following: Have circle theorems always featured on the UK curriculum? You only have to glance at an old textbook or O level paper to see that the topics studied at secondary school have changed considerably over the last century. ![]() ![]() What topics would you include? There's a lot of mathematics to choose from. ![]() Imagine designing a mathematics GCSE from scratch. In teaching this topic, we have the pleasure of exploring a set of theorems - a small selection of the many fascinating properties of circles - then teaching students how to apply their new mathematical knowledge to solve geometrical problems. They are the perfect example of a topic that is well placed in secondary school mathematics. There was no question about whether circle theorems should earn their place on the new mathematics curriculum.
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