Ideas For Instruction Circle Theorems

There was no enquiry nearly whether circle theorems should earn their house on the novel mathematics curriculum. They are the perfect illustration of a theme that is good placed inwards secondary schoolhouse mathematics. In teaching this topic, nosotros receive got the pleasance of exploring a laid upward of theorems - a small-scale alternative of the many fascinating properties of circles - hence teaching students how to apply their novel mathematical noesis to solve geometrical problems.

Background
Imagine designing a mathematics GCSE from scratch. What topics would y'all include? There's a lot of mathematics to select from. You solely receive got to glance at an one-time textbook or O flat newspaper to run across that the topics studied at secondary schoolhouse receive got changed considerably over the final century. Have circle theorems e'er featured on the U.K. curriculum?

Looking dorsum at the 1957 O flat Syllabus, nosotros receive got the following:

Extract from 1957 O flat maths syllabus

Most of this looks real familiar, although the theorem 'if ii circles touch, the betoken of contact is on the trace of piece of work of the centres' is non covered anymore. Note that the term 'cyclic quadrilateral' isn't used, but instead 'angles inwards contrary segments are supplementary'.

In the 1957 O flat exam there were many questions relating to circle theorems, including a expose of proofs. The enquiry below is maybe the easiest, too is similar inwards mode to the questions seen on electrical flow GCSE papers. Our students mightiness relish having a become at questions similar this - the maths that their grandparents did at school.

1957 O flat question

There was also a enquiry inwards the 1957 newspaper relating to the Power of a Point Theorem, which right away features inwards iGCSE (specifically the Intersecting Chords Theorem) but non GCSE.

Interestingly the wording (but non the substance) of the circle theorems content on the 1974 O flat Syllabus differs notably from the 1957 Syllabus. Again, circle theorems characteristic heavily inwards the Geometry department of the 1974 O flat exam. In the enquiry below, our electrical flow GCSE students would live able to do job a, but non job b.

1974 O flat question

Over the next forty years, circle theorems content was reduced (in trace of piece of work amongst a gradual displace to include more topics inwards secondary mathematics, but less depth). The 1984 O level (an splendid examination - similar to today's higher GCSE but to a greater extent than challenging) contained solely ii circle theorem questions. Here's ane of them:

1984 O flat question

The electrical flow GCSE specifications are real specific nearly which theorems are included (most of circle theorems on the 1957 Syllabus receive got stood the examine of time). The extract below is from the 2012 Edexcel GCSE specification.

Extract from Edexcel GCSE 2012 Mathematics Influenza A virus subtype H5N1 specification

The novel Edexcel GCSE maths specification (effective from 2015) is far less specific, only stating:

Thankfully the new AQA specification is less vague, confirming the continuation of the existing content.

The novel OCR specification contains a similar flat of detail.

Source: Wikipedia

So it's long been accepted that circle theorems are worth including inwards secondary mathematics education. Although there's been a small-scale reduction inwards the arrive at of theorems covered over the final lx years, the bulk receive got survived. Circle theorems may receive got petty straight application inwards 'real life', but that doesn't matter. Mathematicians relish applying circle theorems to solve interesting geometrical problems. The trickier problems tin sack actually examine our logical thinking skills too are peculiarly satisfying. I'm pleased that circle theorems are hither to stay. So let's hold off at a few teaching ideas.

Influenza A virus subtype H5N1 proficient house to start when teaching circle theorems is a recap of angle facts relating to triangles too parallel lines. Most teachers hence acquire their students doing some exploration into angles inwards circles. Once each theorem is established, students should do a lot of practice, starting amongst uncomplicated cases hence moving onto to a greater extent than complex questions involving to a greater extent than than ane theorem. We require to hold off at proofs along the way too.

Vocabulary

Paper plate vocab from Mr Collins

There's a lot of novel vocabulary inwards this topic. First nosotros receive got the parts of a circle - your students could explore these using this blog post from Mr Collins.

This is also a proficient chance to verbalize about etymology - tangent originates from the Latin tangere 'to touch' and arc originally referred to the sun's apparent displace inwards the sky. Segment (think oranges) and sector (think pie charts) are both from the Latin secare 'to cut'. Diameter is from the Greek dia- "across, through" and metron "a measure".

As good equally the parts of a circle, nosotros also receive got novel words inwards the descriptions of the theorems. For example vocabulary I brand some suggestions for activities to back upward students when y'all innovate novel fundamental words, such equally vocabulary noesis surveys too literacy walls.

'Name the line'
vocabulary exercise

Investigations
It's prissy to give students the chance to investigate the angles inwards a circle using ICT, geoboards (virtual or real), worksheet investigations or paper folding.  I beloved this serial of Geogebra worksheets from The Maths Teacher. Students tin sack play around amongst the angles on enshroud hence answer the questions provided. In Folding Circles: Exploring Circle Theorems through Paper Folding from the NCTM, students manipulate newspaper circles to explore circle theorems (scroll downward to theorems iii too iv inwards the instructions). I also similar these ii uncomplicated paper-based demonstrations: angle at the centre and angles inwards the same segment.

Extract from Folding Circles (NCTM)

Another view for investigating angles inwards circles involves hula hoops - this originates from the invitee post this post by Kelly Achtymichuk (source of the pictures below) too Mr Collins' post 'Circle Theorems & Hula Hoops!!' to run across how other teachers receive got adapted the idea.

Cyclic Quadrilaterals

I'm non going to become into special nearly each theorem but I do desire to call cyclic quadrilaterals. The give-and-take cyclic is from the Greek kuklos which agency 'circle' or 'wheel'. A cyclic quadrilateral is a quadrilateral whose vertices all prevarication on a unmarried circle (ie the vertices are concyclic).

All triangles receive got a circumcircle but non all quadrilaterals do. Ask your students to sketch an illustration of a quadrilateral amongst circumcircle (eg a square) too ane without (eg a rhombus).

I similar the enquiry below (from this CIMT exercise) to develop understanding of the give-and-take concyclic.

Question from CIMT

Proof

Students should receive got a become at working the proofs out themselves. But they're non accustomed to proof, hence they'll in all probability require some direction. For example, when they're proving that the angle inwards a semicircle is a right-angle, y'all could demo them the pace pictured too inquire 'how would y'all acquire to this step?' And where would y'all become from here?'.

I also actually similar this activity from rhemsley on TES which makes the proofs to a greater extent than accessible past times starting amongst numbers earlier generalising.

The NCETM also gives us this proof worksheet where students create total inwards the missing words.

Tangent proofs are non currently on the GCSE syllabus but if y'all desire to testify the tangent theorems amongst your class, spotter James Tanton's video Tangent Theorems for Circles too the Pythagorean Theorem beforehand - the explanations are real clear.

If y'all desire circle theorems too proofs on your classroom wall hence these posters from danbar1000 on TES are great.

Resources
If you're planning a sequence of lessons on circle theorems hence you'll honor enough of useful worksheets too activities listed inwards my shape resources library. 

In my final gems post I mentioned the costless app Angles? - Let's solve figures problems! by Gakko Net Inc. If your students receive got access to iPads inwards lessons hence this is a vivid app for practice. It starts off amongst triangles too parallel lines but y'all tin sack skip ahead to circle theorems.

Particularly complex circle theorem questions, or 'busy' diagrams (like the enquiry below), could live enlarged on A3 newspaper - this would live a proficient chance to job the acrylic sheets suggested inwards Jeremy Denton's post 'Improving students' collaboration using acrylic sheets'.

Source: CIMT

Memorising the theorems

Circle theorems are ane of those topics that require recollect equally good equally understanding. If students combat to spot the solution to a job hence I encourage them to become through a checklist of circle theorems inwards their head, annotating the diagram where they can. Often students fille the 'angles inwards the same segment are equal' steps because they aren't e'er equally obvious equally others (as inwards the illustration shown below).

Enough practise of using circle theorems mightiness live sufficient to memorise them, but in this post Don Steward suggests that in that location mightiness live do goodness inwards chanting the theorems. I've non tried this yet but dorsum inwards main schoolhouse I learnt my times tables past times chanting - it seemed to move well. He suggests that the instructor taps on a diagram too the degree telephone telephone out the theorem (see his post for associated resources).

Chanting exercise from Don Steward

Off on a tangent
I desire to goal amongst a couplet of curt non-syllabus ideas that would live prissy to hold off at amongst your students if y'all receive got time.

Have y'all ever looked at a regular This post nearly Thales Theorem describes how the presenter of children's TV programme Art Attack demonstrated how to honor the centre of a circle. If it featured on Art Attack hence why non do it inwards a maths lesson? There is a prissy interactive demonstration of the method here including printable instructions.

If you're teaching circle theorems for the kickoff fourth dimension or y'all precisely wanted some fresh ideas, I promise this post has been helpful. Please part your ideas too favourite resources inwards the comments below, or through Twitter. 

This was a rather long post hence if y'all made it this far too it's however Nov 2014, delight vote for me inwards the U.K. Blog Awards! Thank you.

All O flat cloth inwards this post was taken from Cambridge Assessment.