Ideas For Education Circle Theorems

There was no inquiry nearly whether circle theorems should earn their house on the novel mathematics curriculum. They are the perfect representative of a theme that is good placed inward secondary schoolhouse mathematics. In teaching this topic, nosotros receive got the pleasance of exploring a prepare of theorems - a pocket-size alternative of the many fascinating properties of circles - thence teaching students how to apply their novel mathematical noesis to solve geometrical problems.

Background
Imagine designing a mathematics GCSE from scratch. What topics would yous include? There's a lot of mathematics to select from. You exclusively receive got to glance at an quondam textbook or O score newspaper to come across that the topics studied at secondary schoolhouse receive got changed considerably over the final century. Have circle theorems ever featured on the United Kingdom of Great Britain together with Northern Ireland of Britain together with Northern Republic of Ireland curriculum?

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

Extract from 1957 O score maths syllabus

Most of this looks really familiar, although the theorem 'if 2 circles touch, the indicate of contact is on the business of the centres' is non covered anymore. Note that the term 'cyclic quadrilateral' isn't used, but instead 'angles inward contrary segments are supplementary'.

In the 1957 O score exam there were many questions relating to circle theorems, including a seat out of proofs. The inquiry below is perchance the easiest, together with is similar inward 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 score question

There was also a inquiry inward the 1957 newspaper relating to the Power of a Point Theorem, which at in ane lawsuit features inward 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 score Syllabus differs notably from the 1957 Syllabus. Again, circle theorems characteristic heavily inward the Geometry department of the 1974 O score exam. In the inquiry below, our electrical flow GCSE students would live able to do purpose a, but non purpose b.

1974 O score question

Over the next forty years, circle theorems content was reduced (in business amongst a gradual displace to include more topics inward secondary mathematics, but less depth). The 1984 O level (an splendid seek - similar to today's higher GCSE but to a greater extent than challenging) contained exclusively 2 circle theorem questions. Here's ane of them:

1984 O score question

The electrical flow GCSE specifications are really specific nearly which theorems are included (most of circle theorems on the 1957 Syllabus receive got stood the seek 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 score of detail.

Source: Wikipedia

So it's long been accepted that circle theorems are worth including inward secondary mathematics education. Although there's been a pocket-size reduction inward the attain of theorems covered over the final lx years, the bulk receive got survived. Circle theorems may receive got petty straight application inward 'real life', but that doesn't matter. Mathematicians relish applying circle theorems to solve interesting geometrical problems. The trickier problems tin actually seek our logical thinking skills together with are especially satisfying. I'm pleased that circle theorems are hither to stay. So let's expect at a few teaching ideas.

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

Vocabulary

Paper plate vocab from Mr Collins

There's a lot of novel vocabulary inward 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 skillful chance to verbalize about etymology - tangent originates from the Latin tangere 'to touch' and arc originally referred to the sun's apparent motility inward 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 every bit the parts of a circle, nosotros also receive got novel words inward the descriptions of the theorems. For example vocabulary I brand some suggestions for activities to back upwards students when yous innovate novel cardinal words, such every bit vocabulary noesis surveys together with literacy walls.

'Name the line'
vocabulary exercise

Investigations
It's squeamish to give students the chance to investigate the angles inward a circle using ICT, geoboards (virtual or real), worksheet investigations or paper folding.  I honey this serial of Geogebra worksheets from The Maths Teacher. Students tin play around amongst the angles on enshroud thence respond the questions provided. In Folding Circles: Exploring Circle Theorems through Paper Folding from the NCTM, students manipulate newspaper circles to explore circle theorems (scroll downwardly to theorems 3 together with iv inward the instructions). I also similar these 2 uncomplicated paper-based demonstrations: angle at the centre and angles inward the same segment.

Extract from Folding Circles (NCTM)

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

Cyclic Quadrilaterals

I'm non going to become into item 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 representative of a quadrilateral amongst circumcircle (eg a square) together with ane without (eg a rhombus).

I similar the inquiry 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, thence they'll in all likelihood ask some direction. For example, when they're proving that the angle inward a semicircle is a right-angle, yous could exhibit them the pace pictured together with inquire 'how would yous acquire to this step?' And where would yous become from here?'.

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

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

Tangent proofs are non currently on the GCSE syllabus but if yous desire to evidence the tangent theorems amongst your class, scout James Tanton's video Tangent Theorems for Circles together with the Pythagorean Theorem beforehand - the explanations are really clear.

If yous desire circle theorems together with proofs on your classroom wall thence these posters from danbar1000 on TES are great.

Resources
If you're planning a sequence of lessons on circle theorems thence you'll detect enough of useful worksheets together with activities listed inward 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 inward lessons thence this is a bright app for practice. It starts off amongst triangles together with parallel lines but yous tin skip ahead to circle theorems.

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

Source: CIMT

Memorising the theorems

Circle theorems are ane of those topics that require think every bit good every bit understanding. If students acquire by to spot the solution to a job thence I encourage them to become through a checklist of circle theorems inward their head, annotating the diagram where they can. Often students missy the 'angles inward the same segment are equal' steps because they aren't ever every bit obvious every bit others (as inward the representative shown below).

Enough practise of using circle theorems mightiness live sufficient to memorise them, but in this post Don Steward suggests that at that spot mightiness live do goodness inward chanting the theorems. I've non tried this yet but dorsum inward chief schoolhouse I learnt my times tables yesteryear chanting - it seemed to operate well. He suggests that the instructor taps on a diagram together with the cast telephone phone out the theorem (see his post for associated resources).

Chanting exercise from Don Steward

Off on a tangent
I desire to cease amongst a twain of curt non-syllabus ideas that would live squeamish to expect at amongst your students if yous receive got time.

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

If you're teaching circle theorems for the start fourth dimension or yous precisely wanted some fresh ideas, I promise this post has been helpful. Please percentage your ideas together with favourite resources inward the comments below, or through Twitter. 

This was a rather long post thence if yous made it this far together with it's nonetheless Nov 2014, delight vote for me inward the United Kingdom of Great Britain together with Northern Ireland of Britain together with Northern Republic of Ireland Blog Awards! Thank you.

All O score cloth inward this post was taken from Cambridge Assessment.