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.

5 Maths Gems #16

It's December! Time for students as well as teachers to kickoff getting overly excited nearly the upcoming Christmas holidays. There's yet iii to a greater extent than weeks of teaching to learn through though, as well as most schools do mock GCSEs inward Dec or January, thence things are equally busy equally ever. Let's expect at or thence of the ideas that receive got been shared on Twitter this week. I've got lots of overnice geometry for y'all today.

1. Frozen Fractals

I'm non a fan of Disney princess films, only I receive got daughters at nowadays thence I've been forced to picket Frozen many many times. I truly quite similar it. Being a mathematician, my favourite business office is when Elsa sings nearly 'frozen fractals' inward the vocal Let it Go. I had no stance what the give-and-take fractal meant until a few years agone thence it seems strange to remove heed it inward a children's film. It got me thinking - is in that place whatever value inward teaching our students nearly fractals at secondary school? I squall upwards in that place is. 

What are Fractals as well as why should I care? by George Dallas. This is an incredibly clear as well as accessible article. The best means to empathise fractals is to clitoris 1 - the article describes how to clitoris a Von Koch curve.

Give your students or thence dotted or triangle graph newspaper as well as they could follow the instructions inward this worksheet to clitoris a 'Koch snowflake'. While they're creating their fractals, exhibit them this lovely microscopic timelapse video of existent snowflakes (thanks to @misseyfs for sharing this).

Because I'm writing this post on 1st Dec I'm thinking of Christmassy snowflakes, only nosotros tin flame expect at other fractals too. The Sierpinski Triangle is also a overnice introduction to agreement fractals. The awesome website fractalfoundation.org has an fantabulous worksheet (a 'fractivity') in which each of your students brand their ain Sierpinski Triangle - they as well as then stick them together to brand a 'class fractal'. Show the video on the page Fractal Trianglethon to wow your students alongside giant Sierpinski Triangles!

Alternatively your students could create Fractal Cutouts like the 1 pictured below.

And this Mathematical Christmas Tree is equally fractally equally they come.

If y'all desire to farther explore fractals, your students mightiness hold out interested inward this Numberphile video 'Measuring Coastline'.

Whether y'all brand snowflakes, triangles or cutouts, the signal is that students come upwards away from the lesson agreement what a fractal is. It may non hold out on the syllabus only it's interesting mathematics nonetheless. And your students volition hold out able to country their friends that they empathise the mathematical reference inward Frozen.

2. Angles
I honey this angles interrogation from @j_lanier. I'd role it inward a GCSE revision lesson. Students receive got to apply a number of rules to uncovering all 4 angles - it's a proficient chance for them to practise articulating their reasoning.

3. Pythagorean Triples
I was looking at recent posts on my favourite blog, the ever-brilliant Median yesteryear Don Steward, when I noticed a link to a weblog I'd non seen earlier - Pythagorean Triple Tasks yesteryear Hannah Jones. There's or thence lovely activities here.

By the way, speaking of Don Steward, did y'all know he also has a blog for exercise resources? These are equally good designed equally his rich tasks. There's loads of proficient questions for GCSE revision, which y'all mightiness uncovering useful this calendar week if your Year 11s receive got mocks coming up. I especially similar these grids of 'practice makes perfect' questions. Also, I spotted this fantastic work inward his A & A* questions - a lovely starter for my Year 12s:

4. Semi-circle

Did y'all run across Ed Southall's awesome semi-circle problem? I'd role this equally an extension activeness inward a Pythagoras lesson, only there's loads of methods for solving it - others receive got used similar triangles, trigonometry as well as coordinate grids. It's both accessible as well as challenging for students from Key Stage 3 upwards to Influenza A virus subtype H5N1 level.

5. Christmassy Maths
I started to write a seasonal maths resources page only lots of other lovely bloggers receive got done something similar. So rather than reinvent the wheel, I'll signal y'all inward the administration of:

• @missradder's post Christmaths
• @Ms_Kmp's post ChrisMaths Cheer
• @solvemymaths' post 2014 Christmas Maths Lessons

There are hundreds of fantabulous Christmas-themed maths resources. I'd similar to recommend a few inward add-on to those inward the posts above. There are 2 from Teachit Maths that I especially similar - the 12 days of Christmas budget activity which involves work solving using percentages, as well as Mistletoe as well as Lines which is proficient exercise of forthwith business graphs. I also similar these resources from Oxford University Press. Most Christmas resources are aimed at Key Stage 3 students, only your Year 11s mightiness similar this Christmas Shading Graphical Inequalities by frickard on TES. And how nearly this lovely Christmas tree logic problem?

Logic work from ageofpuzzles.com

That's it for this week. Please don't forget to expire along tweeting or emailing me photos of misconceptions or mutual mistakes inward your students' work. Thanks!

5 Maths Gems #18

Most of my readers in all probability already subscribe to Chris Smith's lovely maths newsletter. Full of instruction ideas, puzzles, jokes together with mathematical trivia, it's a joy to have every week. Chris has been producing the newsletter since 2007 when he was a wee NQT. Amongst other things, the newsletter is a fantastic machinery for sharing expert practice.

Chris instantly has over 900 newsletter subscribers. This calendar week he published his 303^rd issue. I've exclusively been subscribing for a twosome of months but Chris was variety plenty to part his entire dorsum catalogue alongside me, which I've actually enjoyed reading. The ideas inwards my weekly gems posts are usually sourced from Twitter, but today I characteristic my v favourite ideas from the kickoff 100 issues of Chris's newsletters. Huge cheers to Chris for allowing me to create this!

1. Factorising
Here's a lovely thought for introducing factorising.

2. Connecting numbers
This is a actually squeamish starter action to prepare ideas relating to number properties.

3. Learning yesteryear Rope
This is neat for exploring polygons.

4. Finding a Route
Another squeamish starter activity.

5. Revision notes

In Issue 27, Chris featured an thought from TES which is designed to encourage students to revise for tests.

“I gave each pupil an A4 sail of newspaper alongside their lift together with my signature on it together with told them they had a essay out coming upwards together with they were allowed to choose the blank page inwards alongside them. It was upwards to them what they did alongside the page. If they wrote on it, fine, if they didn't, fine. Immediately ane of the to a greater extent than troublesome pupils shouted out, “Great! I'm going to re-create downward all the examples from my book.” Objective ane achieved. They were going to revise.

At the destination of the test, I collected their sheets to await at them inwards to a greater extent than detail. It was really rewarding when pupils pretty much demanded that they got them back. Objective 2 achieved. My pupils instantly value all of their revision notes."

Puzzles

Each of Chris's newsletters features a puzzle - here's 2 that I peculiarly liked (answers are at the bottom of this post):

And finally, 2 bits of trivia:

That's it from Chris's newsletters for now, but I volition choose yous to a greater extent than or less to a greater extent than newsletter gems later I've read the adjacent 100 issues. Don't forget to follow Chris on Twitter together with email him if you'd similar to have his weekly newsletters.

What I've been upwards to
I've had a busy together with emotional week. My motherhood operate out is coming to an destination together with I'm keen to start planning lessons. I don't accept my timetable yet, then am eagerly awaiting tidings of what classes I'll last instruction inwards 3 weeks. On Tuesday I hosted #mathscpdchat which was all most mutual mathematical misconceptions - yous tin plough over the sack read a summary here. On Wed I had a labor interview inwards which I taught a lesson on indices to a Year 8 course of report - it was fun to last dorsum inwards the classroom together with I'm pleased to tell I got the job, then volition last moving schools inwards May to drib dead an Aspiring Lead Practitioner. I was too really proud to accept an article published inwards Academies Week most the novel maths GCSE together with the associated problems alongside recruitment. Feedback together with comments are really welcome.

Puzzle answers: 1. r = 29  2. Pi/2

5 Maths Gems #23

It seems similar ages since my final gems post. School has been busier than I anticipated. This calendar week I had reports to write in addition to a Parents' Evening to attend - both tasks were peculiarly challenging because I've alone been teaching these classes for a few weeks. I've also been mark at dwelling every eventide therefore it's difficult to detect fourth dimension for Twitter in addition to blogging. I receive got 3 other posts inward draft - ane almost teaching quadratics, ane almost spontaneity in addition to ane featuring some other batch of highlights from Chris Smith's newsletter dorsum catalogue. So sentinel this space, they're all coming soon! In the meantime, here's 5 ideas yous powerfulness detect useful.

1. Perspective
I've been teaching 3D visualisation to my Year 7s this calendar week (isometric drawing, plans, elevations in addition to nets). I've decided that this is ane of my top 3 'worst topics to teach' amongst constructions in addition to transformations. But it did give me an chance to locomote the flick below, which I got from Twitter a few months agone (original source unknown - delight say me if yous know!). I set it on the board when my students arrived for their lesson on plans in addition to elevations.

2. Etymathology

My novel favourite Twitter line of piece of work concern human relationship is Gems 10.

This week @BedtimeMath shared this fact almost seconds, which I'd never heard before:

It's funny how nosotros tin give the sack locomote a discussion similar 'seconds' our whole lives in addition to never give whatever thought to where it came from. Etymathology followed this tweet with the origins of the discussion minute:

And at that spot yous become - some other lawsuit of Twitter's contribution to education. I acquired novel cognition to overstep straight to my students. I expect frontward to learning more!

3. Match My Parabola
I've written almost Desmos many times, most late inward my postal service Polygraph Rocks. I'm ever nifty to remove heed novel ideas for teaching with Desmos therefore I loved Michael Fenton's (@mjfenton) post 'Match My Parabola'.

The postal service describes a lesson suitable for Higher GCSE or Core AS students studying quadratics. The thought is that students are given a laid of coordinates in addition to receive got to detect a quadratic business office that passes through them. Michael rattling helpfully provides 10 activities - all your students receive got to produce is click on the links in addition to consummate the tasks (here's an example for yous to receive got a become at - locomote inward your business office inward jail cellular telephone ix on the left to encounter if you're right).

Michael has previously written almost 'Match My Line' - a similar lesson but for linear graphs, suitable for Key Stage 3 or 4. I've establish that students of whatever historic menses dearest playing with Desmos therefore I'm looking frontward to using this ane too. Really clever ideas from Michael.

4. Valentines Relay
It's alone 2 weeks until half-term! Fri 13th Feb would locomote a adept 24-hour interval to produce Chris Smith's (@aap03102) lovely Valentines Maths Relay Race.

5. War on Error
Do your students proceed making the same mistake? Highlight mutual misconceptions inward a display using Mr Taylor's (post for total details. This calendar week ane of my students wrote (x + 5)² = x² + 25 therefore that would locomote my outset entry to the display!

That's it for my gems this week, but deport with me - I receive got a few to a greater extent than things I desire to say yous about...

Multiple Choice Tests
I wrote almost multiple alternative tests in addition to Quick Key inward Gems 11. This calendar week I tried it out for the outset time. Like my outset endeavour at using Plickers, it wasn't rattling successful! But I produce recall to essay both apps in ane lawsuit again - I recall they require a fleck of practice.

Quick Key is an app that scans multiple alternative tests in addition to gives yous instant feedback on results. I establish it rattling slowly to ready - I wrote a short quiz on quadratics in addition to exactly entered the right answers to Quick Key every bit I wrote it. In the lesson, I gave my students twenty minutes to brand themselves a ane page summary of everything we'd covered on quadratics, in addition to and then I gave them a twenty infinitesimal quiz inward which they were allowed to refer to their revision sheet.

I was excited almost mark the tests using Quick Key but I establish it rattling difficult to detect a shadow-free, glare-free identify to scan. In the destination it took exactly every bit long every bit if I'd marked them past times hand. The essay out results varied from 20% to 100%, with most students getting 80% or 90% (the most mutual error was choosing H5N1 instead of C inward the enquiry below). So it was a successful exercise inward assessing the agreement of each pupil in addition to confirming that the bulk of students were create to movement onto the adjacent topic, in addition to at in ane lawsuit I exactly demand to figure out how to address the gaps with those who got a depression score.

So inward summary, the lesson inward which students made a revision summary in addition to then took a brusque multiple alternative essay out was successful, but I'll demand to acquire the hang of Quick Key.

World's Hardest Geometry Problem
I've exactly finished teaching angles to Year vii in addition to was actually impressed when ane of my students told me that she'd establish an angles occupation online in addition to had spent 2 hours working on it over the weekend, but fifty-fifty with the assistance of all her household unit of measurement she couldn't solve it. It turned out to locomote the 'World's Hardest Easy Geometry Problem' which I've featured before... I promised her I'd receive got a become at it this weekend - I suspect I may destination upwardly bespeak Twitter for help!

Don Watch
Have yous been keeping an oculus on all the bright novel resources on Jumping along a line', suggested using the 0^th term to piece of work out the n^th term of a sequence. This isn't how I ordinarily learn sequences but I'll essay it adjacent time.

I also spotted the 0^th term on a Pret homework that Grant Barker (@AccessMaths) sent me this week. We've at in ane lawsuit got to a greater extent than than 100 homeworks on prethomework.weebly.com so if you're ever looking for novel homework ideas in addition to then produce cheque it out.

Mr Maths
Finally, I'll locomote out yous with ane of the fantastic images made by @solvemymaths this calendar week - cheque out his blog for the total 'Math Mr Men Series'.