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.

5 Maths Gems #16

It's December! Time for students as well as teachers to outset getting overly excited almost the upcoming Christmas holidays. There's even thus 3 to a greater extent than weeks of teaching to larn through though, as well as most schools do mock GCSEs inward Dec or January, thus things are every bit busy every bit ever. Let's expect at roughly of the ideas that receive got been shared on Twitter this week. I've got lots of dainty geometry for you lot today.

1. Frozen Fractals

I'm non a fan of Disney princess films, merely I receive got daughters at nowadays thus I've been forced to scout Frozen many many times. I genuinely quite similar it. Being a mathematician, my favourite business office is when Elsa sings almost 'frozen fractals' inward the vocal Let it Go. I had no persuasion what the give-and-take fractal meant until a few years agone thus it seems strange to listen it inward a children's film. It got me thinking - is at that spot whatever value inward teaching our students almost fractals at secondary school? I remember at that spot 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 agency to sympathize fractals is to push clit i - the article describes how to push clit a Von Koch curve.

Give your students roughly dotted or triangle graph newspaper as well as they could follow the instructions inward this worksheet to push clit 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, merely nosotros tin expect at other fractals too. The Sierpinski Triangle is also a dainty 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 thus 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 i pictured below.

And this Mathematical Christmas Tree is every bit fractally every bit they come.

If you lot desire to farther explore fractals, your students mightiness survive interested inward this Numberphile video 'Measuring Coastline'.

Whether you lot brand snowflakes, triangles or cutouts, the betoken is that students come upwardly away from the lesson agreement what a fractal is. It may non survive on the syllabus merely it's interesting mathematics nonetheless. And your students volition survive able to say their friends that they sympathize the mathematical reference inward Frozen.

2. Angles
I dearest this angles interrogation from @j_lanier. I'd travel it inward a GCSE revision lesson. Students receive got to apply a number of rules to honor all 4 angles - it's a expert 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 past times Don Steward, when I noticed a link to a weblog I'd non seen earlier - Pythagorean Triple Tasks past times Hannah Jones. There's roughly lovely activities here.

By the way, speaking of Don Steward, did you lot know he also has a blog for do resources? These are every bit good designed every bit his rich tasks. There's loads of expert questions for GCSE revision, which you lot mightiness honor 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 you lot run across Ed Southall's awesome semi-circle problem? I'd travel this every bit an extension action inward a Pythagoras lesson, merely 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 upwardly to H5N1 level.

5. Christmassy Maths
I started to write a seasonal maths resources page merely lots of other lovely bloggers receive got done something similar. So rather than reinvent the wheel, I'll betoken you lot inward the direction 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 expert do of at nowadays business graphs. I also similar these resources from Oxford University Press. Most Christmas resources are aimed at Key Stage 3 students, merely your Year 11s mightiness similar this Christmas Shading Graphical Inequalities by frickard on TES. And how almost this lovely Christmas tree logic problem?

Logic work from ageofpuzzles.com

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

5 Maths Gems #18

Most of my readers likely already subscribe to Chris Smith's lovely maths newsletter. Full of instruction ideas, puzzles, jokes in addition to 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 directly has over 900 newsletter subscribers. This calendar week he published his 303^rd issue. I've solely been subscribing for a twosome of months but Chris was sort plenty to part his entire dorsum catalogue alongside me, which I've actually enjoyed reading. The ideas inward my weekly gems posts are commonly sourced from Twitter, but today I characteristic my v favourite ideas from the get-go 100 issues of Chris's newsletters. Huge thank you lot 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 prissy starter action to railroad train ideas relating to number properties.

3. Learning past times Rope
This is keen for exploring polygons.

4. Finding a Route
Another prissy 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 mention in addition to my signature on it in addition to told them they had a essay coming upward in addition to they were allowed to select the blank page inward alongside them. It was upward 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 downwards all the examples from my book.” Objective ane achieved. They were going to revise.

At the terminate of the test, I collected their sheets to expect at them inward to a greater extent than detail. It was real rewarding when pupils pretty much demanded that they got them back. Objective 2 achieved. My pupils directly 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 select you lot to a greater extent than or less to a greater extent than newsletter gems afterwards I've read the side past times side 100 issues. Don't forget to follow Chris on Twitter in addition to email him if you'd similar to have his weekly newsletters.

What I've been upward to
I've had a busy in addition to emotional week. My motherhood larn out is coming to an terminate in addition to I'm keen to start planning lessons. I don't select my timetable yet, in addition to thence am eagerly awaiting intelligence of what classes I'll hold upward instruction inward 3 weeks. On Tuesday I hosted #mathscpdchat which was all most mutual mathematical misconceptions - you lot tin read a summary here. On Wed I had a project interview inward which I taught a lesson on indices to a Year 8 shape - it was fun to hold upward dorsum inward the classroom in addition to I'm pleased to nation I got the job, in addition to thence volition hold upward moving schools inward May to perish an Aspiring Lead Practitioner. I was likewise real proud to select an article published inward Academies Week most the novel maths GCSE in addition to the associated problems alongside recruitment. Feedback in addition to comments are real welcome.

Puzzle answers: 1. r = 29  2. Pi/2

5 Maths Gems #23

It seems similar ages since my concluding gems post. School has been busier than I anticipated. This calendar week I had reports to write together with a Parents' Evening to attend - both tasks were especially challenging because I've solely been teaching these classes for a few weeks. I've also been mark at domicile every even out thus it's difficult to discovery fourth dimension for Twitter together with blogging. I receive got 3 other posts inwards draft - i virtually teaching quadratics, i virtually spontaneity together with i featuring some other batch of highlights from Chris Smith's newsletter dorsum catalogue. So spotter this space, they're all coming soon! In the meantime, here's v ideas you lot mightiness discovery useful.

1. Perspective
I've been teaching 3D visualisation to my Year 7s this calendar week (isometric drawing, plans, elevations together with nets). I've decided that this is i of my top 3 'worst topics to teach' with constructions together with transformations. But it did give me an chance to role the moving-picture exhibit below, which I got from Twitter a few months agone (original source unknown - delight nation me if you lot know!). I pose it on the board when my students arrived for their lesson on plans together with elevations.

2. Etymathology

My novel favourite Twitter concern human relationship is Gems 10.

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

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

And at that topographic point you lot acquire - some other instance of Twitter's contribution to education. I acquired novel cognition to locomote past times direct to my students. I await forrad to learning more!

3. Match My Parabola
I've written virtually Desmos many times, most lately inwards my shipping service Polygraph Rocks. I'm e'er non bad to listen novel ideas for teaching with Desmos thus I loved Michael Fenton's (@mjfenton) post 'Match My Parabola'.

The shipping 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 together with receive got to discovery a quadratic business office that passes through them. Michael rattling helpfully provides 10 activities - all your students receive got to create is click on the links together with consummate the tasks (here's an example for you lot to receive got a acquire at - acquire into your business office inwards jail cellular telephone ix on the left to meet if you're right).

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

4. Valentines Relay
It's solely ii weeks until half-term! Fri 13th Feb would last a proficient twenty-four hours to create Chris Smith's (@aap03102) lovely Valentines Maths Relay Race.

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

That's it for my gems this week, but comport with me - I receive got a few to a greater extent than things I desire to nation you lot about...

Multiple Choice Tests
I wrote virtually multiple selection tests together with Quick Key inwards Gems 11. This calendar week I tried it out for the outset time. Like my outset assay at using Plickers, it wasn't rattling successful! But I create intend to assay both apps i time again - I remember they require a combat of practice.

Quick Key is an app that scans multiple selection tests together with gives you lot instant feedback on results. I institute it rattling slow to gear upwards - I wrote a short quiz on quadratics together with only entered the right answers to Quick Key every bit I wrote it. In the lesson, I gave my students 20 minutes to brand themselves a i page summary of everything we'd covered on quadratics, together with and thus I gave them a 20 infinitesimal quiz inwards which they were allowed to refer to their revision sheet.

I was excited virtually mark the tests using Quick Key but I institute it rattling difficult to discovery a shadow-free, glare-free identify to scan. In the terminate it took exactly every bit long every bit if I'd marked them past times hand. The examine results varied from 20% to 100%, with most students getting 80% or 90% (the most mutual error was choosing Influenza A virus subtype H5N1 instead of C inwards the inquiry below). So it was a successful exercise inwards assessing the agreement of each pupil together with confirming that the bulk of students were create to motion onto the adjacent topic, together with forthwith I exactly demand to figure out how to address the gaps with those who got a depression score.

So inwards summary, the lesson inwards which students made a revision summary together with thus took a brusk multiple selection examine 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 seven together with was actually impressed when i of my students told me that she'd institute an angles work online together with had spent ii hours working on it over the weekend, but fifty-fifty with the assistance of all her identify unit of measurement she couldn't solve it. It turned out to last the 'World's Hardest Easy Geometry Problem' which I've featured before... I promised her I'd receive got a acquire at it this weekend - I suspect I may terminate upwards bespeak Twitter for help!

Don Watch
Have you lot been keeping an optic on all the bright novel resources on Jumping along a line', suggested using the 0^th term to locomote out the n^th term of a sequence. This isn't how I commonly instruct sequences but I'll assay it adjacent time.

I also spotted the 0^th term on a Pret homework that Grant Barker (@AccessMaths) sent me this week. We've forthwith got to a greater extent than than 100 homeworks on prethomework.weebly.com so if you're ever looking for novel homework ideas together with thus create banking concern gibe it out.

Mr Maths
Finally, I'll locomote out you lot with i of the fantastic images made by @solvemymaths this calendar week - banking concern gibe out his blog for the total 'Math Mr Men Series'.