easy way to learn mathematics: January 2020

5 Maths Gems #11

I've got roughly vivid gems for y'all this week. As commons Twitter has been on transcend form.

1. Quick Key
Last calendar week I wrote close Plickers, which is an app y'all tin role inwards the classoom for formative assessment - y'all ask a device only your students don't. Today I'm suggesting roughly other app that y'all tin role for assessment - Quick Key. With this app y'all tin laid multiple alternative homeworks or tests as well as grade them inwards seconds past times scanning the respond sheets ('Quick Tickets') with your phone.

Quick Key is the excogitation of American instructor Walter Duncan (@4_teachers). It "turns your band into a scanner as well as eliminates hand-grading of assessments, fifty-fifty for teachers working inwards paper-based classrooms without a calculator or internet. Analytics as well as information exports are easy, thus y'all tin focus on your students".

My initial thoughts were that although this app could live on rattling useful (less fourth dimension spent marking = to a greater extent than fourth dimension spent planning character lessons), it won't allow me to banking concern friction match students' methods, which limits opportunities for useful feedback. But for straightforward cognition checks, to inform me how much my students know, it would operate rattling well.

But as well as then I read this post close it. Find the Factors is designed by @IvaSallay who says it's an fantabulous agency for children as well as adults to review multiplication facts, role logic, as well as strengthen encephalon power. The puzzles are based on a multiplication grid where most of the numbers own got been removed. The grid tin live on completed using logic as well as cognition of factors - inwards the event below y'all could foremost past times completing the 2 circled cells. The puzzles come upward inwards varying levels of difficulty, as explained here. Find the Factors has already been rattling pop with tweeters who've tried it this week.

4. Chalk Pens

Kev Lister ‏(@ListerKev) tweeted close chalk pens. I've written close writing on desks, windows as well as walls before, only these are novel to me. Kev says the chalk wipes off non-porous surfaces easily with a damp cloth. He uses these pens to write targeted questions or supply scaffolding for selected students. I intend my boss would live on happier with me writing on desks with these rather than whiteboard markers.

5. Alternative Number Line
I'm reluctant to include 2 ideas from Ed Southall (@solvemymaths) inwards i post, only he's a legend thus I'll allow it... In the same agency the maths clock inwards my kitchen makes a nifty talking betoken for guests to my house, his alternative let on line should live on displayed inwards Influenza A virus subtype H5N1 grade classrooms - students (and teachers!) volition live on intrigued. Mr G Walton (@mr_g_walton) has made a fantastic header to larn with it.

Stuart Price (@sxpmaths) suggests making an activity out of this - cutting the let on business upward as well as give it to your Year 13s thus they tin kind it into order. They may non live on familiar with roughly of the annotation only it's a skillful chance for interesting discussions (I flora out how to read ternary to sympathise Ed's alternative for let on 5. I'd also non seen flooring as well as ceiling annotation before).

Stuart also suggests that students pattern their ain let on lines. I honey this thought - it would operate with students of whatever age, from principal schoolhouse to university. It's like to an activity I suggested inwards my reflection stickers. Students stick a 'Not nonetheless yeti' or 'I got it' sticker inwards their majority to allow y'all know whether they've understood the work.

Speaking of stickers, Adil Jaffer (@adil_3) shared his fantastic feedback stickers thus y'all don't own got to proceed writing the same affair all the fourth dimension when you're marking.

One of the post pop tweets I saw this calendar week was from @MathedUp who shared his fantastic 'request a selfie' idea. I tin encounter this actually improving the character of students' operate - many of my students would honey the chance to print their parents. If y'all role iDoceo as well as then it's slow to own got a movie on your iPad as well as e-mail it to a parent.

Did y'all encounter these practice books, tweeted by @dandesignthink? The boxes at the bottom of the pages incorporate a checklist (accuracy, spelling, grammar), cardinal words, farther questions, cardinal points, what went well, as well as feedback. An 'Exercise majority 2.0' costs 50p - amount details are available here. There's also a maths version - Maths majority 2.0 is as awesome. I wishing I had a budget to pass because I'm desperate to gild these!

The to a greater extent than I role Twitter, the to a greater extent than inclined I am to hashtag everything I write (or say!). I own got to bound myself from putting hashtags inwards text messages to my hubby ('what do y'all desire for dinner tonight? #pleasesaytakeaway'). It's prissy to encounter that I'm non lone - Cathy Yenca (post close it. Find the Factors is designed by @IvaSallay who says it's an fantabulous agency for children as well as adults to review multiplication facts, role logic, as well as strengthen encephalon power. The puzzles are based on a multiplication grid where most of the numbers own got been removed. The grid tin live on completed using logic as well as cognition of factors - inwards the event below y'all could foremost past times completing the 2 circled cells. The puzzles come upward inwards varying levels of difficulty, as explained here. Find the Factors has already been rattling pop with tweeters who've tried it this week.

4. Chalk Pens

Stuart Price (@sxpmaths) suggests making an activity out of this - cutting the let on business upward as well as give it to your Year 13s thus they tin kind it into order. They may non live on familiar with roughly of the annotation only it's a skillful chance for interesting discussions (I flora out how to read ternary to sympathise Ed's alternative for let on 5. I'd also non seen flooring as well as ceiling annotation before).

Stuart also suggests that students pattern their ain let on lines. I honey this thought - it would operate with students of whatever age, from principal schoolhouse to university. It's like to an activity I suggested inwards my Open Evening post where y'all concur a 'design a maths clock' competition. The clock pictures below are all taken from sbcrafts.net where y'all tin banking concern friction match out a big attain of clock let on ideas that could also live on used on let on lines.

And finally...
Algebraic long segmentation is fun when y'all know how to do it, only it tin live on a chip tricky to learn to Year 12. It's prissy to know at that spot are alternative methods. I've factorised polynomials past times inspection earlier only non using a grid as suggested by post close it. Find the Factors is designed by @IvaSallay who says it's an fantabulous agency for children as well as adults to review multiplication facts, role logic, as well as strengthen encephalon power. The puzzles are based on a multiplication grid where most of the numbers own got been removed. The grid tin live on completed using logic as well as cognition of factors - inwards the event below y'all could foremost past times completing the 2 circled cells. The puzzles come upward inwards varying levels of difficulty, as explained here. Find the Factors has already been rattling pop with tweeters who've tried it this week.

4. Chalk Pens

Stuart Price (@sxpmaths) suggests making an activity out of this - cutting the let on business upward as well as give it to your Year 13s thus they tin kind it into order. They may non live on familiar with roughly of the annotation only it's a skillful chance for interesting discussions (I flora out how to read ternary to sympathise Ed's alternative for let on 5. I'd also non seen flooring as well as ceiling annotation before).

Stuart also suggests that students pattern their ain let on lines. I honey this thought - it would operate with students of whatever age, from principal schoolhouse to university. It's like to an activity I suggested inwards my The Art of Collaboration', it's e'er nifty to take heed when these ideas larn far into classrooms. Have a nifty half-term intermission everyone!

A House For Gimmicks?

A friend latterly used the give-and-take 'gimmicks' inwards reference to only about of the ideas I portion on this website. I got a fleck defensive but I retrieve it raises an of import point.

I'm good aware that role of teachers is to impart knowledge. I'm paid to fill upward children's minds amongst mathematics. I demand no equipment, no facilities, no tools - I only demand my brain, my vocalisation as well as an audience. I haven't lost sight of that. But if I receive got tools at my disposal that aid brand my explanations clearer, aid inspire as well as engage my students, improve my assessment as well as feedback, as well as brand my teaching experience to a greater extent than enjoyable, thence I volition utilise those tools.

What is a gimmick?
The give-and-take gimmick originally meant a slice of magician's apparatus. It similar a shot refers to 'a play a trick on or device intended to attract attention, publicity, or trade'. In teaching you lot could come across it every bit anything that is at that topographic point primarily for the role of engagement, enjoyment or novelty value. Examples may include writing on desks, post-it banker's complaint activities, Plickers, stickers as well as stamps. So I acknowledge that only about of the ideas as well as resources I characteristic inwards my weekly gems posts are a fleck gimmicky. I convey that. Some of my posts focus on the procedure of imparting mathematical noesis (like my posts on trigonometry, Pythagoras, calculus and fractions) as well as others are to a greater extent than concerned amongst techniques to engage students as well as brand the learning procedure to a greater extent than enjoyable.

Gimmicks gone wrong
In my NQT year, I took on a smart but chatty Year eleven class. We finished the GCSE syllabus past times Christmas as well as they were all on rails for A*s. After their mock exams, I had to notice a way to move along them engaged as well as focussed for a whole term. I started past times reviewing topics from Key Stage 3 but I felt that I was patronising them. We did only about past times papers but they didn't focus. By Feb half-term I was dreading every lesson thence I asked my colleagues for advice. I was advised to acquire my students to learn only about revision lessons. So I did - they worked inwards groups as well as took it inwards turns to deliver maths lessons over a 3 calendar week period. To live frank, these lessons turned out to live absolutely useless. No pupil made whatever progress whatsoever. At the terminate of the procedure no-one inwards the shape knew whatever to a greater extent than mathematics than they'd known 3 weeks earlier. The exclusively someone who'd learnt anything was me - I learnt what it felt similar to sit down inwards a truly badly taught lesson. This was a classic event of a gimmick gone incorrect - I'd prioritised amusement over mathematics.

I was real aware that the whole endeavor had been a huge waste materials of time. Even worse, it may receive got truly been damaging - I'd managed to undermine my ain potency past times delegating my role. I never managed to acquire them to focus again. Quite a few students from that shape got a bird H5N1 inwards their GCSE when they were capable of getting an A*. I yet experience responsible for that.

Fast-forward a duet of years as well as I institute myself inwards the same predicament. This fourth dimension I'd done only about vivid revision lessons amongst my novel Year eleven class, peculiarly inwards reviewing the topics that are ofttimes overlooked inwards GCSE revision. But i time to a greater extent than I had a lot of lessons to fill, as well as i time to a greater extent than I was advised past times my colleagues to acquire the students to learn the lessons. I was told that the argue it hadn't worked before was because I hadn't done it right. I decided to seek it i time to a greater extent than but amongst a novel approach. I shared the criteria against which I'd assess the lessons (for event they had to pattern their ain worksheet) as well as I gave my students a listing of things to consider:

I gave them to a greater extent than fourth dimension to project design their lessons as well as to a greater extent than back upward as well as guidance inwards the planning process. Did it work? Well, it was ameliorate than the get-go time. The lessons as well as resources were a much higher quality. The students enjoyed doing the teaching as well as everyone was engaged inwards the lessons. Some progress was made (a negligible sum - as well as that was mainly when I interrupted lessons to highlight misconceptions). They learnt far far less maths inwards the 3 weeks of student-taught lessons than they would receive got learnt if I'd taught those lessons.

I produce know teachers who insist that this approach industrial plant good if properly managed. They retrieve these lessons operate good because the students are engaged as well as enjoying themselves as well as developing 'soft skills'. I state these lessons don't operate good because no mathematics is learnt.

These ii things are non mutually exclusive - students tin hand notice both bask maths lessons as well as larn mathematics. The pleasance of mathematics comes inwards learning something novel as well as applying it to solve a problem.

So what should I receive got done amongst these classes to fill upward that fourth dimension betwixt the mocks as well as their lastly exams? Well I could receive got given them only about rich tasks that consolidate skills as well as noesis across a break of topics (Don Steward has only about first-class examples). In add-on I could receive got enriched their instruction amongst interesting mathematics that is non on the GCSE syllabus. I could receive got talked almost the history of mathematics, etymology, peculiarities... I could receive got explored break systems, Fermat's Last Theorem, primes, matrices, gear upward theory... Such a wasted opportunity. Next twelvemonth - no to a greater extent than gimmicks for me.

Don Steward GCSE revision

A house for gimmicks?
If I had to selection a side inwards the traditional vs progressive teaching debate, I'd selection traditional. That agency I come across a lot of value inwards teachers giving clear explanations as well as students developing fluency through independent practice. I come across less value inwards grouping operate as well as enquiry approaches. However, i of the things I bask almost teaching is that I receive got opportunities to live creative inwards my approaches. I bask experimenting amongst innovative teaching tools as well as technology. But inwards doing thence I don't lose sight of my primary priority, which is teaching mathematics.

If I desire to work dots as well as stickers for marking, it may or may non live adding value but it's non obstructing learning is it? If I work Plickers for assessment, it doesn't hateful I've stopped imparting noesis does it? These 'gimmicks' may live useful. They're for certain non detrimental. Other gimmicks, similar getting students to produce my undertaking for me, are detrimental.

I'm trying to notice my way to becoming an 'excellent' instructor as well as I'll seek a broad attain of tools as well as teaching approaches until I acquire there. I'm learning from my mistakes every bit I go.

So, I've decided - I'll move along writing my regular gems posts, where I portion teaching ideas I've seen on Twitter. And yes, only about of them volition live gimmicky. Some ideas volition live to a greater extent than useful than others. But I'll also focus my thinking (and my writing) on the of import questions: what mathematical concepts should nosotros live teaching as well as how tin hand notice nosotros best explicate those concepts? First up: circle theorems. Watch this space.

Too traditional...?

5 Maths Gems #12

This is going to live on a relatively brusk gems ship service because I'm exhausted from entertaining xix toddlers this afternoon at my daughter's 3rd birthday party. So I'll live on brief today, but facial expression a lengthier ship service side yesteryear side calendar week when I've recovered.

1. Area of a circle (again)
In my seventh gems post I shared an applet that beautifully demonstrates how to derive the surface area of a circle yesteryear cutting it into sectors. This calendar week I establish a vivid gif that shows the same thing.

This gif is taken from the animatics (animations + mathematics) section of @MathsChallenges' website. The thinking behind this method is neatly explained in Geometry: What is the intuition behind the formula for the surface area of a circle? which was shared this calendar week by @GeometryDad.

2. Standard form

'resources library has or too thence corking ideas.

3. H5N1 grade resources

I saw this tweet from @DrTrapezio...

too thence I checked this out too establish that the Cambridge Mathematics Education Project is inviting schools to register every bit affiliates to gain early on access to their gratuitous H5N1 grade maths resources nether development. I'm e'er on the facial expression out for H5N1 grade resources too thence I registered. There's loads of corking materials on the website too I facial expression forrad to seeing the collection develop.

Some of the CMEP's resources are forthwith featured on NRICH. For example, the question below is vivid (and non every bit difficult every bit it looks...)

I too similar the 'always, sometimes, never' bill of fare form 'Discriminating'.

Extract from Discriminating

And Parabella is a lovely work for your Year 12s studying coordinate geometry.

4. Operation Countdown

H5N1 combination of ideas too inventiveness from @LWmathsmatters, @EJmaths too @taylorda01 led to this first-class 'Operation Countdown'. I frequently role the 'four threes' activeness every bit a plenary inward an monastic say of operations lesson amongst Year 7 but it industrial plant amongst whatever historic menstruum group. I in i lawsuit used it every bit a Year xiii starter too they actually enjoyed it. The version below involves 5 threes but it's the same idea.

This film is from @taylorda01. @EJmaths also made this PowerPoint which includes a dark too white version.

5. #units #ruler #workings

I read the ship service 'Hashtag closure' yesteryear A identify for gimmicks?'). David's ship service is close using a hashtag to summarise a lesson.

Then A Frenzy of Marking: 'Progress Over Time' close feedback stickers. When I mark, I create notice that I write the same thing over too over again, too thence stickers would salve me a lot of time. Problem is, I'm non organised plenty to role stickers. I create my mark inward a multifariousness of locations too unremarkably but receive got a reddish pen to hand. Hashtags volition function but every bit good every bit stickers, every bit long every bit my students empathize the important of the hashtags.

When I grade I'm non succinct enough. For lawsuit I frequently write 'please larn to assistance companionship at 1.30pm on Midweek inward Room 111 for assistance amongst this topic'. Instead, I could write #helpclub. I too write things similar 'Look at the enquiry - yous were meant to circular your reply to i important figure'. I could but write #rounding. My students volition larn what I hateful (especially if they receive got this key inward their practise books) too it volition salve me loads of time. "What's the quest inward the # symbol?" I take away heed yous inquire - well, yes, the symbol isn't essential, but #MrsMorganLikesHashtags too they're non doing whatever terms are they? :)

So that's it - 5 quick gems from a busy mathematical mum. I was planning to characteristic or too thence thoughts close the value (or lack of value) of iPads inward maths didactics today but didn't larn fourth dimension to create the enquiry required, too thence volition salve that for my side yesteryear side gems post.

I'm too planning to write a ship service close circle theorems side yesteryear side week. I've got or too thence ideas but if you

receive got an interesting means of teaching circle theorems or whatever amazing resources, delight allow me know.

Also, don't forget to cheque out my other website, prethomework.weebly.com - the extensive collection has grown over half-term.

I'll larn out yous amongst a base of operations eleven abacus which I squall upwards is cute but volition annoy @solvemymaths (see his ship service 'My Base eleven Abacus Bugbear').

5 Maths Gems #13

In my gems posts I portion approximately of the best teaching ideas I've seen on Twitter each week. This is my 13th gems post. I promise you're non triskaidekaphobic.

1. Flying Numbers
At Julia Smith's (@tessmaths) creativity workshop at the lastly maths conference, she shared an activeness inward which students brand 'Flying Numbers' to hang from the ceiling inward the classroom or corridor. It industrial plant similar this: they select a discover together with brand that discover from forest inward Design & Technology (drill a hole at the overstep for hanging). They pigment their discover hence purpose fine markers or chalk pens to decorate it alongside facts almost that number. This is a lovely chance for students to purpose the cyberspace to explore discover properties together with discovery out all sorts of wonderful mathematical things that are non on the curriculum. The examples below were made at the maths conference - it was a actually enjoyable exercise.

Ed Southall (post almost teaching a binary lesson).

Alice inward FractalLand is a stunning animated slideshow. Show it to your degree together with explore sequences, fractals, the golden ratio together with more. The text at the bottom explains each concept real clearly. I love this! This is precisely the variety of affair I was referring to inward my post service 'A house for gimmicks?' when I said that nosotros should brand fourth dimension to enrich our students' teaching alongside interesting mathematics that is non on the GCSE syllabus.

The Primary and Secondary Treasure Hunts are real good designed. Full instructions are included together with at that topographic point are questions on areas of mathematics such equally cryptography, primes together with probability. The questions are rather challenging for the intended historic menses groups - if I was organising an inter-school maths challenger hence I mightiness purpose these questions.

The video collection is due to grow - an exciting prospect if this 2 infinitesimal wordless video exploring triangles is a gustation of things to come.

3. Imbalance Problems
If you're non familiar alongside Don Steward's resources hence halt reading this post service straightaway together with depository fiscal establishment friction match out his mobiles', hence 'mobile inequalities' together with finally 'mobile moments'.

In mobiles, the activities foremost alongside discover problems that learn progressively harder, hence motion on to algebraic balancing problems. Here's a numerical instance which tin displace live solved alongside arithmetics together with logic (fill inward the missing weights hence that everything balances correctly).

There are lots of variations, such equally this 1 where the weights must add together upward to the discover inward the triangle (this is a straightforward example):

The algebraic problems educate skills inward writing algebraically together with simplifying expressions. In the work below, students accept to consummate the missing weights to brand the combined hanging weight equal to 8n.

These are swell problems. Check out all 3 posts to run across how Mr Steward developed the idea. It's equally good worth reading Paul Salomon's (last week's gems post has paid off - both Debbie Hart ‏(@Debbie_Hart_UK) and Jane Appleton (‏@JaneAppleton24) accept tried it.

I've been busy writing a post service almost circle theorems which volition live cook soon. Also, later seeing approximately swell Halloween together with Guy Fawkes maths resources, I've started to collate a 'seasonal resources' page. I've seen hence many skillful Christmas resources, it's difficult to select the best simply I'm getting there.

So at that topographic point you lot go, my 13th gems post. It wasn't scary at all.

Image source: guernseydonkey.com

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.