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 #19

Hello in addition to welcome to my 19th gems post. This is where I percentage but about of the best education ideas I've seen on Twitter each week. Most of my gems accept come upward from across the Atlantic today.

1. Polygraph
There's been a lot of buzz almost Desmos' novel Polygraph activities. They're mathematical versions of Guess Who, designed to 'foster the pleasance in addition to the ability of words without the drudgery of the lists'. There are currently 4 Polygraph activities:

Let's expect at Polygraph: Parabolas. Students operate inwards pairs (a picker in addition to a guesser) on split tablets, phones or computers. The picker selects a graph from a prepare of parabolas. The guesser sees the same prepare of parabolas - their employment is to position which graph the picker has chosen past times quest a serial of yes/no questions. For instance they mightiness inquire almost the release of roots or the place of the vertex or y-intercept. The guesser types their questions in addition to the picker responds amongst yes, no, or don't know. The guesser keeps quest questions until they are able to position the chosen graph.

Students hence accept the chance to analyse in addition to refine their questioning, in addition to yous accept a degree discussion to formalise the mathematical vocabulary.

As amongst Desmos' awesome Gems 13, it's incredibly quick in addition to slow to larn started. I'm going to endure education quadratics to Year 10 presently in addition to can't hold off to purpose the Parabolas activity. And I programme to purpose the Linear Graphs version amongst my Year 9s too.

Extract from Polygraph: Lines

2. Point, Point, Gradient

Tina Cardone (@crstn85) shared this lovely activity 'Point, Point, Slope' past times Michael Fenton (@mjfenton). It's a elementary but effective lesson view inwards which students accept to purpose whatsoever digit from 2 to nine to create 2 points - yous could laissez passer them post-its or release tiles to shuffle around. You furnish criteria, such equally 'create 2 points which create upward one's heed a draw amongst the greatest possible gradient'. Read the amount post for to a greater extent than ideas. There's loads yous could create amongst this activity, for instance yous could inquire your students to use 6 unlike digits (from 0 - 9) to create 3 points which prevarication on the same line. It's a expert means to educate agreement of linear graphs in addition to I recall it would operate good at both GCSE in addition to AS level.

Michael's spider web log is amount of splendid ideas. For instance in Two Fractions students house digits in addition to operators inwards the boxes below to brand an facial expression amongst the greatest possible value.

This reminds me of the Blanks activities I featured inwards Gems 17, which yous tin laissez passer notice read to a greater extent than almost inwards Sarah Aldous' post Gems 13 I talked almost making your students 'Number Experts'. I shared 2 websites that are useful for finding out properties of numbers (numbergossip.com in addition to numdic.com). Both websites are splendid - for example, numbergossip.com tells me that 28 is a perfect, composite, happy number, and numdic.com (which wins the prize for funniest URL) gives me the binary in addition to Roman Numeral representations of 28. I've right away discovered a novel website - the Number Property Calculator - which I similar because of the means it gives data almost the important of the release properties. Here's a small extract from what comes upward when I search for 28:

I constitute this website when I was browsing through onetime posts on Math Munch. Math Munch is a 'Weekly Digest of the Mathematical Internet'. It's packed amount of fantastic resources in addition to ideas hence it's definitely worth subscribing to have weekly e-mail updates, in addition to yous tin laissez passer notice follow them on Twitter likewise (@MathMunch).

5. Drawing Euclid

Andy Shaw (@Squidworm74) pointed me inwards the direction of these lovely Euclid videos on YouTube past times Shoo Rayner (@shoorayner). They are fantastic videos in addition to I can't hold off to percentage them amongst my Year vii students. Check out the amount playlist here. Here's an instance almost unlike types of triangles:



That's it for this week. I promise you've constitute but about inspiration inwards this post. I'll travel out yous amongst this bright video past times the always awesome Vi Hart - The Gauss Christmath Special. Thanks to Chris Smith (@app03102) for sharing this.




Have a wonderful Christmas!

5 Maths Gems #32

Hello too welcome to my 32^nd gems post. This is where I part teaching ideas too resources I've seen on Twitter.

1. Expression of the Day

You mightiness already last familiar with Jeremy Denton's (Number of the Day. He's right away created Expression of the Day too. These activities tin last used to ensure that students oft practise their position out too algebra skills. Do banking firm tally out Jeremy's weblog post Starter Activity - Expression of the Day for the PowerPoint file too an illustration of how to piece of job this fantastic resource.

2. Maths Documentaries

In the final lesson or 2 of the schoolhouse twelvemonth some teachers present videos during maths lessons, which I intend is fine every bit long every bit students are learning nigh maths from those videos. Not certain what to show? Thanks to @AlgebraFact for sharing this page of Mathematics Documentaries from CosmoLearning - enough to pick out from here.

For to a greater extent than ideas for halt of term lessons, run across my post 'End of term!'.

3. Angles inwards a Polygon
Thanks to @sporteredu for sharing these pictures showing a practical demonstration of the outside angles of a polygon summing to 360^o.  

@ProfSmudge shared a similar laid of pictures of an irregular pentagon.

I've ever used an animation to demonstrate this too convey never thought of doing this lovely paper-based alternative.

For to a greater extent than animations run across my post Animations too Simulations.

4. Reasoning Statements

Thanks to Mel (@Just_Maths) for sharing this awesome circle theorems activeness 'Great angle chase'. I honey activities similar this - it's my favourite form of maths problem.

I was real pleased that Mel shared this link because it led me to a resources website I'd never seen before. I've right away added lots of resources from Top Drawer Teachers to my resources libraries. 

I especially liked these reasoning statements -  a groovy listing of propositions for class discussion. The instructor presents a suggestion too asks questions such every bit "Is it true?", "Is it simply sometimes true, or is it or ever true?", "When is it true? How do y'all know?",  "How could nosotros demonstrate/show/prove that it is true?".  Examples include:

• The hateful of iii consecutive numbers is the middle number.
• Triangles tessellate. 
• Volume = expanse of the halt × length 
• If the H2O inwards our dams increases past times 10% this month, too thence decreases past times 10% adjacent month, it volition last dorsum to where it started.

5. Zoolander

Mary Pardoe (blog post past times Simon Gregg (@Simon_Gregg) nigh teaching Year 4 nigh scaling. It features a Zoolander enquiry from Robert Kaplinsky (@robertkaplinksy). Fans of Zoolander volition similar this!

Students demand to intend nigh how many times would y'all demand to scale a model upwards to brand a existent school. Questions to enquire too possible approaches are detailed on robertkaplinksy.com.

Update

here).

Did y'all run across my page 'New GCSE Support' which links to resources for the novel GCSE topics? I'll last adding a lot to a greater extent than resources over the adjacent duet of months. You tin access this page from the 'Extras' card on my weblog (top correct - unless you're viewing the mobile site inwards which illustration you'll demand to click on 'view spider web version' to run across this menu).

The London Maths Hubs are joining forces to run a conference on Th ninth July. It looks similar a practiced pick of speakers too workshops thence if you're based inwards or close London it would last worth bespeak for a twenty-four hr catamenia out of schoolhouse thence y'all tin attend.

There's some lovely novel resources coming from my favourite websites lately - do banking firm tally out the recent offerings from MathsPad, Teachit Maths too Median Don Steward. 

This article nigh the relative difficulty of maths exams to a greater extent than or less the footing is interesting. I similar these box plots:

• Finally, Billy Adamson asked people to depict maths inwards 1 word too here's the fantastic results:

Have a groovy calendar week everyone!

5 Maths Gems #38

Hello in addition to welcome to my 38^th gems post. This is where I portion 5 teaching ideas I've seen on Twitter. As teachers start making plans in addition to preparing resources for September, there's been a flurry of inspiring ideas on Twitter - I tin barely pop off along up!

1. Tessellation
It's quite rare that there's exciting word inwards the basis of mathematics, simply we've late seen the uncovering of a novel type of Surrounding a Point activity from Don Steward pictured below. Mathigon has a lovely tessellation department featuring overnice pictures in addition to an online tool.

I've never seen tessellation animations earlier then thank y'all to @MathsMastery for sharing these, I'll role them the adjacent fourth dimension I learn tessellation.

2. Displays

There's been lots of tweets in addition to posts almost displays this week. I similar these lovely vinyl corridor displays from @cOmplexnumber.

I also similar this increment mindset display from English linguistic communication teacher Rebecca Foster (Gems 3, simply would jibe on a smaller noticeboard. 

One of my favourite classroom display ideas is the judgement stems display from Gems 26. It's practiced to bring displays that are of existent role to students throughout the year.

It's worth reading Paul Collins' (post almost the novel displays he's set upward inwards his classroom in addition to corridors.

3. History of Maths
I don't know really much almost the history of maths in addition to that's a nifty shame. It's a priority for me to ameliorate my dependent area cognition inwards this area. I intend to start past times aid @Kris_Boulton's workshop at the adjacent maths conference.

Kris' post A History of Maths features a timeline showing people in addition to societies of telephone substitution influence inwards mathematics. It also shows telephone substitution publications in addition to ideas. The picture below is alone a really pocket-size extract - bring a facial expression through the total timeline. It's fascinating to come across the social club of things. Perhaps display this inwards your classroom in addition to say your students almost the origins of the topics they're studying.

4. Challenge Questions for Year 12
Stuart Price ‏(@sxpmaths) has collated to a greater extent than or less AS stretch in addition to challenge questions for total topics. They tin locomote accessed here. Thanks Stuart for this helpful resource.

If y'all learn H5N1 score in addition to are coming to the maths conference inwards Sheffield on 26^th September, produce come upward along to my workshop almost H5N1 score resources.

5. Exit Tickets
Many maths teachers role locomote out tickets inwards every lesson. The stance is that y'all laid a unmarried inquiry or brusk laid of questions at the halt of the lesson in addition to analyse the results. Typically students would manus their locomote out ticket to y'all every bit they leave of absence the classroom. It's of import to pop off along the questions elementary – if students larn them incorrect y'all involve to know why. It's an effective agency of identifying misconceptions in addition to deciding whether students are create to motility on. 

Lesley Hall (RAG123.

Update
I've published 4 weblog posts inwards the concluding calendar week in addition to I bring to a greater extent than or less other 4 posts inwards draft - I actually involve to halt all the blogging in addition to pass to a greater extent than or less fourth dimension preparing my showtime lessons! Here's what I've been upward to since my concluding gems post:

• I wrote an article almost conferences for the NCETM Secondary Magazine.
• I wrote a post Behaviour Management for Beginners which features a few resources I've prepared for September - my expectations videos bring been popular.
  

• I was chuffed that my weblog appeared on Vuelio's Words to Avoid page - these are definitely worth a read.
• I was really pleased that @oldandrewuk in addition to @Just_Maths set upward an Echo Chamber specifically for United Kingdom of Great Britain in addition to Northern Ireland of Britain in addition to Northern Republic of Ireland maths didactics blogs. Follow @MathsEcho to pop off along up-to-date amongst all the latest posts.
• I popped into schoolhouse and, borrowing a lovely stance from Danielle Bartram (@MissBsResources), set Maths Mr Men stickers from @solvemymaths on my new calculators then I tin pop off along rail of them.

Finally, 4 reminders:

• If y'all haven't already done so, y'all tin 'like' my new Facebook page for updates.
• #mathstlp volition furnish at 7pm on Dominicus 30^th August for lesson planning advice.
• My tangents in addition to areas is worth reading if you're teaching GCSE adjacent year.
• Check out my hashtag #mathsgems for inspiring ideas from previous gems posts.

That's it from me. I'll leave of absence y'all amongst this overnice puzzle from the ATM (@ATMMathematics).

5 Maths Gems #42

Hello in addition to welcome to my 42^nd gems post. This is where I part roughly of the best instruction ideas I've seen on Twitter.

1. Resources
To start, a few novel resources that yous mightiness non cause got seen.

Miss Norledge (@MissNorledge) drew inspiration from a Maths Feast problem to brand 3 coordinate geometry problems of her own. These lovely 'Triangle Regions' tasks are suitable for GCSE students or Year 12s.

The clever folks at MathsPad croak along to create fantabulous resources - this Always/Sometimes/Never True activity for Fractional Indices is excellent. The solutions render rattling helpful explanations too. I've added this to my resource library.

Here's roughly other corking novel resources from MathsPad - How many ways tin dismiss nosotros write 1 million? These are fantabulous activities for consolidating cognition of indices in addition to criterion form.

Check out mathspad.co.uk for to a greater extent than fantastic resources. And if you're on Twitter, Nicola (@MathsPadNicola) in addition to James (@MathsPadJames) are good worth following.

2. Enrichment for Sixth Form Mathematicians
Thanks to Graham Colman (@colmanweb) for sharing his listing of Maths Beyond the Curriculum activities. This listing is written for Year 12/13 - it would last especially useful for a pupil who intends to study maths at university. It could easily last adapted for other twelvemonth groups too. Here's an extract:

3. m4ths.com
Steve from m4ths.com teaches at Thomas Clarkson Academy inwards Wisbech. He has been sharing roughly bright novel resources recently. Here's a few examples.

His 'A flat page too.

4. Target Stickers
Target grades are an obsession inwards many schools. I'm non a fan. I tell my students to aim for hap grades regardless of what their target says. After all, targets are largely based on sweeping generalisations in addition to dodgy data.

Most teachers (myself included) are regularly required to study on progress against targets. I similar the sticker persuasion below from Charmayne Bailey (@MissBaileyMaths). The scale makes it actually clear to students where they stand, what their minimum target degree is, in addition to where they're heading.

5. GCSE Revision Ideas
I honey this persuasion from Mel (Some Things I've Tried inwards which I shared roughly novel instruction ideas I've been trying this term

In Case You Missed It... in which I shared roughly ideas in addition to resources that cause got been good received on Twitter

Five to Follow #ff in which I recommended 5 creative maths teachers to follow on Twitter.

I enjoyed participating inwards the bit Maths Journal Club chat - yous tin dismiss read a write-up of the intelligence here.

I used the fractions activity below alongside my Year 7s (I outset featured this activity inwards my post Thoughts on Teaching Fractions). I was quite taken aback yesteryear how revealing this chore was - it uncovered roughly serious misconceptions.

Ticket sales for my exciting Christmaths Party are going well.  There's solely 25 twenty-four hours tickets left in addition to then majority now! It's going to last a fantastic consequence - run into christmaths.weebly.com for details.