Tricks Too Tips 1: Hcf

I lately presented a workshop at the National Mathematics Teacher Conference (#mathsconf2015) entitled 'Tricks as well as Tips: Clever Methods for Explaining Mathematical Concepts'. This postal service summarises the content of that workshop for those who were unable to attend. I receive got quite a lot to comprehend therefore I human face I'll demand to write 3 or 4 weblog posts. In this i I'm going to explicate the rationale for the workshop as well as depict option methods for finding a Highest Common Factor. In subsequent posts I'll comprehend sequences, linear graphs, surds, quadratics, chemical compound measures as well as a few to a greater extent than bits as well as pieces.

Workshop aim
How create you uncovering the Highest Common Factor of 2 numbers? Do yous piece of occupation a Venn method? List the factors? Use the Euclidean Algorithm? Are in that location 'better' methods that yous don't know about? (how create nosotros define 'better'?). These are the variety of questions I desire to explore. The aim of my workshop was to encourage people to reverberate on their dependent area noesis as well as the effectiveness of their explanations. I also hoped that everyone would larn novel methods that they mightiness consider using at school.

What determines the agency nosotros conduct the explicate things?

Most teachers flora pedagogy habits during their grooming as well as NQT year. The method they piece of occupation the start fourth dimension they learn a topic volition in all likelihood remain with them throughout their career - unless they brand a concious effort to assay a novel method. 

One of the few things I recall nearly GCSE maths was that I solved equations yesteryear 'moving price over the equals sign' (the 'magic portal method'). This method is straightaway considered to hold out a shortcut which stands inwards the agency of conceptual understanding. These days it is to a greater extent than acceptable to learn students to solve equations using inverse operations (ie residue the equation yesteryear 'doing the same thing' to both sides). It's lucky that during my PGCE someone told me non to piece of occupation the magic portal method because earlier therefore I was convinced that it was 'the' agency to solve equations. 

Source: https://mrjasonto.wordpress.com

When nosotros were pupil teachers, nosotros drew our methods from a multifariousness of sources. ITT courses don't comprehend much inwards the agency of mathematical methods, therefore nosotros were left to get together ideas from examples inwards textbooks, our memories from school, observations of teachers as well as conversations with colleagues. It's these conversations with colleagues that are vital, but nosotros only don't acquire plenty fourth dimension for them. Most of the novel methods I've encountered over the years receive got non been through organised CPD (ie 'collaboration sessions') inside my department, but instead through jeopardy encounters. My start always weblog postal service was nearly an option method for matrix multiplication that I'd happened to stumble across online.

During my PGCE I was asked to learn a lesson on Highest Common Factor. I recall my mentor showing me the 'Venn Method'. As a final result of that conversation I used the Venn Method for years, until yesteryear jeopardy a colleague mentioned an alternative. Let's human face at that option method now, as well as a release of others. Will yous stick with the method yous know as well as love, or volition yous assay something new?

Highest Common Factor as well as Lowest Common Multiple

I've identified 6 methods for finding the HCF as well as LCM of 2 numbers. I'll explicate each method hither as well as position whatever pros as well as cons.

1. Listing

Source: mathx.net

There's no impairment inwards the listing method. It's brilliant inwards price of underlying conceptual agreement - students tin view precisely what they're trying to arrive at here. List all the factors of the 2 numbers as well as uncovering the biggest release that's inwards both lists - that's your HCF. List all the multiples of 2 numbers as well as uncovering the smallest release that's inwards both lists - that's your LCM. Simple! Shame it's therefore time-consuming. And, inwards my experience, students sometimes immature lady factors from their list. One agency to avoid this is yesteryear listing factors using a pairing method similar this:

Factor rainbows are a pretty option (see this article from the NCTM).

2. The Venn Method

This is a pop method inwards the UK. First, nosotros demand to create a prime number gene breakdown. By the way, if you're pedagogy prime number factorisation therefore yous mightiness similar these lovely magic portal method'). This method is straightaway considered to hold out a shortcut which stands inwards the agency of conceptual understanding. These days it is to a greater extent than acceptable to learn students to solve equations using inverse operations (ie residue the equation yesteryear 'doing the same thing' to both sides). It's lucky that during my PGCE someone told me non to piece of occupation the magic portal method because earlier therefore I was convinced that it was 'the' agency to solve equations. 

Source: https://mrjasonto.wordpress.com

When nosotros were pupil teachers, nosotros drew our methods from a multifariousness of sources. ITT courses don't comprehend much inwards the agency of mathematical methods, therefore nosotros were left to get together ideas from examples inwards textbooks, our memories from school, observations of teachers as well as conversations with colleagues. It's these conversations with colleagues that are vital, but nosotros only don't acquire plenty fourth dimension for them. Most of the novel methods I've encountered over the years receive got non been through organised CPD (ie 'collaboration sessions') inside my department, but instead through jeopardy encounters. My start always weblog postal service was nearly an option method for factor tree activities from Don Steward.

Once yous receive got the prime number factors of each number, push clitoris a Venn diagram as well as house the mutual factors inwards the intersection of 2 sets, as shown inwards the illustration below.

tutorvista.com

The HCF is the production of the elements inwards H5N1 intersection B (ie 2 x 2 x 2 x 2 inwards the illustration above) as well as the LCM is the the production of all the elements inwards H5N1 matrimony B (ie 2 x 2 x 2 x 2 x 2 x 2 x 5). Note that United Kingdom of Great Britain as well as Northern Ireland of Britain as well as Northern Republic of Ireland GCSE students are non even therefore familiar with this terminology (ie intersection as well as union), but they volition hold out nether the novel GCSE syllabus.

Even though I taught the Venn method for years, I'm non a huge fan of it. In my experience, students are ok with filling inwards the Venn diagram but therefore they oft can't recall which 'bit' is the HCF. If they create recall the method therefore they in all likelihood don't receive got a clue why it works.

Confusingly, it seems that some people piece of occupation a dissimilar Venn method which involves putting all factors (not prime number factors) into a Venn diagram as well as identifying the highest gene inwards the intersection (see illustration below). This is some other cast of the listing method described higher upwards - it's just a dissimilar agency of organising the list. Let's telephone band this Lenn Method - it's a hybrid of Venn as well as Listing.

http://edtech2.boisestate.edu/brianroska/506/finalproject/gcf.html

3. Prime Factor Pairing

An alternative to the Venn Method is to create the prime factorisation but therefore skip the Venn. Write the prime number factors of each release out as shown inwards the illustration below therefore it's slow to view which factors appear inwards both release - the production of these is the HCF. This method is featured inwards magic portal method'). This method is straightaway considered to hold out a shortcut which stands inwards the agency of conceptual understanding. These days it is to a greater extent than acceptable to learn students to solve equations using inverse operations (ie residue the equation yesteryear 'doing the same thing' to both sides). It's lucky that during my PGCE someone told me non to piece of occupation the magic portal method because earlier therefore I was convinced that it was 'the' agency to solve equations. 

Source: https://mrjasonto.wordpress.com

When nosotros were pupil teachers, nosotros drew our methods from a multifariousness of sources. ITT courses don't comprehend much inwards the agency of mathematical methods, therefore nosotros were left to get together ideas from examples inwards textbooks, our memories from school, observations of teachers as well as conversations with colleagues. It's these conversations with colleagues that are vital, but nosotros only don't acquire plenty fourth dimension for them. Most of the novel methods I've encountered over the years receive got non been through organised CPD (ie 'collaboration sessions') inside my department, but instead through jeopardy encounters. My start always weblog postal service was nearly an option method for this post yesteryear Don Steward.

4. Euclidean Algorithm

I dear the subtraction-based Euclidean Algorithm. It sounds complicated but it's incredibly easy. Try a few examples yourself to view how straightforward it is.

The method (including why it works) is explained inwards James Tanton's video below. I actually similar this method but for some argue I'm hesitant to piece of occupation it with students... Would you?

Note that this method doesn't plough over yous the Lowest Common Multiple, but it's easily found i time you've got the Highest Common Factor. 

This looks similar a pain, but cancelling helps - inwards the illustration higher upwards I found the HCF of sixty as well as 84, therefore to uncovering the LCM I multiply sixty yesteryear 84 as well as dissever yesteryear the HCF.

5. The Indian Method

I've stopped using the Venn Method at schoolhouse as well as straightaway piece of occupation this instead. I'm non certain it's actually called the Indian Method, I'm solely calling it that because of this video. At my schoolhouse nosotros telephone band it the Korean Method because a Korean pupil introduced it to us! I've found that my students actually similar it. It's difficult to acquire wrong. It's slow to explore why it industrial plant too.

Say nosotros desire to uncovering the Highest Common Factor as well as Lowest Common Multiple of 315 as well as 420. Write downward the 2 numbers, therefore (to the left, as inwards my illustration above) write downward any mutual factor. I've chosen 5. Now dissever 315 as well as 420 yesteryear v as well as write the answers underneath (63 as well as 84 inwards this case). Keep repeating this procedure until the 2 numbers receive got no mutual factors (ie 3 as well as 4 above). Now, your Highest Common Factor is only the production of numbers on the left. And for the Lowest Common Multiple, uncovering the production of the numbers on the left as well as the numbers inwards the bottom row (to uncovering the LCM, human face for the L shape).

6. The 'Upside Down Birthday Cake Method'

I should scream this method because I proceed spotting it online (video here). The solely departure betwixt this as well as the Indian Method is that hither nosotros tin solely take prime factors. This seems unnecessary - the Indian Method is quicker. In the illustration below, why dissever yesteryear 2 if yous location larger mutual factors? Why non start yesteryear dividing yesteryear 4, 6 or 12?

Source: https://uk.pinterest.com/pin/13229392628819982/

So that's it - 6 methods for finding a HCF as well as LCM.

Integers
I actually similar this magic portal method'). This method is straightaway considered to hold out a shortcut which stands inwards the agency of conceptual understanding. These days it is to a greater extent than acceptable to learn students to solve equations using inverse operations (ie residue the equation yesteryear 'doing the same thing' to both sides). It's lucky that during my PGCE someone told me non to piece of occupation the magic portal method because earlier therefore I was convinced that it was 'the' agency to solve equations. 

Source: https://mrjasonto.wordpress.com

When nosotros were pupil teachers, nosotros drew our methods from a multifariousness of sources. ITT courses don't comprehend much inwards the agency of mathematical methods, therefore nosotros were left to get together ideas from examples inwards textbooks, our memories from school, observations of teachers as well as conversations with colleagues. It's these conversations with colleagues that are vital, but nosotros only don't acquire plenty fourth dimension for them. Most of the novel methods I've encountered over the years receive got non been through organised CPD (ie 'collaboration sessions') inside my department, but instead through jeopardy encounters. My start always weblog postal service was nearly an option method for set of questions from Don Steward. The next query confused i of my brightest Year 10s:

"A individual has a rectangular plot of dry reason measuring 8.4m yesteryear 5.6m. To survey the release of dandelions they desire to dissever it every bit into the minimum release of foursquare plots. What is the size of each foursquare plot as well as how many such squares volition in that location be?"

My student's approach to this query was to effort to uncovering the HCF of 8.4 as well as 5.6 using the Indian Method. This is what she did:

But she realised that her reply made no sense. 28 can't hold out the HCF of 8.4 as well as 5.6. Can yous view where she went wrong? Although yous tin dissever 8.4 yesteryear seven (indeed, yous tin dissever 8.4 yesteryear anything), it doesn't hateful than seven is a gene of 8.4. The numbers on the left must hold out factors of the numbers at the top. Non-integers don't receive got factors. H5N1 improve approach would receive got been to convert the measurements to centimetres as shown below. 

Factors vs Multiples

If your students fighting to recall the departure betwixt factors as well as multiples therefore they mightiness uncovering this helpful: for factors, mean value of a factory (where separate parts are set together). For multiples, mean value of multi-packs eg if cokes are sold inwards multi-packs of 6 therefore I tin purchase 6, 12, eighteen etc. These ideas are taken from this resource from the thechalkface.net.

Conclusion

Did yous know all of these methods? Will yous assay something new? Please allow me know of whatever skillful methods that I've missed.

Even if you create upwards one's hear to stick with the method you're currently using, at to the lowest degree you've now reflected on how yous learn this topic. Teachers rarely receive got the chance to interruption as well as reflect.

This postal service should hold out read amongst Ed Southall's post Complements #9 LCM as well as HCF which explains the underlying concepts.

The whole presentation from my workshop is here. In my adjacent postal service (give me a few days to write it!) I'll comprehend methods for pedagogy sequences, linear graphs as well as surds.

Cpd - Maths Teachers Pb The Way

Maths is a genuinely hard dependent plain to teach. Not alone produce nosotros remove maintain to tackle negative perceptions - maths is difficult, maths is boring, maths is pointless - but nosotros every bit good remove maintain to explicate some genuinely tricky concepts. We necessitate to develop skills inward students such every bit work solving, visualisation in addition to reasoning. These skills don't come upward easily. Teaching maths is a huge challenge, in addition to it's genuinely of import to acquire it right.

Sometimes it's helpful to bounce ideas roughly amongst our beau maths teachers. We all produce goodness from sharing skillful practice. Sometimes nosotros necessitate inspiration. Up until recently, at that topographic point were few opportunities to acquire this sort of support. But there's a revolution happening inward maths education. It's called #mathsTLP.

What is #mathsTLP?

TLP is Twitter Lesson Planning. This was Ed Southall's clever sentiment - you lot tin read his original send service close it here. It takes house every Dominicus eventide betwixt 7pm in addition to 8pm. Let me explicate how it works. If you're looking for an sentiment or a resource, tweet your asking in addition to include the hashtag #mathsTLP. Something similar this:

People who are monitoring the #mathsTLP hashtag volition encounter your asking in addition to response to it - similar this:

At the terminate of the chat, Ed (@solvemymaths) collates the resources on his blog.

#mathsTLP isn't merely close sharing resources in addition to lesson ideas, it's a forum inward which maths teachers verbalize to each other close teaching maths. What could endure improve than that?

Is it working?

It's bloody brilliant. We're working together for the sake of our students. I cannot intend of a to a greater extent than effective utilisation of social media.

#mathsTLP has been running for iv weeks instantly in addition to we've covered a huge attain of topics including sequences, inequalities, circle theorems, simultaneous equations in addition to probability. Through #mathsTLP I've connected amongst lots of novel people in addition to remove maintain discovered many fantastic novel resources. 

For a build novel hashtag, it's getting a lot of tweets (the orangish occupation below represents #mathsTLP, blueish in addition to dark-green are the well-established hashtags #mathschat in addition to #mathscpdchat respectively).

Beyond maths?

Subject-specific CPD is considerably to a greater extent than helpful than the sort of full general 'teaching in addition to learning' CPD that is oftentimes delivered at schoolhouse Insets. At some schools - mine included - no fourth dimension or coin is allocated to subject-specific learning. Since my NQT year, I haven't received whatever preparation that's genuinely had an send upon on my teaching. Twitter is my sole source of evolution - without it I'd endure working inward a bubble.

Twitter's maths instructor community is vivid - its members remove maintain genuinely mastered the effective utilisation of Twitter every bit a platform for collaboration. I wonder whether other subjects are heading the same way. Do other subjects remove maintain an equivalent to #mathsTLP, a chat specifically for sharing lesson ideas in addition to resources?

Outside of Twitter, at that topographic point are initiatives which offering hands-on help, advice, noesis in addition to inspiration. H5N1 fantastic illustration is The Physics Factory, a 'grassroots network of teachers working to pose the fizz dorsum into physics'. Much similar those who participate inward #mathsTLP, the people behind The Physics Factory recognise that teachers necessitate to operate together. They portion ideas in addition to resources, they develop dependent plain knowledge. They don't say teachers how to teach, but they inspire teachers to beloved what they do.

Subject-specific initiatives are essential. To remove maintain an send upon inward the classroom, teachers necessitate to endure inward beloved amongst their subject, in addition to derive potency from it. Immersion inward their dependent plain is vital.

I'm feeling rattling positive close all this collaboration. If teachers proceed to operate together similar this, we're heading inward the correct direction. I'm proud to endure role of #mathsTLP in addition to I promise other subjects are inspired to follow our lead.

Tricks In Addition To Tips 3: Quadratics

highest mutual factor and methods for sequences, linear graphs too surds.

The vertex of a quadratic graph
This comes upward inwards the novel maths GCSE, inwards questions similar this which is taken from the Pearson Edexcel GCSE (9-1) Mathematics Sample Assessment Materials:

Pause for a infinitesimal too expect at this question, because it's a non bad instance of the modify inwards difficulty inwards the novel GCSE. Note the utilisation of business office annotation too the term turning point. The equation doesn't factorise. The graph doesn't cross the x-axis. The coordinates of the turning betoken aren't integers. This is a notable footstep upward from the type of questions asked inwards electrical current GCSE exams.

For the purpose of this post, let's consider the business office y = x² - 6x + 10. There are a give away of ways to notice the coordinates of the turning betoken - how would y'all create it?

1. Vertex Form
The term 'vertex form' is non unremarkably used inwards the UK. Vertex shape is what y'all acquire when y'all consummate the square. So we'd write the business office equally follows:

y = (x - 3)² + 1

Now nosotros tin sack position the turning betoken at in 1 trial away. I've e'er explained it a chip similar this:

"The (x - 3)²  is squared thence it tin sack never move negative. The lowest it tin sack move is zero. It's null when x is 3. The lowest possible y value is 0 + 1. So nosotros know that the minimum is at (3,1)..."

This explanation is inwards draw alongside my idea procedure - it's the agency I position the turning betoken - but my students actually create out alongside it. Of all the things I teach, this is the explanation that gets the most blank looks! So this yr I tried a dissimilar approach when revising this theme alongside my Year 11s. This fourth dimension I relied on my students' noesis of graph transformations. I told them to intend of the graph y = (x - 3)² + 1 equally a transformation of the graph y = x². It's been translated iii units correct too 1 unit of measurement up. The vertex moves from (0,0) to (3,1). They constitute this approach actually slowly - it made a lot to a greater extent than feel to them. Suddenly all my students were able to notice the vertex of a quadratic function.

As long equally students own got studied graph transformations thence this approach seems to work. This instruction gild is worth bearing inwards hear when designing a Scheme of Work. 

From at nowadays on, I'm going to utilisation the transformation method. But at that topographic point are alternatives...

2. H5N1 formula
In unopen to countries, students only memorise a formula. They larn that the x coordinate of the vertex is -b/2a. They thence notice the y coordinate yesteryear substituting that value into the equation.

By memorising this formula, y'all tin sack notice the coordinates of the turning betoken of whatever quadratic function without completing the square. At my conference session I showed the video below - lookout it to encounter how the method is explained. I'm non a fan of this approach. I don't desire my students to only memorise a formula - there's no conceptual agreement here.


3. Differentiation
Differentiation is e'er a pleasure. We don't create calculus at GCSE, but I idea it worth mentioning hither that unopen to other method to notice a turning betoken of a business office is to laid upward the derivative equal to zero. As y'all tin sack encounter below, for a quadratic that volition e'er hand us x = -b/2a.

4. Symmetry
For a quadratic that intercepts the x axis, the vertex is the midpoint of the 2 roots. This plant because parabolas are symmetrical. Up until late I idea this approach wasn't possible for quadratics that don't intercept the x axis, but thence I discovered James Tanton's method. It's described below for the business office  y = x² + 4x + 5  - for to a greater extent than exceptional too examples, encounter this Algebra too Core AS resource libraries are packed total of recommended resources for instruction this fantastic topic, such as this activity from Susan Wall.

Preparing for the novel GCSE
As I was writing this postal service it occurred to me that there's a lot of actually of import things that maths departments demand to do this term to cook for the novel GCSE. Writing novel Schemes of Work is a huge job, equally is finding suitable resources for instruction the novel GCSE topics.

CPD for maths teachers is likewise actually important. All maths teachers demand to move familiar alongside the novel GCSE content - they demand to know what's been added too what's been removed. They demand to expect at lots of instance questions.

The other affair that all teachers demand to create at nowadays is a patch of written report noesis banking concern gibe - are at that topographic point whatever topics on the novel GCSE syllabus that you're non familiar with? This is specially relevant for teachers who've never taught H5N1 degree maths. Has everyone inwards your subdivision idea close how to learn the novel GCSE topics? It's fourth dimension for unopen to vital maths subdivision CPD.

Mei Conference 2015

Last calendar week was an incredibly busy calendar week for me. On overstep of a number of occupation solid unit of measurement events too personal commitments, I did the following:

• Attended #mathsconf4 inwards London too helped run the TweetUp
• Hosted #mathsTLP on Lord's Day nighttime too #mathscpdchat (on behalf of the NCETM) on Tuesday night
• Had my source SLT lesson observation at my novel school
• Did my source e'er TeachMeet presentation at #TMEpsom
• Wrote 3 weblog posts too processed a dozen novel Pret Homeworks
• Travelled to Bath to attend too introduce at the MEI Conference.

It was a slap-up calendar week but I'm therefore pleased it's over! I've already written virtually #mathsconf4, the #mathscpdchat virtually Problem Solving too #TMEpsom virtually Pret Homeworks, therefore this is my lastly ship service virtually the events of my busy calendar week - this ship service is virtually my sense of the MEI Conference.

Maths CPD

subject specific CPD. Although I produce pick upwards closed to ideas from generic T&L events, my instruction benefits the most from conversations virtually maths - both the pedagogy too the subject. Unfortunately, aside from on Twitter, opportunities for such conversations are few too far between. My previous school's budget for external courses was null (I'm non exaggerating! No external CPD allowed at all). So at the MEI Conference it was refreshing to run across that closed to schools produce withal let their teachers to attend conferences. I chop-chop discovered that the chance to larn virtually maths too educate equally a instructor was immense.

Because I was presenting on the Sat I got to attend that hateful solar daytime for free, too I was staggered past times both how much I learnt too past times how much I enjoyed learning. The alternative of sessions was huge - over the 3 days at that spot were 12 sessions too inwards each session delegates could select from 8 options. Many, but non all, sessions were focussed on Influenza A virus subtype H5N1 flat instruction and/or the purpose of technology.

Ideas for Influenza A virus subtype H5N1 Level

The source session I attended was "The aspirational flat due east student" past times David Philip too Claire Phillips. The session was virtually how to back upwards AS maths students who convey an due east equally their baseline/target grade. We started past times discussing the reasons that therefore many students neglect AS flat maths too therefore flora out what David too Claire convey done at their schoolhouse to improve results. I've persuasion virtually this number earlier (see my post Bridging the Gap to Influenza A virus subtype H5N1 level) therefore it was slap-up to take away heed what they've been doing, including:

• They run an algebra bootcamp inwards the 3 days earlier term starts for flat B GCSE students.
• They number students alongside a self-assessment workbook containing the MEI exercise for each chapter (and links to solutions).
• They purpose a flipped learning approach for the basics (ie students spotter videos at dwelling occupation solid inwards advance of closed to lessons). Feedback from students has been good.
• They convey a department Twitter account (@mathsnewvic) which is the exclusively agency they communicate sure enough pieces of vital data (eg when the side past times side evidence is).
• They make students information virtually what questions volition live inwards assessments inwards advance (eg 'the side past times side evidence volition consist questions from 2011 past times papers') - students therefore produce all those questions to ready for the assessment, which is amend than doing zip at all.
• They convey whiteboard pigment on the walls. I wrote virtually Vertical Whiteboards in Gems 7 and i time again I've been convinced of their effectiveness. There are lots of alternatives (eg magic whiteboards or writing on windows) if yous can't purpose whiteboard paint. David too Claire said that oft their students brand it inwards lessons too start solving problems on the walls conduct away - sometimes they don't fifty-fifty pose their bags down! If they acquire stuck they are much to a greater extent than probable to inquire closed to other pupil for aid than if they were sitting down. Another delegate said that this approach had transformed her teaching.

There's likewise a few things that David too Claire innovation to implement side past times side year. They recall to brand assessment results (broken downward past times question) world therefore students know who the practiced is inwards each topic, too they tin therefore acquire to their peers for help. They are likewise going to brand purpose of the MathsWatch App which is gratis for students when the schoolhouse subscribes to MathsWatch (the Influenza A virus subtype H5N1 flat video collection is growing).

Problem Solving
I likewise attended a session on Problem Solving at Key Stage 3 too four alongside Phil Chaffé. During this session I had fun trying out a make of maths problems.

My favourite problem, which I recall would live peculiarly suitable for Year 8 or 9, was 'Zag Area 1', which is shown below too available to download (with solution) from Integral (login required).

The session notes likewise featured these concern carte problems from the FMSP - these volition live available online from September.

All the problems featured inwards this session were drawn from Integral, the FMSP too Nrich. If you're looking for occupation solving resources therefore produce likewise banking venture check out my recent ship service Favourite Problems.

Mathematical Methods
National Mathematics Teacher Conference inwards March. It was virtually alternative mathematical approaches (eg all the unlike methods nosotros tin purpose to discovery the highest mutual constituent of 2 numbers). If yous await my 'Methods' page then you'll run across the posts I wrote after the lastly conference explaining the content of my workshop.

This fourth dimension I added an extra subdivision to my presentation - I spoke virtually how to factorise harder quadratics. I demonstrated 3 methods. The source was the 'guess too test' method (commonly used past times anyone over the historic catamenia of 20), the 2d was the 'grouping' method (used past times many U.K. teachers) too the tertiary was the 'slide too divide' method (sometimes used inwards the US, but considered a 'trick' past times many). We talked virtually advantages, disadvantages too alternatives. If you're interested inwards finding out to a greater extent than therefore run across my ship service Factorising Harder Quadratics, which I've but updated. An interesting alternative approach that many delegates had non seen earlier was 'the monic method', which is demonstrated inwards these 2 examples:

I promise that delegates flora my session useful. I did acquire closed to prissy encouraging feedback subsequently therefore I recall it went well.

Summary

The plenary 'Bouncing Bombs too Boomerangs' past times Dr Hugh Hunt, Senior Lecturer inwards the Department of Engineering at Cambridge University, was entertaining too fascinating. Overall my whole hateful solar daytime at the MEI Conference was fantastic too I regret non beingness able to attend to a greater extent than sessions.

So at that spot ends my busy week. I've fallen agency behind alongside my to-do listing therefore forthwith it's fourth dimension to acquire dorsum on track. First up, I'll proceed to locomote on my page of resources for the new GCSE specification.

Thanks to MEI for organising such a high character conference - I actually promise I'm able to attend i time again side past times side year.

High Expectations

During Wednesday's Inset I delivered a 5 infinitesimal presentation to all teaching staff most high expectations.

At my novel school, teachers are encouraged to engage inwards enquiry - in that place is a comprehensive programme of internal CPD too a seat out of teachers are task masters degrees. The school, to a greater extent than hence than some others, has high expectations of its teaching staff. So inwards my presentation I had to travel careful non to Blue Planet the obvious. I gave a real curt summary of the Pygmalion effect (ie that having high expectations of students volition Pb to higher achievement). I also spoke briefly most the comport on of using encouraging linguistic communication alongside students, too how teachers alongside depression expectations oftentimes neglect to give their students constructive feedback.

I did desire to include at to the lowest degree ane practical thought inwards my presentation, hence I shared @MathedUp's 'Request a Work Selfie' which went downwards well. This is a peachy agency to signal high expectations from the outset. It industrial plant good inwards whatsoever subject.

I've been doing a lot of thinking most expectations lately. Teachers receive got to bargain alongside conflicts hither - practise I actually receive got high expectations of a educatee who I acquire into for Foundation GCSE? It presents a moral dilemma. Sometimes it's useful to footstep dorsum too aspect at the bigger picture. What should children experience at school? What's the bespeak of it all?

Why practise y'all sentinel University Challenge?
I bask watching University Challenge. I marvel at the noesis of the contestants too experience a peachy feel of achievement if I acquire fifty-fifty simply ane or 2 questions right. I don't know most of the answers, simply similar some students inwards my lessons, but I nevertheless bask watching it. Why? Because University Challenge has high expectations of its audience. The questions are hard, in that place are no gimmicks, too I respond good to that. The makers don't dumb it down. They aspect a lot of me. And I similar it that they aspect a lot of me.

The argue Brian Cox is hence successful alongside serial such equally Wonders of the Universe is that these programmes receive got high expectations of their audience. We rising to these expectations.

If nosotros receive got high expectations of our students hence they volition rising to them.

What would y'all prefer - an hr alongside Joey Essex or 5 minutes alongside Stephen Hawking? I'm sure we'd all pick out the latter. Even if everything he said went over our heads, we'd practise goodness from beingness inwards an intellectually stimulating environment.

I attended the MEI Conference inwards June. There was a lecture called 'Bouncing Bombs too Boomerangs' yesteryear Dr Hugh Hunt, Senior Lecturer inwards the Department of Engineering at Cambridge University. I wasn't familiar alongside much of the maths too physics he spoke about. Nevertheless, I thoroughly enjoyed the lecture too got a lot out of the experience. Dr Hunt spoke alongside passion, engaged his audience, too had high expectations.

'The biggest comport on on a educatee is from a instructor inwards honey alongside their subject'
People who are actually involved alongside their dependent area never tell ‘I’d honey to tell y'all most this but I can't because y'all won’t empathize it’.

I ane time attended a lecture yesteryear Simon Singh most The Simpsons too their Mathematical Secrets. I learnt lots of novel things too I came away to a greater extent than excited most my dependent area than I'd e'er been before. I rushed domicile to my husband, grabbed a pen too paper, too talked him through Fermat's Last Theorem. I excitedly showed him the 'near miss' solution that appeared on The Simpsons. Did he empathize what I was talking about? Of course of written report he did. In fact, I after heard him telling mortal else most it. Here's what's interesting: my hubby got an E inwards his maths GCSE. He retook it inwards Year 12 too got an E again. Over the course of written report of our 17 twelvemonth relationship, I'd never tried to explicate anything mathematical to him before. But my excitement most sharing this actually interesting mathematics overrode whatsoever expectations.

This is why the best teachers are those who are passionate most their subject. They don't say, 'It mightiness travel likewise difficult for them to understand'. They say, 'I can't hold off to tell them most it'.

It's all most narrative
Adding depth doesn't hateful making something to a greater extent than complicated. To add together depth to a topic nosotros verbalize most historical perspective, social dimensions too practical applications. The fundamental to making complex ideas manageable is narrative. For example, the causes of the Second World War are deeply complex too whole libraries could travel filled alongside books analysing those causes. It's a complex topic but practise history teachers tell it's likewise difficult to teach? No, they learn it yesteryear telling stories. It is those stories that spark the involvement of their students.

We must receive got high expectations of ourselves too what nosotros are capable of explaining.

I was lately teaching a marker of depression attaining Year 8s. I decided to learn them Pythagoras' Theorem, too was warned that it mightiness travel likewise much for them. I started alongside stories of the eccentric Pythagoras of Samos - they listened too learnt because these stories are interesting. By the halt of the lesson perhaps solely one-half the marker could confidently solve a work using Pythagoras' Theorem, but the lesson nevertheless had a far improve resultant than if I'd had depression expectations of the marker too had non tried to learn the topic at all.

Pitch it right, but don't dumb it down
If a topic is hard, don’t dorsum down. If y'all aim high too receive got mixed success hence y'all volition nevertheless accomplish to a greater extent than than if you'd aimed low. If your students fifty-fifty acquire 5% of what y'all say, hence improve they were inwards that top flying academic surroundings too got solely 5% than they were inwards a 'dumbed down' lesson. Why? Because educational activity should accept students from the known to the unknown. It should give them noesis that they couldn't maybe brand it their everyday lives. If nosotros trammel the intellectual horizon of our lessons, nosotros missy the bespeak completely.

If nosotros receive got high expectations of our pupils, nosotros are relaxed that they mightiness non 'get' all of it, because nosotros are showing them the "best that has been thought too said" inwards this area. They deserve that.

I did lots of reading over summertime to ready for my presentation. I especially enjoyed this post from @HuntingEnglish most the comport on of a maths instructor alongside high expectations - it's good worth a read.

I promise I've managed to give y'all some nutrient for thought most high expectations. My operate on this has sure had an comport on on the agency I've started off the twelvemonth alongside my novel classes. You tin discovery the slides from my presentation here. Thanks for reading.

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