easy way to learn mathematics: Conference

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5 Maths Gems #8

I enjoyed coming together many of yous at La Salle Education's National Mathematics Teacher Conference yesterday. It was a fantastic chance for sharing ideas. But 500 maths teachers out of 350,000 is a drib inward the ocean. It's such a shame that thence many schools weren't represented.

Although at that spot are some pockets of collaboration throughout the UK, for event through regional TeachMeets, the fact is that the bulk of U.K. maths teachers alive inward a bubble. Through no error of their own, they know really niggling almost approaches taken inward other schools - including pedagogy, curriculum pattern together with novel technologies. Twitter is purpose of the solution, but it doesn't acquire inward enough.

this post, "in high performing countries teachers pass 30% of their fourth dimension teaching, alongside the residual spent planning, preparing together with assessing every bit good every bit working on improving their practice. Compared to the UK, where it’s 90%".

This weekend, 500 maths teachers gave upward a precious Sabbatum to larn from each other. I'm really grateful to La Salle Education for providing that opportunity. To me it farther highlighted the quest for cross-school subject-specific collaboration during term-time. Instead of spending our Inset days hearing almost our schools' latest mark policies, nosotros should all live on out having mini-conferences of our own. Let's promise that when our novel Maths Hubs acquire dorsum from their trips to Shanghai, improving collaboration is spill out of their listing of priorities - together with yesteryear that I hateful collaboration alongside all schools, non simply the schools alongside tweeting teachers or a proactive Head of Maths.

Now for my weekly maths gems. For those who receive got simply started reading my blog, this is where I part 5 non bad ideas that I've seen on Twitter each week. If yous desire to read whatsoever of my previous gems posts, they're all here.

1. Paint! And my ideal classroom

smartwallpaint.com

I was talking to a friend almost the 'Vertical Non-Permanent Surfaces', magic whiteboards together with acrylic sheets I featured inward this post, "in high performing countries teachers pass 30% of their fourth dimension teaching, alongside the residual spent planning, preparing together with assessing every bit good every bit working on improving their practice. Compared to the UK, where it’s 90%".

This weekend, 500 maths teachers gave upward a precious Sabbatum to larn from each other. I'm really grateful to La Salle Education for providing that opportunity. To me it farther highlighted the quest for cross-school subject-specific collaboration during term-time. Instead of spending our Inset days hearing almost our schools' latest mark policies, nosotros should all live on out having mini-conferences of our own. Let's promise that when our novel Maths Hubs acquire dorsum from their trips to Shanghai, improving collaboration is spill out of their listing of priorities - together with yesteryear that I hateful collaboration alongside all schools, non simply the schools alongside tweeting teachers or a proactive Head of Maths.

Now for my weekly maths gems. For those who receive got simply started reading my blog, this is where I part 5 non bad ideas that I've seen on Twitter each week. If yous desire to read whatsoever of my previous gems posts, they're all my final gems post and he told me almost something wonderful. He plant inward a novel schoolhouse where all the classroom walls are painted alongside whiteboard paint. So he has no whiteboards inward his room (other than the interactive board), but he - together with his students - tin write on whatsoever wall.

This got me thinking almost my ideal classroom. I'd receive got all the walls painted alongside whiteboard pigment (student operate would live on proudly displayed inward the corridors instead of inward the classroom). I similar the thought of students sitting inward a horseshoe when I teach, similar inward the motion-picture demo below. I intend this would halt low-level disruption together with would encourage meliorate course of written report discussion. They could also do independent practise inward these chairs.

hyperspacellc.com

Separate to this horseshoe I'd receive got a collaboration area, alongside grouping tables that tin live on written on (whiteboard pigment plant on desks too!), or this post, "in high performing countries teachers pass 30% of their fourth dimension teaching, alongside the residual spent planning, preparing together with assessing every bit good every bit working on improving their practice. Compared to the UK, where it’s 90%".

This weekend, 500 maths teachers gave upward a precious Sabbatum to larn from each other. I'm really grateful to La Salle Education for providing that opportunity. To me it farther highlighted the quest for cross-school subject-specific collaboration during term-time. Instead of spending our Inset days hearing almost our schools' latest mark policies, nosotros should all live on out having mini-conferences of our own. Let's promise that when our novel Maths Hubs acquire dorsum from their trips to Shanghai, improving collaboration is spill out of their listing of priorities - together with yesteryear that I hateful collaboration alongside all schools, non simply the schools alongside tweeting teachers or a proactive Head of Maths.

Now for my weekly maths gems. For those who receive got simply started reading my blog, this is where I part 5 non bad ideas that I've seen on Twitter each week. If yous desire to read whatsoever of my previous gems posts, they're all acrylic sheets.

The rooms I currently learn inward are every bit good minor for me to easily accomplish every pupil piece they're working, thence fifty-fifty simply a niggling to a greater extent than infinite would operate wonders.

2. Indices
If you're looking for a novel approach to teaching index laws, yous mightiness taste 'this post, "in high performing countries teachers pass 30% of their fourth dimension teaching, alongside the residual spent planning, preparing together with assessing every bit good every bit working on improving their practice. Compared to the UK, where it’s 90%".

This weekend, 500 maths teachers gave upward a precious Sabbatum to larn from each other. I'm really grateful to La Salle Education for providing that opportunity. To me it farther highlighted the quest for cross-school subject-specific collaboration during term-time. Instead of spending our Inset days hearing almost our schools' latest mark policies, nosotros should all live on out having mini-conferences of our own. Let's promise that when our novel Maths Hubs acquire dorsum from their trips to Shanghai, improving collaboration is spill out of their listing of priorities - together with yesteryear that I hateful collaboration alongside all schools, non simply the schools alongside tweeting teachers or a proactive Head of Maths.

Now for my weekly maths gems. For those who receive got simply started reading my blog, this is where I part 5 non bad ideas that I've seen on Twitter each week. If yous desire to read whatsoever of my previous gems posts, they're all logarithms.

Once you've taught indices, endeavor my index laws homework together with fractional/negative indices homework.

3. ThingLink
this post, "in high performing countries teachers pass 30% of their fourth dimension teaching, alongside the residual spent planning, preparing together with assessing every bit good every bit working on improving their practice. Compared to the UK, where it’s 90%".

This weekend, 500 maths teachers gave upward a precious Sabbatum to larn from each other. I'm really grateful to La Salle Education for providing that opportunity. To me it farther highlighted the quest for cross-school subject-specific collaboration during term-time. Instead of spending our Inset days hearing almost our schools' latest mark policies, nosotros should all live on out having mini-conferences of our own. Let's promise that when our novel Maths Hubs acquire dorsum from their trips to Shanghai, improving collaboration is spill out of their listing of priorities - together with yesteryear that I hateful collaboration alongside all schools, non simply the schools alongside tweeting teachers or a proactive Head of Maths.

Now for my weekly maths gems. For those who receive got simply started reading my blog, this is where I part 5 non bad ideas that I've seen on Twitter each week. If yous desire to read whatsoever of my previous gems posts, they're all Mathematical Concepts Wall' yesteryear this post, "in high performing countries teachers pass 30% of their fourth dimension teaching, alongside the residual spent planning, preparing together with assessing every bit good every bit working on improving their practice. Compared to the UK, where it’s 90%".

This weekend, 500 maths teachers gave upward a precious Sabbatum to larn from each other. I'm really grateful to La Salle Education for providing that opportunity. To me it farther highlighted the quest for cross-school subject-specific collaboration during term-time. Instead of spending our Inset days hearing almost our schools' latest mark policies, nosotros should all live on out having mini-conferences of our own. Let's promise that when our novel Maths Hubs acquire dorsum from their trips to Shanghai, improving collaboration is spill out of their listing of priorities - together with yesteryear that I hateful collaboration alongside all schools, non simply the schools alongside tweeting teachers or a proactive Head of Maths.

Now for my weekly maths gems. For those who receive got simply started reading my blog, this is where I part 5 non bad ideas that I've seen on Twitter each week. If yous desire to read whatsoever of my previous gems posts, they're all Samuel S. Wilks said,"Statistical thinking volition ane twenty-four hr menstruation live on every bit necessary for efficient citizenship every bit the powerfulness to read together with write!" (he was paraphrasing HG Wells). Although through our curriculum nosotros learn students how to do together with translate a diverseness of graphs, I'm non convinced that 'the full general public' are able to brand feel of many of the graphs they see inward the media. I intend it's worth spending some fourth dimension looking at both misleading graphs together with interesting graphs.

Ridiculously misleading graph

Interesting graphs promote non bad discussions. Give your students some fourth dimension to expect at the graph, operate out what it shows together with and thence intend almost the underlying trends or causes. Here's a distich to acquire yous started.

Source: 'How nosotros believe' yesteryear Micheal Shermer

Source: ONS

The 2nd graph is from '8 Facts almost Young People' yesteryear the ONS, shared this calendar week by @statshan.

Speaking of interesting graphs that spark discussion, yous must read this post, "in high performing countries teachers pass 30% of their fourth dimension teaching, alongside the residual spent planning, preparing together with assessing every bit good every bit working on improving their practice. Compared to the UK, where it’s 90%".

This weekend, 500 maths teachers gave upward a precious Sabbatum to larn from each other. I'm really grateful to La Salle Education for providing that opportunity. To me it farther highlighted the quest for cross-school subject-specific collaboration during term-time. Instead of spending our Inset days hearing almost our schools' latest mark policies, nosotros should all live on out having mini-conferences of our own. Let's promise that when our novel Maths Hubs acquire dorsum from their trips to Shanghai, improving collaboration is spill out of their listing of priorities - together with yesteryear that I hateful collaboration alongside all schools, non simply the schools alongside tweeting teachers or a proactive Head of Maths.

Now for my weekly maths gems. For those who receive got simply started reading my blog, this is where I part 5 non bad ideas that I've seen on Twitter each week. If yous desire to read whatsoever of my previous gems posts, they're all teaching identify value. I honey posts alongside lots of ideas together with resources recommendations for a specific topic - they're really useful for lesson planning (I've written a few topic specific posts too).

Second, I desire to elevate this post, "in high performing countries teachers pass 30% of their fourth dimension teaching, alongside the residual spent planning, preparing together with assessing every bit good every bit working on improving their practice. Compared to the UK, where it’s 90%".

This weekend, 500 maths teachers gave upward a precious Sabbatum to larn from each other. I'm really grateful to La Salle Education for providing that opportunity. To me it farther highlighted the quest for cross-school subject-specific collaboration during term-time. Instead of spending our Inset days hearing almost our schools' latest mark policies, nosotros should all live on out having mini-conferences of our own. Let's promise that when our novel Maths Hubs acquire dorsum from their trips to Shanghai, improving collaboration is spill out of their listing of priorities - together with yesteryear that I hateful collaboration alongside all schools, non simply the schools alongside tweeting teachers or a proactive Head of Maths.

Now for my weekly maths gems. For those who receive got simply started reading my blog, this is where I part 5 non bad ideas that I've seen on Twitter each week. If yous desire to read whatsoever of my previous gems posts, they're all revision races. This is ane of those posts that non alone provides a fantastic resources but also clearly explains how to utilisation it effectively.

Finally, I desire to recommend @fmaths42's post service almost times tables starters which provides a railroad train of progressively challenging practise grids, addition a proposition to impress these grids onto stickers. Times tables are rather helpful so I'd urge yous to ignore Jo Boaler's advice almost non ensuring that children larn their times tables - sure she must receive got said that inward a 2nd of madness. Read @websofsubstance's post service The Educators - Jo Boaler for some interesting (in my see - really sensible) views on this.

Finally, receive got a acquire at this query from Tony Gardiner's session at yesterday's conference, shared yesteryear @mrsdenyer. I'll live on using this every bit a quick starter alongside both Year eleven together with Year 12.

PS - I'm off to Center Parcs tomorrow (term-time holidays are a huge wages of beingness on motherhood leave!) thence at that spot won't live on a gems post service side yesteryear side Saturday. I'll live on dorsum the next calendar week though.

#Mathsconf2015

I had a nifty fourth dimension at La Salle Education's National Maths Teacher Conference inward Birmingham yesterday. It was good worth going upwards at that topographic point on Fri nighttime - I had a fantastic evening, enjoying a delicious dinner as well as the society of swain tweeters @solvemym aths, @tessmaths, @7puzzle, @MrPatelMaths and @DocendoTim. I was inward the bar at midnight to regard the inaugural of all of Pi Day but upwards vivid as well as early on the adjacent morning time to brand the in conclusion preparations for my workshop (with a slightly sore caput due to besides much vino the nighttime before...). I was a purse of nerves over breakfast pastries but thankfully the lovely Craig Barton (an experienced as well as pop conference presenter) reassured me amongst words of wisdom.

@MissBsResources, me as well as @MrPatelMaths at 12.01am on Pi Day -
cheers to @El_Timbre for the posters and @jennypeek for the photo

At the conference, I actually enjoyed David Thomas' session '3 Techniques You Should Know (but in all likelihood don't)'. It was helpful to regard expert examples of bar modelling, algebra tiles as well as double lay out lines. I already role some aspects of these methods inward my pedagogy but the presentation gave me enough of nutrient for thought as well as at that topographic point were moments where I could actually regard the advantages of these approaches. In the 2nd session I saw Robert Wilne speak close developing reasoning. He was an entertaining speaker amongst lots of expert ideas including his 'olympic podium' of operations.

The TweetUp at luncheon was fun - Bruno Reddy did a nifty chore organising it. It was amazing to regard so many teachers trying to solve @solvemymaths' puzzles on 'the dorsum of an envelope' so to speak. His puzzles are awesome but are all pretty challenging. The few that I've tried to solve bring involved a long menstruum of concentration as well as frustration (followed yesteryear satisfaction, so good worth it!). I was actually impressed to regard teachers getting stuck into these tricky problems.

TweetUp: @MistryMan6's selfie amongst me as well as the lovely @MissBsResources

I presented inward the in conclusion session of the twenty-four hours to an audience of some 100. The aim of the session was to utter over unlike approaches to pedagogy topics such every bit sequences, linear graphs, surds as well as quadratics. I had lots of ideas to portion as well as I was pleased that I learnt fifty-fifty to a greater extent than methods from delegates afterward the session. It was a wonderful chance to swap pedagogy ideas (this is the kind of collaboration nosotros should all produce at schoolhouse but never acquire time). I was careful non to label whatsoever methods every bit correct or wrong, but but to advise alternatives as well as - hopefully - encourage people to mean value close their dependent champaign noesis as well as delivery. I was pleased to regard @danicquinn as well as @BodilUK enthusiastically discussing methods at the cease of the session.

Me presenting at MathsConf III - thanks @DrBennison for the photo

If y'all were at my session so you'll already bring a re-create of this workbook which is total of links, as well as here's the presentation as promised. If y'all weren't at my session so the presentation won't brand much sense, but I'll write a weblog postal service before long to explicate all. Give me a calendar week or 2 to acquire it written though!

It was squeamish to cease the twenty-four hours amongst a quick sip at the bar amongst @letsgetmathing, @MrPatelMaths as well as @MrKMorrison. One of the best things close these conferences is coming together upwards amongst like-minded maths teachers who enthusiastically hand upwards a Sabbatum for professional person development. Over the weekend it was nifty to select grip of upwards amongst people I met at the in conclusion conference as well as I enjoyed coming together lots of novel people too.

Many cheers to the La Salle squad for the tremendous attempt they lay into organising the conference. I'm distressing that I in all likelihood won't hold upwards at the adjacent i (it's on 20^th June - the weekend of my 10^th marriage ceremony anniversary) but I volition definitely hold upwards at MathsConf V inward September as well as I'm already looking forrard to it.

Me inward the conference programme :)

Best mug ever? Thanks @solvemymaths

******************************
Blog posts that review/mention my workshop:
@BodilUK
@DrBennison
@WorkEdgeChaos
@mrprcollins

Tricks Together With 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 shipping service summarises the content of that workshop for those who were unable to attend. I direct maintain quite a lot to embrace thus I await I'll demand to write iii or 4 spider web log posts. In this 1 I'm going to explicate the rationale for the workshop as well as depict choice methods for finding a Highest Common Factor. In subsequent posts I'll embrace 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 produce you detect the Highest Common Factor of 2 numbers? Do you lot piece of employment a Venn method? List the factors? Use the Euclidean Algorithm? Are at that topographic point 'better' methods that you lot don't know about? (how produce nosotros define 'better'?). These are the form of questions I desire to explore. The aim of my workshop was to encourage people to reverberate on their acre of written report cognition 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 pick out the explicate things?

Most teachers constitute instruction habits during their grooming as well as NQT year. The method they piece of employment the outset fourth dimension they instruct a theme volition belike remain with them throughout their career - unless they brand a concious elbow grease to essay a novel method.

One of the few things I recollect virtually GCSE maths was that I solved equations past times 'moving price over the equals sign' (the 'magic portal method'). This method is instantly considered to last a shortcut which stands inwards the agency of conceptual understanding. These days it is to a greater extent than acceptable to instruct students to solve equations using inverse operations (ie residuum the equation past times 'doing the same thing' to both sides). It's lucky that during my PGCE someone told me non to piece of employment the magic portal method because earlier as well as then 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 embrace much inwards the agency of mathematical methods, thus 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 direct maintain non been through organised CPD (ie 'collaboration sessions') inside my department, but instead through risk encounters. My outset always spider web log shipping service was virtually an choice method for matrix multiplication that I'd happened to stumble across online.

During my PGCE I was asked to instruct a lesson on Highest Common Factor. I recollect my mentor showing me the 'Venn Method'. As a final result of that conversation I used the Venn Method for years, until past times risk a colleague mentioned an alternative. Let's aspect at that choice method now, as well as a position out of others. Will you lot stick with the method you lot know as well as love, or volition you lot essay something new?

Highest Common Factor as well as Lowest Common Multiple

I've identified vi methods for finding the HCF as well as LCM of 2 numbers. I'll explicate each method hither as well as position whatsoever 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 reckon precisely what they're trying to hit here. List all the factors of the 2 numbers as well as detect the biggest position out that's inwards both lists - that's your HCF. List all the multiples of 2 numbers as well as detect the smallest position out that's inwards both lists - that's your LCM. Simple! Shame it's thus time-consuming. And, inwards my experience, students sometimes fille factors from their list. One agency to avoid this is past times listing factors using a pairing method similar this:

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

2. The Venn Method

This is a pop method inwards the UK. First, nosotros demand to produce a prime number cistron breakdown. By the way, if you're instruction prime number factorisation as well as then you lot mightiness similar these lovely magic portal method'). This method is instantly considered to last a shortcut which stands inwards the agency of conceptual understanding. These days it is to a greater extent than acceptable to instruct students to solve equations using inverse operations (ie residuum the equation past times 'doing the same thing' to both sides). It's lucky that during my PGCE someone told me non to piece of employment the magic portal method because earlier as well as then I was convinced that it was 'the' agency to solve equations.

Source: https://mrjasonto.wordpress.com

Once you lot direct maintain the prime number factors of each number, clitoris a Venn diagram as well as house the mutual factors inwards the intersection of 2 sets, as shown inwards the instance below.

tutorvista.com

The HCF is the production of the elements inwards Influenza A virus subtype H5N1 intersection B (ie 2 x 2 x 2 x 2 inwards the instance above) as well as the LCM is the the production of all the elements inwards Influenza A virus subtype H5N1 spousal human relationship B (ie 2 x 2 x 2 x 2 x 2 x 2 x 5). Note that U.K. GCSE students are non withal familiar with this terminology (ie intersection as well as union), but they volition last 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 as well as then they oftentimes can't recollect which 'bit' is the HCF. If they produce recollect the method as well as then they belike don't direct maintain a clue why it works.

Confusingly, it seems that some people piece of employment a dissimilar Venn method which involves putting all factors (not prime number factors) into a Venn diagram as well as identifying the highest cistron inwards the intersection (see instance below). This is some other shape of the listing method described inwards a higher house - it's just a dissimilar agency of organising the list. Let's telephone telephone 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 produce the prime factorisation but as well as then skip the Venn. Write the prime number factors of each position out out as shown inwards the instance below thus it's slowly to reckon which factors appear inwards both position out - the production of these is the HCF. This method is featured inwards magic portal method'). This method is instantly considered to last a shortcut which stands inwards the agency of conceptual understanding. These days it is to a greater extent than acceptable to instruct students to solve equations using inverse operations (ie residuum the equation past times 'doing the same thing' to both sides). It's lucky that during my PGCE someone told me non to piece of employment the magic portal method because earlier as well as then I was convinced that it was 'the' agency to solve equations.

Source: https://mrjasonto.wordpress.com

4. Euclidean Algorithm

I honey the subtraction-based Euclidean Algorithm. It sounds complicated but it's incredibly easy. Try a few examples yourself to reckon 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 employment it with students... Would you?

Note that this method doesn't hand you lot the Lowest Common Multiple, but it's easily found 1 time you've got the Highest Common Factor.

This looks similar a pain, but cancelling helps - inwards the instance inwards a higher house I found the HCF of lx as well as 84, thus to detect the LCM I multiply lx past times 84 as well as separate past times the HCF.

5. The Indian Method

I've stopped using the Venn Method at schoolhouse as well as instantly piece of employment this instead. I'm non certain it's actually called the Indian Method, I'm entirely calling it that because of this video. At my schoolhouse nosotros telephone telephone 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 slowly to explore why it plant too.

Say nosotros desire to detect the Highest Common Factor as well as Lowest Common Multiple of 315 as well as 420. Write downward the 2 numbers, as well as then (to the left, as inwards my instance above) write downward any mutual factor. I've chosen 5. Now separate 315 as well as 420 past times five as well as write the answers underneath (63 as well as 84 inwards this case). Keep repeating this procedure until the 2 numbers direct maintain 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, detect the production of the numbers on the left as well as the numbers inwards the bottom row (to detect the LCM, aspect for the L shape).

6. The 'Upside Down Birthday Cake Method'

I should holler this method because I proceed spotting it online (video here). The entirely divergence betwixt this as well as the Indian Method is that hither nosotros tin entirely withdraw prime factors. This seems unnecessary - the Indian Method is quicker. In the instance below, why separate past times 2 if you lot spot larger mutual factors? Why non start past times dividing past times 4, 6 or 12?

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

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

Integers
I actually similar this magic portal method'). This method is instantly considered to last a shortcut which stands inwards the agency of conceptual understanding. These days it is to a greater extent than acceptable to instruct students to solve equations using inverse operations (ie residuum the equation past times 'doing the same thing' to both sides). It's lucky that during my PGCE someone told me non to piece of employment the magic portal method because earlier as well as then 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 embrace much inwards the agency of mathematical methods, thus 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 direct maintain non been through organised CPD (ie 'collaboration sessions') inside my department, but instead through risk encounters. My outset always spider web log shipping service was virtually an choice method for set of questions from Don Steward. The next query confused 1 of my brightest Year 10s:

"A soul has a rectangular plot of dry soil measuring 8.4m past times 5.6m. To survey the position out of dandelions they desire to separate it as into the minimum position out of foursquare plots. What is the size of each foursquare plot as well as how many such squares volition at that topographic point be?"

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

But she realised that her respond made no sense. 28 can't last the HCF of 8.4 as well as 5.6. Can you lot reckon where she went wrong? Although you lot tin separate 8.4 past times vii (indeed, you lot tin separate 8.4 past times anything), it doesn't hateful than vii is a cistron of 8.4. The numbers on the left must last factors of the numbers at the top. Non-integers don't direct maintain factors. Influenza A virus subtype H5N1 improve approach would direct maintain been to convert the measurements to centimetres as shown below.

Factors vs Multiples

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

Conclusion

Did you lot know all of these methods? Will you lot essay something new? Please permit me know of whatsoever proficient methods that I've missed.

Even if you create upwardly one's heed to stick with the method you're currently using, at to the lowest degree you've now reflected on how you lot instruct this topic. Teachers rarely direct maintain the chance to interruption as well as reflect.

This shipping service should last read with 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 shipping service (give me a few days to write it!) I'll embrace methods for instruction sequences, linear graphs as well as surds.

Tricks In Addition To Tips 2: Sequences, Linear Graphs In Addition To Surds

I latterly presented a workshop at the National Mathematics Teacher Conference (Highest Common Factor.

Linear Sequences

What's the n^th term of the sequence below?

5, 8, 11, 14, 17, ...

It mightiness exclusively accept y'all a minute to operate it out - how did y'all produce it? This is roughly other 1 of those topics where there's lots of unlike approaches. Ask the same inquiry to a random sample of teenagers from across the United Kingdom of Great Britain together with Northern Ireland of Britain together with Northern Republic of Ireland together with you'll reckon a broad hit of methods used.

Here I'm going to depict 4 unlike methods (there may endure more!). As y'all read them, consider whether you're going to stick amongst the method y'all currently use, or seek something new.

1. Zeroth term
Inspired past times this post on Don Steward's blog, I taught the 0^th term method for the outset fourth dimension this year. To operate out the value of q inwards the n^th term U_n = pn + q, nosotros merely footstep dorsum from the outset term inwards the sequence (ie q = U₀).

In the illustration 5, 8, 11, 14, 17..., subtract 3 from the outset term to larn U₀ = 2. So the n^th term is 3n + 2. This method is thence quick that it's at nowadays my preferred method for working out n^th terms.

When I taught this method it went well, although a misconception did surface after on. When students were asked to 'write downward the outset 3 price of the sequence 4n + 3', roughly of them gave the respond 3, 7, xi (ie they started their sequences from n = 0 instead of n = 1). This is something to endure aware of side past times side fourth dimension I learn this method.

Jumping along a business - Median Don Steward

2. Shifting times tables
'Shifting times tables' is a pop method. The catch is that nosotros compare our sequence to a times table. In the illustration below, compare the sequence to the 3 times table. Ask your students how to shift the 3 times tabular array to larn the sequence together with they'll location that they demand to add together two, thence the n^th term is 3n + 2.

If y'all project design to role this method for the outset fourth dimension together with thence I recommend this NRICH article Shifting Times Tables, which comes amongst an interactive tool. If y'all subscribe to MathsPad, they also stimulate got an interactive tool for identifying shifts.

This method plant good for quadratic sequences as good (a theme on the novel GCSE syllabus). For illustration to notice the n^thterm of the sequence 4, 7, 12, 19, 28 nosotros tin compare it to n² together with notice that it is shifted upwardly past times 3 (ie the n^th term is n² + 3).

3. Formula
It's fairly straightforward to derive the next formula for the n^th term:

Why produce nosotros confine this formula to Influenza A virus subtype H5N1 level? I 1 time took on a GCSE shape inwards Year xi together with asked them to notice roughly n^th terms inwards a revision lesson. I was surprised to reckon them all using 'the Influenza A virus subtype H5N1 score formula' - their previous instructor had taught them it, together with why not? They seemed quite happy amongst it. I wouldn't role it amongst my Year 7s though - their algebra skills are real basic when they outset reckon sequences. In our example, this is what they'd stimulate got to produce to notice the n^th term:

Perhaps this is to a greater extent than endeavor than necessary. The 0^th term method is considerably quicker.

4. Substitution
This is the agency I taught sequences for years. To operate out the value of q inwards the n^th term U_n = pn + q, nosotros substitute a value for U_n, p together with n, together with thence solve for q. For illustration inwards the sequence below, nosotros know that U_n = 3n + q. We substitute the outset term to larn v = 3 + q. Therefore q = 2.

Thinking virtually it, this is the same method I'd role to notice the value of c inwards y = mx + c if I know the slope together with a indicate on the line. In fact given that linear graphs are merely graphical representations of linear sequences, whatever methods for finding the equation of a at 1 time business operate for finding the n^th term of a sequence. I don't produce plenty to brand this connective amongst my students.

Linear Graphs

Influenza A virus subtype H5N1 at 1 time business through the indicate (5, 7) has slope 4. How would y'all notice the equation of the line?

As I said inwards my transportation service virtually linear graphs, I learn this differently at GCSE together with Influenza A virus subtype H5N1 level. Wouldn't it endure improve to selection my preferred approach together with stick amongst it?

At GCSE my students write downward y = mx + c together with substitute values for y, one thousand together with x, together with thence solve for c (let's telephone vociferation upwardly this Method 1). At Influenza A virus subtype H5N1 score my pupils role the formula y - y₁ = m(x - x₁) (let's telephone vociferation upwardly this Method 2). Look at the steps involved - for this question, the methods are as efficient.

My electrical current Year 10s had an American instructor final term - she covered linear graphs amongst them. When nosotros came to revise this topic, they took a piece to figure out where to start. Evetually I heard 1 say, "is this the point-slope thing? What was that formula again?".

OK, thence they'd been taught to telephone vociferation upwardly slope 'slope', that's easily fixed. But they'd also been taught Method 2 together with they couldn't recall the formula. This is a problem. Given that there's less to memorise, I intend they'd endure improve off amongst Method 1.

It's worth reading Nix the Tricks (essential reading for all maths teachers), we're shown a didactics play tricks called 'Jailbreak Radicals'. Thankfully I've never seen this 'zero conceptual agreement method' used inwards the UK.

Instead, most teachers tell students to position a foursquare element together with and thence separate the surd accordingly eg √45 = √9√5 = 3√5.

Sometimes students fighting to location foursquare factors. In this illustration they mightiness prefer to simplify using prime number factors.

Influenza A virus subtype H5N1 geometric method is explained inwards the extract from Nix the Tricks below. Try to simplify a few surds similar this yourself together with reckon what y'all think. Notice that y'all all the same demand to position a foursquare factor.

Source: Nix the Tricks

The feedback from my conference workshop suggests that roughly people notice this unnecessarily complicated. But if you're going to seek it, I recommend this post by @ChrisHunter36 and this resource (pages v - 6) past times @314Piman.

That's it for today's post. I promise you've flora it helpful. Please comment below or tweet me if y'all know of whatever choice methods for didactics these topics. My side past times side transportation service volition endure all virtually quadratics - sketching, expanding together with factorising.