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.

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

I lately 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 powerfulness solely guide keep you lot a instant to operate it out - how did you lot create it? This is roughly other 1 of those topics where there's lots of dissimilar approaches. Ask the same interrogation to a random sample of teenagers from across the United Kingdom of Great Britain in addition to Northern Ireland in addition to you'll regard a broad arrive at of methods used.

Here I'm going to depict 4 dissimilar methods (there may travel more!). As you lot read them, consider whether you're going to stick alongside the method you lot currently use, or bear witness something new.

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

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

When I taught this method it went well, although a misconception did surface afterwards on. When students were asked to 'write downwards the starting fourth dimension 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 travel aware of side past times side fourth dimension I instruct this method.

Jumping along a trace of piece of job - Median Don Steward

2. Shifting times tables
'Shifting times tables' is a pop method. The see 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 iii times tabular array to learn the sequence in addition to they'll topographic point that they demand to add together two, thence the  n^th term is 3n + 2.

If you lot project design to role this method for the starting fourth dimension time in addition to thence I recommend this NRICH article Shifting Times Tables, which comes alongside an interactive tool. If you lot subscribe to MathsPad, they also guide keep an interactive tool for identifying shifts.

This method plant good for quadratic sequences also (a topic on the novel GCSE syllabus). For illustration to notice the n^th term of the sequence 4, 7, 12, 19, 28 nosotros tin compare it to n² in addition to notice that it is shifted upwards 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 create nosotros confine this formula to Influenza A virus subtype H5N1 level? I 1 time took on a GCSE flat inwards Year xi in addition to asked them to notice roughly n^th terms inwards a revision lesson. I was surprised to regard them all using 'the Influenza A virus subtype H5N1 flat formula' - their previous instructor had taught them it, in addition to why not? They seemed quite happy alongside it. I wouldn't role it alongside my Year 7s though - their algebra skills are really basic when they starting fourth dimension encounter sequences. In our example, this is what they'd guide keep to create to notice the n^th term:

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

4. Substitution
This is the means 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 in addition to n, in addition to thence solve for q. For illustration inwards the sequence below, nosotros know that U_n = 3n + q. We substitute the starting fourth dimension term to learn v = 3 + q. Therefore q = 2.

Thinking almost it, this is the same method I'd role to notice the value of c inwards y = mx + c if I know the slope in addition to a signal on the line. In fact given that linear graphs are precisely graphical representations of linear sequences, whatever methods for finding the equation of a direct trace of piece of job operate for finding the n^th term of a sequence. I don't create plenty to brand this connectedness alongside my students.

Linear Graphs

Influenza A virus subtype H5N1 direct trace of piece of job through the signal (5, 7) has slope 4. How would you lot notice the equation of the line?

As I said inwards my ship almost linear graphs, I instruct this differently at GCSE in addition to Influenza A virus subtype H5N1 level. Wouldn't it travel amend to alternative my preferred approach in addition to stick alongside it? 

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

My electrical flow Year 10s had an American instructor terminal term - she covered linear graphs alongside 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 outcry upwards slope 'slope', that's easily fixed. But they'd also been taught Method 2 in addition to they couldn't retrieve the formula. This is a problem. Given that there's less to memorise, I intend they'd travel amend off alongside Method 1. 

It's worth reading Nix the Tricks (essential reading for all maths teachers), we're shown a pedagogy clit a fast 1 on called 'Jailbreak Radicals'. Thankfully I've never seen this 'zero conceptual agreement method' used inwards the UK.

Instead, most teachers tell students to pose a foursquare component in addition to and thence dissever the surd accordingly eg √45 = √9√5 = 3√5.

Sometimes students scrap to topographic point foursquare factors. In this illustration they powerfulness 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 in addition to regard what you lot think. Notice that you lot nonetheless demand to pose 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 bear witness 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 establish it helpful. Please comment below or tweet me if you lot know of whatever alternative methods for pedagogy these topics. My side past times side ship volition travel all almost quadratics - sketching, expanding in addition to factorising.

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.

A Tour Of Resourceaholic.Com

There's a lot going on hither on resourceaholic.com. I thought it mightiness travel helpful to pose out a guide to assistance novel visitors discovery their means around.

Resources

My resource libraries tin travel accessed using the tabs at the top of the page. I update these libraries whenever I discovery a especially goodness complimentary resource. I also produce regular maintenance to take away broken links. I promise that my libraries assistance to cut teachers' workload past times giving them quick, tardily access to goodness resources.

The most visited page is Algebra which lists algebra resources for Key Stage 3 as well as 4. The resources are grouped past times topic.

It's a shame that my Core A2 page doesn't larn to a greater extent than visits - at that spot aren't many online banks of Influenza A virus subtype H5N1 grade resources as well as I intend there's some actually useful materials here. I travel it all the fourth dimension when I'm planning Year thirteen lessons. In 2 years it has exclusively had 7,000 visits (compared to 42,000 for the algebra page) which suggests that most Influenza A virus subtype H5N1 grade teachers either don't know most it or don't need it.

Posts

I've published 173 weblog posts thence far, averaging 7 a month. Browse through my blog archive.

My top v weblog posts (based on the issue of views), are:

1. Tricks as well as Tips 1: HCF 
2. Ideas for Teaching Circle Theorems
3. Five things you lot mightiness non know most the novel GCSE content #1
4. Teaching Indices
5. Words of Wisdom: Teaching Foundation GCSE

If you lot haven't read Ideas from Shanghai then delight produce own got a look. I wrote it at a fourth dimension of much opposition (on Twitter, at least) to Maths Hubs.

My favourite weblog post - because I enjoyed writing it thence much - is The Joy of Planning. I wrote it at a fourth dimension when some prominent bloggers were suggesting that teachers should deliver scripted lessons.

Most of my posts are most ideas as well as resources that you lot tin travel inwards the classroom only sometimes I write persuasion pieces, for lawsuit my post Worries is most recruitment concerns as well as curriculum change. Occasionally I write posts that aren't maths-specific, for lawsuit my posts High Expectations as well as Behaviour Management for Beginners.

If you're an Influenza A virus subtype H5N1 grade instructor as well as thence you lot mightiness savour my posts A Level Reforms: First Thoughts, Bridging the Gap to Influenza A virus subtype H5N1 Level as well as Kicking off Year 12. I also own got a issue of posts most Influenza A virus subtype H5N1 grade resources as well as topics.

Gem Awards 2015 where I shared some highlights. I'll travel writing Gem Awards 2016 shortly to grade the instant anniversary of my blog.

Some of my weblog posts comprise collections of resources, including Stretching Practice, Higher GCSE Revision as well as Multiple Choice Questions.

My post Open Evening provides ideas as well as resources for opened upward evenings, Enrichment provides a listing of in-school speakers as well as Favourite Problems provides links to work solving websites. I own got a issue of Teaching Trigonometry as well as Long Live Stem as well as Leaf) as well as posts most Factorising Harder Quadratics as well as Algebraic Division).

Information as well as Support

.com provides data for maths teachers including conference listings, Twitter hashtags as well as links for primary teachers. These pages tin travel accessed through the 'extras' bill of fare on the right-hand side (this isn't visible on the mobile site).

I desire my weblog to travel a primary root of back upward for teaching the novel GCSE specification thence I've worked difficult to get together together resources as well as back upward on this real life graphs, tangents as well as areas, sequences, inequalities and quadratic graphs.

On a light-hearted note, my pages Mathsy Gifts as well as Words to Avoid always heighten a smile.

Spin-offs
post.

Another projection was my Misconceptions website. The thought was that people ship me photos of educatee misconceptions which I collate as well as publish. This would travel useful for novel teachers, for lawsuit if it's your commencement fourth dimension teaching indices you lot tin hold off at all the indices misconceptions to assistance you lot project design effectively. People sent me loads of nifty examples only I had to surrender on the projection really early because it's impossible to piece of work part-time, woman raise 2 children as well as run 3 websites! It could travel automated (ie people could upload photos directly) only I don't own got the fourth dimension or expertise to brand that happen. If anyone wants to own got command of this projection I'm happy to travel past times it over.

Another projection involved pulling together collections of problems relating to specific topics - for lawsuit a gear upward of problems that involve Pythagoras' Theorem. There's v sets here and I promise to brand some to a greater extent than over summer.

Since I started writing resourceaholic.com I've been involved inwards organising and/or speaking at diverse events - you lot tin encounter a listing of my achievements as well as engagements on my About Me page.

Updates
If you'd similar to travel along up-to-date amongst my blog, you lot tin subscribe past times electronic mail using the 'Follow past times Email' box on the right-hand side. Once you've confirmed your electronic mail address, you'll have an electronic mail whenever I pose out a novel post. I currently own got around 200 subscribers.

Alternatively you lot tin similar my Facebook page or follow me on Twitter. I'm on Twitter a lot!

The bulk of my audience is from the UK of Britain as well as Northern Republic of Ireland only I produce own got a few international readers too, especially from the US.

I promise this has been a helpful introduction to what you lot tin discovery on resourceaholic.com. I'm happy to respond whatsoever questions - delight tweet or email me, or comment below. I'm also happy to own got requests for weblog posts (but delight don't inquire me to promote products!).

Thanks for reading!

A cake to celebrate my 50th gems post
(with cheers to @tessmaths as well as my lovely Twitter friends)

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.