Matrix Multiplication

When I teach matrix multiplication I discovery that pupils tin rate the axe top away a fleck lost inward their calculations.  I constitute a fantastic novel method on this blog.

Let's tell nosotros accept matrix A together with matrix B.  We desire to calculate AB.  Write downwardly the 2 matrices side-by-side.  Then move matrix Influenza A virus subtype H5N1 down. Now write your answer inward the infinite at the bottom right.  I intend the side past times side measuring is self-explanatory from the film below, merely read the master copy blog post for a amount explanation.

I honey this method - it's together with thence unproblematic to retrieve together with at in 1 trial most impossible to top away lost when multiplying matrices.

And - on the dependent area of matrices - this made me chuckle:

Highest Mutual Ingredient Together With Lowest Mutual Multiple

There are many methods for finding the Highest Common Factor together with Lowest Common Multiple of ii or to a greater extent than numbers.  For years I've been didactics pupils to lay the prime number factors into a Venn Diagram, every bit described here. 

I recently discovered an option method that is impressively quick together with simple.  It is described inwards this video every bit the 'Indian Method'. It's like to the 'upside downwardly birthday cake method' but it's much quicker because in that place is no requirement to role primes.

Say nosotros desire to detect the Highest Common Factor together with Lowest Common Multiple of 24 together with 36.
Write down the ii numbers, then (to the left, every bit inwards my illustration below) write downwardly whatever mutual element (ie 2, 3, 4, vi or 12).  I've chosen 6.  Now dissever 24 together with 36 past times vi together with write the answers underneath (4 together with vi inwards this case).  Keep repeating this procedure until the ii numbers have no mutual factors (ie 2 together with iii below).  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 together with the numbers inwards the bottom row. It's slow to hollo back which is which - to detect the LCM, await for the L shape.

It's thus quick!  And simple!  Try it.

Don Steward features an option method inwards this spider web log post.  He mentions that you lot tin detect a LCM past times dividing the production of the ii numbers past times their HCF ie inwards this example, (24 x 36)/12 = 72.

See my Resources Library for resources for didactics HCF together with LCM.

Factorising Harder Quadratics

My pupils panic at the sight of a quadratic alongside a leading coefficient greater than one.  I factorise these quadratics yesteryear inspection (the 'guess as well as test' method) but my pupils aren't satisfied alongside this proposition - they desire a to a greater extent than structured approach.

Influenza A virus subtype H5N1 normally taught method inwards the Great Britain involves splitting the middle term inwards 2 (sometimes called the 'Grouping Method'). This is explained really clearly here (thanks to SRWhitehouse for this resource). Teachitmaths.co.uk has a PowerPoint explaining this method. It's worth watching James Tanton's video 'Splitting the Middle Term' too. He's non a fan!

'Grouping'. Source: Flat World Education

An alternative, which seems slow at get-go but paves the agency for a large reveal of misconceptions, is the 'slide as well as divide' method. The method, as well as its associated problems, are nicely described inwards Nix the Tricks.

Nix the Tricks offers an interesting choice - I've provided 2 examples hither but it's worth reading the majority for the total explanation.

It's also worth looking at this post yesteryear Don Steward to run across his tap live yesteryear method for finding factors as well as for lots of helpful exercise questions.

Also worth a mention: when I get-go start instruction quadratic factorisation, I similar to purpose this well-designed sum-product worksheet from greatmathsteachingideas.com as a starter. It's skillful exercise of an essential skill.

Large Foursquare Roots

In C1, the non-calculator H5N1 degree module, my pupils sometimes conk stuck solving quadratic equations that are difficult to factorise. This mightiness live on because the leading coefficient is greater than i (see my earlier post) or it may only live on because the numbers inwards the quadratic are large.  I usually propose that they persevere amongst factorising, but if they accept to resort to using the quadratic formula together with thence they may need to foursquare origin a large foursquare number. Here's only about elementary ideas for how to produce thence - inwards this example I'm finding the foursquare origin of 324, though these methods travel as good for larger foursquare numbers.

Note that if the quadratic equation can't live on factorised inwards the outset house together with thence nosotros tin either role the formula or consummate the foursquare to solve it, together with nosotros volition destination upwardly amongst a surd inwards our finally answer.  My side past times side post service is all most didactics surds!

A Percentages Trick

Just a quick postal service today to portion a non bad tip:

You tin swap the percent in addition to the amount:  16% of £25 is hard to calculate inward your head, only 25% of £16 is just the same amount.

So if you lot desire to rapidly live on out 16% of £25 without a calculator, uncovering 25% of £16 instead (ie split £16 past times 4). Simple!

Think nigh why this works.

Source: www.aperiodical.com

Thoughts On Didactics Fractions

Fraction misconceptions - MathsPad

Yuvraj (@YuviLite) asked me to write a post service virtually teaching fraction addition. I similar a challenge! Fractions are notoriously hard to teach, riddled alongside misconceptions as well as crucial to acquire right. If I were to endure thorough on this topic, I’d demand to create a hell of a lot of research as well as write a dissertation length weblog post.

I for sure haven't been that rigorous, but I accept works life a few ideas. First read what Yuvraj has written virtually difficulties inwards teaching fractions, as well as thus accept a hold off at my ideas at the end.


When I hear students utter the phrase, ‘uh no non fractions’ a pocket-sized phonation buried deep inwards my subconscious feels the same thing. Fractions is 1 of those topics which seems to endure on the system of function for EVERY twelvemonth as well as yet in that place are nevertheless thus many students who are befuddled yesteryear them as well as this is 1 of the reasons I accept been thinking virtually how I learn the topic.

I am a instructor who believes inwards developing agreement as well as non exactly only reducing mathematics to a serial of procedures. I accept e'er thought this as well as it was drummed into me on my PGCE as well as Masters. Yet fractions has brought me to a crossroads. Before I tell you lot why, allow us hold off at dissimilar strategies of teaching the improver as well as subtraction of fractions. 

The Cross Multiplying (Butterfly Method). This is the 1 I was taught at school. I had no agreement of why it worked but it got me the correct answer as well as at that historic menstruum that is all I cared about. To add together 2 thirds as well as 3 quarters nosotros would multiply the denominators together as well as thus cross multiply the numerators and denominators as shown below.

The Grid Method. I of late discovered this option procedural method. Draw a grid equally shown below. The starting fourth dimension fraction is written on the top as well as the instant along the side. A plus sign is set inwards the starting fourth dimension box equally this is the functioning nosotros desire to comport out. We set the products of the other numbers into the grid. We as well as thus depict around the highlighted boxes as well as this would give us our numerator, inwards this instance 8 + nine = 17. The white box would endure our denominator.

Fraction Strips. In NixTheTricks, the role of fractions strips is advocated to learn the improver as well as subtraction of fractions. For representative to add together a one-half as well as a quarter we would start alongside visual representations equally shown below.

When nosotros set the one-half as well as quarter together nosotros can't limited this equally a unmarried fraction without cutting the pieces into equal parts (quarters inwards this case). The thought hither is that the agreement of a mutual denominator is built upward as well as linked to the method for adding fractions.

Paper Folding. This method is from Mike Ollerton’s fantabulous ‘Getting The Buggers To Add Up’ - read the explanation here (or purchase the book!). 

Fraction Wall. I accept also seen teachers role fraction walls to aid students equally a method of identifying equivalent fractions. Some people role this or shapes to acquire the thought of equivalent fractions across.

My Dilemma
I accept e'er gone downwards the conceptual route. I accept championed the conceptual route, until I got to algebraic fractions. The conceptual route e'er seems to Pb dorsum to the to a greater extent than procedural route, especially for fractions where the denominators are 2 dissimilar prime numbers, as well as for algebraic fractions. If students are to a greater extent than used to the procedural route, they should inwards theory endure able to able this physical care for for algebraic fractions.

Which has led me to question: is it worth the hurting of rewiring students who e'er seemed to accept heard of the cross multiplication method fifty-fifty earlier they accept reached me (even the Year 7s)? I accept spent many hours struggling alongside folding bits of newspaper as well as sharing cakes to embed the thought of equivalent fractions. 

Is it plenty exactly to explicate why the cross multiplication or grid method works as well as and thus ensure fluency inwards the skill?


Thanks Yuvraj. I wonder the same thing. In many topics nosotros fighting to respect the correct residuum of conceptual agreement vs procedural methods. Us pathetic maths teachers are forever fretting virtually it.

My preferred procedural method, for both numerical as well as algebraic fractions, is the no-frills version (find the lowest mutual multiple of the 2 denominators, as well as thus convert both fractions to equivalent fractions alongside that denominator, as well as thus add together the numerators). It works for me, but I learn high powerfulness pupils thus I accept a relatively slow life.

Now for a few teaching ideas. I’m non trying to solve the world’s fraction problems, but I promise you lot may respect some interesting activities hither for your fractions lessons.

Don Steward is a Genius
subtracting fractions activity inwards which students hold off at the diagram below as well as figure out what's going on. As Mr Steward says, this gives the instructor fourth dimension to drinkable their tea! It's a truly clear agency to explicate fraction subtraction using mutual denominators.

Misconceptions as well as Assessment
This blog has some vivid ideas for identifying fraction misconceptions which powerfulness endure peculiarly helpful when you lot start tackling fractions alongside a Year vii class. For example, the author asked his students to explicate what's incorrect inwards each of the 3 pictures below as well as got some interesting responses.

You could as well as thus inquire students to role the same shapes as well as respect an accurate agency to correspond the fractions indicated. Other ideas inwards this weblog include bespeak the students to consummate the sentences 'Fractions are...', 'When you lot add together 2 fractions you lot accept to...' as well as 'You role fractions when you...'. Or fifty-fifty inquire pupils to write downwards 'Everything I Know virtually Fractions' as well as write a listing of what they'd similar to learn. These activities volition aid you lot create upward one's hear your starting dot inwards Year 7. The blog is good worth a read.

Manipulatives as well as Visual Aids
Yuvraj already touched on visual aids. I retrieve beingness told during my PGCE that I only must role Cuisenaire Rods, but non truly having a clue what to create alongside them. There's whole books written virtually this! I works life explanations here as well as here - the method seems rather overcomplicated. But if they appeal to you, as well as thus Nrich provides this online tool for demonstrating Cuisenaire Rods on the Interactive Whiteboard.

This website has a lot of skilful materials on fractions as well as visual aids, including downloadable fraction circles.

‘Number 1 inwards disguise’
Look at the 'NOID' poster below. Yes, it is quite ridiculous. But it’s really slow to follow the thinking hither as well as I wonder if this explanation (or similar) would aid railroad train students' understanding. If you’re looking something a niggling less NOIDy for teaching improper fractions as well as thus this weblog post has some fantabulous ideas using manipulatives.

From cindywhitebcms.blogspot.com

Division
'Ours is non to argue why, ours is but to inverse as well as multiply'. This made me express joy because I've never attempted to explicate fraction partitioning to my pupils beyond the 'KFC' dominion (keep, flip, change). Don Steward has written an fantabulous post virtually ways to explicate why it works - this one from moveitmaththesource.com peculiarly appealed to me. I'll top away on this inwards my dorsum bag inwards instance ever asked the question.

Final thoughts
I'm afraid I don't accept the answer to how best to railroad train conceptual understanding, but in that place may endure some ideas hither that you lot haven't seen before. In price of resources, there's tonnes of worksheets, games as well as activities online - run across my resources library for some recommendations. For to a greater extent than teaching ideas, here's some farther reading:

• The Guardian's 'How to learn fractions' 
• Bruno Reddy's 'With Fractions, Common-Core Training Goes Beyond 'Invert as well as Multiply''
• James Tanton's 'Troublesome Fractions'.

Thank you lot thus much to Yuvraj for his helpful input to this post. Comments as well as tweets would endure much appreciated - we'd dearest to hear your ideas.

Fraction hopscotch inwards the maths corridor - from cisdmathmusings.blogspot.com

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.