Matrix Multiplication

When I teach matrix multiplication I respect that pupils tin expire a fighting lost inwards their calculations.  I constitute a fantastic novel method on this blog.

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

I beloved this method - it's together with then unproblematic to retrieve together with instantly virtually impossible to expire lost when multiplying matrices.

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

Highest Mutual Element Together With Lowest Mutual Multiple

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

I recently discovered an option method that is impressively quick as well as simple.  It is described inwards this video every bit the 'Indian Method'. It's like to the 'upside downwards birthday cake method' but it's much quicker because at that spot is no requirement to operate primes.

Say nosotros desire to notice the Highest Common Factor as well as Lowest Common Multiple of 24 as well as 36.
Write down the ii numbers, then (to the left, every bit inwards my representative below) write downwards whatsoever mutual component (ie 2, 3, 4, half dozen or 12).  I've chosen 6.  Now separate 24 as well as 36 past times half dozen as well as write the answers underneath (4 as well as half dozen inwards this case).  Keep repeating this procedure until the ii numbers have no mutual factors (ie 2 as well as iii below).  Now, your Highest Common Factor is merely the production of numbers on the left. And for the Lowest Common Multiple, notice the production of the numbers on the left as well as the numbers inwards the bottom row. It's slowly to retrieve which is which - to notice the LCM, await for the L shape.

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

Don Steward features an option method inwards this weblog post.  He mentions that you lot tin notice 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 as well as LCM.

Factorising Harder Quadratics

My pupils panic at the sight of a quadratic alongside a leading coefficient greater than one.  I factorise these quadratics past times 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 real 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 commencement but paves the agency for a large publish 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 option - 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 past times Don Steward to run into his tap exceed method for finding factors as well as for lots of helpful practise questions.

Also worth a mention: when I commencement start didactics quadratic factorisation, I similar to role this well-designed sum-product worksheet from greatmathsteachingideas.com as a starter. It's proficient practise of an essential skill.

Large Foursquare Roots

In C1, the non-calculator Influenza A virus subtype H5N1 marking module, my pupils sometimes perish stuck solving quadratic equations that are difficult to factorise. This powerfulness live because the leading coefficient is greater than i (see my earlier post) or it may but live because the numbers inward the quadratic are large.  I commonly propose that they persevere amongst factorising, but if they convey to resort to using the quadratic formula in addition to thence they may require to foursquare origin a large foursquare number. Here's roughly uncomplicated ideas for how to create thence - inward this representative I'm finding the foursquare origin of 324, though these methods run every bit good for larger foursquare numbers.

Note that if the quadratic equation can't live factorised inward the commencement house in addition to thence nosotros tin either purpose the formula or consummate the foursquare to solve it, in addition to nosotros volition halt upwards amongst a surd inward our lastly answer.  My adjacent post service is all close didactics surds!

A Percentages Trick

Just a quick post service today to portion a groovy tip:

You tin rate notice swap the per centum as well as the amount:  16% of £25 is hard to calculate inwards your head, just 25% of £16 is exactly the same amount.

So if you lot desire to speedily function out 16% of £25 without a calculator, honour 25% of £16 instead (ie split £16 past times 4). Simple!

Think most why this works.

Source: www.aperiodical.com

Thoughts On Instruction Fractions

Fraction misconceptions - MathsPad

Yuvraj (@YuviLite) asked me to write a post nearly teaching fraction addition. I similar a challenge! Fractions are notoriously hard to teach, riddled amongst misconceptions together with 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 together with write a dissertation length weblog post.

I for certain haven't been that rigorous, but I convey establish a few ideas. First read what Yuvraj has written nearly difficulties inwards teaching fractions, thence convey a expect at my ideas at the end.


When I listen students utter the phrase, ‘uh no non fractions’ a minor vocalization 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 together with yet in that place are nevertheless thence many students who are befuddled past times them together with this is 1 of the reasons I convey been thinking nearly how I learn the topic.

I am a instructor who believes inwards developing agreement together with non exactly only reducing mathematics to a serial of procedures. I convey ever thought this together with it was drummed into me on my PGCE together with Masters. Yet fractions has brought me to a crossroads. Before I tell you lot why, allow us expect at dissimilar strategies of teaching the add-on together with 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 respond together with at that historic current that is all I cared about. To add together 2 thirds together with 3 quarters nosotros would multiply the denominators together thence cross multiply the numerators and denominators as shown below.

The Grid Method. I of late discovered this choice procedural method. Draw a grid equally shown below. The kickoff fraction is written on the overstep together with the minute along the side. A plus sign is set inwards the kickoff box equally this is the functioning nosotros desire to behave out. We set the products of the other numbers into the grid. We thence depict around the highlighted boxes together with this would give us our numerator, inwards this instance 8 + ix = 17. The white box would endure our denominator.

Fraction Strips. In NixTheTricks, the purpose of fractions strips is advocated to learn the add-on together with subtraction of fractions. For illustration to add together a one-half together with a quarter we would start amongst visual representations equally shown below.

When nosotros set the one-half together with 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 together with linked to the method for adding fractions.

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

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

My Dilemma
I convey ever gone downwards the conceptual route. I convey championed the conceptual route, until I got to algebraic fractions. The conceptual route ever seems to Pb dorsum to the to a greater extent than procedural route, especially for fractions where the denominators are 2 dissimilar prime numbers, together with 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 ever seemed to convey heard of the cross multiplication method fifty-fifty earlier they convey reached me (even the Year 7s)? I convey spent many hours struggling amongst folding bits of newspaper together with sharing cakes to embed the thought of equivalent fractions. 

Is it plenty exactly to explicate why the cross multiplication or grid method plant together with thence ensure fluency inwards the skill?


Thanks Yuvraj. I wonder the same thing. In many topics nosotros combat to notice the correct residuum of conceptual agreement vs procedural methods. Us pitiful maths teachers are forever fretting nearly it.

My preferred procedural method, for both numerical together with algebraic fractions, is the no-frills version (find the lowest mutual multiple of the 2 denominators, thence convert both fractions to equivalent fractions amongst that denominator, thence add together the numerators). It plant for me, but I learn high powerfulness pupils thence I convey 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 notice some interesting activities hither for your fractions lessons.

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

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

You could thence inquire students to purpose the same shapes together with notice an accurate means 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 convey to...' together with 'You purpose fractions when you...'. Or fifty-fifty inquire pupils to write downwards 'Everything I Know nearly Fractions' together with write a listing of what they'd similar to learn. These activities volition aid you lot create upward one's heed your starting dot inwards Year 7. The blog is good worth a read.

Manipulatives together with Visual Aids
Yuvraj already touched on visual aids. I recollect beingness told during my PGCE that I only must purpose Cuisenaire Rods, but non genuinely having a clue what to create amongst them. There's whole books written nearly this! I establish explanations here together with here - the method seems rather overcomplicated. But if they appeal to you, thence Nrich provides this online tool for demonstrating Cuisenaire Rods on the Interactive Whiteboard.

This website has a lot of proficient materials on fractions together with 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 together with I wonder if this explanation (or similar) would aid prepare students' understanding. If you’re looking something a trivial less NOIDy for teaching improper fractions thence this weblog post has some first-class ideas using manipulatives.

From cindywhitebcms.blogspot.com

Division
'Ours is non to argue why, ours is but to inverse together with multiply'. This made me express joy because I've never attempted to explicate fraction sectionalization to my pupils beyond the 'KFC' dominion (keep, flip, change). Don Steward has written an first-class post nearly ways to explicate why it plant - this one from moveitmaththesource.com especially appealed to me. I'll leave of absence along this inwards my dorsum steal inwards instance ever asked the question.

Final thoughts
I'm afraid I don't convey the respond to how best to prepare 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 together with activities online - run into 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 together with Multiply''
• James Tanton's 'Troublesome Fractions'.

Thank you lot thence much to Yuvraj for his helpful input to this post. Comments together with tweets would endure much appreciated - we'd beloved to listen your ideas.

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

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

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 factor tree activities from Don Steward.

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

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 this post past times Don Steward.

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^th term 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.