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!

5 Maths Gems #5

Twitter is a unlike identify straightaway everyone's dorsum at school. During the solar daytime it's actually tranquillity because teachers are difficult at work, merely inwards the eve it comes live on again. Despite existence the outset calendar week of term, the United Kingdom of Great Britain as well as Northern Ireland of Britain as well as Northern Republic of Ireland maths twitter chats (#mathscpdchat on Tuesdays as well as #mathschat on Wednesdays) were equally busy equally e'er - equally park they gave me enough of inspiration.

This calendar week likewise saw the supply of instructor blogs. I actually bask reading reflections of what teachers convey been trying inwards their lessons, what went good as well as what didn't. These blogs are a fantastic source of ideas.

This 5th weekly gems shipping is (unintentionally) brought to yous past times the missive of the alphabet P. It features patterns, puzzles, primes, projects, plenaries as well as posters.

1. Patterns as well as Puzzles

Fawn Nguyen's (blog post close a successful lesson inwards which her students were actually engaged inwards exploring these sequences.

Sticking amongst visual maths, Denise Gaskins (@letsplaymath) shared a puzzle from the website 1001 Visual Puzzles. I explored this site as well as constitute that it's full of rattling odd as well as master copy puzzles. I likewise learnt a piddling close topology - what a bizarre branch of mathematics. Here's an event of a topological puzzle (students would demand to advert to a map of South America for this):

2. Primes

Primes are actually interesting, aren't they? I've seen to a greater extent than or less slap-up materials on primes this week.

Richard Green (@edfromo) is a mathematician who posts regularly on Google+. He writes actually interesting things close a broad gain of mathematical topics. He frequently features animations which you'll know I'm a fan of if yous read my shipping close animations as well as simulations. For event this is a rattling clear explanation of Mersenne primes as well as perfect numbers.

If you're thinking of doing something on primes amongst your students, @the_chalkface shared this first-class true or fake activity which students volition detect interesting as well as stretching.

I likewise similar this activeness 'Further Factor Properties' from @TeachitMaths, mainly because it involves sexy primes as well as I'm a chip young similar that.

4. Projects

I've never sentiment of textbook redesign before, merely it could live on a actually skilful mini-project at the destination of a topic. On blog post close a successful lesson inwards which her students were actually engaged inwards exploring these sequences.

Sticking amongst visual maths, Denise Gaskins (@letsplaymath) shared a puzzle from the website 1001 Visual Puzzles. I explored this site as well as constitute that it's full of rattling odd as well as master copy puzzles. I likewise learnt a piddling close topology - what a bizarre branch of mathematics. Here's an event of a topological puzzle (students would demand to advert to a map of South America for this):

2. Primes

Primes are actually interesting, aren't they? I've seen to a greater extent than or less slap-up materials on primes this week.

Richard Green (@edfromo) is a mathematician who posts regularly on Google+. He writes actually interesting things close a broad gain of mathematical topics. He frequently features animations which you'll know I'm a fan of if yous read my shipping close Pythagoras' Theorem as well as added a topic specials page as well as thence yous tin rapidly detect similar posts. I likewise had a rant close student targets inwards which I questioned whether they are effective or fifty-fifty necessary. After Ed Southall (@edsouthall) shared these awesome wall decals, I updated my mathsy gifts page. I've likewise added a duo to a greater extent than homeworks to the Pret homeworks page.

How skilful are yous at drawing lines of best fit? Test yourself here (thanks to @MARitchings) - fun for teachers as well as likewise a useful lesson resource.

Finally - look, blog post close a successful lesson inwards which her students were actually engaged inwards exploring these sequences.

Sticking amongst visual maths, Denise Gaskins (@letsplaymath) shared a puzzle from the website 1001 Visual Puzzles. I explored this site as well as constitute that it's full of rattling odd as well as master copy puzzles. I likewise learnt a piddling close topology - what a bizarre branch of mathematics. Here's an event of a topological puzzle (students would demand to advert to a map of South America for this):

2. Primes

Primes are actually interesting, aren't they? I've seen to a greater extent than or less slap-up materials on primes this week.

Richard Green (@edfromo) is a mathematician who posts regularly on Google+. He writes actually interesting things close a broad gain of mathematical topics. He frequently features animations which you'll know I'm a fan of if yous read my shipping close my end maths gems post.

5 Maths Gems #11

I've got roughly vivid gems for y'all this week. As commons Twitter has been on transcend form.

1. Quick Key
Last calendar week I wrote close Plickers, which is an app y'all tin role inwards the classoom for formative assessment - y'all ask a device only your students don't. Today I'm suggesting roughly other app that y'all tin role for assessment - Quick Key. With this app y'all tin laid multiple alternative homeworks or tests as well as grade them inwards seconds past times scanning the respond sheets ('Quick Tickets') with your phone.

Quick Key is the excogitation of American instructor Walter Duncan (@4_teachers). It "turns your band into a scanner as well as eliminates hand-grading of assessments, fifty-fifty for teachers working inwards paper-based classrooms without a calculator or internet. Analytics as well as information exports are easy, thus y'all tin focus on your students".

My initial thoughts were that although this app could live on rattling useful (less fourth dimension spent marking = to a greater extent than fourth dimension spent planning character lessons), it won't allow me to banking concern friction match students' methods, which limits opportunities for useful feedback. But for straightforward cognition checks, to inform me how much my students know, it would operate rattling well.

But as well as then I read this post close it. Find the Factors is designed by @IvaSallay who says it's an fantabulous agency for children as well as adults to review multiplication facts, role logic, as well as strengthen encephalon power. The puzzles are based on a multiplication grid where most of the numbers own got been removed. The grid tin live on completed using logic as well as cognition of factors - inwards the event below y'all could foremost past times completing the 2 circled cells. The puzzles come upward inwards varying levels of difficulty, as explained here. Find the Factors has already been rattling pop with tweeters who've tried it this week.

4. Chalk Pens

Kev Lister ‏(@ListerKev) tweeted close chalk pens. I've written close writing on desks, windows as well as walls before, only these are novel to me. Kev says the chalk wipes off non-porous surfaces easily with a damp cloth. He uses these pens to write targeted questions or supply scaffolding for selected students. I intend my boss would live on happier with me writing on desks with these rather than whiteboard markers.

5. Alternative Number Line
I'm reluctant to include 2 ideas from Ed Southall (@solvemymaths) inwards i post, only he's a legend thus I'll allow it... In the same agency the maths clock inwards my kitchen makes a nifty talking betoken for guests to my house, his alternative let on line should live on displayed inwards Influenza A virus subtype H5N1 grade classrooms - students (and teachers!) volition live on intrigued. Mr G Walton (@mr_g_walton) has made a fantastic header to larn with it.

Stuart Price (@sxpmaths) suggests making an activity out of this - cutting the let on business upward as well as give it to your Year 13s thus they tin kind it into order. They may non live on familiar with roughly of the annotation only it's a skillful chance for interesting discussions (I flora out how to read ternary to sympathise Ed's alternative for let on 5. I'd also non seen flooring as well as ceiling annotation before).

Stuart also suggests that students pattern their ain let on lines. I honey this thought - it would operate with students of whatever age, from principal schoolhouse to university. It's like to an activity I suggested inwards my reflection stickers. Students stick a 'Not nonetheless yeti' or 'I got it' sticker inwards their majority to allow y'all know whether they've understood the work.

Speaking of stickers, Adil Jaffer (@adil_3) shared his fantastic feedback stickers thus y'all don't own got to proceed writing the same affair all the fourth dimension when you're marking.

One of the post pop tweets I saw this calendar week was from @MathedUp who shared his fantastic 'request a selfie' idea. I tin encounter this actually improving the character of students' operate - many of my students would honey the chance to print their parents. If y'all role iDoceo as well as then it's slow to own got a movie on your iPad as well as e-mail it to a parent.

Did y'all encounter these practice books, tweeted by @dandesignthink? The boxes at the bottom of the pages incorporate a checklist (accuracy, spelling, grammar), cardinal words, farther questions, cardinal points, what went well, as well as feedback. An 'Exercise majority 2.0' costs 50p - amount details are available here. There's also a maths version - Maths majority 2.0 is as awesome. I wishing I had a budget to pass because I'm desperate to gild these!

The to a greater extent than I role Twitter, the to a greater extent than inclined I am to hashtag everything I write (or say!). I own got to bound myself from putting hashtags inwards text messages to my hubby ('what do y'all desire for dinner tonight? #pleasesaytakeaway'). It's prissy to encounter that I'm non lone - Cathy Yenca (post close it. Find the Factors is designed by @IvaSallay who says it's an fantabulous agency for children as well as adults to review multiplication facts, role logic, as well as strengthen encephalon power. The puzzles are based on a multiplication grid where most of the numbers own got been removed. The grid tin live on completed using logic as well as cognition of factors - inwards the event below y'all could foremost past times completing the 2 circled cells. The puzzles come upward inwards varying levels of difficulty, as explained here. Find the Factors has already been rattling pop with tweeters who've tried it this week.

4. Chalk Pens

Kev Lister ‏(@ListerKev) tweeted close chalk pens. I've written close writing on desks, windows as well as walls before, only these are novel to me. Kev says the chalk wipes off non-porous surfaces easily with a damp cloth. He uses these pens to write targeted questions or supply scaffolding for selected students. I intend my boss would live on happier with me writing on desks with these rather than whiteboard markers.

5. Alternative Number Line
I'm reluctant to include 2 ideas from Ed Southall (@solvemymaths) inwards i post, only he's a legend thus I'll allow it... In the same agency the maths clock inwards my kitchen makes a nifty talking betoken for guests to my house, his alternative let on line should live on displayed inwards Influenza A virus subtype H5N1 grade classrooms - students (and teachers!) volition live on intrigued. Mr G Walton (@mr_g_walton) has made a fantastic header to larn with it.

Stuart Price (@sxpmaths) suggests making an activity out of this - cutting the let on business upward as well as give it to your Year 13s thus they tin kind it into order. They may non live on familiar with roughly of the annotation only it's a skillful chance for interesting discussions (I flora out how to read ternary to sympathise Ed's alternative for let on 5. I'd also non seen flooring as well as ceiling annotation before).

Stuart also suggests that students pattern their ain let on lines. I honey this thought - it would operate with students of whatever age, from principal schoolhouse to university. It's like to an activity I suggested inwards my Open Evening post where y'all concur a 'design a maths clock' competition. The clock pictures below are all taken from sbcrafts.net where y'all tin banking concern friction match out a big attain of clock let on ideas that could also live on used on let on lines.

And finally...
Algebraic long segmentation is fun when y'all know how to do it, only it tin live on a chip tricky to learn to Year 12. It's prissy to know at that spot are alternative methods. I've factorised polynomials past times inspection earlier only non using a grid as suggested by post close it. Find the Factors is designed by @IvaSallay who says it's an fantabulous agency for children as well as adults to review multiplication facts, role logic, as well as strengthen encephalon power. The puzzles are based on a multiplication grid where most of the numbers own got been removed. The grid tin live on completed using logic as well as cognition of factors - inwards the event below y'all could foremost past times completing the 2 circled cells. The puzzles come upward inwards varying levels of difficulty, as explained here. Find the Factors has already been rattling pop with tweeters who've tried it this week.

4. Chalk Pens

Kev Lister ‏(@ListerKev) tweeted close chalk pens. I've written close writing on desks, windows as well as walls before, only these are novel to me. Kev says the chalk wipes off non-porous surfaces easily with a damp cloth. He uses these pens to write targeted questions or supply scaffolding for selected students. I intend my boss would live on happier with me writing on desks with these rather than whiteboard markers.

5. Alternative Number Line
I'm reluctant to include 2 ideas from Ed Southall (@solvemymaths) inwards i post, only he's a legend thus I'll allow it... In the same agency the maths clock inwards my kitchen makes a nifty talking betoken for guests to my house, his alternative let on line should live on displayed inwards Influenza A virus subtype H5N1 grade classrooms - students (and teachers!) volition live on intrigued. Mr G Walton (@mr_g_walton) has made a fantastic header to larn with it.

Stuart Price (@sxpmaths) suggests making an activity out of this - cutting the let on business upward as well as give it to your Year 13s thus they tin kind it into order. They may non live on familiar with roughly of the annotation only it's a skillful chance for interesting discussions (I flora out how to read ternary to sympathise Ed's alternative for let on 5. I'd also non seen flooring as well as ceiling annotation before).

Stuart also suggests that students pattern their ain let on lines. I honey this thought - it would operate with students of whatever age, from principal schoolhouse to university. It's like to an activity I suggested inwards my The Art of Collaboration', it's e'er nifty to take heed when these ideas larn far into classrooms. Have a nifty half-term intermission everyone!

5 Maths Gems #18

Most of my readers likely already subscribe to Chris Smith's lovely maths newsletter. Full of instruction ideas, puzzles, jokes in addition to mathematical trivia, it's a joy to have every week. Chris has been producing the newsletter since 2007 when he was a wee NQT. Amongst other things, the newsletter is a fantastic machinery for sharing expert practice.

Chris directly has over 900 newsletter subscribers. This calendar week he published his 303^rd issue. I've solely been subscribing for a twosome of months but Chris was sort plenty to part his entire dorsum catalogue alongside me, which I've actually enjoyed reading. The ideas inward my weekly gems posts are commonly sourced from Twitter, but today I characteristic my v favourite ideas from the get-go 100 issues of Chris's newsletters. Huge thank you lot to Chris for allowing me to create this!

1. Factorising
Here's a lovely thought for introducing factorising.

2. Connecting numbers
This is a actually prissy starter action to railroad train ideas relating to number properties.

3. Learning past times Rope
This is keen for exploring polygons.

4. Finding a Route
Another prissy starter activity.

5. Revision notes

In Issue 27, Chris featured an thought from TES which is designed to encourage students to revise for tests.

“I gave each pupil an A4 sail of newspaper alongside their mention in addition to my signature on it in addition to told them they had a essay coming upward in addition to they were allowed to select the blank page inward alongside them. It was upward to them what they did alongside the page. If they wrote on it, fine, if they didn't, fine. Immediately ane of the to a greater extent than troublesome pupils shouted out, “Great! I'm going to re-create downwards all the examples from my book.” Objective ane achieved. They were going to revise.

At the terminate of the test, I collected their sheets to expect at them inward to a greater extent than detail. It was real rewarding when pupils pretty much demanded that they got them back. Objective 2 achieved. My pupils directly value all of their revision notes."

Puzzles

Each of Chris's newsletters features a puzzle - here's 2 that I peculiarly liked (answers are at the bottom of this post):

And finally, 2 bits of trivia:

That's it from Chris's newsletters for now, but I volition select you lot to a greater extent than or less to a greater extent than newsletter gems afterwards I've read the side past times side 100 issues. Don't forget to follow Chris on Twitter in addition to email him if you'd similar to have his weekly newsletters.

What I've been upward to
I've had a busy in addition to emotional week. My motherhood larn out is coming to an terminate in addition to I'm keen to start planning lessons. I don't select my timetable yet, in addition to thence am eagerly awaiting intelligence of what classes I'll hold upward instruction inward 3 weeks. On Tuesday I hosted #mathscpdchat which was all most mutual mathematical misconceptions - you lot tin read a summary here. On Wed I had a project interview inward which I taught a lesson on indices to a Year 8 shape - it was fun to hold upward dorsum inward the classroom in addition to I'm pleased to nation I got the job, in addition to thence volition hold upward moving schools inward May to perish an Aspiring Lead Practitioner. I was likewise real proud to select an article published inward Academies Week most the novel maths GCSE in addition to the associated problems alongside recruitment. Feedback in addition to comments are real welcome.

Puzzle answers: 1. r = 29  2. Pi/2

5 Maths Gems #20

Hello. I'm pleased to introduce my 20th gems post today - a milestone! Twitter has been repose this calendar week because Christmas took our minds off operate for a duet of days, but I've even therefore got around corking ideas for you. The Christmas holidays are brusk in addition to busy therefore it won't live on long earlier you're dorsum to planning lessons - I promise this post gives y'all around inspiration for the Spring term.

1. Factorising Quadratics
You know questions like this:

In Question 1 above, y'all tin come across that the move past times left jail mobile telephone has to comprise a three because it's a constituent of both three in addition to 6. The residue of the cells tin live on completed speedily i time i mutual constituent has been established. This is a uncomplicated instance but Don develops the questions to acquire increasingly difficult, around having multiple solutions. 

The adjacent ready of activities extend the same see to algebra, starting alongside this:

There's lots of these to consummate - first-class exercise of factorising linear expressions.

The adjacent phase of this exercise is factorising quadratics.

Conveniently I'm currently planning a Year 10 lesson on factorising quadratics. I desire my students to create a lot of exercise therefore volition definitely live on using this activity. The questions construct upwards to a suitably challenging marking of difficulty, ending alongside this one:

I desire to encourage my students to factorise 'harder' quadratics (ie a > 1) past times inspection. This is my preferred method (ie 'guess in addition to test') but my students ever need that I instruct them a to a greater extent than structured approach (eg 'the Grouping Method') which frustrates me. Their insistence on next an algorithm suggests a lack of confidence. I mean value the enquiry higher upwards turns factorising quadratics into a sort of logic problem. Tackling this enquiry without an algorithm mightiness aid my students prepare the confidence to factorise harder quadratics past times inspection.

One final see for a lesson on factorising quadratics - I similar the work below from openmiddle.com. There are a unwrap of possible solutions therefore y'all could challenge students to discovery a dissimilar solution to the someone adjacent to them. 

2. Angle Sense alongside the Interactive Whiteboard
I've been planning a Year vii lesson on angles inwards which I'd similar my students to approximate angle sizes. If y'all were asked to to clitoris an angle of 210^o freehand, how would y'all create it? I'd mean value of it every bit a direct delineate of piece of work addition a 3rd of a right-angle. If y'all choose proportional reasoning skills in addition to therefore it's pretty slow to brand an educated guess. An angle estimation activity would operate perfectly good without technology (read out a serial of angle sizes in addition to inquire your students to clitoris their freehand estimates on paper. They in addition to therefore banking firm represent their estimates using a protractor - around other useful skill). But if y'all desire a similar activity for the interactive whiteboard in addition to therefore y'all mightiness similar this fun here. Some of these angle games are quite funny - Banana Hunt in exceptional made me chuckle.

If y'all similar these interactive whiteboard games in addition to therefore you'll discovery loads at Sheppard Software. It's amusing that there's an Absolute Value Number Balls game - this concept isn't covered until Year thirteen inwards the U.K. but I bet my students would beloved to play this game - 5 minutes lite relief inwards a C3 lesson!

FlashMaths.co.uk is around other corking website for interactive whiteboard activities. Flash Maths was created past times Jonathan Hall (@studymaths) who brings us a plethora of fantastic tools on StudyMaths.co.uk. If y'all haven't seen it before, banking firm represent out MathsBot.com which is his uncomplicated (but brilliant) worksheet generator.

3. Big Questions

Billy Adamson (@Billyads_47) shared a fantastic ready of mathematical thinking prompts 'The Big Questions'. Here's a few examples: 

Lovely opened upwards questions from Billy to generate give-and-take in addition to prepare understanding. There's around to a greater extent than adept examples of opened upwards questions here:

4. Trigonometric Problem Solving

Our Year 13s' work solving skills are tested when they're asked to simplify expressions involving trigonometric identities inwards C3 (like the example below).

I discovery that my students acquire frustrated when they can't location a 'way in' direct away. They surrender quickly. There's genuinely a pretty criterion ready of starting points, every bit described on www.intmath.com (@intmath). 

I combat to aid my students experience confident inwards tackling these problems, therefore I genuinely similar this activity from @mjfenton. Here's an extract:

The see is that nosotros starting fourth dimension alongside a lot of construction in addition to gradually give fewer hints until students are able to solve the problems themselves. The steps mightiness appear logical to us, but we're experienced work solvers. 

It's a adept see for maths teachers to seek to solve unfamiliar problems every instantly in addition to and therefore (like the instance below from ‏@dannytybrown) to remind ourselves that mathematical work solving oftentimes requires patience, inventiveness in addition to multiple attempts. We all experience frustration inwards work solving, only similar our students do, but nosotros know that the satisfaction of eventually finding the solution is good worth it.

5. Dividing alongside Decimals

I've mentioned earlier that I beloved MathsPad's resources - enough of them are costless in addition to the residue come upwards at a toll of exclusively £3 per month. Whether your schoolhouse subscribes or not, it's worth registering for e-mail updates inwards lodge to continue runway of all the novel resources. This month, the interactive resources on Decimal Calculations caught my eye. It ever surprises me how many students volition happily tell that xl divided past times ½ equals 20. Activities similar the i shown below volition aid tackle this misconception in addition to encourage students to mean value earlier they answer.

That's it for this week. I'll exit y'all alongside a video from 1977 - 'Congruent Triangles' past times Bruce in addition to Katharine Cornwell (another jewel found on @MathMunch). Happy New Year!