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!

5 Maths Gems #5

Twitter is a dissimilar identify similar a shot everyone's dorsum at school. During the 24-hour interval it's actually tranquillity because teachers are difficult at work, only inwards the eventide it comes hold upwards again. Despite beingness the kickoff calendar week of term, the Great Britain maths twitter chats (#mathscpdchat on Tuesdays as well as #mathschat on Wednesdays) were every bit busy every bit always - every bit park they gave me enough of inspiration.

This calendar week every bit good saw the provide of instructor blogs. I actually taste reading reflections of what teachers accept 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 transportation is (unintentionally) brought to you lot 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 virtually 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 real odd as well as master puzzles. I every bit good learnt a footling virtually topology - what a bizarre branch of mathematics. Here's an trial of a topological puzzle (students would demand to cite 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 peachy materials on primes this week.

Richard Green (@edfromo) is a mathematician who posts regularly on Google+. He writes actually interesting things virtually a broad arrive at of mathematical topics. He frequently features animations which you'll know I'm a fan of if you lot read my transportation virtually animations as well as simulations. For trial this is a real 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 splendid true or fake activity which students volition detect interesting as well as stretching.

I every bit good similar this action '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 take in of textbook redesign before, only it could hold upwards a actually adept mini-project at the halt of a topic. On blog post virtually 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 real odd as well as master puzzles. I every bit good learnt a footling virtually topology - what a bizarre branch of mathematics. Here's an trial of a topological puzzle (students would demand to cite 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 peachy materials on primes this week.

Richard Green (@edfromo) is a mathematician who posts regularly on Google+. He writes actually interesting things virtually a broad arrive at of mathematical topics. He frequently features animations which you'll know I'm a fan of if you lot read my transportation virtually Pythagoras' Theorem as well as added a topic specials page as well as thus you lot tin chop-chop detect similar posts. I every bit good had a rant virtually 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 every bit good added a brace to a greater extent than homeworks to the Pret homeworks page.

How adept are you lot at drawing lines of best fit? Test yourself here (thanks to @MARitchings) - fun for teachers as well as every bit good a useful lesson resource.

Finally - look, blog post virtually 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 real odd as well as master puzzles. I every bit good learnt a footling virtually topology - what a bizarre branch of mathematics. Here's an trial of a topological puzzle (students would demand to cite 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 peachy materials on primes this week.

Richard Green (@edfromo) is a mathematician who posts regularly on Google+. He writes actually interesting things virtually a broad arrive at of mathematical topics. He frequently features animations which you'll know I'm a fan of if you lot read my transportation virtually my final maths gems post.

5 Maths Gems #11

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

1. Quick Key
Last calendar week I wrote almost Plickers, which is an app y'all tin role inward the classoom for formative assessment - y'all need a device merely 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 selection homeworks or tests in addition to grade them inward seconds past times scanning the answer sheets ('Quick Tickets') with your phone.

Quick Key is the conception of American instructor Walter Duncan (@4_teachers). It "turns your cry upwardly into a scanner in addition to eliminates hand-grading of assessments, fifty-fifty for teachers working inward paper-based classrooms without a estimator or internet. Analytics in addition to information exports are easy, thence y'all tin focus on your students".

My initial thoughts were that although this app could last real 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 gibe students' methods, which limits opportunities for useful feedback. But for straightforward cognition checks, to inform me how much my students know, it would function real well.

But thence I read this post almost it. Find the Factors is designed by @IvaSallay who says it's an splendid means for children in addition to adults to review multiplication facts, role logic, in addition to strengthen encephalon power. The puzzles are based on a multiplication grid where most of the numbers possess got been removed. The grid tin last completed using logic in addition to cognition of factors - inward the illustration below y'all could commencement past times completing the 2 circled cells. The puzzles come upwardly inward varying levels of difficulty, as explained here. Find the Factors has already been real pop with tweeters who've tried it this week.

4. Chalk Pens

Kev Lister ‏(@ListerKev) tweeted almost chalk pens. I've written almost writing on desks, windows in addition to walls before, merely 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 render scaffolding for selected students. I mean value my boss would last 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) inward 1 post, merely he's a legend thence I'll allow it... In the same means the maths clock inward my kitchen makes a neat talking betoken for guests to my house, his alternative break line should last displayed inward Influenza A virus subtype H5N1 degree classrooms - students (and teachers!) volition last 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 break job upwardly in addition to give it to your Year 13s thence they tin variety it into order. They may non last familiar with roughly of the tone merely it's a adept chance for interesting discussions (I constitute out how to read ternary to sympathise Ed's selection for break 5. I'd also non seen flooring in addition to ceiling tone before).

Stuart also suggests that students pattern their ain break lines. I honey this thought - it would function with students of whatever age, from main schoolhouse to university. It's like to an activity I suggested inward my reflection stickers. Students stick a 'Not withal yeti' or 'I got it' sticker inward their mass to permit y'all know whether they've understood the work.

Speaking of stickers, Adil Jaffer (@adil_3) shared his fantastic feedback stickers thence y'all don't possess got to buy the farm on 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 come across this actually improving the character of students' function - many of my students would honey the chance to print their parents. If y'all role iDoceo thence it's slowly to accept a motion painting on your iPad in addition to electronic mail it to a parent.

Did y'all come across these practise books, tweeted by @dandesignthink? The boxes at the bottom of the pages incorporate a checklist (accuracy, spelling, grammar), telephone commutation words, farther questions, telephone commutation points, what went well, in addition to feedback. An 'Exercise mass 2.0' costs 50p - amount details are available here. There's also a maths version - Maths mass 2.0 is every bit awesome. I wishing I had a budget to pass because I'm desperate to guild 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 possess got to confine myself from putting hashtags inward text messages to my hubby ('what do y'all desire for dinner tonight? #pleasesaytakeaway'). It's overnice to come across that I'm non lonely - Cathy Yenca (post almost it. Find the Factors is designed by @IvaSallay who says it's an splendid means for children in addition to adults to review multiplication facts, role logic, in addition to strengthen encephalon power. The puzzles are based on a multiplication grid where most of the numbers possess got been removed. The grid tin last completed using logic in addition to cognition of factors - inward the illustration below y'all could commencement past times completing the 2 circled cells. The puzzles come upwardly inward varying levels of difficulty, as explained here. Find the Factors has already been real pop with tweeters who've tried it this week.

4. Chalk Pens

Kev Lister ‏(@ListerKev) tweeted almost chalk pens. I've written almost writing on desks, windows in addition to walls before, merely 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 render scaffolding for selected students. I mean value my boss would last 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) inward 1 post, merely he's a legend thence I'll allow it... In the same means the maths clock inward my kitchen makes a neat talking betoken for guests to my house, his alternative break line should last displayed inward Influenza A virus subtype H5N1 degree classrooms - students (and teachers!) volition last 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 break job upwardly in addition to give it to your Year 13s thence they tin variety it into order. They may non last familiar with roughly of the tone merely it's a adept chance for interesting discussions (I constitute out how to read ternary to sympathise Ed's selection for break 5. I'd also non seen flooring in addition to ceiling tone before).

Stuart also suggests that students pattern their ain break lines. I honey this thought - it would function with students of whatever age, from main schoolhouse to university. It's like to an activity I suggested inward 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 gibe out a big arrive at of clock break ideas that could also last used on break lines.

And finally...
Algebraic long partition is fun when y'all know how to do it, merely it tin last a flake tricky to learn to Year 12. It's overnice to know in that place are option methods. I've factorised polynomials past times inspection earlier merely non using a grid as suggested by post almost it. Find the Factors is designed by @IvaSallay who says it's an splendid means for children in addition to adults to review multiplication facts, role logic, in addition to strengthen encephalon power. The puzzles are based on a multiplication grid where most of the numbers possess got been removed. The grid tin last completed using logic in addition to cognition of factors - inward the illustration below y'all could commencement past times completing the 2 circled cells. The puzzles come upwardly inward varying levels of difficulty, as explained here. Find the Factors has already been real pop with tweeters who've tried it this week.

4. Chalk Pens

Kev Lister ‏(@ListerKev) tweeted almost chalk pens. I've written almost writing on desks, windows in addition to walls before, merely 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 render scaffolding for selected students. I mean value my boss would last 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) inward 1 post, merely he's a legend thence I'll allow it... In the same means the maths clock inward my kitchen makes a neat talking betoken for guests to my house, his alternative break line should last displayed inward Influenza A virus subtype H5N1 degree classrooms - students (and teachers!) volition last 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 break job upwardly in addition to give it to your Year 13s thence they tin variety it into order. They may non last familiar with roughly of the tone merely it's a adept chance for interesting discussions (I constitute out how to read ternary to sympathise Ed's selection for break 5. I'd also non seen flooring in addition to ceiling tone before).

Stuart also suggests that students pattern their ain break lines. I honey this thought - it would function with students of whatever age, from main schoolhouse to university. It's like to an activity I suggested inward my The Art of Collaboration', it's ever neat to listen when these ideas larn far into classrooms. Have a neat half-term interruption everyone!

5 Maths Gems #18

Most of my readers in all probability already subscribe to Chris Smith's lovely maths newsletter. Full of instruction ideas, puzzles, jokes together with 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 instantly has over 900 newsletter subscribers. This calendar week he published his 303^rd issue. I've exclusively been subscribing for a twosome of months but Chris was variety plenty to part his entire dorsum catalogue alongside me, which I've actually enjoyed reading. The ideas inwards my weekly gems posts are usually sourced from Twitter, but today I characteristic my v favourite ideas from the kickoff 100 issues of Chris's newsletters. Huge cheers 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 squeamish starter action to prepare ideas relating to number properties.

3. Learning yesteryear Rope
This is neat for exploring polygons.

4. Finding a Route
Another squeamish 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 lift together with my signature on it together with told them they had a essay out coming upwards together with they were allowed to choose the blank page inwards alongside them. It was upwards 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 downward all the examples from my book.” Objective ane achieved. They were going to revise.

At the destination of the test, I collected their sheets to await at them inwards to a greater extent than detail. It was really rewarding when pupils pretty much demanded that they got them back. Objective 2 achieved. My pupils instantly 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 choose yous to a greater extent than or less to a greater extent than newsletter gems later I've read the adjacent 100 issues. Don't forget to follow Chris on Twitter together with email him if you'd similar to have his weekly newsletters.

What I've been upwards to
I've had a busy together with emotional week. My motherhood operate out is coming to an destination together with I'm keen to start planning lessons. I don't accept my timetable yet, then am eagerly awaiting tidings of what classes I'll last instruction inwards 3 weeks. On Tuesday I hosted #mathscpdchat which was all most mutual mathematical misconceptions - yous tin plough over the sack read a summary here. On Wed I had a labor interview inwards which I taught a lesson on indices to a Year 8 course of report - it was fun to last dorsum inwards the classroom together with I'm pleased to tell I got the job, then volition last moving schools inwards May to drib dead an Aspiring Lead Practitioner. I was too really proud to accept an article published inwards Academies Week most the novel maths GCSE together with the associated problems alongside recruitment. Feedback together with comments are really 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 travel for a span of days, but I've nevertheless got around nifty ideas for you. The Christmas holidays are curt in addition to busy therefore it won't live on long earlier you're dorsum to planning lessons - I promise this post gives you lot around inspiration for the Spring term.

1. Factorising Quadratics
You know questions like this:

In Question 1 above, you lot tin give notice run across that the travel past times left jail cellular telephone has to incorporate a iii because it's a element of both iii in addition to 6. The residual of the cells tin give notice live on completed speedily 1 time 1 mutual element has been established. This is a unproblematic instance but Don develops the questions to choke increasingly difficult, around having multiple solutions. 

The adjacent prepare of activities extend the same sentiment to algebra, starting amongst this:

There's lots of these to consummate - first-class practise 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 practise therefore volition definitely live on using this activity. The questions construct upward to a suitably challenging degree of difficulty, ending amongst 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 intend the inquiry inward a higher house turns factorising quadratics into a form of logic problem. Tackling this inquiry without an algorithm powerfulness aid my students educate the confidence to factorise harder quadratics past times inspection.

One final sentiment for a lesson on factorising quadratics - I similar the occupation below from openmiddle.com. There are a publish of possible solutions therefore you lot could challenge students to discovery a dissimilar solution to the someone adjacent to them. 

2. Angle Sense amongst the Interactive Whiteboard
I've been planning a Year vii lesson on angles inward which I'd similar my students to approximate angle sizes. If you lot were asked to to pull an angle of 210^o freehand, how would you lot create it? I'd intend of it equally a guide business addition a 3rd of a right-angle. If you lot choose proportional reasoning skills therefore it's pretty slowly to brand an educated guess. An angle estimation activity would travel perfectly good without applied scientific discipline (read out a serial of angle sizes in addition to enquire your students to pull their freehand estimates on paper. They therefore depository fiscal establishment jibe their estimates using a protractor - around other useful skill). But if you lot desire a similar activity for the interactive whiteboard therefore you lot powerfulness similar this fun here. Some of these angle games are quite funny - Banana Hunt in special made me chuckle.

If you lot similar these interactive whiteboard games 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 inward the United Kingdom of Great Britain in addition to Northern Ireland but I bet my students would honey to play this game - 5 minutes lite relief inward a C3 lesson!

FlashMaths.co.uk is around other nifty 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 you lot haven't seen it before, depository fiscal establishment jibe out MathsBot.com which is his unproblematic (but brilliant) worksheet generator.

3. Big Questions

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

Lovely opened upward questions from Billy to generate intelligence in addition to educate understanding. There's around to a greater extent than expert examples of opened upward questions here:

4. Trigonometric Problem Solving

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

I discovery that my students acquire frustrated when they can't topographic point a 'way in' guide away. They surrender quickly. There's genuinely a pretty criterion prepare of starting points, equally described on www.intmath.com (@intmath). 

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

The sentiment is that nosotros offset amongst a lot of construction in addition to gradually give fewer hints until students are able to solve the problems themselves. The steps powerfulness appear logical to us, but we're experienced occupation solvers. 

It's a expert sentiment for maths teachers to attempt to solve unfamiliar problems every at 1 time in addition to therefore (like the instance below from ‏@dannytybrown) to remind ourselves that mathematical occupation solving oftentimes requires patience, inventiveness in addition to multiple attempts. We all experience frustration inward occupation solving, merely similar our students do, but nosotros know that the satisfaction of eventually finding the solution is good worth it.

5. Dividing amongst Decimals

I've mentioned earlier that I honey MathsPad's resources - enough of them are complimentary in addition to the residual come upward at a terms of alone £3 per month. Whether your schoolhouse subscribes or not, it's worth registering for electronic mail updates inward companionship to proceed rail 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 nation that forty divided past times ½ equals 20. Activities similar the 1 shown below volition aid tackle this misconception in addition to encourage students to intend earlier they answer.

That's it for this week. I'll travel out you lot amongst a video from 1977 - 'Congruent Triangles' past times Bruce in addition to Katharine Cornwell (another precious rock found on @MathMunch). Happy New Year!