Linear Simultaneous Equations

Solving linear simultaneous equations is rather satisfying. In this postal service I advise approximately techniques too resources for pedagogy this topic.

Equality too Lincoln
When I saw the cinema Lincoln I was struck yesteryear Abraham Lincoln’s reference to Euclid’s First Axiom ‘Things which are equal to the same matter are also equal to ane another’. Influenza A virus subtype H5N1 well-made betoken inward a give-and-take nearly racial equality, but also an interesting claw into a method for solving simultaneous equations such as

y = 4x + 6

y = 2x + 3

Both 4x + six too 2x + three are equal to y, thus logically nosotros convey 4x + six = 2x + three (this is essentially merely a exchange method). Why non present this this blog gives a adept explanation of Lincoln's point)

Elimination too 'SSS'
When solving simultaneous equations yesteryear elimination, my pupils abide by it really slow to recall the ‘SSS’ dominion ie ‘Same Sign Subtract’. For example, inward the equations 3x + 5y = 59; 2x + 5y = 56, this dominion says that because the 5y inward each equation is positive (‘same sign’), we subtract one equation from the other.

However, I noticed that when solving simultaneous equations such every minute -3x + 5y = 41; 2x + 5y = 56, approximately pupils saw that the -3x too the 2x were different signs too incorrectly decided that this meant they should add the 2 equations. So it’s worth clarifying which signs nosotros are referring to inward the ‘Same Sign Subtract’/’Different Sign Add’ approach.

Graphical Solutions
It’s then of import for pupils sympathise that whenever they solve linear simultaneous equations, they are inward fact finding the betoken of intersection of 2 direct lines. It’s essential to position a query involving parallel lines (eg solve 3x + 4y = 10; 6x + 8y = 15) too perish them thinking nearly why at that spot are no solutions.  I similar to present them this cute picture too.

Wordy Equations
This is ane of my favourite simultaneous equation questions: “A zoo has several ostriches too several giraffes. In full they convey xxx eyes too 44 legs. How many ostriches too how many giraffes are inward the zoo?”. Make certain you lot discourage pupils from using the missive of the alphabet O for ostrich, because it looks similar a zero.

Resources
There's lots of resources online – here’s a few I’ve picked out:

• Don Steward’s Median blog has a huge make of fantastic questions too activities.
• TeachitMaths has a swell ‘wordy question’ worksheet too clear slides on elimination too substitution.
• UEA has a worksheet on linear simultaneous equations amongst a overnice progression of questions.
• There is a useful worksheet on Solving Simultaneous Equations Graphically yesteryear Tristan Jones on TES. 
• This Geogebra worksheet yesteryear themathsteacher.com is a adept practise inward solving simultaneous equations graphically amongst ICT.
• Peter Bland and JustMaths both render assay questions on this topic.
• MEI has a overnice extension activity. 

Linear Graphs

One of the topics that my pupils fighting amongst the most, from Key Stage iii correct through to Year 12, is linear graphs.

My brilliant Year 11s know the dominion for finding the slope of a perpendicular line, but can't operate out how to respond a GCSE interrogation similar this:

Strangely, I learn this differently at GCSE too Influenza A virus subtype H5N1 level.  At GCSE my pupils write down

y = mx + c

too substitute values for y, thousand too x, too thus solve for c.  Whereas at Influenza A virus subtype H5N1 marker my pupils utilization the equation

y - y₁ = m(x - x₁)

Essentially the same method but a slightly dissimilar approach. Perhaps I should hold out to a greater extent than consistent too innovate the 'A marker method' at GCSE. I'd similar to remove heed what other teachers do.

In this postal service I’ll advise closed to ideas too resources for didactics linear graphs.

Key words

Let's showtime amongst the basics. There's a lot of novel vocabulary inward coordinate geometry, closed to of which pupils volition accept encountered at primary school. This is a overnice vocabulary noesis survey every bit suggested inward my earlier post. 

Here's a overnice play a trick on for remembering the words parallel too perpendicular (this could hold out helpful for spellings - parallel is oftentimes misspelt): 

too this activity is suitable for Year six or 7. 

Introducing the shape y = mx + c
I'd unremarkably showtime this topic inward Year seven or 8 amongst plotting conduct work graphs (Teachit Maths provides an excellent Drawing Straight Line Graphs booklet for this purpose), before exploring the shape y = mx + c.

This topic provides an splendid chance for pupil investigation using Desmos. Use a worksheet to guide pupils through the lesson:

• Investigating Straight Lines amongst Desmos (by Tristan Jones on TES)
• Straight Line Graphs inward Desmos (by me!)

Once pupils are able to translate the values of thousand too c too thus they tin endeavour this exercise or this true/false activity (both from rogradymaths.blogspot.co.uk). Teachit Maths also has a overnice 'Understanding y = mx+c' worksheet and two activities on horizontal too vertical lines.

My weblog postal service 'All nigh gradient' features ideas too resources for didactics slope (the postal service focuses mainly on real-life applications of the concept of steepness, but also features links to slope worksheets too activities).

Understanding parallel too perpendicular gradients

So here's the primal data that pupils demand to understand:

(source)

If they know how to operate out a slope every bit 'rise over run' (or equivalent) too thus pupils should hold out able to derive the 'negative reciprocal' dominion for perpendicular gradients from an activeness similar this ane from MathsPad.

Here's a few activities for a lesson on perpendicular lines:

• Finding Equations of Parallel too Perpendicular Lines past times Mathematics Assessment Project
• Perpendicular Lines Worksheet (by me!)
• AS marker Parallel too Perpendicular Lines  (suitable for GCSE extension)
• Perpendicular lines: equations past times toticity.co.uk

Bringing it all together
When students written report linear graphs at AS level, they don't actually acquire anything novel - my GCSE pupils should accept the necessary skills too noesis to respond C1 coordinate geometry questions. I gave my Year xi pupils questions 5 - seven from this AS worksheet final twelvemonth - it took them ages to respond each question, but they were for sure capable. Essentially it's merely the wording of the questions that throws them.

Here's closed to to a greater extent than recommended resources suitable for GCSE students:

• Foundation: Linear Graphs 'I Can' from Mathsbox
• Mathattack Linear Graphs homework sheet
• Topic checklist
• MathsPad Linear Graphs Problems
• Rich activities from Five Triangles
• Rich activities from Median Don Steward

Finally, fun for puzzle-lovers in linear relationships sudoku and a overnice Key Stage iii Christmas activeness on plotting linear graphs from Teachit Maths - Mistletoe too Lines.

Finally, always wondered which countries utilization y = mx + c annotation too which countries utilization something different? Check out this map.

Introducing Algebra

Building a corporation conceptual agreement of algebra is absolutely fundamental. As amongst fractions, it's essential that nosotros acquire it correct because it underpins too thus much of what follows. But, every bit amongst fractions, pupils tin actually combat here.

Sometimes proficient didactics resources inspire us when we're looking for ways to construction too deliver our teaching. This postal service contains to a greater extent than or less recommendations that may plough over you lot nutrient for persuasion when you’re planning your lessons.

Misconceptions
Before didactics whatever theme for the kickoff fourth dimension it's actually of import to intend near potential misconceptions. The best affair a educatee instructor or NQT tin do is inquire an experienced teacher. This algebra misconception summary is also helpful. 

The kickoff fourth dimension I taught algebra, I used this true/false activity to divulge whatever outstanding misconceptions inward the concluding lesson of the sequence. As a newbie teacher, I was astounded past times the mistakes pupils made, similar putting both of these cards inward the 'true' column:

This activity provided a proficient chance to possess got private discussions amongst pupils too assistance them brand sense of it all. So it’s useful both every bit a learning sense too for formative assessment.

I'm non stalking him, promise
I've featured resources from Don Steward inward almost every weblog postal service I've written. I brand no apologies for this. His website is amazing. Here's a few highlights from his algebra collection:

• Lengthy expressions is a dainty means to acquire pupils used to forming expressions containing letters too numbers.
• Expression pyramids are an engaging means to exercise simplifying too this activity is a perfect extension.
• In Steps, pupils possess got to create upward one's demand heed what pace has been taken to acquire from 1 aspect to the adjacent (see below).
• Six expressions is a rich activity for pupils who've been taught how to expand unmarried brackets.
• Three sets of first-class activities assistance students explore concepts relating to simplifying too equivalence: 'equivalence', 'same too transformed' and 'equivalent things'.

Steps - Median Don Steward

There's lots to a greater extent than on Median Don Steward - await nether the labels algebra, simplifying, expressions too substitution.

Fawn Nguyen’s Noah’s Ark
This is a lovely activity which gets pupils used to the persuasion of representing values amongst letters (or pictures inward this case).

Noah's Ark past times Fawn Nguyen

Visual representations
'I'm thinking of a number' questions are a proficient means to acquire started amongst solving equations. MathsPad had a dainty card sort that makes the link betwixt words too algebra. MathsPad also features a worksheet that uses visual representations to construct an agreement of how to solve equations. The same approach is taken to rearranging formulae - this lesson plan features their arrive at of associated resources.

MathsPad

More ideas

• These slides from newmrsc (@_z_0_e) on TES are actually proficient for a kickoff await at algebra.
• For substitution, I similar this code breaker from Teachit Maths (and this minute code breaker, which involves harder expressions). Substitution scenarios is dainty too.
• algebra resources library. Do comment below or tweet me if you lot possess got whatever to a greater extent than resources or ideas to add.

Image: crotchetycomics.blogspot.com

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 #13

In my gems posts I portion roughly of the best teaching ideas I've seen on Twitter each week. This is my 13th gems post. I promise you're non triskaidekaphobic.

1. Flying Numbers
At Julia Smith's (@tessmaths) creativity workshop at the lastly maths conference, she shared an activeness inwards which students brand 'Flying Numbers' to hang from the ceiling inwards the classroom or corridor. It industrial plant similar this: they direct a release in addition to brand that release from forest inwards Design & Technology (drill a hole at the overstep for hanging). They pigment their release in addition to then role fine markers or chalk pens to decorate it amongst facts virtually that number. This is a lovely chance for students to role the cyberspace to explore release properties in addition to detect out all sorts of wonderful mathematical things that are non on the curriculum. The examples below were made at the maths conference - it was a actually enjoyable exercise.

Ed Southall (post virtually teaching a binary lesson).

Alice inwards FractalLand is a stunning animated slideshow. Show it to your shape in addition to explore sequences, fractals, the golden ratio in addition to more. The text at the bottom explains each concept really clearly. I love this! This is just the kind of affair I was referring to inwards my postal service 'A house for gimmicks?' when I said that nosotros should brand fourth dimension to enrich our students' instruction amongst interesting mathematics that is non on the GCSE syllabus.

The Primary and Secondary Treasure Hunts are really good designed. Full instructions are included in addition to in that place are questions on areas of mathematics such every bit cryptography, primes in addition to probability. The questions are rather challenging for the intended historic catamenia groups - if I was organising an inter-school maths contender in addition to then I mightiness role these questions.

The video collection is due to grow - an exciting prospect if this 2 infinitesimal wordless video exploring triangles is a gustatory modality of things to come. 

3. Imbalance Problems
If you're non familiar amongst Don Steward's resources in addition to then halt reading this postal service straight off in addition to banking concern lucifer out his mobiles', in addition to then 'mobile inequalities' in addition to finally 'mobile moments'.

In mobiles, the activities kickoff amongst release problems that larn progressively harder, in addition to then motion on to algebraic balancing problems. Here's a numerical instance which tin live solved amongst arithmetics in addition to logic (fill inwards the missing weights thence that everything balances correctly). 

There are lots of variations, such every bit this 1 where the weights must add together upwardly to the release inwards the triangle (this is a straightforward example):

The algebraic problems railroad train skills inwards writing algebraically in addition to simplifying expressions. In the occupation below, students accept to consummate the missing weights to brand the combined hanging weight equal to 8n.

These are dandy problems. Check out all 3 posts to run into how Mr Steward developed the idea. It's every bit good worth reading Paul Salomon's (last week's gems post has paid off - both Debbie Hart ‏(@Debbie_Hart_UK) and Jane Appleton (‏@JaneAppleton24) accept tried it.

I've been busy writing a postal service virtually circle theorems which volition live attain soon. Also, later seeing roughly dandy Halloween in addition to Guy Fawkes maths resources, I've started to collate a 'seasonal resources' page. I've seen thence many adept Christmas resources, it's difficult to direct the best only I'm getting there.

So in that place yous go, my 13th gems post. It wasn't scary at all.

Image source: guernseydonkey.com

5 Maths Gems #15

Welcome to my 15th weekly gems post - this is where I characteristic to a greater extent than or less of the best teaching ideas I've seen on Twitter each week.

1. Slice The Pie

Paul Rehmet (@prehmet) has built an interactive tool to help students construct numerical fluency. It's a uncomplicated fraction game - Slice The Pie - which helps students railroad train an intuitive sense of the relative sizes of numbers. It's non compatible alongside mobiles yet but would piece of work on your interactive whiteboard. I honey this. Have a play alongside it, I think you'll honey it too. And follow Paul on Twitter because there's to a greater extent than to come upwardly from him.

2. Remembering Inequalities

In my post virtually Ideas from Shanghai I shared this visualisation which helps students cry back which inequality sign is which.

All students demand to exercise is visualise numbers equally blocks - the wider halt of the inequality symbol fits over the bigger stack of blocks. So if x > 2, nosotros know x must last a divulge bigger than 2. 

Chris Smith (@aap03102) pointed out that this doesn't piece of work for negative numbers. 

I'm sure as shooting our friends inwards Shanghai possess got a solution for this but that data hasn't filtered through to us, therefore let's come upwardly up alongside our ain alternative that industrial plant for both positive together with negative numbers. We only demand a agency for students to cry back which symbol is which.

I'm actually grateful to all the lovely creative tweeters who offered ideas - my favourite was this calligram from Chris. 

So if x < - 8, nosotros encounter the < symbol together with think of the alphabetic character L, therefore nosotros know x is less than -8.

Eventually students volition automatically know the important of the symbols on sight, without having to visualise blocks or words.

These methods are far meliorate than thinking of crocodiles - this tin confuse students, equally described inwards Nix the Tricks (extract below).

3. Squares together with Cubes
I'm enjoying La Salle Education's (@LaSalleEd) novel 'Problem of the Day' series.  You tin search the collection past times historic menses or topic. The instance below (Square Boxes) is a prissy logic occupation consolidating noesis of foursquare numbers.

If you're teaching foursquare numbers, you lot powerfulness also similar this 'Square together with Cube Numbers Puzzle' from MathsPad...

... together with this lovely square together with cube numbers display from Clarissa Grandi (@c0mplexnumber).

4. Mathsticks
I lately discovered the brilliant Mathsticks website which is created past times John Duffty (@johnduffty). I registered together with explored - it's to a greater extent than oftentimes than non primary materials but to a greater extent than or less of the ideas powerfulness last helpful at Key Stage 3. Here's a pair of examples:

This Parallel & Perpendicular Slope. Once you've taught students virtually the gradients of parallel together with perpendicular lines, larn them inwards a calculator room (or larn the iPads out, if you lot possess got them) together with give them this worksheet, which I think volition last really slow for them to follow independently. They are given sure as shooting lines which they plot (in Desmos, though I suppose this could last done past times hand) together with therefore they add together the right parallel or perpendicular lines to brand the required shape. It's a actually prissy activity which volition assistance you lot cheque their agreement of this topic.

Extract from Desmos Graph Challenge

Here's my respond to the minute interrogation inwards the extract above.

To extend this task, you lot could possess got students trammel the domains of the functions, equally I've done below. If you lot exercise this, it's of import that they figure out how to calculate the coordinates of the vertices.

If you're teaching parallel together with perpendicular lines therefore also cheque out my post linear graphs for to a greater extent than ideas together with resources.

A asking for misconceptions
When I project design lessons, I abide by it helpful to endeavour to predict the misconceptions that powerfulness come upwardly up. Sometimes I'm surprised past times the mistakes that students make, which are oftentimes the effect of my explanations non beingness clear enough. Knowledge of mutual misconceptions comes alongside sense but I think it would exercise goodness novel maths teachers if they could consult a website to encounter examples of the form of things students larn wrong. 

A not-so-common misconception from 1 of Chris Smith's students

American instructor Michael Pershan (@mpershan) has a bright website Math Mistakes where he compiles, analyses together with discusses the mathematical errors that students make. I desire to prepare something similar for the Great Britain (because our curricula together with vocabulary are therefore different), but I won't possess got fourth dimension to exercise the first-class analysis that Michael does. Instead it would only last a collection of misconceptions, tagged past times topic. But I even therefore think it powerfulness last useful.

So I possess got a asking - could you lot outset taking photos of misconceptions inwards your students' piece of work together with emailing or tweeting them to me? Once I've got a modest alternative I'll outset a website together with I hope, similar my Pret homework website, it volition grow speedily together with attempt to last a helpful resource. Thank you lot inwards advance!

That's it for today. Please don't forget to vote for me! :), I'll larn out you lot alongside this cute 'Math nosotros know drama', shared past times @FascinatingVids.