Resource Library: Complex Numbers

Resources Anonymous, I admitted I accept a serious resources addiction. Based on the popularity of that post, I mean value I powerfulness brand this weblog primarily a resources library. So you lot tin come upwardly hither to await for resources together with ideas when you're planning lessons. Would that live on helpful? I similar to live on helpful. I suppose that's why I became a teacher.

Today I'm focusing on complex numbers. Although non widely studied inwards Great Britain secondary schools (only past times Further Maths students), there's sure as shooting no shortage of education resources online - here's around highlights:

• First, my overstep discovery - Complex Number Operations Teaching & Learning Plans from the Project Maths Development Team. This is packed sum of lesson ideas together with at that spot are tonnes of bright pupil activities inwards the appendices. I'm excited to get-go using these. There's likewise a PowerPoint version.
• I highly recommend the activities provided past times the Further Maths Support Network/MEI (school login required). Here's an event of their topic notes. Their arrive at of resources likewise includes worksheets, activities together with assessments.  
• SRWhitehouse on TES produces splendid Influenza A virus subtype H5N1 marking resources, including a alternative on complex numbers. I especially similar this introductory exercise.
• Illustrative Mathematics gives us 7 splendid activities: 
• Venn activities, this resource sheet is nearly classifying types of numbers.  There's a related activeness on page 32 of Complex Number Operations Teaching & Learning Plans.  



Finally, fun for teachers together with students alike - here's a nice Complex Adding Maze and Nrich has a Complex Countdown game! Enjoy.

What theme would you lot similar me to characteristic next? Please tweet me your requests!

Linear Graphs

One of the topics that my pupils combat 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 function out how to respond a GCSE enquiry similar this:

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

y = mx + c

as well as substitute values for y, k as well as x, as well as hence solve for c.  Whereas at Influenza A virus subtype H5N1 marking my pupils occupation the equation

y - y₁ = m(x - x₁)

Essentially the same method but a slightly dissimilar approach. Perhaps I should endure to a greater extent than consistent as well as innovate the 'A marking method' at GCSE. I'd similar to take away heed what other teachers do.

In this shipping service I’ll propose roughly ideas as well as resources for instruction linear graphs.

Key words

Let's kickoff amongst the basics. There's a lot of novel vocabulary inwards coordinate geometry, roughly of which pupils volition convey encountered at primary school. This is a overnice vocabulary cognition survey every bit suggested inwards my earlier post. 

Here's a overnice play tricks for remembering the words parallel as well as perpendicular (this could endure helpful for spellings - parallel is oft misspelt): 

as well as this activity is suitable for Year vi or 7. 

Introducing the shape y = mx + c
I'd unremarkably kickoff this theme inwards Year seven or 8 amongst plotting at nowadays occupation graphs (Teachit Maths provides an excellent Drawing Straight Line Graphs booklet for this purpose), before exploring the shape y = mx + c.

This theme provides an first-class chance for pupil investigation using Desmos. Use a worksheet to direct pupils through the lesson:

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

Once pupils are able to translate the values of k as well as c as well as hence they tin shipping away 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 as well as vertical lines.

My weblog shipping service 'All most gradient' features ideas as well as resources for instruction slope (the shipping service focuses mainly on real-life applications of the concept of steepness, but also features links to slope worksheets as well as activities).

Understanding parallel as well as perpendicular gradients

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

(source)

If they know how to function out a slope every bit 'rise over run' (or equivalent) as well as hence pupils should endure able to derive the 'negative reciprocal' dominion for perpendicular gradients from an activeness similar this i from MathsPad.

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

• Finding Equations of Parallel as well as Perpendicular Lines yesteryear Mathematics Assessment Project
• Perpendicular Lines Worksheet (by me!)
• AS marking Parallel as well as Perpendicular Lines  (suitable for GCSE extension)
• Perpendicular lines: equations yesteryear 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 convey the necessary skills as well as cognition 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 certain capable. Essentially it's simply the wording of the questions that throws them.

Here's roughly 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 as well as Lines.

Finally, e'er wondered which countries occupation y = mx + c annotation as well as which countries occupation something different? Check out this map.

End Of Term!

The summertime holidays are fast approaching. Although I'm a believer inward pedagogy maths correct upwards to the halt of term, it tin live difficult to learn a topic properly when lessons are beingness interrupted yesteryear sports day, assemblies, schoolhouse trips etc. But if nosotros don't deliver worthwhile lessons until the final solar daytime of term, as well as thus nosotros can't complain when parents desire to conduct maintain their children on vacation during term-time.

I'm non against the sentiment of watching a video or playing games inward maths lesson, exactly let's brand certain our students are either learning something novel or becoming to a greater extent than engaged as well as excited yesteryear mathematics.

My schoolhouse has a criterion prepare of challenges, games, projects as well as investigations that nosotros purpose every year, exactly here's a few of my choice suggestions for halt of term activities:

Fermat's Last Theorem as well as Pythagoras
Have you lot read The Simpsons as well as their Mathematical Secrets by Simon Singh? It's a bright book. Influenza A virus subtype H5N1 overnice halt of term lesson could involve you lot explaining Fermat's Last Theorem (or purpose this BBC video to explicate it for you) as well as showing your flat how it features inward the Simpsons (it's actually rather clever that Homer seems to conduct maintain disproved it - inward fact it's a 'near miss' solution - every bit explained yesteryear Simon Singh in this video from Numberphile).

In this lesson your pupils could exercise approximately query into Fermat, or maybe other famous mathematicians - they could create a poster every bit suggested inward this lovely activity from Interactive Maths.  Alternatively, they could await at related Pythagorean problems, such every bit proving Pythagoras' Theorem (which tin live done alongside Origami!), finding Pythagorean triples or investigating Pythagoras alongside other polygons.

BBC iPlayer is currently showing the Fermat's Last Theorem episode of Horizon - although the maths is rather advanced (it made my encephalon hurt!), it's an interesting episode as well as gives an insight into the travel of mathematicians.

If you lot similar these Fermat lesson ideas, you lot mightiness also similar my postal service on teaching binary which features to a greater extent than ideas from Simon Singh's book.

Pascal Investigations
Pascal's Triangle contains lots of amazing patterns as well as you lot tin easily fill upwards a lesson exploring them. Teachit Maths has a swell PowerPoint as well as activity sheet for this lesson. In this short video a maths instructor briefly explains this history of the triangle. For older students you lot could also innovate 'choose' tone as well as link the triangle to binomial expansions.

Pi week
My colleague i time had a Pi themed final calendar week of term alongside i of his classes. The students all brought inward cakes as well as pies, they watched Life of Pi (a lovely cinema exactly non mathematical!) as well as they did Pi-based activities including a Pi recital competition, a freehand circle drawing contest as well as making a giant Pi paperchain (a unlike coloring per digit).

If you lot conduct maintain whatever other creative ideas for halt of term lessons, I'd dear to involve heed them.

Inspiration

I was really lucky to encounter Simon Singh verbalize at the Maths inward Action lectures inward London final year. My schoolhouse took twenty Year 12s to the outcome exactly this yr nosotros conception to conduct maintain xl because it was such a bright solar daytime of lectures. I came away feeling inspired. I actually recommend these lectures, which are for both Influenza A virus subtype H5N1 flat as well as GCSE students.

Do you lot know what makes the numbers on this jumbotron interesting?  Find out inward this article.

Bringing Babies Into A Statistics Lesson

I’m non especially into babies (apart from my ain lilliputian terrors darlings) merely I learn at a girls’ schoolhouse together with the bulk of my students response rather good to babe materials (apologies for the stereotyping). It’s surprisingly tardily to characteristic babies inward your statistics lessons if yous thence please. I’ve already blogged virtually my Bibonacci starter. There’s besides this pregnant moo-cow evidence query (part d is the exclusively S1 evidence query I've always seen which requires students to know that 68.3% of unremarkably distributed information is inside i measure divergence of the mean).

Edexcel S1 Jan 2002, Q5

I used this query inward an S1 lesson when I was heavily pregnant, much to the entertainment of my students. It led to a tidings virtually the gestation periods of other mammals (those misfortunate elephants!). We’d clearly await a fairly strong correlation betwixt size together with gestation - when I learn a conduct chances I'll run into if I tin pattern an interesting activeness using this information (I wonder whether anything similar is done inward biology).

Speaking of gestation periods - I recollect when my due appointment for my starting fourth dimension immature lady came together with went, I tried to discovery statistics online to tell me how far overdue I mightiness go. I wanted to know the probability of babe arriving at 42 weeks. I didn’t discovery much online at the fourth dimension merely i time to a greater extent than I experience it could brand for an interesting lesson.

If yous are a nurture together with thence yous may live on familiar alongside ‘the ruddy book’, which is the Personal Child Health Record issued to all babies born inward the UK. This is a child's principal tape of immunisations, development, wellness reviews together with thence on. At the dorsum of the ruddy mass are the ‘UK-WHO Growth Charts’ where scatter graphs are plotted of the child’s tiptop together with weight against their age. On these scatter graphs are centile lines, to allow parents together with wellness professionals to compare the child’s measurements against other children of the same age. On a few occasions, I’ve been asked to explicate the pregnant of these charts to friends alongside newborn babies, thence it occurred to me that the increment charts could assistance me explicate the concept of percentiles to my S1 classes.

The ruddy mass contains this description of the centile lines on the increment charts:

More information virtually the charts is available inward this fact sheet from the RCPCH.

While I was thinking virtually how to pattern an activeness for my S1 bird based on these charts, I discovered that the writer of The Chalk Face has beaten me to it alongside this prissy 'Quartiles together with Percentiles' activity. And @srcav has already written this spider web log ship service 'the mathematics of parenthood' virtually the potential to role increment charts inward his teaching.

Incidentally, the ruddy mass says ‘someone who has been appropriately trained should consummate the increment chart’. Hmm. I promise they're non suggesting that members of the populace are incapable of adding a dot to a uncomplicated scatter graph?!

Right, my babe is awake similar a shot thence I'll proceed to develop these ideas approximately other time...  Do permit me know if yous tin think of whatever other baby-related information to add together to my theme.

My mini-mathematicians.  July 2014.

Resources, Resources, Resources...

Inequalities activeness - Project Maths

I've been working actually difficult on my resources libraries lately, hence I'm pleased to tell I'm live fix to portion them! They tin flame live accessed through the tabs at the move past times of the page.

This is where I've lay all my resources recommendations for you lot to browse when you're planning lessons. The A' degree pages volition in all likelihood live the most helpful equally it tin flame live difficult to notice practiced resources for Key Stage 5. 

My recommendations are past times no agency exhaustive - there's loads of practiced resources I haven't discovered nonetheless (the meshing is likewise big, I can't proceed upward alongside it!). But I volition update my listings whenever I come upward across something peculiarly good. To clarify what I hateful past times 'good' - my recommendations include well-designed activities, rich tasks and worksheets for topics where I retrieve it tin flame live difficult to notice suitable practise questions. 

My Top three Websites

The resources I've recommended are all gratis as well as come upward from a broad arrive at of sources.  Here's my electrical current favourite websites, inwards no detail order.

1. Roof Truss task (trigonometry) and Bee Aware (scatter graphs) - at that topographic point are hundreds to a greater extent than as well as they're all awesome.  
2. MathsPad has enough of first-class gratis resources. Subscription (£3 per month) gives access to their total range. I dear the way their resources are designed, including their interactive activities which are all really engaging as well as slowly to use. Example resources include this lovely Square as well as Cube Numbers Puzzle, and for older pupils I actually like Is it Rational?. 
3. Teachit Maths has large, well-organised resources libraries.  Free registration gives access to resources inwards pdf form, which is fine for all their printable activities as well as worksheets. You'll demand a subscription (£25 per year) if you lot also desire to access PowerPoints, interactive activities, pedagogy tools etc. Worksheets are well-designed, engaging as well as include solutions. Examples of their resources include this Mental Powers Code Breaker as well as this Transforming Graphs carte kind activity, which I accept used successfully alongside both my Year 11s as well as Year 12s.

Special get upward to The Chalk Face and Mathematics Assessment Project - both of these websites are total of first-class ideas. I also larn a lot of resources from TES. One of my favourite TES contributors is SRWhitehouse who creates first-class H5N1 degree activities. For a total (and ever-expanding) listing of recommended resources websites, meet my links page.

Factor Trees - Median Don Steward

My Blog
Now I've finished creating my resources libraries, I tin flame larn dorsum to focussing on my blog. Here's or hence of the highlights hence far...

• My Area as well as Volume post service - featuring a giant lawn tennis ball - was really popular.
• My suggestions for halt of term lessons on Binary as well as Fermat also went downward well.
• Have you lot tried this method for finding inwards HCFs as well as LCMs?  It's hence much quicker than the Venn method.
• Have you lot read the mass post near it.
• I lay a lot of persuasion into my post service All About Gradient. I'm certain that pupils would relish the lesson ideas I've suggested for investigating steepness - accept you lot watched the video of the cyclist attempting a 38% gradient? My legs wound only thinking near it!
• I've written a break of posts focussing on specific topics or concepts such equally dividing past times zero, angle facts, skewness as well as simultaneous equations.  I've also discussed pedagogy tools such equally Desmos, comic strips, blogs as well as methods for developing vocabulary. 
• It was my uncovering of a novel matrix multiplication method that brought me to blogging - if you lot instruct matrices hence create banking concern gibe it out.
• Finally, something fun for the teachers - evidence the Maths Movie Quiz.

For a total listing of my weblog posts meet the blog archive.  Not bad going for three months!

'Find the Coordinates' chore past times Susan Wall

Long Alive Stalk Too Leaf

I planned to write a postal service close pedagogy stalk together with foliage diagrams. But so I realised that many maths teachers reckon them to survive pointless, a waste materials of pedagogy fourth dimension together with a welcome removal from the curriculum.

Now let’s cry upwards close this for a minute. Admittedly stalk together with foliage diagrams are rarely used inwards ‘real life’. In fact the exclusively mutual examples nosotros tin abide by are timetables, such every bit the Japanese develop timetable shown inwards the moving painting below (more examples here). But does this ‘uselessness’ actually hateful that stalk together with foliage diagrams shouldn’t survive taught? If that’s the illustration so a release of the topics nosotros learn inwards maths should survive taken off the curriculum. 

Japanese develop timetable. Source: Wikipedia

Much of the maths nosotros learn at Key Stage three helps pupils access to a greater extent than advanced topics (all roads Pb to calculus?!). Maths begets maths. So exercise stalk together with foliage diagrams Pb us anywhere? Do they assist pupils develop whatever useful skills together with understanding?

I'm going to tell (in a sheepish whisper) that they do. Stem together with foliage diagrams are 1 of a prepare of tools that could survive used to develop agreement of the concepts of median, mode, quartiles, spread, skew together with outliers. I cry upwards they are accessible to depression powerfulness pupils. I cry upwards that pupils bask creating them. I cry upwards they Pb to interesting discussions when comparing ii sets of data. I sure enough don't cry upwards they should survive a compulsory theme at whatever age, but I run into no terms inwards teachers using them if they like. Isn't it prissy to conduct maintain that freedom?

John Tukey, the inventor of stalk together with foliage diagrams, wrote "If nosotros are going to brand a mark, it may every bit good survive a meaningful one. The simplest - together with most useful - meaningful grade is a digit" (source). Excellent indicate Mr Tukey. Did you lot know he also invented the box together with whisker plot?

John Tukey is also quoted every bit saying "There is no information that tin survive displayed inwards a pie chart, that cannot survive displayed improve inwards another type of chart". I agree. In the same agency many of you lot don't similar stalk together with foliage diagrams, I don't similar pie charts (this excellent spider web log postal service close the misuse of pie charts is worth a read).

Right, straight off I've attempted to justify this spider web log post, here's my lesson ideas:

Starting Point
You could commencement past times presenting your shape amongst a large prepare of information (like the 1 below) together with cry for them what the trend is. It's hard to spot. Ask your pupils for ideas on how could nosotros organise this information so nosotros tin easily run into the mode.

12, 54, 21, 56, 32, 14, 42, 23, 20, 48, 21, 34, 12, 15, 16, 42, 47, 46, 32, 21, 45, 43, 21, 34, 26, 32, 17, 18, 45, 21, 44, 32, 12, 23, 45, 24, 35, 38, 28, 18, 29, 20, 10, 36, 12, 42, 47, 32, 47, 37, 12, 54, 23, 17, 29, 38, 32, 12, 17, 13, 21, 34, 12, 15, 16, 42, 47, 46, 12, 16, 43, 23, 16, 46, 52

I abide by stalk together with foliage plots rather pleasing to the eye. Much nicer to hold off at than a dull listing of unordered data.

An option starting indicate is a 'beat the teacher' game, which I've adapted from an thought past times the NCETM. In this activity, pupils cry upwards of a release betwixt 10 together with 40. Each release is called out inwards turn, spell pupils brand a listing inwards their books. Meanwhile the instructor also makes their ain list, but organises it every bit a stalk together with foliage diagram. Once all the numbers are listed, the instructor asks the shape to run out median, trend together with range. The instructor volition exercise it much quicker from their stalk together with foliage diagram, thus he/she wins the race.

Physical Activities
This spider web log post is close introducing stalk together with foliage diagrams using post-it notes. It's a lovely thought for an engaging together with accessible lesson. Mr Collins shared a similar thought inwards this spider web log post, but he uses students together with desks instead of post-its. If you lot similar the thought of human stalk together with foliage diagrams so you lot powerfulness also similar these breath-holding activities - 1 from Don Steward together with a similar 1 from the NCETM.

Data Types
Encourage pupils to cry upwards close dissimilar types of information - what is suitable for a stalk together with foliage diagram together with what isn't? For example, could the full release of goals scored per game inwards a flavor of football game matches survive displayed inwards a stalk together with foliage diagram? Probably not, because the information would by together with large survive unmarried digit numbers. Try my worksheet for to a greater extent than on this.

Mini Projects
Don Steward has around prissy activities involving stalk together with foliage diagrams, such every bit this this Presidents Longevity Project. I also similar Tim Jefferson's skewness).

Back-to-back stalk together with foliage diagrams are useful for comparing data. Writing comparisons betwixt information sets - whether represented past times stalk together with foliage diagrams or box together with whisker plots - is something that pupils (especially EAL pupils) ofttimes abide by difficult.

There's lots of S1 (A level) stalk together with foliage questions - similar the 1 below - that could survive explored at Key Stage three (which begs the question, why were they considered suitably challenging for H5N1 level?).

Edexcel S1 Nov 2004, Question 1

I rarely disagree amongst people or play devil's advocate so this postal service is a chip risky for me! Be nice. If you lot exercise learn stalk together with foliage diagrams so I promise you lot abide by these ideas helpful. Thanks to those on twitter who shared their views together with ideas (especially @srcav - who is non a fan of stalk together with foliage diagrams!).

Your Commencement Maths Lesson Amongst Twelvemonth 7

A Year 7's first ever maths lesson inwards secondary schoolhouse is rather daunting. For both the students and the teacher. As this is the fourth dimension of twelvemonth when many maths teachers volition travel planning those initiative of all lessons, I thought it powerfulness travel helpful to advise about ideas.

With a novel Year seven class, the initiative of all lesson is non but almost setting expectations almost demeanor as well as the like, it’s also almost getting your students excited almost maths as well as edifice their confidence. It’s almost helping them to rapidly settle into their novel environment. And - on a to a greater extent than practical Federal Reserve annotation - it’s almost learning their names.

First I’ll speak yous through my suggestions for that initiative of all Year seven lesson, thence I’ll portion about ideas suggested past times other teachers. And I’d dearest to withdraw heed your ideas too.

A release lesson
As students arrive, greet them at your classroom door as well as manus them a carte du jour or sticker amongst a release on it. I advise the numbers 1 – xxx but whatever numbers volition work. While they notice their seats, an educational activity on the board tells them to holler back almost the properties of their release (bear inwards heed they powerfulness non know the important of the word belongings inwards this context).

Tell students to stand upward up as well as concord their carte du jour upward if they convey a multiple of 5. Each pupil standing upward says their holler as well as release inwards plough (one of the benefits of this activeness is that yous - as well as their classmates - acquire their names). Ask i of the students standing to define the word multiple.

Now produce the same for diverse release properties eg ‘stand upward if yous convey a portion of 24’, ‘stand upward if yous convey a prime number’, ‘stand upward if yous convey a foursquare number’ as well as thence on. There are diverse give-and-take points relating to the definitions, such equally whether i is a prime number. You acquire the idea. Try to adjust it thence each pupil stands upward at to the lowest degree twice, to assist amongst the holler learning.

There’s diverse other activities yous could purpose amongst students when they convey release cards. Perhaps acquire the pupils amongst numbers three – 8 to line of piece of occupation upward at the front end as well as enquire the flat to holler the polygons amongst each release of sides. It volition travel encouraging if yous start the twelvemonth past times letting students portion amongst yous about of the maths they already know. But yous could also give them a gustatory modality of the exciting novel maths to come. For event yous could enquire the students amongst the cards 1, 2, 3, 5, 8, thirteen as well as 21 to stand upward inwards companionship at the front end of the flat as well as enquire their classmates to location the blueprint (add an extra 1 at the beginning!). It's unlikely they would convey seen Fibonacci before.

Sticking amongst the release theme, yous could try:

• this Number Teams game from Teachit Maths 
• these 4 interactive whiteboard tasks, from MathsPad. 
• These number riddles from Transum. 

30 Maths Starters - Puzzle 2

If yous desire the flat to settle amongst about private or duo activities, endeavor about of these '30 Maths Starters'. I advise activities 2 as well as nine.

And how almost a squeamish game of Fizz Buzz to terminate the lesson?

If yous convey administrative bits to produce amongst the class, such equally handing out books, they could hold off until side past times side lesson.

More ideas
Here are a few ideas I've constitute online:

• Try these Ordering past times Numbers activities past times Frank Tapson 
• I similar 'Back to School' past times Transum
• Suffolk Maths suggests lots of proficient ideas for a Year seven transition lesson. This includes 'Stand Up' - which is similar to my suggestion. Whenever I holler back I've been creative as well as master copy I notice that someone's already thought of the same ideas! :)

While looking for these lessons, I constitute this 'Moving on inwards my Mathematics - Year six to Year 7' booklet that yous may travel interested in.

Craig Barton's blog post on transition inwards secondary maths is a must-read for novel Year seven teachers. I know that my Key Stage three teaching would travel improved if I knew to a greater extent than almost what children encompass at principal school.

And finally, here's about to a greater extent than ideas for learning names!

A level
I promise that’s been helpful. Please comment if yous convey your ain ideas to share. My side past times side post service volition travel almost the initiative of all Year 12 lesson of the year.

Kicking Off Yr 12

Following on from my final post on ideas for the outset Year seven lesson of the year, here's to a greater extent than or less ideas for the start of Year 12.

Unfortunately nosotros don't actually induce got whatsoever fourth dimension to spare inwards our 6th shape teaching. It's a challenge merely to fit plenty lessons into the schoolhouse twelvemonth to encompass all the class textile earlier our students become off to accept their exams. So, sadly, there's no fourth dimension for enrichment or getting to know each other - students induce got to striking the reason running.

Entry Assessment
At my school, equally at many others, nosotros enquire our students to laid upward for Year 12 yesteryear doing to a greater extent than or less summertime work. We hold back them to induce got a corporation agreement of GCSE topics when they start the course, peculiarly surds, indices as well as algebra. Last year, for the outset time, nosotros introduced a Year 12 entry assessment. The piece of job of the entry assessment was to assistance us position the students who needed additional back upward from the start of the course. We equally good hoped it would position whatsoever students who shouldn't live on taking AS maths at all, inwards the promise that they mightiness reckon choosing an option course.

Extract from Year 12 Entry Assessment

The results from our entry assessments were rather worrying. H5N1 large pose out of the cohort got less than 50% fifty-fifty though the bulk achieved a degree A* at GCSE. The unproblematic enquiry 'Calculate 238 x 6.3' was answered real badly, showing a lack of basic maths skills.

I had a huge amount of students come upward to my extra back upward sessions - I believe the entry assessment was a skilful 'wake upward call' for them. It motivated these students to focus as well as piece of job difficult correct from the start. All students improved their score considerably inwards the subsequent re-test, but even then to a greater extent than or less of the marks were quite low. On H5N1 bird results solar daytime inwards a few weeks I'll live on actually interested to compare AS grades achieved to performance inwards the entry assessment final September.

Here's the files inwards illustration you lot desire to borrow them:

• Year 12 Entry Assessment   (answers) 
• Feedback sheet  (students who did good got a 'welcome to AS Maths' message!)
• Year 12 Entry Assessment re-test 

Student Handbooks

We paw out student handbooks to every Year 12 inwards the outset lesson of the year. They're asked to continue these inwards the forepart of their maths folder. It's peculiarly helpful that they induce got their ain re-create of the formula canvass as well as a guide to reading examination score schemes.  

Activities

Here's to a greater extent than or less prissy resources for the outset duo of maths lessons:

• What's the Difference - run across below. This give-and-take activity reminds students of the importance of next the social club of operations.

• Introduction to C1 Algebra matching activity - from SRWhitehouse on TES
• Ever wondered why? - an engaging as well as challenging indices puzzle from JustMaths 
• Indices activity - from SRWhitehouse on TES

My Core AS and Statistics AS pages induce got lots to a greater extent than resources recommendations for the residue of the course.

If you'd similar Word copies of my pdfs to arrange for your ain school, delight ask.

Scatter Graphs

This weblog is becoming a fleck statistics-heavy, which is ironic given that I don’t peculiarly similar instruction statistics at Key Stage three in addition to 4.  I much prefer instruction algebra. I’m also quite dandy on surds in addition to indices. And I similar trigonometry. In fact, apart from the strange annoying-to-teach topic similar loci in addition to constructions, I’d seat most information treatment topics correct at the bottom of my ‘things I relish teaching’ list. It never fails to amaze me how long roughly teenagers accept to plot a graph. What a faff.

Anyway, this ship service is all nigh ideas for instruction scatter graphs. And I intend to follow it upward amongst a ship service looking at applications of box in addition to whisker plots exterior the maths classroom.  But in addition to hence I’ll halt amongst all the statistics.

Practical activities

Unusually for a maths lesson, there's opportunities for roughly practical activities when nosotros innovate scatter graphs. The obvious watch is to receive got students plot pinnacle or manus bridge against shoe size. If you lot desire to acquire fifty-fifty to a greater extent than practical, endeavor a human scatter graph inwards the classroom or inwards the playground (using chalk axes).

Outliers 

I similar to purpose the ii scatter graphs shown below (stolen from a Ten Ticks worksheet) inwards a class give-and-take nigh outliers.  Ask your class what could receive got caused these extreme values in addition to you'll in all probability acquire roughly actually creative suggestions!

Line of Best Fit

I learn my students to depict a employment of best check past times trying to minimise the distances betwixt all the points in addition to the line. That’s what a to the lowest degree squares regression does. Contrary to pop belief, in that place doesn't receive got to live the same seat out of points to a higher house in addition to below the line. Nor does the employment receive got to become through whatsoever of the points. And the employment for certain doesn't receive got to become through the origin. Beware of dodgy methods taught inwards other subjects. This misconceptions activity past times frickard on TES is helpful - I've used this inwards a 'hands on caput if you lot retrieve it's true, hands on hips if you lot retrieve it's false' activity.

Impossible Predictions

'Extrapolation' from regression lines is something that comes upward at Influenza A virus subtype H5N1 level, simply the watch tin brand for interesting discussions at Key Stage three or GCSE. The cardinal shout out for is that fifty-fifty if nosotros bring out a linear human relationship inwards the information we're examining, nosotros can't assume that human relationship extends beyond the attain of the data. Here's a prissy instance for class discussion:

The graph below shows a negative correlation betwixt yr in addition to fourth dimension taken past times athletes to run 100m. It suggests that the speed of 100m runners has increased over time. Influenza A virus subtype H5N1 employment of best check has been drawn. Why produce you lot retrieve 100m runners are getting faster? Do you lot retrieve this tendency volition continue?

Possible reasons for the increment inwards speed include: advances inwards technology (ie to a greater extent than lightweight or streamlined wearing clothing in addition to trainers), query into optimal diets in addition to grooming plans, sponsorship important athletes receive got to a greater extent than fourth dimension to train, the introduction of starting blocks inwards the 1940s in addition to an improvement inwards the accuracy of timing the race.

If the tendency continues indefinitely (ie if nosotros depict the employment of best check into the future) in addition to hence at roughly shout out for people would live able to run 100m inwards null seconds, which is impossible. There must live a shout out for at which the fourth dimension volition plateau, due to limits inwards humans' physical capabilities. Although in that place is a linear correlation, nosotros must live careful non to brand predictions inwards the past times or time to come because the linear human relationship is over a express fourth dimension period.

The graph comes from this Minitab Blog post in addition to the watch was seat into my caput past times Kings College London PGCE tutor Chris Olley. 

"Wind is caused past times the rotation of windmills"

I love the causation vs correlation discussion.  It's a really of import learning shout out for for students who volition come upward across all sorts of dodgy statistics inwards everyday life.

doubt.it

Influenza A virus subtype H5N1 typical instance is shoe size in addition to reading age. The ii variables are positively correlated, simply apparently the size of someone's feet has no comport upon on their reading ability. For to a greater extent than ideas depository fiscal establishment check out this spurious correlations website.

If you're planning a class give-and-take on causation in addition to hence receive got a quick read of 'Correlation does non imply causation'on Wikipedia (not e'er the best source simply gives roughly skillful examples).

Correlation does non imply causation, simply it does waggle its eyebrows suggestively in addition to gesture furtively land mouthing "look over there". – xkcd 

Resources

Finally, here's a few resources that you lot could purpose inwards your lessons:

• I love this Bee Aware action from Don Steward. It could easily live made into a mini-project/display move nigh the drive in addition to comport upon of the demise of dearest bees.
• The fantabulous Mathematics Assessment Project gives us ii tasks - Scatter Diagram and Birds' Eggs
• There's no shortage of real-life applications of scatter graphs. This worksheet 'Shake It Up amongst Scatterplots' is i of many examples (here's the solutions).
• Nrich gives us this nice David in addition to Goliath activity.
• 'Foldables' seem to live a big affair inwards the US math lessons, simply I've never seen them inwards the UK. I retrieve my students would quite similar them. Here's an watch for correlation foldables from the weblog 'Everybody is a Genius'.
• I similar this online tool for checking the accuracy of lines of best check from Mark Ritchings
• Here's an interesting group project. I similar the watch of giving each pair/group ii options hence they tin pick out the variables they prefer, in addition to that everyone inwards the class is working on something different. Also, the 'assigned tasks' for each grouping fellow member (see lastly page) is an watch I haven't seen earlier for organising grouping work. These item tasks may non live suitable for a U.K. lesson because nosotros don't await students to calculate the equations of lines of best fit, simply the watch could live adapted.

I promise this ship service has been helpful.  Please comment if you lot receive got whatsoever ideas to share. 

By the way, this cool moving scatter graph is worth a look.

xkcd.com

Consistent Mathematics

‘But Miss, that’s non what our in conclusion instructor said’

I read an interesting Multiplication methods are a adept instance of this. There’s been to a greater extent than or less tidings nigh multiplication methods lately on twitter - most people concord that if nosotros acquire the right answer too then it shouldn't thing what method nosotros use.

As teachers nosotros withdraw keep our ain preferences. And pupils may withdraw keep their ain methods too, learnt from teachers, parents, tutors, Kumon (grrr...) too so on. Influenza A virus subtype H5N1 adept instance is the agency nosotros discovery a highest mutual factor - there’s many dissimilar methods, all as valid (I'm convinced I've finally establish the Poorly Executed Mnemonics Definitely Addle Students' by Bill Shillito - so I won’t go through it all again, but let’s think nigh how to cook it.

As a minimum, nosotros involve all teachers inward a schoolhouse to hold upward on board amongst a novel approach. In an ideal province of affairs we’d acquire all maths teachers inward the dry reason – no actually, inward the basis – on board. Now, inward a rather tiny pace towards changing the basis – I’ll start past times discussing this inward my division coming together on the outset twenty-four hours of term inward September.

If our pupils withdraw keep met BODMAS at principal school, we’ll withdraw keep to tell them to disregard it, too we’ll withdraw keep to exhibit them why past times demonstrating the misconceptions it may Pb to. Like many others, I’m an advocate of teaching GEMA or GEMS instead. Some American maths teachers withdraw keep already written blog posts nigh this approach. It’s a hurting that most of my lovely resources reference the term BODMAS though. If I acquire fourth dimension over summertime I’ll brand to a greater extent than or less choice GEMA resources.

Changing quartile terminology
Now I don’t think I’m going to brand whatsoever progress amongst this ane because it doesn’t final result inward whatsoever serious misconceptions. Am I the solely individual bothered past times the fact that nosotros utilisation the term ‘lower quartile’ to depict both the bottom 25% of the information too the cut-off value 25% of the agency through the data? John Tukey, inventor of the box plot, used the tidings ‘hinge’ instead ie the lower quartile consists of the values betwixt the minimum value too the lower hinge. I prefer this terminology, too I’ve seen it used inward a few places, but I doubtfulness it volition hold upward widely adopted.

Maths nosotros desire to modify but can’t
When nosotros innovate the thought of squaring sinx at Influenza A virus subtype H5N1 level, nosotros tell pupils to write sin²x instead of sinx² (to clarify that it’s non the x that’s existence squared). But nosotros withdraw keep to carefully explicate that sin^-1x doesn't hateful the reciprocal of sinx, but truly the inverse ie arcsin. The obvious solution to this confusing annotation is to halt using -1 to betoken inverse functions – but too then we’d involve to redesign estimator buttons! Why didn’t estimator designers write arcsin, arccos too arctan on the inverse trig buttons?

The utilisation of -1 to hateful both a reciprocal too an inverse business office is ane of the many mathematical annoyances discussed inward this post service on Reddit ‘What Mathematical Conventions would you lot change?’. You mightiness also hold upward interested inward this article ‘The most mutual errors inward undergraduate mathematics’.

Inconsistent mathematical methods
I don’t think it’s a work if pupils utilisation a diverseness of methods to answer the same question. Multiplication methods are a adept instance of this. There’s been to a greater extent than or less tidings nigh multiplication methods lately on twitter - most people concord that if nosotros acquire the right answer too then it shouldn't thing what method nosotros use.

As teachers nosotros withdraw keep our ain preferences. And pupils may withdraw keep their ain methods too, learnt from teachers, parents, tutors, Kumon (grrr...) too so on. Influenza A virus subtype H5N1 adept instance is the agency nosotros discovery a highest mutual factor - there’s many dissimilar methods, all as valid (I'm convinced I've finally establish the best method though!). Another instance is the approach nosotros accept to commutation too rearranging. When I utilisation the Cosine Rule to discovery an angle, I substitute the numbers into the formula a² = b² + c² - 2bcCosA then I rearrange. But I’ve noticed that most of my colleagues tell students to rearrange the algebra before they substitute. It mightiness confuse students when they outset consider a novel approach, but this opens upward a useful dialogue nigh the equivalence of dissimilar methods.

Here’s a few to a greater extent than examples of where teachers accept dissimilar approaches:

• When teaching trigonometry inward right-angled triangles, non everyone says ‘SOHCAHTOA’. At schoolhouse I learnt a Multiplication methods are a adept instance of this. There’s been to a greater extent than or less tidings nigh multiplication methods lately on twitter - most people concord that if nosotros acquire the right answer too then it shouldn't thing what method nosotros use.
  
  As teachers nosotros withdraw keep our ain preferences. And pupils may withdraw keep their ain methods too, learnt from teachers, parents, tutors, Kumon (grrr...) too so on. Influenza A virus subtype H5N1 adept instance is the agency nosotros discovery a highest mutual factor - there’s many dissimilar methods, all as valid (I'm convinced I've finally establish the trigonometry triangles too.
• I’ve already written nigh differing methods for finding the equation of a immediately line inward my Multiplication methods are a adept instance of this. There’s been to a greater extent than or less tidings nigh multiplication methods lately on twitter - most people concord that if nosotros acquire the right answer too then it shouldn't thing what method nosotros use.
  
  As teachers nosotros withdraw keep our ain preferences. And pupils may withdraw keep their ain methods too, learnt from teachers, parents, tutors, Kumon (grrr...) too so on. Influenza A virus subtype H5N1 adept instance is the agency nosotros discovery a highest mutual factor - there’s many dissimilar methods, all as valid (I'm convinced I've finally establish the linear graphs post.
• I ane time had a Year eleven cast who’d been taught past times their previous instructor to discovery the nth term of a sequence using the a + (n-1)d formula that nosotros usually learn at Influenza A virus subtype H5N1 level. It worked well.
• I don’t utilisation the ‘FOIL’ guild when I expand brackets on the board too this confuses to a greater extent than or less of my pupils who appear to think that FOIL is the solely right method.

Inconsistent vocabulary too pronunciation
I country parab-ola, my colleague says para-bola. I’m pretty certain I country it right (in fact, I merely checked here to brand sure!). But I had a 2d of doubtfulness when my pupils all laughed at my pronunciation because they’d heard it pronounced differently the previous year. Now this truly isn’t the terminate of the world, but it made me wonder nigh my pronunciation of other words, such as cyclic (as inward cyclic quadrilateral), alternate (as inward alternate angles) too ln (as inward natural log).

As for vocabulary, I switch betwixt the damage 'index' too 'power' all the time. This is merely ane instance of my inconsistencies. I don't think it's a work to utilisation a diverseness of words, inward fact it mightiness hold upward a adept thing, as long as all novel vocabulary is explained to students.

Inconsistent standards
I taught a Year 12 in conclusion twelvemonth whose piece of work was laid out so badly it was almost impossible to follow. She scattered her workings all over the page, sometimes showing me where to expect side past times side amongst flowery arrows. She seat multiple equals signs on the same line of working, too her terminal answer was ever hidden away somewhere. Her piece of work was also mathematically incorrect. Although I commented - many times - on the province of her workings, I decided it was to a greater extent than of import to focus our express fourth dimension together on fixing her mathematical misconceptions. I felt frustrated that her appalling approach to setting out her piece of work hadn’t been dealt amongst at an before age.

I shout out upward when I was a pupil inward the Upper Sixth Form, my maths instructor wrote on a homework that I’d used the symbols for 'equals' too 'therefore' interchangeably. I’d seen her utilisation a double arrow symbol on the board too misunderstood its meaning. I was utterly confused (and embarrassed!) because I’d been doing it wrong for over a twelvemonth too she’d never mentioned it before!

I also think nosotros could create a lot to a greater extent than to encourage clearer mathematical handwriting (both numbers too algebra). We all withdraw keep our ain styles though. I’m guilty of writing diagonal lines inward fractions on the board fifty-fifty though I know I shouldn’t.

A instance study
It’s worth reading this good exercise study from Ofsted: “In many schools, students’ long-term progress inward mathematics is slowed past times inconsistent teaching. At Archbishop Temple School, real effective leadership of a rigid squad of teachers ensures that students sense teaching that is consistently adept or better. Students inward all groups create goodness because dissimilar teachers utilisation mutual approaches too teaching styles”. There are lots of adept ideas inward this instance study, such as the formal sharing of resources betwixt teachers too ‘book swapping’ – non inward the critical ‘how much mark create you lot do?’ form of way, but amongst a persuasion to encouraging consistent practice.

I think that novel products too technologies similar Complete Mathematics past times La Salle Education volition create a lot to back upward teachers, improve consistency too enhance teaching standards. The flipped classroom approach, which has been gaining popularity recently, may also withdraw keep an behave upon on consistency.

I’d hold upward interested to withdraw heed your views on whether nosotros involve to improve consistency inward maths teaching too how nosotros could become nigh doing so.

Image: mentalfloss.com

Deliberate Mistakes

Image: weandserendipity.com

I'm ever on the sentinel for novel ideas for GCSE revision lessons. If I write most this persuasion at nowadays in addition to hence hopefully I'll retrieve it come upwardly Spring.

This twelvemonth I had non bad success alongside relay races in addition to carousels inward my revision lessons. I likewise did some pupil-led lessons which were a lot less successful.

This 'deliberate mistakes' persuasion is something I've adapted from this blog post, hence give cheers you lot to @bkdidact for sharing the link in addition to @kellyoshea for the inspiration. I've never seen this inward maths earlier but it may endure that many of you lot already create it, if hence in addition to hence I'd similar to hear your feedback.

The Mistake Game
Students, inward pairs or pocket-sized groups, are given a employment to solve in addition to explicate to the class. In their explanation, they must include a deliberate mistake. Not exactly a numerical slip, but something that highlights a possible misconception that their classmates powerfulness have.

If I role this for GCSE revision in addition to hence by essay questions (the longer, unstructured types) would operate best - there's some examples inward this first-class laid of questions from Suffolk Maths. But this persuasion could likewise operate when educational activity specific topics. I mean value I may endeavour it this twelvemonth when educational activity both trigonometry in addition to probability trees.

For example, if you lot role the Mistake Game when educational activity Pythagoras' Theorem, you lot could role this inquiry (solve for x):

The mutual error hither is to non foursquare the four (though students may come upwardly up alongside something else).

Here's some other example, from this worksheet from thechalkface.net.

There's numerous things they could (deliberately) create incorrect inward this one!

When each grouping goes through their inquiry on the board, their classmates volition endure listening carefully to location the mistake, hence it's non bad for getting students to focus. I'm non certain how best to organise the responses. Perhaps students could lay their paw upwardly every bit presently every bit they location the error hence it tin endure corrected similar a shot in addition to and hence the residue of the solution tin endure done correctly. Or peradventure when the grouping cause got finished presenting their amount explanation, a volunteer could come upwardly up to the board in addition to right their work. Or maybe later each presentation the students could utter over the employment inward their groups, create upwardly one's hear what was incorrect in addition to re-do the inquiry correctly.

This is a chip similar 'tick or trash' exercises - it's helpful for pupils to endure aware of mutual mistakes. I've likewise used these 'Classic Mistakes' inward previous revision lessons.

The Mistake Game is likewise a practiced chance for students to encounter each other’s approaches to solving problems in addition to utter over the effectiveness of dissimilar methods.

I mean value this action volition engage the whole shape - I'll endeavour it out in addition to allow you lot know! If you've ever tried anything similar this in addition to hence delight part your thoughts. I recommend reading most the sense of the instructor who wrote the original post. As she says, ‘we all larn to a greater extent than when a incorrect respond goes upwardly than when we’re exactly watching perfection’.

Image: bigthink.com