Introducing Differentiation

Influenza A virus subtype H5N1 spell agone I was asked by @_JPMason to write a shipping service near how I innovate differentiation to my Year 12s. I've been procrastinating for weeks but hither I am, large drinking glass of vino inwards hand, attempting to tackle this challenge. I've alone introduced calculus a few times as well as I don't intend I've e'er done it especially well. So this is a practiced chance for me to hit some enquiry as well as some thinking which volition hopefully improve my ain teaching practice.

This shipping service is primarily aimed at teachers who are teaching calculus for the outset fourth dimension - I promise y'all discovery some useful ideas here.

An Instantaneous Rate of Change
As good every bit knowing how to differentiate, students demand an intuitive agreement of the slope of a bend at a point.

Students are familiar alongside the concept of gradient as a charge per unit of measurement of change, but alone inwards the context of the steepness of direct lines. Their side past times side footstep is to sympathise that they tin give notice also create upwards one's hear the slope of a curve. But different a direct line, the steepness of a bend changes constantly. Hence nosotros conduct hold a gradient function - set inwards a value of x as well as nosotros discovery the derivative at a specific betoken - an 'instantaneous charge per unit of measurement of change'.

You could start past times showing your students a elementary distance-time graph. Discuss how the slope changes inwards each department of the graph - your students volition easily position positive, negative as well as null gradients. Ask them how to assign a let on to those gradients as well as they volition in all likelihood recollect that slope is calculated every bit 'rise over run' ie distance divided past times time. In a distance-time graph, the value of the slope represents speed. Now utter over the persuasion that direct lines correspond constant speed. Ask your students if they tin give notice run or bike at a constant speed. Perhaps seek it inwards the classroom - conduct hold your students walk across the room simultaneously, attempting to keep constant speed. Impossible? If your schoolhouse is lucky plenty to ain a motion sensor then this is a practiced chance to utilisation it - graph matching activities are nifty fun. The betoken is that speed varies as well as thus it should hold out represented past times a curve. It's slowly to create upwards one's hear the slope of a direct draw but how tin give notice nosotros create upwards one's hear the slope of a curve? That's the large question.

'Making feel of differentiation' by Project Maths describes the persuasion of an instantaneous charge per unit of measurement of modify inwards a agency your students volition understand: "The speed of an object at an instant inwards fourth dimension is surprisingly hard to define precisely. Consider the statement: 'At the instant he crossed the complete draw inwards 2009 Usain Bolt was travelling at 28 mph'. How tin give notice such a claim hold out substantiated? Influenza A virus subtype H5N1 photograph taken at that instant would hold out no assist at all every bit it would demo Usain Bolt motionless. There is some paradox inwards trying to written report Usain’s displace at a detail instant inwards fourth dimension since, to focus on a unmarried instant, y'all effectively halt the motion! Problems of displace were of fundamental work organisation to Zeno as well as other philosophers every bit early on every bit the fifth century B.C. The modern approach, made famous past times Newton's calculus, is to halt looking for a elementary notion of speed at an instant, as well as instead to hold off at speed over pocket-sized fourth dimension intervals containing the instant". Khan Academy has a truly practiced Introduction to Differential Calculus video near Usain Bolt as well as instantaneous rates of change.

Once you've given students a sense of savor of what calculus involves, seek a elementary action similar this to acquire them thinking near the changing gradients of curves:

Source: Project Maths

They should hold out able to consummate the tabular array without whatsoever instruction, based on what they know near gradients from Key Stage 3/4. I've produced a worksheet that covers these introductory ideas. This could hold out exist gear upwards every bit a homework inwards advance of the outset calculus lesson.

First Principles
Once students start differentiating using a set of rules, this topic is fairly straightforward. They apply a elementary physical care for as well as acquire the answers correct - hi presto, they're doing calculus. But it's essential that we show them where the rules come upwards from, so let's hold off at that.

Like most Influenza A virus subtype H5N1 degree teachers, my outset calculus lesson focuses on differentiation from outset principles. In the nifty calculus contention of the 17th Century, Newton as well as Leibniz independently invented calculus to allow the notion of charge per unit of measurement of modify to hold out made mathematically precise. To quote Jonny Griffiths, "I similar the persuasion that every bit mathematics teachers nosotros are trying to rerun the history of the dependent area within our students' heads, alone quicker! Maybe discovering the dominion for differentiating xⁿ is such an of import lawsuit inwards this history that this action wins its position inwards an AS class where fourth dimension is tight".

In my lesson on differentiating from outset principles, I plot y = x² on the board as well as justice the slope at x = iii using a serial of increasingly brusk chords. Once we've established that the slope of the tangent (and hence the curve) at this betoken is 6, I hit the same for a twain to a greater extent than points. I as well as then become through the algebra for the full general betoken (x, x²). We found that the slope component is 2x as well as utter near how nosotros tin give notice utilisation that component to discovery the slope at whatsoever point.

Then I acquire the students to conduct hold a become alongside y = x³ and y = x⁴. They hold off for patterns as well as figure out the dominion for y = xⁿ. Sounds straightforward but the limits as well as deltas may confuse your students, so don't rush them.

betterexplained.com

This outset principles method is clearly explained inwards this article from Math Centre. If y'all haven't taught a outset principles lesson earlier as well as are a chip unsure as well as then picket this video to come across how it's explained. Most teachers I've spoken to innovate differentiation inwards a similar way. Some hit a lot less explaining than me, instead getting the pupils to hit most of the piece of work themselves past times manus or using ICT. If you're looking for a resources to guide y'all inwards this outset principles lesson as well as then this worksheet from SRWhitehouse on TES mightiness hold out helpful. If you'd rather students utilisation ICT (Desmos or Autograph), as well as then this RISP gives some guidance.

Resources

I truly savour teaching differentiation in 1 lawsuit we've got past times that tricky outset lesson. Optimisation is especially satisfying.

I've listed a few teaching resources here.

For activities that assist students practise the basic rules, in that location are simple polynomials dominoes and factorised quadratic dominoes from Teachit Maths. WJEC gives us a differentiation pelmanism for the interactive whiteboard as well as the Standards Unit has a Matching Functions as well as Derivatives activity.

SRWhitehouse on TES gives us pick of fantabulous differentiaton exercises.

Once y'all acquire onto stationary points, here's a few resources to try:

• Finding stationary points of cubic functions - Standards Unit
• Optimisation - The Chalk Face
• Maxima as well as Minima problems (notes & slides) - Nuffield Foundation

Project Maths is a fantastic website alongside lots of practiced resources, many of which are featured inwards my resource libraries. It’s based on the Irish Gaelic schoolhouse organisation so is a chip hard to navigate if you’re non familiar alongside their curriculum. The best bits are the activities at the dorsum of their T&L plans - for lawsuit Introduction to Calculus is packed total of fantabulous resources for introducing differentiation as well as explaining concepts relating to rates of change. It fifty-fifty features a Usain Bolt action which ties inwards nicely alongside the ideas above.

Final thoughts
Differentiation truly exposes gaps inwards students' indices as well as algebra skills. If you're looking for resources to assist y'all learn the other topics covered inwards Year 12, come across my Core AS Resources Library.

I highly recommend that all maths teachers read the articles near calculus on betterexplained.com, starting alongside Calculus: Building Intuition for the Derivative.

Jo or Gottfried?

Calculus is a rare branch of mathematics inwards which y'all truly desire a educatee to inquire the enquiry 'But Miss, when volition nosotros e'er demand to know this?'. There are endless applications in a broad let on of fields including engineering, architecture, medicine, physics, finance as well as economics.

As usual, I'm truly keen to hear your teaching ideas - delight comment below or tweet me.

Apologies for whatsoever mistakes inwards my terminology inwards this post. I'm no Leibniz. If I've made whatsoever glaring errors as well as then hit allow me know.

If all else fails as well as then inquire your students to purchase the Cartoon Guide to Calculus. Who needs teachers?

5 Maths Gems #22

Hello as well as welcome to my 22^nd gems postal service - this is where I percentage v didactics ideas I've seen on Twitter.

I've discovered at schoolhouse this calendar week that alter is hard. It's tardily to nation 'I'm non going to learn it similar that anymore' or 'I cause got a fantastic novel thought for didactics this topic' but when it comes to genuinely delivering lessons it's tardily to stick with tried as well as tested methods as well as resources. We desire our students to accept risks but how oft do nosotros accept risks equally teachers? This has been on my heed a lot this calendar week - I'll write a postal service nearly this soon.

1. 2015

Every twelvemonth Chris Smith features the Math Forum Year Game inward his newsletter. The idea is to role the digits inward the twelvemonth 2015 as well as the operations +, -, x, ÷, ^ (raised to a power), sqrt (square root), as well as ! (factorial), to write expressions for the integers 1 to 100. Answers tin endure submitted through the Math Forum website, or perchance you lot could brand a classroom/corridor display similar these examples:

2015 Challenge classroom display by @Jeremy_Denton

Challenge 2015 display by @c0mplexnumber

post nearly the 2014 challenge is too worth a read if you're going to drive this.

2. The Constant Character
Teaching indefinite integration? I loved this tweet from @inFinnityPi.

This lovely thought originated from @edenspresence who shared the student-decorated constant characters below. When students forget the 'plus C' they cause got to decorate a graphic symbol - a proficient agency to learn far stick!

3. 'I helped'
Influenza A virus subtype H5N1 squeamish thought from @GiftedBA - 'I helped' stickers to encourage students to offering each other meaningful aid as well as to recognise those who do so. Download a sticker template here.

4. What's the question?
Influenza A virus subtype H5N1 genuinely squeamish thought from Marcus Fleet is to write an reply on the board as well as enquire students for possible questions.

It's squeamish that the shape cause got chosen a winning question!

I'm quite surprised to encounter the variety of algebra students are doing inward Year half dozen - it's to a greater extent than advanced than I realised.

In the conversation that followed this tweet, Miss_Ren said that she likes to write a issue on the board as well as enquire what the enquiry could cause got been. It links to all areas of maths as well as requires no printed resources. Paul Godding shared simply about similar ideas - such equally writing 5, ix as well as 10 on the board as well as bespeak for the strange i out (students cause got to come upward up with 2 reasons for each answer). There's charge to a greater extent than fantastic problems similar this on Paul's website 7puzzleblog.com.

5. What the examiner sees
If you're didactics a Year eleven shape I remember it's good worth showing them this motion painting from @TheMathsMagpie to ensure they are aware that they shouldn't run exterior the allocated spaces inward their exams.

This post gives to a greater extent than data nearly the motion painting above. Use it amongst this really helpful laid upward of slides GCSE Maths - Easy Ways to Make Sure You Don't Lose Marks, too from @TheMathsMagpie.

WWW/EBI

I'm resisting the urge to write a weekly postal service reflecting on what went good (and what didn't learn well) at school. Instead I'll simply characteristic i personal reflection inward each of my gems posts.

I'm didactics angles to Year vii at the moment. I gave them a homework inward which they were required to hand reasons for their answers. Some students gave bright reasons as well as I was pleased to encounter the right role of mathematical damage I'd taught them such equally 'supplementary'. But many students simply wrote out their calculations inward words instead of stating the angle facts they'd used.

In their defence, nosotros hadn't practised 'giving reasons' inward class, as well as so I decided to address this yesteryear setting simply about angle problems as well as bespeak for total reasons with each answer. Conveniently Don Steward has simply published a laid upward of angle problems that were the right score of challenge.

It seemed similar a proficient thought but I wouldn't nation the lesson went well. What happened was that my students were writing such detailed reasons inward total sentences that it became genuinely time-consuming (for event solving enquiry 8 to a higher identify involves using 3 unlike angle facts, thus 3 sentences). They were doing as well as so much writing that they weren't doing plenty maths for my liking. Plus you lot could listen a pivot driblet inward my classroom, as well as that makes me uncomfortable. In the halt I told them to halt writing total sentences as well as write one-word bullet points instead (eg isoscoles/triangle/straight line/complementary etc). They were genuinely relieved! I'd dear to listen my readers' thoughts on this - should I cause got persevered with the tedious but of import sentence-writing practice? I'm non sure. Please tweet me your opinion!

5 Maths Gems #27

Hello in addition to welcome to my 27^th gems post. This is where I part v of the best pedagogy ideas I've seen on Twitter recently. I'll last brief today because I experience similar I'm drowning inwards marking, reports in addition to lesson planning at the moment! Thankfully it's exclusively i calendar week until the Easter holidays, thence there's lite at the destination of the tunnel.

1. Which One Doesn't Belong?
This website is pure genius from @MaryBourassa. 'Which One Doesn't Belong?' is 'a website dedicated to providing thought-provoking puzzles for math teachers in addition to students alike. There are no answers provided every bit in that place are many different, right ways of choosing which i doesn't belong'.  Here's a twain of examples:

Ask students to uncovering a argue why each i doesn't belong. What a fantastic tool for generating interesting aeroplane discussion. Mary welcomes contributions to the website - details in addition to templates are available Pythagoras Lesson. In this lesson Pythagoras problems were pose unopen to the room in addition to students worked inwards pairs to solve the problems - they had to introduce their instructor amongst every bit many right answers every bit they could inwards an hour. They had 3 ‘hint’ tokens that they could merchandise inwards for deal at whatever point. I similar this take in - I receive got a habit of offering likewise much help, which tin last counterproductive. The express provide of hint tokens volition encourage perseverance inwards students - they'll exclusively inquire for deal when they really demand it.

Spiderbox from illustrativemathematics.org

4. Surds
Two lovely surd resources were shared on Twitter this week. First upward was @aap03102's excellent surds magic square.

And then @MathsPadJames (of awesome website mathspad.co.uk) shared this bright activity:

Followed past times some other laid upward of examples:

5. Sticky Maths

5 Maths Gems #30

Hi together with welcome to my 30^th gems post. This is where I percentage only about of the best teaching ideas I've seen on Twitter.

1. Construction Comics
Back inwards Gems 4 I featured Paul Collins' (comic resources for teaching constructions. In his post he suggests only about opportunities to role these resources inwards the classroom.

One of Paul Collins' structure comic strips 

I haven't taught constructions this twelvemonth hence I've non nevertheless had a run a hazard to role these resources, or the awesome Gems 21).

I loved the operate shared yesteryear my namesake Mrs Morgan (@em0rgan) on Twitter this week. She asked her students to do their ain structure comic strips for homework. The results facial expression excellent. I'm non unremarkably a fan of homeworks similar this because they're hence fourth dimension consuming (both for students to gain together with teachers to mark) together with oft they don't add together much value. But inwards this instance I intend at that topographic point is value inwards students learning the procedures yesteryear documenting the steps, inwards improver to practising the constructions.

2. Influenza A virus subtype H5N1 Level Resources

Thanks to @OxfordEdMaths for sharing a gear upwards of excellent A degree assessments for C1 - C4, D1, D2, FP1, FP2, M1, M2, S1 together with S2. It tin endure difficult to honor sources of expert Influenza A virus subtype H5N1 degree questions. These gratis resources are going to endure incredibly helpful adjacent twelvemonth for theme assessments, homeworks or revision.

Speaking of Influenza A virus subtype H5N1 degree resources, do banking concern agree out the website 0a explains: Calculus (with pics together with gifs) -thanks to @mathhombre together with @solvemymaths for sharing this. I dear the pictures together with gifs on this website - they mightiness assistance teachers explicate concepts inwards calculus.

3. Homework Expectations
I've only discovered MathFour.com from Bon Crowder - it's total of creative ideas, only about of which I excogitation to characteristic inwards hereafter gems posts. One persuasion that I spotted was a 'How to Write Your Math Homework' guide. I'm going to brand a version of this for my novel classes inwards September. I can't complain nearly my students' operate non coming together my expectations if I haven't made those expectations clear from the outset.

4. Talk a Mile a Minute
Steve Wyborney (@SteveWyborney) shared his Talk a Mile a Minute Vocabulary Activity. The video below describes the activeness inwards detail. It involves students describing sets of words to their partner together with and hence discussing what those words relate to. I intend this would endure a cracking activeness for an terminate of twelvemonth lesson. Have a facial expression at Steve's post for resources. If you're looking for words for this activeness together with hence try this Vocabulary Checklist for Key Stage 3.  

5. Activities

I've seen only about lovely activities together with questions on Twitter this week.

@MathedUp shared this 'Negative numbers code-breaker activeness amongst a twist'.

Gems 29.

@DJUdall shared this activeness which links solving equations amongst expanse together with perimeter.

@DJUdall likewise shared a query he made which combines Pythagoras' Theorem together with Standard Form. 

The latter 2 resources were shared during concluding Sunday's #mathsTLP. This is a cracking identify for maths teachers to collaborate. Do bring together inwards on Sundays at 7pm to enquire for assistance amongst lessons or to offering ideas.

Update
Because my examination classes are directly on report acquire out I had fourth dimension to excogitation only about actually prissy lessons this week. With my Year 10s I did a lesson on similar triangles using resources from the Mathematics Assessment Project. They had to kind 8 pairs of triangles into 'similar', 'not similar' or 'can't tell', drawing on their cognition of angle facts. This project is a lot harder than it sounds together with takes nearly 45 minutes to complete. I actually recommend it - here's an instance of the operate they produced hence you lot tin see what it involves.

I talked to all my students nearly palindromic numbers this calendar week (I remembered to yell it every fourth dimension I wrote the appointment on the board, because the dates were palindromic when written inwards the format m/dd/yy). Although many of my students could plough over me an instance of a discussion that is the same frontward together with backward (eg racecar), most were non familiar amongst the discussion palindrome.

Those of you lot who role Gems 10) for formative assessment volition endure pleased to listen that you lot tin directly insert pictures. For instance you lot mightiness desire to insert questions from www.diagnosticquestions.com.

That's it from me for now, but at that topographic point volition endure only about other gems post presently because there's been loads of cracking materials on Twitter lately. Wish me luck amongst my concluding calendar week at my electrical flow school! My novel project starts afterward half-term.

I'll acquire out you lot amongst this GCSE resource by thornn on TES (shared by @eggarsmaths on Twitter) which you lot mightiness honor useful for concluding infinitesimal examination tips over the coming weeks.