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?