Introducing the ideas of calculus to students with not much knowledge beyond GCSE

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#1
We have to do these mini lectures (I call them lecturettes) in our Physics class, they're supposed to be about anything we want them to be. They should take about 15 mins, give or take. And I wanted to do mine on calculus. I want to go over integration, differentiation, and the fundamental theorem of calculus.

I don't want it to be rigorous, so I'll skip any algebraic manipulation and give the final answers. I do want it to be engaging, e.g. ask for people to give ideas of what could be done in this situation or that.

I'll be using simple classical mechanics problems (e.g. a free falling object) to introduce calculus, and show how it must be used to solve these problems.

Is this realistic (within the time frame given)? I don't really think it is. I mean I could take up more time, but it's definitely not going to be more than 30 mins. Should I concentrate on one topic only, as in just do integration? (I'd rather do that than differentiation.)
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6 years ago
#2
Not realistic. You may just be able to demonstrate Riemann integration in that time. I doubt you'd be able to do anything like prove the fundamental theorem of calculus or get anywhere close.

There's no point just telling people results with no derivation.
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6 years ago
#3
(Original post by gagafacea1)
Is this realistic (within the time frame given)? I don't really think it is. I mean I could take up more time, but it's definitely not going to be more than 30 mins. Should I concentrate on one topic only, as in just do integration? (I'd rather do that than differentiation.)
I don't think that it's realistic. That's too much to do. I would do differentiation, since you would need to talk about anti-derivatives to discuss integration properly, and you would thus need to discuss both.
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#4
(Original post by alow)
Not realistic. You may just be able to demonstrate Riemann integration in that time. I doubt you'd be able to do anything like prove the fundamental theorem of calculus or get anywhere close.

There's no point just telling people results with no derivation.
Yeah I always do this, I just think that it takes me 10 mins to understand to it should be easy enough. I'm of course wrong for doing that.
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6 years ago
#5
Differentiation is probably the one you should go for. They teach it at GCSE (I'm in year 11 atm) in our school to the people who do Further Maths and people dislike it, so integration might be a bit too complex a topic, especially for 15 minutes.
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6 years ago
#6
You could try and link some of the concepts with those that would already be familiar to a competent GCSE candidate. For example: at GCSE, students are taught to calculate the gradient of a straight line using the formula where means "change in..." (although many GCSE students may not be familiar with this notation, they will be used to the ideas behind this). You could then explain that differentiation is a method of evaluating the gradient at a point on a curve, by taking the tangent (a straight line) and applying the formula. I certainly wouldn't go as far as a rigorous proof by first principles. Integration is a little trickier, but again you could try and link the idea with evaluating the areas under straight lines and take it from there. I always found diagrams help when explaining integration (e.g. one where the area under the curve is split into small strips) as the students are probably more likely to swallow a complex idea if there is some visible evidence that it works.
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6 years ago
#7
I'd say this is a good idea. You won't be able to do it rigorously(even if you had the time, rigorous treatment would be too advanced and too abstracted from A-level physics), but you can give some motivations and fundamental ideas and do some basic derivations. You don't have enough time in 15 min to give any reasonable treatment of integration, differentiation and the fundamental theorem of calculus and these ideas need time to sink in so I think it'd be best if you stick to differentiation and maybe just mention integration at the end for those interested. Some algebraic manipulation should be given, otherwise it'll be hard for some to understand what you're doing, but you should find a balance: you shouldn't be writing algebra for the whole thing because some people will lose interest.

Bear in mind that you'll have to explain any other results you use eg. binomial theorem fairly well.

Edit: I say do differentiation rather than integration because the motivation and ideas and basic proofs are all easier for differentiation.
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6 years ago
#8
I think you'd do well to introduce the idea of differentiation as the gradient with a really small x-difference, relate it to a velocity-time, or distance-time graph. Khanacademy has a really good video on this, if you want any ideas.
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#9
(Original post by Zacken)
I think you'd do well to introduce the idea of differentiation as the gradient with a really small x-difference, relate it to a velocity-time, or distance-time graph. Khanacademy has a really good video on this, if you want any ideas.
I was thinking I could start with average velocity, and then use the idea that as gets smaller and smaller, the average velocity starts to get closer to the instantaneous velocity. I just don't want to use ideas from maths, I want to show them that calculus isn't one of those "useless maths stuff". I probably should have mentioned that this is an A2 class, but some people on it don't do A -level maths, and even then, I'm sure more than 90% don't even know the idea I just discussed above. ( I totally blame the a levels btw, I mean it's definitely not their fault)

Edit: Btw I'm going to use Latex to make the presentation!
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6 years ago
#10
(Original post by gagafacea1)
I was thinking I could start with average velocity, and then use the idea that as gets smaller and smaller, the average velocity starts to get closer to the instantaneous velocity. I just don't want to use ideas from maths, I want to show them that calculus isn't one of those "useless maths stuff". I probably should have mentioned that this is an A2 class, but some people on it don't do A -level maths, and even then, I'm sure more than 90% don't even know the idea I just discussed above. ( I totally blame the a levels btw, I mean it's definitely not their fault)

Posted from TSR Mobile

This might help, skip the first minute or two.

Edit: Latex? I thought it would be more of a powerpoint thing, and in any case, I feel like this kind of discussion is best done on a whiteboard and on the spot, but each to his own...
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6 years ago
#11
(Original post by gagafacea1)
Edit: Btw I'm going to use Latex to make the presentation!
Good for you! Do you know about LaTeX Beamer? Very good for slide presentations using LaTeX.
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#12
(Original post by Gregorius)
Good for you! Do you know about LaTeX Beamer? Very good for slide presentations using LaTeX.
Yeah I'm using that plain old template (with the title in a blue rectangle, do you know it?), I'm way too much of a newbie (and too lazy) to be "constructing" one from scratch.
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6 years ago
#13
(Original post by gagafacea1)
Yeah I'm using that plain old template (with the title in a blue rectangle, do you know it?), I'm way too much of a newbie (and too lazy) to be "constructing" one from scratch.
No need to start from scratch - there are loads of pre-defined themes. For example...

\documentclass{beamer}
\usecolortheme{beaver}
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#14
(Original post by Gregorius)
No need to start from scratch - there are loads of pre-defined themes. For example...

\documentclass{beamer}
\usecolortheme{beaver}
Wait, this IS what I'm using! The Madrid one to be precise. The template helped with the outline and sections and stuff, I didn't know how any of that worked, before I saw the template.
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#15
(Original post by Zacken)

This might help, skip the first minute or two.

Edit: Latex? I thought it would be more of a powerpoint thing, and in any case, I feel like this kind of discussion is best done on a whiteboard and on the spot, but each to his own...
Oh don't get me wrong, I wasn't planning on going there without a whiteboard marker. But he wanted a powerpoint, so I'm giving him one. I'm using Latex because I like the way Maths looks in it, and it isn't all that hard to use. And you can actually make presentations, albeit they'll be outputted in pdf form.
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