CAS in ME: Theory and practise. Paul Drijvers Freudenthal Institute Utrecht University P.Drijvers@fi.uu.nl. Theories concerning CAS use. 1. Specific local theories, originating from CAS research 2. Originating from ME research in general, and applied to CAS use
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CAS in ME: Theory and practise
1. Specific local theories, originating from CAS research
2. Originating from ME research in general, and applied to CAS use
(or somewhere in between the two …)
P:For a = 0 you have a straight line. Can you see this in the formula, too?
P:That’s a pity.
M:Yeah, but with the calculator, I think it is much more clumsy, because normally I understand it very well, but such a formula, I don’t see much in it if I just enter it into the calculator and it draws the graph.
P:And if you just look at it, without calculator, you take x, add a times the square root of x^2+1, what happens then if a = 0?
M:Well then it gets straight but I really don’t know why, no idea.
P:What happens with a times that square root if a equals zero?
M:Ehm, well then the square root will be zero as well?
P:Yeah, so what will be left of the formula in fact?
M:x + a times x^2 +1, isn’t it?
P:But a was zero, remember?
P:And in this case
M:Let’s look, well then, … well the square root is then zero en the square, yes zero squared is also zero, so in fact, then I think this complete part is skipped, or not?
P:And what will remain?
M:Eh, x + a times … +1 or something?
P:No x isn’t zero but a equals zero, isn’t it?
M:… O yeah … well then, then I think the square root is dropped.
M:And the rest remains.
P:Yes, and what is the rest then?
M:Well x + a times x^2 +1, .. , or not?
P:But a was zero?
M:O then it is eh x + x^2 +1
P:No, because eh it says, for this a you should read a zero in this case,
P:If a = 0, then you get x + 0 times, a whole part.
P:But how much is zero times a whole part?
P:Yes. So what will be dropped?
M:In fact the complete last part?
P:So what will remain?
M:x + a?
P:No, because a = 0, yes, so
P:Yes. Are you guessing now or eh?
M:No, I really think so.
P:OK, I also really think so.
M:Then it is only x.
M:O I understand it, that’s why it is so!
M:Yeah but I think it is a bit strange because normally you have a graph and you draw from point to point but here you suddenly have for each a a different graph.
M:Whereas as you draw yourself this never happens.