CAS in ME: Theory and practise. Paul Drijvers Freudenthal Institute Utrecht University [email protected] 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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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?
M: Eh, no.
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.