This MitPY is a request from frequent commenter, Red Five:
From frequent commenter, SRK:
A question for commenters: how to explain / teach integration by substitution? To organise discussion, consider the simple case
Here are some options.
1) Let . This gives , hence . So our integral becomes . Benefits: the abuse of notation here helps students get their integral in the correct form. Worry: I am uncomfortable with this because students generally just look at this and think “ok, so dy/dx is a fraction cancel top and bottom hey ho away we go”. I’m also unclear on whether, or the extent to which, I should penalise students for using this method in their work.
2) Let . This gives . So our integral becomes . Benefit. This last equality can be justified using chain rule. Worry: students find it more difficult to get their integral in the correct form.
3) has the form where and . Hence, the antiderivative is . This is just the antidifferentiation version of chain rule. Benefit. I find this method crystal clear, and – at least conceptually – so do the students. Worry. Students often aren’t able to recognise the correct structure of the functions to make this work.
So I’m curious how other commenters approach this, what they’ve found has been effective / successful, and what other pros / cons there are with various methods.
Following on from David’s comment below, and at the risk of splitting the discussion in two, we’ve posted a companion WitCH.
A question from Michaela, a Year 11 student:
How do I find the implied (maximal) domain, and corresponding range of a function?
(Commenters, please try to use LaTex, or clear code, if you can.)
A question from frequent commenter, Steve R:
Hi, interested to know how other teachers/tutors/academics …give their students a feel for what the scalar and vector products represent in the physical world of and respectively. One attempt explaining the difference between them is given here. The Australian curriculum gives a couple of geometric examples of the use of scalar product in a plane, around quadrilaterals, parallelograms and their diagonals .
Regards, Steve R
A question from frequent commenter, RF:
If you were teaching a (very) mixed ability Year 7 class in their first term of secondary school and had COMPLETE control over the curriculum, what would you start with as the first topic/lesson sequence?