Sixth Sense
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In my last article, I touched on the strategy of guess and check to integrate functions:
Question: What did I differentiate to get ?
Guess:
Check: The derivative of this is
Thought: So I need to multiply this guess by 2 to get the answer I’m looking for
Therefore:
Developing this thought process as early as possible in your students is well worth doing, because they can return to it time and again during their A level studies. One can start immediately after teaching differentiation:
Question 1: What did I differentiate to get
Guess: The power of 3 makes me think of
Check: When I differentiate , I get
Thought: This is 4 multiplied by the answer I was hoping for
New guess: I’ll divide my first guess by 4 , to get
Check: The derivative of this is
Therefore: The answer to the question is
[with constants still to be discussed!]
Question 2: What did I differentiate to get ?
Guess: Given what I’ve just done, I’ll try
Check: If I differentiate, I get
Therefore: Right first time!
Having practised lots of ASlevel problems of the form , your students are now ready for the types they’ll meet at A2, such as the one at the top of the article. In time, most of the method should take place in the students’ heads – they need to develop procedural fluency, and not have to write a short script every time they integrate  but they must always be able to vocalise the process when asked, so that you can check that they are developing conceptual understanding.
Question: ?
Guess: I see a power of 4, so I’ll try
Check: This differentiates to
Thought: Ahha: if I factorise the second bracket, this is 10 times what I want, so I need to divide my guess by 10.
Therefore:
Now, once the existence and significance of the function have been discussed, and your students are happy and confident with the result (as explored last time) that , then they can approach integrals such as and using the guess and check structure :
Guess:
Check: This differentiates to , so I need to divide by 7 and multiply by 2.
Therefore:
Guess:
Check: This differentiates to , so the guess is 8 times too big.
Therefore:
It is, of course, important to see that this method has its limitations – in the same way that the “rule” for differentiating does not apply to , the guess and check procedure will not work easily for certain integrals:
Question: ?
Guess: I see a power of 7, so I’ll try
Check: The derivative of this is
Thought: I need to divide by 16 and multiply by
New guess:
Check: Hang on a minute, I can’t differentiate this without expanding, or using the product rule…
Therefore: … I’m stuck
Teacher: For the time being …
However, having seen further examples of the chain rule, students can successfully guess and check increasingly complicated integrals:
?
?
(and it’s interesting to ask them to compare this with )
Now for the payoff! Every time I teach this next bit I am struck by how quickly most students see what’s happening – they pick up the idea much faster than I did when I was at school and was taught from the textbook “the formal method”.
So, guessing and checking for grownups, as my first Head of Department called it (and I’m sure he took it from his previous Head of Department!), works like this (and happens in the scheme after the product rule has been explored and mastered!):
Question: ?
Guess: I’ll try and see what happens
Check: The derivative of this is
Thought: Hmmm. Tricky. One step at a time. Let's sort the first term out.
New guess:
Check: This differentiates to
Thought: Now I need to include in my guess an extra term which, when differentiated, cancels the second term, which I don’t want
Guess again:
Check: This now differentiates to
Therefore:
Getting the first guess may take a while, and it is important to praise, and explore, the alternative suggestion:
Question:?
Guess: I'll try and see what happens
Check:
Thought: Now I need to subtract something but I have no idea what, because is a harder question than the one I started with...
Therefore: Back to the drawing board!
By the time the students have tried two or three of these they should be able to conclude that, with this type of question, there are two “natural” guesses, but one ends up with a second integral that is easier than the original, whereas the other produces a second integral that is harder than the original: clearly the skill is in picking the better first guess.
Once they’re feeling confident, do get them to try, and is an important example to cover at this point.
By the second or third lesson on this method, having also looked at definite integrals, students need to generalise:
Question:
Guess:
Check: This has derivative
Thought: This is what I want, with an extra term, so I need to subtract from the guess a term which, when differentiated, gives
New guess:
Check:
Therefore:
Hence, informal guessing and checking for grownups becomes traditional integration by parts, but specialising before generalising helps enormously. Repeatedly applying an “out of the blue” formula will probably give students procedural fluency, but their conceptual understanding is very unlikely to develop too. I’m reasonably sure that the socalled explanation which my sixth form self was given started with the last line of the above argument, and it took me a long while to understand what on earth was happening!
Image credit
Page header by takomabibelot (adapted), some rights reserved
