Marking the FP1 test papers, I noticed that Lola started Step 2 of her induction proofs like this (for example):
Prove true for n=k+1 if true for n=k
To prove: uk+1 = 4k+1 + 3(k+1) + 1
True for n=k ⇒ uk = 4k + 3k + 1
More exactly, she just wrote uk = 4k + 3k + 1 rather than:
“True for n=k ⇒ uk = 4k + 3k + 1″
or (what would be just another way of writing the same thing)
“If true for n=k, then uk = 4k + 3k + 1″.
(Remember, X ⇒ Y means the claim X implies the claim Y, or, in other words, the claim Y is true if the claim X is true).
And she didn’t write “To prove”, but just wrote the thing to prove in purple pen and marked it off with asterisks.
But the basic idea is to state at the beginning of Step 2:
- first, what supposing for the sake of argument “true for n=k” tells us
- second, what we want to prove in Step 2 – what we want to get at the end of Step 2
I’ve suggested that when we start Step 2 we should leave some blank space for the working, write the thing we want to prove – uk+1 = 4k+1 + 3(k+1) + 1 – below that blank space, and then work at putting a valid logical argument in that blank space which leads to that conclusion.
Most students don’t like doing that. Students who take pride in presenting their work neatly and clearly are worried that the blank space will be too small, or too big.
I think Lola’s method is better than mine. It makes it clear both to us, doing the proof, and to the reader, exactly where we’re heading in Step 2, without any of the worries.
What do you think?
You do have to adapt Lola’s method a bit – you do need to write “To prove:….” rather than using purple pen to indicate that it’s something you don’t know yet, but are trying to prove – because in the exam you’re not allowed to use coloured pens.
(Why not? Because Edexcel don’t send your actual paper to the marker, but a scan of it. And scanning colour is a bit more expensive than scanning black and white).