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An Introduction to Proof by Contradiction
An introduction to proof by contradiction, a powerful method of mathematical proof.
Why do this problem?
This problem involves a significant 'final challenge' which is broken down into a sequence of three groups of mini-challenges. The mini-challenges are not arranged in any particular order within each group and the problem is structured such that students are likely to 'discover' some of the mini-challenges for themselves as they strive to solve other mini-challenges.
These notes are designed for classes who are able to work in groups.
At the outset all challenges are hidden to the learner to maximise the chance of discovery for the learners.
The purpose of this is two-fold: first to scaffold learners to help them solve a difficult challenge; second to show that mathematics is a natural subject where certain questions naturally arise through the consideration of other questions. This will firstly help students to structure their mathematical thinking and secondly to help them to realise that mathematics is not externally or meaninglessly imposed.
There are three groups of 3 mini-challenges and the final challenge. Leave these all hidden to begin with.
Throughout the challenge the focus will be on constructing clear, concise proof and on thinking of possible extension questions.
Very able students might wish to start on the Final Challenge, but it will be good to give them a single mini-challenge and see where their thinking and invention takes them. Indeed, the best and most inventive students might even 'discover' the final challenge for themselves.
It is suggested that the following approach be taken
1) (10 minutes) Students individually given one of the first mini-challenges to think about and work on. Spread the different mini-challenges amongst the group -- don't allow students to see any of the other challenges or talk about this with their neighbours. Students are to think about their challenges and explicitly write down any other thoughts or questions that arise. Encourage those that think they have an answer to construct the clearest proof possible or to think about possible extensions.
2) (5 minutes) A selection of students to describe their challenge, proofs and other questions arising. It is likely that some of the questions arising will be the problems other students were working on directly.
3) (10 minutes) Ask the class to organise themselves in pairs so as to get insight into solving their mini-challenge or proposed extensions. It might be that students pair with people working on the same mini-challenge or pair with someone who was thinking about a related problem. Give the pairs a new challenge from the second grouping to work on.
4) Repeat step 2 and group into 4s, whilst giving the fours a mini-challenge from the third grouping to work on.
5) Throughout encourage the class to propose their own extension questions.
At some point students might solve their challenges or pose the final challenge for themselves. When appropriate move the discussion onto the construction of a clear proof of the final challenge. This will be ideally a group effort.
As you think about your mini-challenge, what questions and extensions arise?
Complete the sentence: I am finding this task difficult because ...
Complete the sentence: I wonder if ....
Complete the sentence: I would be more able to solve my challenge if I knew ...
Can you explain your proof clearly in words?
Solution of the final challenge on its own is a tough challenge.
This task is designed for group work -- encourage groups not to move on until all in the group understand.
Some students might be uncomforable with posing their own questions or verbalizing their difficulties. Encourage an atmosphere where all questions and difficulties are valid.
Summation of series
Mathematical reasoning & proof
Making and proving conjectures
Meet the team
The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities can be found here.
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