'That's true! But, how do you know? Convince yourself, and others, of some interesting results by making some proofs.'
QED. You might have seen those three letters and wondered what they mean. ‘Quod erat demonstrandum’ is latin for ‘that which was to be demonstrated’. In mathematics we don’t like uncertainty; we like to be absolutely sure of the results we have found, which is why we always like to prove our conjectures. Proof is not always easy, but it is very satisfying! In this activity you will be looking at consecutive numbers and what happens when you add them up. Be prepared for some surprising and neat results!
Students require the following worksheet Consecutive Proofs.
There are also some Consecutive Proofs Teacher Notes.
The following image embodies what this investigation is about.
Here follows an outline of what the task is.
- Students should understand what consecutive numbers are.
- The teacher will help the students fully appreciate what divisibility is and remind them about factors.
- Students will investigate the sum of two consecutive numbers.
- They should try lots of examples to convince themselves of the conjecture, ‘The sum of two consecutive numbers cannot be divided by two.’
- Work through the proof of this on the worksheet.
- Now it’s up to the students! They should use the worksheet to investigate sums of three, four, five, … consecutive numbers.
- Students should form their own conjectures and try to prove them.
- Is there an overall rule about consecutive sums?