Is there an efficient way to work out how many factors a large number has?
Helen made the conjecture that "every multiple of six has more
factors than the two numbers either side of it". Is this conjecture
Find the number which has 8 divisors, such that the product of the
divisors is 331776.
"What totals do we get by adding the consecutive numbers in these sets?"
Write + signs in between the lists of numbers.
"These totals are all examples of numbers that can be written as the sum of consecutive numbers. Do you think all numbers can be written in this way?"
"How about trying to write the numbers from $1$ to $30$ as sums of consecutive numbers?"
Give students time to work in pairs on filling in the gaps from $1$ to $30$. While they are working, write the numbers from $1$ to $30$ on the board ready to collect together the sums the class have found.
If students ask about negative numbers, one possible answer is: "Stick to positive numbers for now, and then perhaps investigate negative numbers later."
Once most pairs have filled in most of the gaps, collect their results on the board.
"Spend a minute looking at these results and then be prepared to talk about anything interesting that you notice."
Give them time to think on their own at first and then share ideas with their partner, before discussion with the whole class.
Next, collect together any noticings, and write them on the board in the form of questions or conjectures. If such conjectures are not forthcoming, there are some suggested lines of enquiry in the problem.
Allow pairs time to work on the conjectures of their choosing, reminding them that they will need to provide convincing arguments to explain any of their conclusions.
If appropriate, bring the class together to spend some time discussing algebraic representations of consecutive numbers ($n, n+1, n+2...$) to give students the tools to create algebraic proofs.
Finally, students could create a poster, a presentation or a short report explaining one key conclusion that they came to, together with the convincing arguments they used to explain it.