Why do this problem?
gives an opportunity to try something out, notice a pattern, make a conjecture, and then prove it. There are several approaches to the proof, both algebraic and visual, which can lead to fruitful discussion about different methods of proof.
Proving the statements in both directions (if and only if) will be quite challenging for some students, and they will need to persevere and show resilience. To help them to develop these skills, we have provided two "proof sorter" activities which can be used to scaffold the task and offer the support that students need in order to complete the problem.
"Choose a triangular number, multiply it by 8 and add 1. What did you get?"
Collect a variety of answers on the board, and invite students to comment on anything they notice:
"The answer is always odd."
"The answer is always a square number."
Once they have made a conjecture, there are different routes to a proof. Some students may prefer a visual approach, looking for ways to show the triangular numbers fitting together to make a square. Others may prefer to work algebraically.
Once students have had a go at proving their conjecture, you may wish to offer them this Proof Sorter
, which has the steps of a possible proof mixed up for them to put in the correct order. There is a second Proof Sorter
which proves the result that
if $8k+1$ is square, $k$ is triangular. This could lead to a discussion about the difference between "If" and "If and only if", which can be explored further in the problem Iffy Logic
There is an interactive version of each proof sorter here:
Interactive Proof Sorter
Interactive Proof Sorter (Converse)
What do you notice when you work out $8T+1$ for a triangular number $T$?
Will this always happen?
Can you prove it will always happen?
If $8k+1$ is square, does that necessarily mean $k$ is triangular?
Can you prove it?
has a selection of similar results for which students could try to construct proofs.
The problem Picturing Triangular Numbers
is a good introduction to visual proof, and can be used as a foundation to a pictorial proof of this result.