This article is part of Enriching the Secondary Curriculum.

This article contains three videos showing parts of a 75 minute lesson in which a group of Year 9 students (aged 13-14) worked with us on the problem Tilted Squares.

We have written a few bullet points about each video to draw attention to teaching points which we consider to be important.

The first video shows the start of the lesson.

First Video

• Setting the scene; preparing students for being stuck.
• Beginning with a familiar context (squaring numbers)
• Establishing a linked activity (moving from 'normal' squares to 'tilted' squares)
• Introducing a 'big picture' challenge that formed the focus for the rest of the lesson
• Working together to check area skills before letting students loose on the activity
• Responding to students' ideas rather than imposing a particular method
• Being prepared to take a detour from the main focus of the lesson to explore trial and improvement and finding square roots.
• Letting students set the pace; not racing on to new ideas until everyone is ready
• Introducing a method that the students hadn't come up with, and discussing elegance and efficiency
• Developing a shared vocabulary - inviting students to suggest a way to describe tilted squares.
• Introducing students to the notion of mathematicians working collaboratively, to set the scene for group work.

• Introducing the idea of a conjecture
• Gathering results together and recording them in a systematic way
• Using results to make predictions (including the same bigger numbers used at the beginning of the lesson)
• Making a link between these predictions and the 'Big Picture' challenge
• Introducing algebra - n across and 1 up, to formalise the conjecture
• Use of question mark to emphasise status as conjecture rather than fact
• Speculation on what would happen if we looked at n across and 2 up
• A need for more data to verify (or reject) the new conjecture

Third Video

• Circulating while the class works and listening to groups as they discuss their ideas
• Joining a group at their table and kneeling or sitting to be at their level
• Lots of student talk and not too much teacher talk during this phase
• It's OK if a conjecture turns out to be false, we modify conjectures in light of new results
• Using the same numbers as before, made predictions using the new conjecture
• Moving students from conjecture to convincing arguments
• Proof took the approach students had used for numerical examples and developed it into the general case.
• Proof for tilt of 1 modified to also prove the next case. (If time had allowed, we could have gone on to the general case.)
• Summary drawing attention to the mathematical journey they had undergone, and the importance of asking good questions.