Tilted squares - teaching using rich tasks

This article is part of our Enriching the Secondary Curriculum feature.

This article is centred around 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 pulled out some clips which showcase strategies for fostering positive attitudes towards mathematics and for guiding learners through different types of mathematical thinking.

Video 1

This video shows the setting up of the activity at the start of the lesson.

• 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 mathematical 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.

Video 2

In this video clip, the results are being collected approximately 5 minutes later.

• 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 the conjecture's status as not being 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.

Video 3

This video clip shows the last part of the lesson, where students made the generalisation that leads to Pythagoras's Theorem.

• 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 okay if a conjecture turns out to be false; we modify conjectures in light of new results.
• Using the same numbers as before, making predictions using the new conjecture.
• Moving students from conjecture to convincing arguments.
• Proof takes the approach students have used for numerical examples and develops 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.