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Article by Charlie Gilderdale and Alison Kiddle# Tilted Squares - Teaching Using Rich Tasks

*Here is a second video clip, showing the results being collected (approximately 5 minutes later) in the lesson.*

Here is the final video clip, showing the last part of the lesson, when students made the generalisation that leads to Pythagoras's Theorem.

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Age 11 to 18

Published 2014 Revised 2021

*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.*

- 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

Here is the final video clip, showing the last part of the lesson, when 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 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.