# Lofty Lipservice – Achieving Visions through Pythagoras.

This is the third chapter in Lofty Lipservice.

1. First I focussed on the issues I have been struggling with
3. Now I will try and put one vision into practice.

My vision is that students direct and own their learning. The challenge is to plan a lesson that achieves this, while also teaching Pythagoras’ Theorem (all maths teachers are designing a Pythag lesson based on varying visions).

I chose this because, although my lessons are well-sequenced and carefully pitched at student level, the only choice normally available to a student is to skip a few questions if they find too easy. The lack of choice leads to lethargy and questions of why bother, in double lessons.

RF excellently reminded us of the difference between planning (thinking deeply about probing questions and how to ensure learning happens) and resourcing (finding worksheets, aligning your font, thinking about background colours). To that end, I will provide no slides – this blog is the lesson plan (Dan Meyer’s blog subtitle – “Less Helpful” –  starts to make sense)

Essential Question: What is the nicest triangle you can build?

Rationale: Pythagoras’ Theorem allows us to quickly find triangles where all sides have integer length and one angle is a right-angle (using Pythagorean triples). We are flipping the theorem – normally you think that if the triangle is right-angled then the theorem applies.

• I wanted to be deliberately vague about what a nice triangle is. Maybe a student will decide that nice triangles have 1 (or 3) lines of symmetry. Or they will focus on integer angles rather than integer sides. Or they will search for a triangle with integer sides and angles. Student choice in interpreting the question.
• I want to allow the students to use a variety of methods for answering their question. Some might use algebra to create an infinite number of Pythagorean triples. Some might use a compass to build a triangle if they have already chosen the side-lengths (and I would actively encourage this to link to constructions and motivate why we use them). Some might use an app to instantly create triangles given three bits of data. Some might decide they need a quick way of calculating the area of the triangle, and could be given Heron’s formula. I do not mind if their choice of question and choice of answer do not require Pythagoras – as long as some students need to use it then we can share our findings with each other. Student choice in answering the question.
• There should be space for students to present and share their ideas. A video? A talk? A written-up paper? Student choice in presenting their answer.
• Edit: MG reminds me that I should think more carefully about how to encourage productive student-definitions of the word “nice”. It is possible to provide them structures (peer-critiques, timelines) without foisting upon them my ideas of what a nice triangle is.

Other possible routes of student-led enquiry using Pythagoras’ Theorem:

• There are an infinite number of integer solutions to one equation, but no integer solutions to any one of an infinite number of equations. Can you convince yourself this is true? Why is this not a counterexample below? • If I know two sides of a right-angled triangle, can I always find the third? Can you show me by building it explicitly?
• How do I know that Pythagoras’ Theorem is true for any right-angled triangle? The bounty of delicious proofs that exist is excellent fodder for a sequence of lessons where each group is understanding and explaining a different proof.
• How far away is the horizon? I worked this out when walking along the Pembrokeshire coast-path, and Don has also thought about it.
• Nrich’s excellent tilted squares problem
• How many rational points (where both co-ordinates are rational) are there on the unit circle? (One of Nick’s undergraduate students asked this question – great application of Pythagorean triples)
• How many right-angled triangles are there with hypotenuse 3125?
• What about quadrilaterals? If I know three side lengths of a quadrilateral, how much information do I need about the angles before I can tell you the fourth length?

Side-note. Pythagoras’ Theorem is usually drawn with squares attached to each length. It is important to notice that any shape will do. P.S. Is it possible to have student-owned learning without the tasks becoming ridiculously open, or the teacher planning 24 different pathways in the lesson?

P.P.S Last year I started each class by asking the students to fill out a questionnaire asking about their interests outside of Maths. I learnt such fascinating facts as “M has never vomited, and her favourite film is Paranormal Activity” and “A has loads of pets and is proud of graduating into Yr9”. Have I ever used this knowledge when teaching? Do I want to base a lesson around Fast and Furious?

Next year, I will design a new questionnaire, one based on views of Maths. Why do you think you study Maths? Do you prefer proof or exploration? Open-ended tasks or structured worksheets? Links to other subjects or for its own sake? This information will be surely more useful.