Sticks and Shadows

Paul Lockhart’s excellent book Measurement (a constructive follow-up to his infamous diatribe about the start of maths education, A Mathematician’s Lament), includes this gem:


I adapted for mixed-ability Yr9 class. Some students were thinking deeply about the shape of the curve, others were practicing measuring angles and lengths and plotting coordinates.

From formal to concrete
Multiple ways to measure the shadow…


Geogebra app proved the most popular, since the least brainpower required to use it. Maybe only give this to the students once they have struggled with a physical metre-ruler or the unit circle first?


Lovely conversations about how a shadow can have a negative length, and why we would want to think of the metre-ruler sticking straight into the ground and still casting a shadow.

Some students found four squares representing 30 degrees a challenge. Which graph is correct?

Spot the difference.
Student explanation

‘Tis but a Scratch

Inspired by this Monty Python sketch, I asked students to find the area of a triangle, if they know the lengths of the three sides. Gradually I chopped off a mathematical limb, removing ipads, calculators and compasses, before only a brain and a pencil remained.

Some great and unexpected learning:

  • The compass was too small to draw an arc of 13cm. So we enlarged the shape by a factor of 0.5, and thought hard about how that would affect area
  • We thought about how to accurately drop a perpendicular from a point to a line, when measuring the perpendicular height of a triangle
  • We wanted to work out the square root of 21 x 8 x 7 x 6, but without using a calculator. Prime factors to the rescue!

I had given the class 5 minutes to try and find a way of calculating the area of the triangle, using only a pencil and a brain. Some wildly inventive methods, that involved inventing an adapted version of Pythagoras got students close to the correct area. Great creativity, but lacking in precision.

The start of understanding the proof:


  •  If a student knows every constituent skill then they can understand the proof. Whole is greater than sum of the parts – the multiple steps are difficult to keep in your working memory
  • Students have well-trained instinct to expand brackets. Sometimes this isn’t helpful!
  • One student spent a long time trying to prove that s = 1/2(a + b + c). The distinction between something you can define and something you have to prove.
4 students collaborated on this. At times inefficient, but some great conversations.
Lovely annotations.
Outstanding clarity and thought

Lofty Lipservice: Reaching for Student Choice

This is the fourth chapter in Lofty Lipservice.

  1. First I focussed on the issues I have been struggling with
  2. Then the department spoke about our visions more broadly
  3. I planned a lesson, to put my vision of student-choice into practice
  4. Now I will reflect

Last term I developed a lesson on Pythagoras’ Theorem, focussing on providing students with the choice of which question to answer, and how to answer it. I have been working, across Yrs 7,8 and 9, to give more productive freedom to students.

  • In Year 7, freedom in ways of working
  • In Year 8, freedom in interacting with objects in the classroom
  • In Year 9, freedom in developing an individual question

In a (dream) mixed-ability class, students should be working on self-generated questions, in their own way. The teacher should be ensuring the questions are rigorous, the groups productive, and the methods mathematically useful.

Year 7: Ways of working

The challenges, from Don
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Suggested scaffolding
Great teamwork from students who would normally be more distracted

The class had a healthy competitive atmosphere. Students were practising substitution, trial and improvement, negative numbers, collaboration.

Year 8: Interacting with objects

Finding the curvy length around a circle is difficult. We scoured the classroom for circles, measured their diameter and curvy length, and tried to spot patterns. Students enjoyed thinking creatively, searching for doorknobs, clocks, watches, stools, buttons on their iPads… Useful to emphasise that Pi is, by definition, the ratio between circumference and diameter.


Recording ratios, sum and products…



Year 9: Developing individual questions

I told the students that a nice triangle, for me, is one with integer side lengths and one 90 degree angle. With-holding Pythagoras’ Theorem, I asked them to search for as many “nice” triangles as possible.

Next, students developed their own definition of nice.

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An attempt to ensure that “nice” has a productive definition

Some student definitions include:

  • Whole-number area
  • Whole-number perimeter (even if individual side-lengths are not whole numbers…)
  • “Any normal simple triangle” (the result of too much freedom is ridiculous definitions)
  • One side is a multiple of the others
  • “Area” (some students find it difficult to ask well-posed questions)
Excellent self-differentiation
This student worked far harder than usual – perhaps because he owns it?
A productive definition

Further questions

  • When everyone is answering a different question it is difficult to provide meaningful targets for quantity (and quality?) of work. A student with a very restricted definition might not find as many triangles as someone who is only searching for triangles with one line of symmetry.
  • I have a fine-tuned notion of what makes a good question. How to train students in this more effectively?
  • Searching for circles is tangible. What about topics that do not lend themselves to physical objects like this?

How many right-angles?

After a week of exploring angle sums in polygons, both through interior and exterior angles, we attacked an excellent problem by Don:

I have a shape with 100 sides. What is the maximum number of right-angles the shape can have?

Fascinating to see different reactions to the problem:

Go straight for the jugular: try and draw a 100-sided shape
A similar (but digital) attempt
Overly keen pattern spotting. (The odds/evens pattern breaks for n = 11,12)

One group came up with a hierarchy for solving the problem, that I would struggle to improve on:

  1. Guess
  2. Draw it out
  3. Understand the pattern and apply it

Students worked in trios:

  • Geogebra master (in charge of using technology to quickly sketch ideas)
  • Pattern spotter (in charge of generalising any results)
  • Organiser (ensuring good communication within the group)