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.
First I focussed on the issues I have been struggling with
Then the department spoke about our visions more broadly
I planned a lesson, to put my vision of student-choice into practice
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 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.
Some student definitions include:
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)
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?