Inventing Triangles

The advantage of multiple classes is you can refine, compare and contrast. Loving it.

This kind of lesson requires, from the teacher:

  • Mathematical agility, to be able to quickly check if a triangle is worth exploring, or if a conjecture is worth exploring
  • General knowledge, to link in unexpected ways (see below)
  • Love of chaos – got no idea where we were going to travel

One group decided the outer diagonals of the triangle would be 1,3,5,7,9… and then used a Pascal-like rule to construct the rest. They abstracted to just odds and evens, and started noticed nested patterns of triangles. 

Another group started with rt(2) as the diagonal, and multiplied rather than added. They too found the nested triangles…

I got very excited at the parallels with Sierpinski’s triangle. The whole class crowded around, awed into silence, to compare and contrast the two. (One made by continuing down an infinite number of rows, the other chops a finite thing infinitely many times).

Later that afternoon I noticed precisely the same pattern in a textbook. After yelling about being the Neil Armstrong of triangles, boldly going to new mathematical lands, I was disappointed to see that someone got there before the kids. Does it matter? Not really. It does matter that I was ignorant of it existing during the lesson – the excitement would have been less genuine.


My Grandkids’ Lung Cancer

“But the project finished two terms ago!”, Rochelle complained after legging it from her home to meet Cameron and I outside the school gates one weekday evening.
That isn’t how truly authentic projects work – they don’t neatly fit into term-schedules. We were off to summarise our report on the air pollution that would be caused by a proposed set of concrete factories, slap-bang in the olympic park.
In two hours of dense, tightly protocoled discussion, Cameron was the only speaker to get a laugh. Finishing his planned speech, he paused, “I just have two more things I want to say”. I couldn’t decide whether to leap in and move things on, but he spoke, eloquently and from the heart, about intergenerational responsibility. “When you are old we are going to look after you” was the punchline that got the chuckle.
Rochelle, finishing our presentation with the line “Do you really want to be responsible for giving my grandkids lung cancer?” (Which we hastily had drafted while in the waiting outside, scribbling up against the wall), cut through the technical jargon (Section 10.3, 2011 Planning Act, B9 usage) to create a phrase that the room remembered. Several people came up at the end to shake her hand and reference that line. 
The committee voted unanimously against the building of the concrete factories. (My) summary of their position – we accept that all cities require building materials, but we are shocked by the poor quality and thoughtlessness of this specific application. The chairman, in his summary, specifically praised the students – “to hear the two youngsters speak was powerful”. 

“Most adults would have struggled doing what you just did” said Terry Paul, local councillor, to Rochelle

The other protestors went to the pub to celebrate. The kids declined the offer of a beer, and instead went home safe in the knowledge that their learning had truly changed something . “Don’t let anyone tell you that you cannot make a difference to the world” one of the local residents joyously yelled as he shook Cameron’s hand. 

A Tentative Lesson Structure

I have been naturally falling into a rhythm in my Further Maths classes. This structure seems to work well when students are already on board with Maths (and therefore follow you through headaches and open exploration), and when they are meeting lots of new mathematical structures for the first time.



Table to clarify thinking, and then a record of the lesson below.



Obviously, any attempt to codify what all good lessons looks like is ridiculous.


In general

In particular – Matrices



Provide students with a headache to wrestle with

Ask students to solve increasingly complex simultaneous equations

Before providing the aspirin, you need to give the headache. Dan Meyer


Give students key information that it would not be reasonable for them to discover themselves

Lecture on how to multiply matrices (link back to multiplying vectors, which in turn links back to multiplying numbers)

Minimise teacher-talk while not wasting time on wishy washy discovery-based learning


Ask students what questions they have. Important to do individually first – no cross-contamination

“What do you want to know about matrices?’

Now that they have a basic idea, questions start flowing. (Without teacher-input, the blank page is too daunting)


Explore! In groups of three or four students?

Structured comparison of matrices and numbers.

Students need to learn how to behave like a mathematician – conjecture, root around in the darkness, play, prove.


Big reveal: you now have the aspirin for your headache

After playing around with matrices and working out what the equivalent of division is, you can solve big systems of simultaneous equations

Provide an ending to the story, reminder of why this new bit of Maths is useful.

A record of the lesson

Headache time
Initial questions
Untitled picture
Explore time

From Apprentice to Master

A year ago I met Desmond a few times in a cafe, to struggle through some STEP problems, in preparation for his university applications.

Now, he is getting ready to leave London, to study Maths at Oxford. Before he left, he came to Six21 and spent two hours with the 11 Further Mathematicians, leading them down his merry path of algebraic problem-solving.

The explanations were sometimes confusing and fast, but this small sacrifice was easily worth it for the benefits:

  • The students responded so well to a leader only a few years their senior (Desmond called the boys “bro”, and could get away with it).
  • The students felt comfortable enough to repeatedly ask for explanations – “I still don’t get it”.
  • Excellent role-model for high uni aspirations. “You all have the GCSE grades to get into Oxbridge for Maths. All you need now is to work hard”
  • Start the exposure to tricky STEP questions and the associated tricks (add and takeaway something disgusting to an expression to reveal something simple after all). Final thing Desmond said to me: “Make sure you do loads of STEP questions with your students”

All 11 students decided to stay for an extra half-hour after the end of school to try and finish the final problem (prove that any number pq, where p and q are primes greater than 2, can be written as difference of two squares in exactly two ways). Great testament to Desmond’s session.

Best of luck for the future Desmond!


Questions: (could also use this)

(20172 – 20162 + 20152 – 20142)/(2017 + 2016 + 2015 + 2014)


Untitled pictureUntitled picture2

First reflections on Further Maths

Originally there were 7 students doing FM, all male. It took one question, literally one question, “Would you like to do Further Maths?”, for four girls to leap at the opportunity to do it. They just hadn’t considered the possibility. Great reminder to ensure everyone knows they are welcome to join the Maths Clique, especially those from groups that have historically been under-represented.


Highlight of the first week – exploring complex numbers.

  1. I introduced complex numbers as natural next step in progression (x + 2 = 0, 3x – 2 = 0, x^2 – 2 = 0, x^2 + 1 = 0).
  2. Students generated a whole load of really excellent questions

    (If students were able to answer their questions by end of lesson, it is ticked)
  3. “Trust your intuition, and follow all the rules you already know”. The group were able to solve quadratic equations with complex solutions, manipulate expressions involving complex numbers, mess about with fractions using their knowledge of surds as analogy.
  4. When looking at powers of i (remember, all I have given the students is the definition of i)
    • “We are trapped in a loop!” yells Wintana
    • “It’s like a circle” chips in Ifte
    • Noemi draws the circle, notices that it looks like the real number line goes through horizontallyIMG_7218.jpg
    • “I guess the y-axis is the imaginary number line” jokes Igoris sarcastically

What beautiful and constructive conversation from the students. Replicating the discovery of the complex plane, inventing it for themselves. Obviously discovery-based learning has its pitfalls (“Now class, you will discover the trigonometric ratios”), but once you give the students the initial seed they can run very far indeed! I am very excited.

BTW, reminds me of a recent podcast by Ben, Ben and Blue about how lecturing is doing worse than doing nothing at all. Inspired by the work of Carl Wieman, physicist turned educationalist. An expert teacher has comfort in chaos, is able to move agilely in response to the class. Lecturing, in comparison is comfortable, easy, reliable.