# Maths I’ve learnt since starting teaching

When I started teaching I thought I was good at maths. Despite success at A-Level and university, I now realise how little I actually knew about lots of the fundamentals of maths. I think this is partly because of the sheer number of connections and insights that maths offers, and partly because I was taught in a way that prioritised the how rather than the why.

Recently I’ve started jotting down some of the basic maths I have learnt or realised since starting teaching. I have stuck to basics as I find these moments of clarity are more profound for the basics. I make no apologies for not knowing what I didn’t know. The more we’re ready to see that there’s always more to learn, the more open we are to learning even more and sharing that with our students. I enjoy sharing lots of these with classes – they’re reaction is sometimes indifference, but often is “wow – I never knew that!” or “well yeah, of course it is!” The list below shares some examples. I have gained this knowledge mostly from discussions and observations with colleagues, trying new things, and through reading books (such as “Yes but why?” by Ed Southall and Jo Morgan’s ‘”Compendium of Mathematical Methods”).

• The division symbol can be seen as a fraction with dots appearing as the numerator and denominator. This can help students reinforce the equivalence of fractions and division
• Dividing fractions: You can do numerator divide by numerator, and then denominator divide by denominator. E.g. 9/8 ÷ 3/4 = 3/2. Most students are blown away when they see this! Of course the numbers don’t usually work out as nicely as this example, but I think it’s important to show students it is a valid method
• More division: The ‘bus stop’ short division method can be seen as two sides of a rectangle of a given area and you are finding its width as you know its height
• Quadratic formula. The quadratic formula is just a rearrangement of x from the general form of a quadratic equation x2 + bx + c = 0. There 6 derivations of this here.
• Despite knowing that quadratics in the form of x2 – a2 can be factored using the ‘difference between two squares’ approach, I didn’t realise that they can be factored using the ‘standard’ way if the expression is written with 0x in between the terms. For example, x2 – 25 can be written as x2 + 0x – 25 then factored into (x + 5)(x – 5) by considering two numbers that multiply to make -25 and sum to zero
• Given a trapezium in its ‘normal’ orientation, if you make the top length (usually called a) equal to zero, then the trapezium becomes a triangle and therefore the area neatly transforms from ½(a + b)h  to  ½bh
• I didn’t know that the HCF(a,b) x LCM(a,b) = ab. For instance, the HCF of 12 and 20 is 4, and the LCM is 60. And, 12×20 = 4×60. This is nice to explore using Venn diagrams.
• Pythagoras’ formula can used to solve 3D problems. Given a cuboid of dimensions a, b and c, the distance from a corner of the cuboid to the far corner is sqrt(a2 + b2 + c2). This is easy to prove using Pythagoras on the base of the cuboid to find its diagonal length.

As I teach more, read more, listen and observe more I am sure that this list will keep growing, as will my mathematical understanding. I’d be interested to hear any examples you have of maths you’ve learnt since starting teaching so please do share!

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