Barney Maunder-Taylor

PARAMETRIC EQUATIONS: A CATTY LADDER PUZZLE!

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AT A-LEVEL: in Year 1 of A-level we seem to be fine with the regular $x,y$ or “Cartesian” equations of curves e.g. $y=x^2$ is a parabola, or $(x-a)^2+(y-b)^2=r^2$ is a circle. And then in Year 2: BAM! along comes the Parametric Equation of a curve! Why bother? Simply because it allows us to describe and analyse a much larger class of curves! Here’s a simple example:

CAT ON LADDER CASE 1: Tiptoes the Cat sees a ladder propped up against a wall, leaps up onto the middle rung (halfway up the ladder) and settles down for a good catnap. To his horror,

HOW TO USE THE EQUALS SIGN CORRECTLY: 6 TOP TIPS!

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Equals sign: top tips

AT GCSE:

  1. DO use = to connect equivalent expressions, but
  2. DON’T use = to connect equivalent equations.

A reminder of the difference: an expression is an arrangement of letters, numbers and other mathematical symbols: but crucially an expression must not contain an equals sign! It might contain surds, $\pi$, indices, fractions, almost anything at all except for an equals sign (or the other

WHAT IS THE “OPPOSITE” IN MATHS?

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What is the Opposite in Maths?

SUGGESTED LEVEL: interesting for everyone; very useful for A-level; essential for TMUA (Test of Maths for Uni Admission)

Oh dear – the word “opposite” is usually best avoided in maths: there is almost always a better word or phrase, and sometimes it’s just plain wrong. In everyday language we might use the word “opposite” to describe “no not that, the other thing!” but in maths it pays to be accurate: read on for how things can go wrong!

AT GCSE: Let’s start with three cases where it is actually correct to use the word “opposite”:

TOTE DOUBLE BET EXPLAINED

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Tote Double maths

I’ve just returned from an evening at the Kinson Social Club, where I was invited to take part in a TOTE DOUBLE competition. I’m not normally a betting man: because a bet usually means that on average you put in more money than you expect to win, meaning that on average you lose money. But this evening I felt obliged to take part. I dutifully paid my £3.30, having no idea of the rules, and was invited to choose up to twelve “lucky” (?) numbers. Here’s my betting card: what on earth do all these numbers mean?

WHAT’S SPECIAL ABOUT THE NUMBER 2025?

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What's special about the number 2025

or: Why is the square of the nth triangular number equal to the sum of the first n cubes?

SUGGESTED LEVEL: Further Maths A-level or anyone at any level if you love numbers!

Every New Year brings a new number for us all to wonder about: is this year a prime number? A square number? Something else?? As I write we are at the start of 2025 and the interest has been far higher than usual. So: why is 2025 such a very cool number?

SHORT ANSWER (read on if this makes no sense – I will explain!):
$2025=(1+2+3+4+5+6+7+8+9)^2=1^3+2^3+3^3+4^3+5^3+6^3+7^3+8^3+9^3$.
You can easily use a calculator to check that it’s true, but my maths brain really wants an intuitive way to see why this works – ideally using geometry (“shapes”) so I can picture the result!

TOP FIVE MATHEMATICAL SPIRALS

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Top 5 Spirals

RECOMMENDED LEVEL: A-level

5) ARCHIMEDEAN SPIRAL: the spiral you see if you roll up a carpet and look side-on. Constant separation distance between each coil. See my video of how to draw one here or use your favourite free graph sketching software (e.g Geogebra or Desmos). For A-level Further Maths students: the polar equation is $r=aθ$ where the parameter a makes the (constant) separation distance between each coil larger or smaller.

4) LOGARITHMIC SPIRAL: a spiral in which

CIRCLE THEOREMS

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Circle Theorems

IN THIS VIDEO: all eight Circle Theorems needed for GCSE maths demonstrated and explained. NOTE: students of IGCSE (“International GCSE”) also need to know the Intersecting Chords Theorem (not covered in this video).

KEY DIFFERENCES BETWEEN GCSE MATHS AND A-LEVEL MATHS?

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GCSE versus A-level maths

Thinking of taking A-level maths? Here are some key differences you’ll find from the style of GCSE maths, all designed to make our life easier not harder:

1)  EMBRACE THE ALGEBRA!

Maths is ultimately about spotting patterns, and algebra is the language we use to write down patterns. A simple example: $A=l \times w$ is a formula to tell you how big a rectangle is. ANY rectangle. At GCSE, it is reasonable to develop clever tools to avoid having to use algebra; but at A-level, we must embrace algebra: it’s our friend and is there to make maths easier! An example: