Events/News > Philip Lloyd Mathematics Teacher

Mathematics teacher Philip Lloyd with a model of his “Phantom Parabolas”

Mathematics teacher Philip Lloyd with a model of his “Phantom Parabolas” showing the real position of imaginary solutions of equations.

Very few people wake excitedly every Sunday at 3am thinking about calculus! But that is what happened to Epsom Girls Grammar teacher Philip Lloyd, who has come up with a new way of showing the real position imaginary solutions of equations.

The teacher of 44 years is now receiving international praise for his concept.

"It came to me at 3am on a Sunday morning," Mr Lloyd said.

"In simple terms, the solutions of an equation are where its graph crosses the x axis. Some graphs do not cross the x axis but we still say they have solutions which people call ‘imaginary’."

It was this "imaginary" concept which many students struggle to accept. Because they can't see it, many tended to find it difficult to believe.

"I found that graphs have ‘extra bits’ on them which I call ‘phantom graphs’ and these actually do cross the x axis. In fact, I found that the imaginary solutions are at real places."

"I'd think of one type of graph one week and the next week something else would pop up. This continued for weeks. It was a very exciting time!”

"I would get up in the morning and I'd start making these models."

After spending most of his school holidays on the perspex models, Mr Lloyd demonstrated the new concept to his students.

Suddenly they could see what he was talking about and they "absolutely loved it".

Mr Lloyd has been using his models for several years now with great success.

Not only do the students love it, but his concept is gaining momentum in mathematical circles, so much so he was invited to share his idea at an international mathematics conference in South Africa in 2011 where a prominent mathematician from Cambridge University described Phantom Graphs as the "Highlight of the conference". Philip has already given several presentations at universities in New Zealand.

A letter from the head of the conference says Mr Lloyd's paper on the concept is "Quite exceptional and exciting! It is a rare thing to see such a new idea in maths education!"

Keynote Speaker at Galway University

In April this year, well known Mathematics teacher, Philip Lloyd who has taught at Epsom Girls Grammar School for the past 10 years, was invited to go to Ireland where he was the KEYNOTE speaker at a Mathematics Symposium at Galway University.

In 2011, Philip presented his discovery called "Phantom Graphs" at an international mathematics conference at Rhodes University in South Africa. As a result of this, the University of Galway invited Philip to be the Keynote speaker at their symposium on "Innovations in Mathematical Education".

Philip's address started with his philosophy of teaching mathematics using REASONS and not just RULES. He emphasized the importance of "Enthusiasm", which is the key to success in every activity, and on ways to motivate students.

Philip showed several innovative models and ideas he had developed over the past 45 years and of course concluded with an account of his special discovery called "Phantom Graphs" which shows the actual positions of imaginary solutions of equations.

The audience consisted of mathematics lecturers, mathematical education lecturers, school inspectors, school principals, teachers of mathematics from all over Ireland and in particular, a group of recent graduates from a new four year course in Mathematical Education at the University of Galway.

The presentation was enthusiastically and warmly received and many requests were made for copies of his presentation material and resources. Philip commented, "It was a huge honour to be asked to be their keynote speaker. This hopefully brought much kudos to Epsom Girls Grammar and it was very good testament to all the innovative work being done in New Zealand's mathematics education".

Details of Philip's discovery can be found at

Click here to see an entertaining TV interview on Phantom Graphs.

Please click here for more information


Exciting New Development!


I have recently reproduced all my "PHANTOM GRAPHS" on the excellent AUTOGRAPH system. This involved a new technique of finding the ACTUAL EQUATIONS of the phantom graphs. I would like to acknowledge the encouragement given to me by Douglas Butler (Director, iCT Training Centre, Oundle) and in particular the considerable enthusiastic expertise of Simon Woodhead (Development Director, Autograph , Eastmond Publishing Ltd.) in helping with technical problems and producing the following links to the website:

“Autograph Activities”.

Just click on any of the items below.

Press START when the word appears.

If the graph does not appear you will be asked to download “autograph player”.

You will only need to do this once then you may view all the items at any time.

The Autograph Activities are available at the following website:

The items listed below each have a very brief description and may be accessed individually as follows:

1. y = x²

This shows the basic parabola and its PHANTOM.

2. y = (x - 1)² + 1

This shows a quadratic which we normally would say does not cross the x axis but we now see that the PHANTOM does cross the x plane.

3. 3 parabolas

This shows the 3 positions a parabola can have on the coordinate plane and the relative positions of the PHANTOMS.

4. y = x^4

This shows the basic graph with its 3 PHANTOMS.

5. y = x³

This shows the basic graph with its 2 PHANTOMS.

6. y = (x - 1)²(x + 1)²

This shows a typical quartic curve with 3 PHANTOMS.

7. y = x(x - 3)²

This is a typical cubic with its 2 PHANTOMS.

8. y² = x² + 25

This hyperbola has a phantom circle joining its two halves.

9. y² = x(x - 3)²

I call this the alpha graph with its 2 PHANTOMS.

10. y = x²/(x - 1)

The 2 halves of this curve are joined by a phantom ellipse.

11. Asymptotic plane

The rational function approaches the plane y = 2 and the PHANTOM also approaches the same plane

12. Curve with 2 vertical asymptotes

This has a similar sort of equation to number 11 but the surprising phantom looks like a sort of bent pear shape.

13. y = cos(x)

The usual cosine curve lies between 1 and – 1 but the phantoms show cos(x) can have any real value.

14. y = exp(x)

This is the most unusual graph because it is the only one in which the phantoms are not joined to the curve.

15. Hyperbola

This typical hyperbola has an ellipse joining its 2 halves.

16. Solutions of cubic crossing 3 planes

This shows that any horizontal plane will cross a cubic 3 times.

17. y = x³/(x² - 1)

This shows that any equation of the form x^3/(x^2 - 1) = c will result in a cubic equation which must have 3 solutions. Notice that this curve will now cross any horizontal plane y = c exactly 3 times.

18. y = x^4/(x² - 1)

This shows that any equation of the form x^4/(x^2 - 1) = c will result in a quartic equation which must have 4 solutions.

Notice that this curve will now cross any horizontal plane y = c exactly 4 times.

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