**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 positions of imaginary solutions of equations.__

The teacher of 47 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.

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**

In April 2012, Philip was the KEYNOTE SPEAKER at a Mathematics Symposium for "Innovations in Mathematics Education" held at GALWAY University, Ireland.

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

http://tvnz.co.nz/breakfast-news/teacher-makes-maths-changing-discovery-8-38-video-3705404

*"Quite exceptional and exciting! It is a rare thing to see such a new idea in maths education!"*In April 2012, Philip was the KEYNOTE SPEAKER at a Mathematics Symposium for "Innovations in Mathematics Education" held at GALWAY University, Ireland.

------------------------------------------------------------------------------------------------------------

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

**or**

http://www.youtube.com/watch?v=ctZ6gICQ4Pg

http://www.youtube.com/watch?v=ctZ6gICQ4Pg

-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

**Philip is available to give presentations for universities/colleges/teachers and students anywhere in the world.**

**You may contact Philip at**

**philiplloyd1@gmail.com**

**The following 6 pages are a brief summary of PHANTOM GRAPHS produced in 2015 for people who would like an overview of the idea without too much complex mathematical theory.**

**.**

The following is the original presentation paper (

The pictures are actually

The following is the original presentation paper (

*before I had even heard of the excellent Autograph program*) given at the International Mathematics Conference in South Africa in 2011 also at the New Zealand Mathematics Conference and at Auckland university.The pictures are actually

__photographs__of the Perspex models I made.

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

http://tvnz.co.nz/breakfast-news/teacher-makes-maths-changing-discovery-8-38-video-3705404

or

http://www.youtube.com/watch?v=ctZ6gICQ4Pg

__Exciting New Development! (2012)__

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”.

(Try not to use CHROME because they are making it very hard to use this "plugin" facility.)

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:

__http://autograph-maths.com/activities/philiplloyd/phantom.html__

The items listed below each have a very brief description and may be accessed individually by clicking on the following equations:

(You will be able to ROTATE the 3D graphs to get a better appreciation of the concept involved.)

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.

_________________________________________________________________________________________________

ALSO see further resources on Autograph resources website

** ****www.tsm-resources.com/autograph**

**_________________________________________________________________________________________________**

**NEWSFLASH!!! (2013)**

**I recently found some very surprising phantoms which occur when a curve does not have any turning points (eg y = x^3 + x).**

The phantoms produced are not joined to the basic graph.

This was a totally unexpected development.

I investigated the idea using the graph of y = x^3 + axThe phantoms produced are not joined to the basic graph.

This was a totally unexpected development.

I investigated the idea using the graph of y = x^3 + ax

**and****I varied the value of****a.**

*When a is zero, we get the basic curve y = x^3.*

When ais negative, we get the usual cubics with 1max and 1 min and 2 phantoms areWhen a

__joined__to the curve at each max/min point.*When**a is positive, the phantoms become detached from the the curve.*

I have explaned it all on the following SCREENCAST VIDEO.

Just click on the link below.

I have explaned it all on the following SCREENCAST VIDEO.

Just click on the link below.

**http://screencast.com/t/1ujgXFbDTu****(It MAY take a few minutes to load. Please be patient, it is worth it!)**

__________________________________________________________________________________________________________________________________

__POWERPOINT PRESENTATIONS USED IN LECTURES__**If you would like to see the PHANTOM GRAPH theory in detail on PowerPoint, I have attached it in TWO PARTS.**

To see PART 1 click

**HERE**.

When downloaded the file will be at the bottom left, click on it then press F5 to play.

To see PART 2 click

**HERE**.

When downloaded the file will be at the bottom left, click on it then press F5 to play.

__ANOTHER NEWSFLASH!!! (2015)__**I have been thinking about graphs such as**

*y = x^(n)*,**where n = 0, 1, 2, 3, 4... and comparing them with graphs**

**where n takes values**

__between__the whole numbers such as

*y = x^(2.1), y = x^(3.5), y = x^(4.25)*and wondering why the left hand half of these graphs disappears when the power is not a whole number.**Click on this link for a screencast video explaining the concept:**

**http://screencast.com/t/yKBLECJdukz**

**Click HERE to see the amazing PowerPoint!**

**Click HERE to see the theory of the discovery.**

**After further research, I have extended the theory to include graphs of the form**

*y = x^(-n)*.**Click on this video:**http://screencast.com/t/TiNZ813k

**! (2015)**

__YET ANOTHER FIRST__I have always wondered what happens to the graph of

*y = x^x*when

*x*is negative. Well I just made a breakthrough!

*I think you can only see the following graph if you have the AUTOGRAPH program on your computer.*

Click

**HERE**

I have also explained how I found the graph on the following Screencast video.

Just click on

**http://screencast.com/t/m4fmGwmkrT9**

If you do not have access to AUTOGRAPH, you can see a simple word document showing different views of the surprising spiral curve for x < 0. Click

**HERE**

**THE GRAPH OF***y = (-1)^x*(2016)If we just choose INTEGER values of

*x*we just get the

*y values +1 and -1*eg

**(0, 1), (1,**

*–*1), (2, 1), (3,*–*1) ... and (*–*1,*–*1), (*–*2, 1), (*–*3,*–*1)...However, if we choose

*x = 0.1**we get*

*y = 0.95 + 0.31i*

*x = 0.6 we get y = –0.31 + 0.95i*

*x = 1.2 we get y = –0.81 – 0.59i*These points have a

__and an__

**real part**__.__

**imaginary part**In order to make sense of this, we need to be able to plot these

**so we need**

*complex y values***another axis**besides the

**.**

*normal x and y axes*I will use only

**on the**

*real x values**x axis*and in order to plot points such as

**I will put the**

*y = 0.95 + 0.31i***real part**(0.95) on the normal

*y axis*and the

**imaginary part**(0.31

*i*) on the

**.**

*z axis*The DOMAIN of this graph is

**all the real numbers**(ie on the real

*x*axis).

But instead of a simple

**we now have a**

*y AXIS***complex**

*y PLANE*.**I plotted several POINTS in this way eg**

*(0.5, 0 + i) and ( 0.8, – .81 + .59i)*

*and produced this beautiful helix.*

**y = (-1)^x**

**THE GRAPH OF***y = (-2)^x*(2016)If we just choose

**INTEGER**values of

*x*we get the following points:

eg

**(0, 1), (1,**

*–*2), (2, 4), (3,*–*8), (4, –16) ... and (*–*1,*–*½ ), (*–*2, ¼ ), (*–*3,*–***⅛**

**)...**

However, the graph does not just exist as a set of these isolated points.

However, the graph does not just exist as a set of these isolated points.

If we choose

*x = 0.25**we get*

*y = 0.84+ 0.84i*

*x = 0.5 we get y = 0 + 1.41i*

*x = 0.75 we get y = –1.19 + 1.19i*

*x = 1.25**we get*

*y = –1.68 – 1.68i*

*x = 1.5 we get y = 0 – 2.83i*

*x = 1.75 we get y = 2.38 – 1.38i*

*x = –0.25 then y = 0.59 – 0.59i*

*x = –0.5 then y = 0 – 0.71i*

*x = –0.75 then y = –0.59 – 0.59i*

*x = –1.25 then y = –0.3 + 0.3i*

*x = –1.5 then y = 0 + 0.35i*

*x = –1.75 then y = 0.2 + 0,2i*

*etc*

These points have a

**REAL PART**

and an

**IMAGINARY PART**.

In order to make sense of this, we need to be able to plot these

**so we need**

*complex y values***another axis**besides the

**.**

*normal x and y axes*I will use only

**on the**

*REAL x VALUES**x axis*and in order to plot points such as

**I will put the**

*y = 0.2 + 0,2i***real part**on the normal

*y axis*and the

**imaginary part**on the

**, using**

*z axis (imaginary y axis)***Autograph**.

The DOMAIN of this graph is all the real numbers (ie on the real

*x*axis).

But instead of a simple

**we now have a**

*y AXIS*

*complex*

*y PLANE*.**I plotted the POINTS listed above and several more**

*and produced this spiral.*I then found the

**equation**of the curve and the shape becomes clearer.

Here is another version without the extra “points”

*y = ( –2)^x*An interesting variation is to change the base of the equation to

*y = ( –1.25)^x*

____

__2016__: In response to questions on various internet sites about how to visualize where the complex roots of cubic functions are, I decided to produce the following explicit explanations.

**This screencast video will give you a visual demonstration of the theory given below.**

http://screencast.com/t/dkAYxFDwH

SEE VIDEO on INTERSECTION POINTS of CIRCLES with PARABOLAS

http://screencast.com/t/3bdbGbI1u

Also to download a copy of the following click HERE

http://screencast.com/t/3bdbGbI1u

Also to download a copy of the following click HERE

__:__

**2017**

**A fascinating extention of the logarithmic graph.**

I wrote this specially because many people requested the explicit relationship between my

__2017:__

Problems with the “Fundamental Theorem of Algebra”.Problems with the “Fundamental Theorem of Algebra”.

I wrote this specially because many people requested the explicit relationship between my

*phantom graphs*and the fundamental theorem of algebra. CLICK**HERE**

**CONTACT PHILIP LLOYD (Specialist Calculus Teacher) by email:**

philiplloyd1@gmail.com

IF YOU DECIDE TO USE ANY OF MY RESOURCES I WOULD BE PLEASED IF YOU COULD SEND ME AN EMAIL.

I would appreciate any feedback.

See web sites:

http://www.linkedin.com/pub/philip-lloyd/2a/787/7a0

http://www.phantomgraphs.weebly.com

http://www.intersectingplanes.weebly.com

http://trigometer.weebly.com

http://mathematicalgems.weebly.com

http://knowingisnotunderstanding.weebly.com

http://calculusresources.weebly.com

http://algebra-and-calculus-resources-year12.weebly.com

http://liveperformances.weebly.com/

http://www.motivational-and-inspirational-sayings.weebly.com

http://motivation-and-self-esteem-cycles.weebly.com

philiplloyd1@gmail.com

IF YOU DECIDE TO USE ANY OF MY RESOURCES I WOULD BE PLEASED IF YOU COULD SEND ME AN EMAIL.

I would appreciate any feedback.

See web sites:

http://www.linkedin.com/pub/philip-lloyd/2a/787/7a0

http://www.phantomgraphs.weebly.com

http://www.intersectingplanes.weebly.com

http://trigometer.weebly.com

http://mathematicalgems.weebly.com

http://knowingisnotunderstanding.weebly.com

http://calculusresources.weebly.com

http://algebra-and-calculus-resources-year12.weebly.com

http://liveperformances.weebly.com/

http://www.motivational-and-inspirational-sayings.weebly.com

http://motivation-and-self-esteem-cycles.weebly.com

**https://www.quora.com/profile/Philip-Lloyd-2**

Enthusiasm is the key to success in every activity!

Enthusiasm is the key to success in every activity!