**Mathematical Gems from Philip Lloyd.**

**Philip Lloyd has been a very successful mathematics teacher in New Zealand since 1967**.

Philip's personal guiding principle or philosophy:

Mathematics must not be taught as “a set of rules to follow”.

It needs be taught with

**To teach Mathematics using**__REASONS__not__RULES__.Mathematics must not be taught as “a set of rules to follow”.

It needs be taught with

**real understanding**.There is a big difference between:

“

For example, most students “

but few “

Click on: http://screencast.com/t/QYWQ9O8DFz

“

**” a thing and “**__Knowing__**” it.**__Understanding__For example, most students “

**know**” that 2 to the power 0 = 1but few “

**understand**” WHY it is 1Click on: http://screencast.com/t/QYWQ9O8DFz

To INSPIRE students the secret is......

A teacher needs to be enthusiastic all the time!

**ENTHUSIASM!**A teacher needs to be enthusiastic all the time!

**ENTHUSIASM**IS CONTAGIOUS!**ENTHUSIASM**CREATES ENERGY!**ENTHUSIASM**IS THE KEY TO SUCCESS IN EVERY ACTIVITY!The following links are a collection of special gems which illuminate real understanding of concepts which are too often glossed over in most texts.

**1.**

**Understanding Differentiation of Powers of***x*.Click HERE for animated PowerPoint which will delight students.

**2.**

**The Derivative of***e*to the power of*x*.Click on: http://screencast.com/t/QdQwHssGqa

Also click on:http://screencast.com/t/6rJEqdOsqp

**4.**

__The RELATIVE SIZES and POSITIONS of EARTH, MOON and SUN.__Click HERE for a simple way of comparing HUGE distances. How many people realise that the distance from the earth to the moon is only about a third of the

**distance across**the sun?

**5.**

__Drawing graphs of gradient functions and drawing functions from____their gradient graphs.__Click HERE for a clear and comprehensive treatment of this quite complex topic. The booklet contains student practice sheets and answer sheets.

**6.**

**The vanishing points of inflection.***The strange*

**function in this section has a second derivative of zero at**

x = 1 and seems to be quite normal, but it does not have a point of inflection! The full story is far more complicated and fascinating.

Click on http://screencast.com/t/vK9HMkewE

Click HERE for a printable version.

**7.**

**Those essential SPECIAL TRIANGLES.**To really grasp these useful ideas, students need to know how they are formed.

Click on http://screencast.com/t/iXuA4jCACUu

Click HERE for a printable version.

**8.**

**Centres of a Triangle.**The centroid, circumcentre and orthocentre are closely related.

Click on http://screencast.com/t/wd6Cc392xdCb

to see a moving demonstration of how these centres can vary and merge.

**9.**

**Clock-face Polygons.**For a demonstration of this instructional investigation for young students of geometry. Click on http://screencast.com/t/jVhzw7X6wY3

Click HERE for a printable version with worksheets for student investigation.

Click HERE for blank clock faces.

Click HERE for answer sheet.

**10a**

**THE FAMOUS "SPIDER AND FLY PROBLEM".**This would be one of my all-time favourite lessons.

Click HERE for a printable version for teaching notes.

**11.**

__LLOYD’S FORMULA for the area of any triangle with sides__*a, b*,*c*.Click HERE for a printable version of this original idea.

**12.**

**A “***tongue in cheek*” article on the misuse of graphics calculators.Click HERE for a printable version of this article.

**13.**

**“CAN’T GET NO SATISFACTION”.**Click HERE for a printable version of this article about doing things the “

**thinking way**” verses the “

**non-thinking way**”.

**14.**

__WHY BOTHER WITH RADIANS IN YEAR 12 MATHEMATICS?__Click HERE for a printable version of this thought provoking article.

**15.**

__INTRIGUING CONTINUED FRACTIONS.__Click HERE to see how a sequence of continued fractions can apparently produce irrational and imaginary numbers!

**16.**

__SUPERB INVESTIGATION SUITABLE AT MANY LEVELS.__Click HERE for teacher’s notes on an investigation that could be started

even with juniors but ends up as a senior calculus problem.

**17.**

__STATIONARY SATELLITES__. Did you know that communications satellites circle the earth at a distance of about 36,000 kilometres above the equator AND they stay above particular points on the equator?**Click HERE for a poster giving the reasoning and information.**

18.

Find how high you can throw a ball just by timing it;

Find how far away the horizon is at different heights;

Find how fast you would drive round a corner before the car rolls;

Find what speed would make you airborne going over a hump-back bridge;

Find how far motor cycles can jump off ramps; and more!

Click

__SOME FORMULAS THAT ACTUALLY DO REAL STUFF!__Find how high you can throw a ball just by timing it;

Find how far away the horizon is at different heights;

Find how fast you would drive round a corner before the car rolls;

Find what speed would make you airborne going over a hump-back bridge;

Find how far motor cycles can jump off ramps; and more!

Click

**HERE****19.**

__ESSENTIAL THEORY OF DIFFERENTIATION__in a form easily understood by students.**A. Click HERE for "Introduction to differentiation".**

**B. Click HERE for "Continuity and Differentiability".**

**C. Click HERE for "Introduction to the Chain Rule".**

**D. Click HERE for "Differentiation of Products".**

**E. Click HERE for "Differentiation of Quotients".**

**F. Click HERE for "Differentiating 2 to the power**

*x".***G. Click HERE for the "Reason Why we like e to the power**

*x*".**H. Click HERE for "Differentiating ln(**

*x*)".**I . Click HERE for "Differentiating Trigonometric Functions".**

**J. Click HERE for "Summary of Differentiation Techniques."**

**20**

**.**

__ESSENTIAL THEORY OF INTEGRATION__in a form easily understood by students.**A. Click HERE for The Fundamental Theorem of Calculus.**

**Click HERE for a PowerPoint on The Fundamental Theorem.**

**B. Click HERE for an interesting investigation on areas under curves.**

**C. Click HERE for Areas Between Curves.**

**D. Click HERE for How Integrals Requiring Substitutions Work.**

**E. Click HERE for "Newton's Law of Cooling" simplified.**

**F. Click HERE for "Forensic Science" application.**

**G. Click HERE for "Human Growth Hormone" application.**

**H. Click HERE for "Cannabis Use" application.**

**I. Click HERE for "How long a medicine is effective".**

**J. Click HERE for "Legal driving with alcohol in the blood".**

**K. Click HERE for "Radioactive Decay" application.**

**21.**

__ESSENTIAL THEORY of REAL and COMPLEX NUMBERS__in a form easily understood by students.**A. Click HERE for Understanding Logarithms.**

**B. Click HERE for Quadratic Theory in brief.**

**C. Click HERE for "Teach yourself roots of quadratics".**

**D. Click HERE for "Do You Believe In**

*i*? ".**E. Click HERE for**

**Roots of Quadratics and Coefficients of the equation.**

**F. Click HERE for**

**Comparing Fractions, Surds, Complex numbers.**

**G. Click HERE for Rectangular and Polar forms of Complex Numbers.**

**H. Click HERE for De Moivre's Discovery.**

**I(i). Click HERE for Solving Complex Equations.**

**(ii) Click HERE for PowerPoint on Solving Complex Equations.**

**J. Click HERE for Real and Complex Parts.**

**K. Click**HERE

**for**"Have you ever seen this point addressed in any text?"

22.

Did YOU know that the cube root of (-8) does not equal -2?

Also view the VIDEO by clicking http://screencast.com/t/iBwFfvzM8

23. From FLATLAND to the 4th DIMENSION.

See video:

Click

24. Finding solutions to y^x =x^y

Click

Click

**Click****HERE****for PRIMARY VALUES OF INDICES.**Did YOU know that the cube root of (-8) does not equal -2?

Also view the VIDEO by clicking http://screencast.com/t/iBwFfvzM8

23. From FLATLAND to the 4th DIMENSION.

See video:

**https://www.screencast.com/t/cQOsIqbk**Click

**HERE**to see the document.24. Finding solutions to y^x =x^y

Click

**HERE**for the theory.Click

**HERE**for a new set of solutions found by Marcelo Arruda from Brazil.

25. Special Angles between the Hour Hand and the

Minute Hand of a clock.

Click HERE

26. Can passenger jets fly backwards?

Click HERE

27.

28

Minute Hand of a clock.

Click HERE

26. Can passenger jets fly backwards?

Click HERE

27.

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

Click**HERE**28

**.****Converting between Fahrenheit and Celsius temperature scales Mentally.**

Click HEREClick HERE

__CONTACT PHILIP LLOYD__(Specialist Calculus Teacher) by email IF YOU DECIDE TO USE ANY OF MY RESOURCES I WOULD BE PLEASED IF YOU COULD SEND ME AN EMAIL.

philiplloyd1@gmail.com

**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**

Enthusiasm is the key to success in every activity!