**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 position 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.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.

**10**

**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.

**13.**

__LLOYD’S FORMULA____for the area of any triangle with sides__*a, b*,*c*.Click

**HERE**for a printable version of this original idea.

**14.**

**A “***tongue in cheek*” article on the misuse of graphics

**calculators.**Click

**HERE**for a printable version of this article.

**15.**

**“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**”.

**16.**

**WHY BOTHER WITH RADIANS IN YEAR 12 MATHEMATICS?**Click

**HERE**for a printable version of this thought provoking

article.

**17.**

**INTRIGUING CONTINUED FRACTIONS.**Click

**HERE**to see how a sequence of continued fractions can apparently produce irrational and imaginary numbers!

**18.**

**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.

**19. 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.**GEO-STATIONARY SATELLITES Video

**https://www.screencast.com/t/BIryx24f**

**20**.

__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**

**21**.

**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

**22**.

__From FLATLAND to the 4th DIMENSION__.

See video:

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

Click

**HERE**to see the document.

**23**.

__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.

**24**. Special Angles between the Hour Hand and the

Minute Hand of a clock.

Click

**HERE**

**25**. Can passenger jets fly backwards?

Click

**HERE**

**26**. The GOLDEN RECTANGLE.

Click

**HERE**

**27**.

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

Click

**HERE**

**28**

**.**

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

Click HERE

Click HERE

**29**. Teaching "Order of Operations" logically.

click HERE

**30**. Teaching DIVISION of FRACTIONS.

click HERE

Here is a short video covering the above ideas…

**http://screencast.com/t/XlESJ3yEqfzJ**

**31**. The NEWTON-RAPHSON Method.

click HERE

**32**. Does √(a×b) always equal √(a) ×√(b) ?

click HERE

**33**. The graph of y = x^x

click HERE

**34**. Why is the cube root of -1 not equal to -1?

click HERE

**35**. Implicit Differentiation explained.

click HERE

**36**. What do I need to know before the BINOMIAL THEOREM?

click HERE

**37**. How can sin(x) equal -2?

click HERE

**38**. How can e^x be negative?

click HERE

**39**. How can there be 4 intersections for the graphs y = x^2 and x^2 + y^2 = 1? click HERE

**40.**Why is it that when f''(x) = o this represents a point of inflection on the curve y=f(x) click HERE

**41. Trisecting an angle!**

click HERE

click HERE

**42. The amazing graph of y = (-1)^x**

**click HERE**

**43. Why can't we differentiate a power of x and get 1/x ?**

click HERE

44. Why is sin(90) equal to 1 ?

click HERE

click HERE

44. Why is sin(90) equal to 1 ?

click HERE

**45.**

**A simple intersection problem that turned out to be not so simple!**

**click HERE**

**46.**

**What is an inverse function?**

**click HERE**

**47.**

**How can there be TWO angles of projection which will result in the same range for a projectile?**

**click HERE**

**48.**

**What’s the point of radians? Why don’t we just write everything in degrees?**

**click HERE**

**49.**

**Why isn’t math taught in a way that if you know how to do the problems you will get it right even if you make a simple mistake?**

**click**

**HERE**

**50.**

**Why can’t we fold a piece of paper more than eight times?**

**click**

**HERE**

**51. Intersecting Circles with an astonishing conclusion!**

**click HERE**

**52. Some probability problems using tree diagrams and Venn**

diagrams.

diagrams.

**(a) If the probability that a problem will be solved by three people is 1/2, 1/3 and 1/6, then what**

is the probability that the problem will be solved? CLICK HERE

(b) A bag contains 5 white and 3 black balls, and two balls are drawn at random. What is the

probability that both are different colours? CLICK HERE

is the probability that the problem will be solved? CLICK HERE

(b) A bag contains 5 white and 3 black balls, and two balls are drawn at random. What is the

probability that both are different colours? CLICK HERE

**(c) If n(A) =22, n(B) =32 and n(AUB) =40, what is the value of n(A∩B) CLICK HERE**

**(d)**

**If P(A) =0.65 and P(B) =0.35 and P(A**

**∩**

**B) =0.1 what is the P(B|A) CLICK HERE**

**(e)**What is the probability of picking a red marble and heads simultaneously from a basket which

has 5 blue marbles.

**CLICK HERE**

**53.**

__THE WORLD’S HARDEST EASY GEOMETRY QUESTION.__ CLICK HERETO BE CONTINUED!!!

__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!