Information about the first use of mathematical Symbols ~ Jeff Miller's Site (*Last revision: March 28, 2014* ~ click here for his Front Page) and Mathematics Department at Tsing Hua University in Taiwan (*Last revision: July 31, 1999*).

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History of Mathematics

Information about the first use of mathematical Symbols ~ Jeff Miller's Site (*Last revision: March 28, 2014* ~ click here for his Front Page) and Mathematics Department at Tsing Hua University in Taiwan (*Last revision: July 31, 1999*).

Number**Adrian Leatherland**'s Pulchritudinous (Pleasing) Primes

Adrian Leatherland from Monash University in Australia has devoted his site to "Visualising the Distribution of Prime Numbers." His curious and stimulating webpage provides a space where number theory meets art! It appears that he was the first person to create a prime number fractal (see the 2006 article by Daniel C Doolan and Sabin Tabirca, Visualising Infinity on a Mobile Device). Visit Pulchritudinous Primes and enjoy!

Geometry## Explore the Wonders of Geometry at **GoGeometry**

## Learn and Explore Geometry and More With Geometer's Sketchpad

## Euclidean Geometry (Book I) ~ Rigorous Proofs

This is a brilliant site for those who wish to explore the wonders of geometry! Even reading the menu on the home page will be an education for most students. There are discussions of plane and solid geometry, topology, geometry and the real world, geometry and cultures, geometry and art ... and the list goes on. You will need to start with the simpler topics first (and they are not all at the top of the page, so you will have to search for them). As with the MathExchange site, you will need to search for gems here but, if you like geometry, you will find many! Just one of the many menu items (the first one, under the ads) opens to over 700 plane geometry problems, each one discussed by many people with a variety of proofs put forward. These discussions will open your eyes and horizons. Go exploring and discover some treasures at GoGeometry!

This is a wonderful and innovative program created by Nicholas Jackiw. Although it may be used to explore and illustrate a very wide variety of topics in mathematics in an interactive way, it is specifically designed for students to learn geometry. The program is extremely well supported and reasonably priced. Visit the (official) Key Curriculum website or the KCP Technologies resource website to read more, purchase the software, or find extra resources. There are many supporting videos on YouTube as well. The program also comes with program files to help deepen students' understanding of such concepts as slope, geometric transformations, and arithmetic using integers.

This is a YouTube channel owned by MathematicsOnLine. It provides rigorous proofs of many of the propositions from Euclid's First Book. The advantage is that they are all illustrated and you can pause the video to consider the steps being taken. I found this to be a very worthwhile resource. You might look at other videos on their channel as well.

Algebra## Thomas M Bösel's History of Polynomial Equations

## Florian Cajori's *History of the Conceptions of Limits and Fluxions in Great Britain* (1919)

## Samuel Jeake's *Compleat Body of Arithmetick in Four Books* (1701)

## W W Sawyer's *The Search for Pattern* (1970)

## Wolfram Alpha

If you have an interest in the History of Polynomial Equations, you will find this site to be very thorough.

This is an online facsimile copy of Florian Cajori's (1859-1930) book, *A History of the Conceptions of Limits and Fluxions in Great Britain, from Newton to Woodhouse* (1919).

This is an online facsimile copy of Samuel Jeake's *A Compleat Body of Arithmetick in Four Books* (1701)

This is an online copy of Warwick Walter Sawyer's *The Search for Pattern*.

If you wish to have equations analysed or graphed, Wolfram Alpha should be able to help you. Give it a try. It is an amazing site.

Graphs## Implicit Differentiation Exercises

## Amazing Way to Graph the Gradient Function

I found a good selection of exercises at Duane Kouba's webpage at the University of California.

Many years ago I read about a clever method for graphing gradient functions in a publication of the NSW Department of Education called *The Mathematics Teacher*. The contributer was G I Miller from Corowa (near the NSW/Victorian border) and his article is here.

Chance and Data## Christiaan Huygens' *De Ratiociniis in Ludo Aleae* (1657)

## Jacob Bernoulli's *Ars Conjectandi* (1713)

## Abraham de Moivre's *The Doctrine of Chances* (1718)

## Pierre-Simon Laplace's *Théorie Analytique des Probabilités *(1812)

This is an online copy of Christiaan Huygens' *De Ratiociniis in Ludo Aleae* (1657). This was the first published book containing a study of probability theory. Unfortunately, it is in Latin.

This is an online English translation of Jacob Bernoulli's seminal treatise on combinatorics and probability, *Ars Conjectandi* (The Art of Conjecturing). Sadly (for him), it was published eight years after his death by his nephew, Nicolaus Bernoulli, in 1713.

This is an online facsimile of Abraham de Moivre's *The Doctrine of Chances* (1718). Because he lived most of his life in England (as a Hugenot exile from France), this book was written in English. It was the first complete probability text book.

This is an online English transation of portions of Pierre-Simon Laplace's book, *Théorie Analytique des Probabilités* (1812) in which he introduced a host of new ideas and mathematical techniques. It was Laplace who first seriously applied principles of probability to scientific and practical matters. Than you to the kind folk at Xavier University in Cincinnati, Ohio who provided this translation.

Iterations (Repetitions)## The Encyclopedia of Integer Sequences

N J A Sloane and Simon Plouffe's The Encyclopedia of Integer Sequences (Academic Press, San Diego, 1995) is now online! It contains over 5,000 integer sequences, arranged for easy reference, and cites the reseach paper or literature that they came from (you can read more about the publication here). Why is it so useful? You might have been researching something fascinating and wondering if your sequence was relevant to any other field of endeavour. This book provided a resource where you could check to see if someone else had been investigating your sequence. The online link is The Encyclopedia of Integer Sequences. Try typing in any sequence you can think of, such as Fibonacci’s Sequence (1, 1, 2, 3, 5, 8, 13, …) and see what the dictionary uncovers!

Mathematical Quotations## "Quotes" for Teachers

## Quotations from W W Sawyer's Books

I value well chosen quotations. They can inspire, encourage, and stimulate thought in students as they learn.

As I stumble upon useful sites I will add them here.

I was quite impressed with this site. It provides a range of well chosen quotations for a variety of subjects (including mathematics). It also provides many quotations addressing attitudes and values associated with learning and human relationships. Rachel describes the contents of her site as "Famous Quotes to Inspire and Motivate your Students!" I think you will find "Quotes" for Teachers well worth a visit.

One of my favourite authors of mathematics books is Walter Warwick Sawyer. As I share in the section below (Useful On-Line Mathematics Books), one of his works greatly inspired me as a teenager. I have recently discovered that the Navnirmiti Learning Foundation in Pune, India, has set up a web site to honour this man who, sadly, died in 2008. One of the pages contains a small collection of quotations from his books. If you have an interest in learning and/or teaching mathematics, I am confident that you will find quotations here that will 'strike a chord' with you.

Being Creative With Mathematics## Be Amazed at Etérea Studios

## The Omnificent English Dictionary In Limerick Form

## Use Fine Arts in the Mathematics Classroom!

## Using Dance to Build Number Concepts in Virginia, USA

I serendipitously stumbled upon this site and loved it! A Spanish fine artist, *Cristóbal Vila*, has produced some truly wonderful images and an inspiring video that presents many aspects of mathematics in a warmly artistic way (the video link I provided is the YouTube one ... to see the video on his site, click here). Please follow the links and enjoy! If you are a teacher, you might consider playing his 3 minute, 40 second video to your classes.

This wonderful site has an amazing goal ... to 'write at least one limerick for each meaning of each and every word in the English language.' A number of the memorable and entertaining limericks on our quotations page were sourced from this site (with permission). Whatever your interest(s) in life, if you enjoy verbal mischief and creativity, we think you will take delight in exploring The Omnificent English Dictionary In Limerick Form. Remember to search for mathematical terms and enjoy the clever contributions.

I stumbled upon this delightful site while searching for songs that people use to remember mathematical formulae. The resources presented on this site involve dance, song, musicals and much more. Enjoy exploring some of the creative ways that people approach their mathematics!

One of our partners recommended this to us and we thought we should share it with you. Carrie Lewis and Kelly Steele seem to have had their fifth grade students' results improve noticeably by using the medium of dance in the classroom. Students are planning, sequencing, measuring time, performing numerical calculations, finding averages, graphing and interpreting results, making predictions and even doing modular arithmetic! Why not visit this PBS website to have a look?

Sites Belonging to Individual Mathematicians## Professor Keith Devlin (The Math Guy)

Professor Keith Devlin is a mathematician at Stanford University (California), an author, the Math Guy on NPR's Weekend Edition, and an avid cyclist.

- Read his blogs at Devlin's Angle.
- Access the Complete Sound Archive (and articles) from his interviews with National Public Radio at The Math Guy.
- Visit his Personal Web Site.
- Read some of his books (especially the one about Fibonacci). Get your library to purchase them.

Useful On-Line Mathematics Books and Their Authors## W W Sawyer and his "Search for Pattern"

I discovered this book when I was 15 years old. It revolutionised my understanding of mathematics and opened my eyes to so much more than that which was revealed through textbooks at school. You may obtain copies of it via Amazon, AbeBooks, or BookFinder.com. I have also discovered that you can download a free PDF copy of it from here. The PDF file is 12.4 Mb. I have much to thank Walter Warwick Sawyer for. He writes with passion and in such an engaging way that you feel that you are joining him in an adventure. He certainly inspired me as a young mathematician. W W Sawyer was an English mathematician who wrote a number of books with the intention of popularising mathematics. His first such book, "Mathematician's Delight" (1943) sold over 5,000,000 copies and is probably still the most popular/successful mathematics book ever written. To learn more of this amazing mathematician and his work, let me encourage you to visit the W W Sawyer web site set up by the Navnirmiti Learning Foundation in Pune, India.

Learning Groups of Which I Am Aware## The Math Circle ~ Near Harvard University

If you live near Harvard University, you might visit The Math Circle. Way back in 1994, disturbed by the poor quality and low level of mathematics education in the USA, Bob and Ellen Kaplan and their colleague, Tomás Guillermo, began The Math Circle. If you enjoy mathematics but would like the exciting challenge of exploring mathematical topics that are normally outside the school curriculum, then you might contact this group. Its teachers appear to be experienced, enthusiastic and committed to quality mathematical education. Their classes encourage a free discussion of ideas. The courses are mathematically rigorous and the atmosphere is friendly and relaxed. Some of their books seem very worth reading ... we intend reading some ourselves as soon as we can manage it. The idea of what they are doing is excellent! We might add that one of our quotations is from Bob Kaplan.

Interesting Mathematical Games## Circuitry — Complete Circuits to Light Up Bulbs

This (slightly addictive) little game appears on the Internet in a variety of forms but I think you will find Michael Barsotti's version is particularly well designed. Have fun playing it for free here. If you like it you may purchase a customisable version surprisingly cheaply from the same page. If you want a very challenging exercise, you might try to calculate how many different possible games there are (i.e. disregarding reflections and rotations) in an m × n grid if the power cube could have 1,2 or 3 states and cannot appear on an edge. This is not a trivial problem!

Helpful General Mathematics Resource Sites## The Math Forum

## Mathletics

## CoolMath

## MathsIsFun

## StackExchange

## WolframMathWorld

## SOSMath

## Cut the Knot

Ask Dr Rick about Elementary to College Mathematics at **The Math Forum. **This site does not have a fancy presentation but contains a considerable array of useful material. I think it is worth you while visiting The Math Forum.

Compete with other students around the world at **Mathletics. **This Australian site currently has over 3.5 million students worldwide. We have encountered quite a number of students who have benefited from participating. Indeed, competitive students can become very addicted! It encourages good drill on concepts and skills across all Australian curricula (and more) and, importantly, encourages students to build up speed as well. Visit the official Mathletics Site and have a look. The great strength of this site appears to be its emphasis on drill and speed and interactivity.

Dive into a vast array of resources at **CoolMath. **This is one of a family of websites that are full of mathematical material of a fun (non-rigorous) kind. The instruction seems to be mostly text based but there are practice activities and games and interactive opportunities provided. The sites are a bit garish but seem to be well worth exploring. We are adding this link because of the **vast array of resources** on offer. They would appear to be particularly useful for the struggling or average-to-competent student. If you are interested in developing your mathematics at a more rigorous level you will probably still enjoy exploring this site. You should, however, visit other sites and seek out other resources to help you learn to set work out and think your way through deeper and more difficult problems. So, click on the image or this text link Cool math at Coolmath.com to visit them and explore what they have to offer!

Interact with worksheets and images at **MathsIsFun**. There is a considerable amount of material from Kindergarten to Year 9 level at this site. Its great strength is that it has interactive diagrams (flash files) and interactive worksheets/questionnaires. Between them you can discover and learn about mathematical concepts and test your knowledge and understanding of them. There is even a forum where some quite sophisticated mathematics is discussed! This would be good for you to explore all kinds of new mathematics as well as new ways of understanding what you already 'know.' They have a useful MathIsFun Links Page, too. Have fun exploring MathsIsFun!

See people discussing mathematics at **StackExchange. **This is a wonderful site that is very much like an open forum with people asking, discussing and answering a huge variety of mathematical questions. Some questions are posed by school students (they seem to be students studying more advanced levels of senior mathematics). The vast majority are from persons working at university level and beyond. You will find this site stimulating if you have a passion for learning about mathematics and are prepared to search through a lot of things that you don't understand to find some really spectacular little jewels! There is even the occasional discussion that a good Year 8 student would understand and find inspiring, but even a quite advanced Year 11 or 12 student might only understand perhaps one article in forty or fifty. Use the menu on the right to search topics that interest you. Once you have found one interesting article, you may find many more in the 'Related Tags' list that replaces the topics menu. These discussions will open your eyes and horizons. Go exploring and discover some treasures at MathExchange!

Expand your mind in a rigorous way at **WolframMathWorld. **WolframMathWorld bills itself as "the web's most extensive mathematics resource," and it is certainly extensive! It is also for the better students only, as the material is dealt with is a very thorough way. Although the topics range from addition and subtraction (look under Number Theory) right through school curricula to university level and recreational mathematics, the discussions of even the simplest topics are thorough, rigorous and extensive. Visit WolframMathWorld if you have time to spare, want to go exploring, and do not mind a challenge.

For a list of useful mathematics websites make **SOSMath** one of your first stops. The header of their home page declares that you have arrived at "S.O.S. Mathematics - Sites of Interest on the World Wide Web." The links that have been collected at this site are sorted into helpful topics and the people who have compiled the list have taken care in selecting sites of value. We have certainly found many of them well worth exploring. There are about 100 links listed so, even if you chose to explore one site per week it would still take you two years to work your way through what SOSMath has to offer.

Why not **Cut the Knot**? You should be fascinated by this site (Cut the Knot). It contains an amazing variety of mathematical resources that should intrigue and stimulate the serious high school student. I stumbled upon this site while researching Pythagoras' Theorem and the work of Elisha Scott Loomis in particular. Their page discussing Pythagoras' Theorem is quite comprehensive. You will be all the richer for having browsed and explored it!

Graeme Henderson tutored me throughout my entire high school mathematics education, and through his expert tutelage and exceptional ability to explain the concepts that would otherwise be very hard to understand, I was able to grasp and excel at topics that I would have otherwise been unable to. He was an excellent tutor, and I would not hesitate to recommend him to anyone looking for help with mathematics problems!

Matthew T (student, c2005-2013)

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