crystal clear mathematics logo


Sign up to my Newsletter


Manipulating Numbers

Image of Two Hands With Numbered Fingers Performing a MultiplicationHow do we count more rapidly?

If we have five buns and are given three more, we can add them to our total … six, seven, eight.  If the numbers are larger, however, that method becomes impractical.  For example, adding 120 in this way, even if you could add an extra two numbers per second, the exercise would take you one minute!  Larger additions would take even longer.

To overcome this problem, people rapidly learned to use calculating tools … fingers, sticks, and pebbles in order to calculate.  In fact, our word calculate comes from the Latin word calculus (meaning small pebble) because that is what citizens of the Roman Empire used at one stage for adding figures.  Later, the abacus was invented and used extensively in many parts of the world.  In time, many other calculating tools were created as well … Napier’s Bones, Genaille-Lucas Rulers, logarithms, and mechanical and electronic calculating devices (and others).

People also tried to calculate mentally or using marks on paper.  These methods were more abstract and often required learning ‘tables’ of common calculations.  For example, memorising 5 + 3 = 8 would make the exercise of adding the buns very much more rapid!

Eventually, algorithms (strategies or methods) were developed for rapid calculation using all four operations (and more).  You will learn about a variety of those (mental and paper-based) algorithms here.

Adding and Subtracting Integers (Whole Numbers)

Powers, Indices, Roots, Radicals and Surds

Our daughter Angelina was home schooled and from the start did not like maths. Over the years I have battled through, trying 5 different programs along the way. Angelina was surviving but not enjoying the journey but, when it came to algebra, the future looked dim. A friend recommended Graeme to me as her three sons had been tutored by him. She could not recommend him more highly. My daughter has just finished year 12 maths and did very well, thanks to Graeme (that would be an understatement). Graeme has a love for his subject and a genuine interest in his students. Graeme seems to meets his students where they are and tailors the lessons to meet their individual needs and interests. Graeme not only explains concepts clearly, but we all found him an interesting, knowledgeable and humble man. We are extremely happy that Graeme was recommended to us. Now our daughter is looking forward to uni with a grateful heart. We also could not recommend Graeme more highly.
Angela K (parent, 2013)

See all Testimonials

Sign up to my Newsletter

Copyright © Crystal Clear Mathematics | All Rights Reserved

Website Design: | Photography: