[b]Proportional Co-Variation extends from the naturals to the reals.[/b]
Multiplication varies a number in the same ratio 1 varies to the multiplier. Line segments are proportional to the real numbers they represent. So this new applet entails a new approach to visualizing multiplication. The initial example of PCV for the worksheet is [b]-ℯ × -π[/b] which cannot be modeled by repeated addition or repeated subtraction. More information about PCV as a meta model for multiplication (and division) is at [url]http://bit.ly/PCVMULTIPLY[/url] and [url]http://bit.ly/PCVHISTORY[/url]
Best wishes
Jonathan Crabtree
Mathematics Researcher: Melbourne Australia
[url]www.jonathancrabtree.com/mathematics[/url]
[url]www.linkedin.com/in/jonathancrabtree[/url]
P.S. Professional or university level mathematicians may also enjoy a paper A Geometric Approach to Defining Multiplication written by Peter F. McLoughlin and Maria Droujkova which can be downloaded from http://arxiv.org/abs/1301.6602
