I recently enjoyed reading an article from the ATM (Association of Teachers of Mathematics) regarding the geometric representation of surds (Colin Jackson and the all-attainment group). The all attainment group were looking at how to provide a geometric representation of addition and multiplication rules such as:
(1) √2 + √3 ≠ √5
(2) √2 × √3 = √6
Of course, you could get around this on Geogebra using the following applet.
Alternatively, you can create a new triangle with lengths of √2 and √3 and use the cosine rule to find the missing length.
Then using the triangle inequality √2 + √3 ≥ 3.315. Since √9 = 3 we can conclude that
√2 + √3 > √9 and therefore √2 + √3 ≠ √5
This can also be shown by looking at the roots of certain quadratic functions in the following way.
You could then go one step further with a slightly more complex diagram and show that
√2 × √2 = √4 = 2
If anyone else has any more ideas on this I’d be fascinated to know more.