By Andrew Freda, posted August 17, 2015 —

As long as algebra and geometry have been separated, their progress have been slow and their uses limited, but when these two sciences have been united, they have lent each mutual forces, and have marched together towards perfection. —Joseph Louis Lagrange (1736–1813)

I find that many students, parents, and even colleagues see Geometry as a “year off” from math or certainly a year where algebraic skills will rust and fade. I urge all teachers to fight these myths! My vision of the complete math student is one who is strong whether working with numbers, shapes, or algebraic expressions. In addition, the key to retaining understanding of mathematical ideas, rather than being part of a “see, nod, and forget” experience, is to relate what we learn to what we have learned before as well as to the world around us.

Studying ratios is an integral part of any Geometry course, and the Golden Ratio is one of the most famous. Should we ask students to memorize that Φ is approximately 1.618, or exactly

If this is all we ask, students will see, nod, and then forget. Should we ask them to look at a diagram of a golden rectangle and memorize that the ratio of the adjacent sides equals Φ? If this is all we ask, students will see, nod, and then forget. Let us instead examine the Golden Ratio by using geometry and algebra, and we will see that because students are asked to treat this ratio as a mathematical idea, they are more likely to retain an understanding of what Φ is. The following approach is inspired by the wonderful book The Geometry of Art and Life (1946), by Matila Ghyka, which contains some of the most elegant algebra that I have seen.

Any proportion may be expressed as a/b = c/d, and this requires four values. A simpler proportion (using fewer values) is a/b = b/c, which requires three values (and involves the geometric mean b). The simplest proportion is a/b = b/(a + b), which has only two values. 

At this point, we can ask our students what the values are for a and b. Can there be more than one answer? Must the ratio a:b be fixed? We turn to algebra:

Next, we can look at a simple geometric construction of Φ. We have square ABCD with midpoint E and a circle with a radius equal to segment BE. 

We need to prove that AGFD is a golden rectangle and that DF:AD is the golden ratio, so our first approach is to let each side of the square ABCD equal 1 unit. Then, BC = 1, EC = 1/2, and (BE)² = 5/4, which leads to 

We then want to prove that the construction holds no matter what side length of square ABCD is used. 

We all know that such things as theorems involving parallelograms can be proved using Euclidean or analytic geometry, but we should show our students that every important topic of geometry can be examined in multiple ways, including an algebraic approach. Analytic or coordinate geometry complements and reinforces understanding of Euclidean geometry and helps our students see that geometry is not an island; it is another way of looking at mathematical ideas.

ANDREW FREDA, afreda@deerfield.edu, teaches at Deerfield Academy in Deerfield, Massachusetts. He has spoken and written on a variety topics, including the importance of visual representations in algebra, teaching Geometry through stand-alone “units,” ways to understand probability through the use of a dice game, the use of fractals in the high school curriculum, and the role of CAS in the mathematics classroom.