Where do topics pertaining to the History of Mathematics fit in our mathematics frameworks? The Fibonacci sequence is a concept rich in problem solving potential and real world application, that too few teachers are aware of. This post will offer some practice and content connections for the Fibonacci sequence, and will include directions for one of my favorite designs, the Fibonacci Quilt.

I recorded a background of the Fibonacci Sequence, which I show students.

When we provide “low threshold, high ceiling” problems for students to solve, they learn to “make sense of problems and persevere in solving them.” (Math Practice 1). Besides the rabbit problem, there is an interesting science application that produces the Fibonacci numbers. It is found in the family tree of a male bee. Turns out, a female bee has two parents: a male and a female, but a male bee has only one parent: a female. When you trace the genealogy of a male bee backwards, the Fibonacci numbers will be revealed.

Ratios and proportions are topics that are foundational to algebra. Beginning in grade 6, students connect ratio and rate to multiplication and division. In grade 7, students solve multi step problems involving ratios, rate, proportion and percent. Grade 7 students recognize proportional relationships from context, equations, tables and graphs. In grade 8, students deepen their knowledge of proportional relationships and explore functions. In high school mathematics, students solve problems involving trigonometric ratios and rate of change. A ratio is a comparison of two numbers.

The Fibonacci numbers are a recursive sequence of numbers, {1, 1, 2, 3, 5, 8, 13, …} in which each new term is the sum of the two previous terms. We can represent a Fibonacci number by Fn, and the next successive Fibonacci number by F(n+1). It is interesting to consider the limit of (Fn) ÷ (Fn+1), as n tends toward infinity. This special ratio is one of the most famous irrational numbers, phi. Also called the Golden Ratio, phi, is approximately 1.61803398…

The so-called Golden Ratio can be found throughout nature. The golden ratio, or the ratio of the length to the width of a golden rectangle, can be found in the nautilus shell, in art, architecture, and faces like Mona Lisa that are considered beautiful to behold. Besides being found in nature, beautiful designs can be constructed using the Fibonacci numbers. The Fibonacci spiral can be constructed using graph paper, a straight edge and a compass. Mathsisfun.com provides directions and information on the Fibonacci spiral. One of my students made the spiral you see below.

Another design I am particularly fond of is the Fibonacci Quilt. My students made the two shown below. The directions for making the Fibonacci Quilt are:

- Use a sheet of graph paper and divide it into four quadrants.
- Start in the first quadrant. On the very first row of squares above the “positive x-axis”, make rectangles of height
**1**, using two alternating colors with widths**1, 1, 2, 3, 5, 8**. - Then, above that row, using alternating colors, make another row of rectangles, still of height
**1**, with widths**1, 1, 2, 3, 5, 8.** - Above that row, make another row of rectangles of height
**2**, with widths**1, 1, 2, 3, 5, 8**. - Above that row, make another row of rectangles with height
**3**, with widths**1, 1, 2, 3, 5, 8.** - Continue changing the height of new rows of rectangles according to the Fibonacci numbers and complete Quadrant I.
- Now move onto Quadrant II. Consider rotating the paper so that Quadrant II is in the position where Quadrant I was.
- Follow the same directions for Quadrant II as you did for Quadrant I. Do the same for Quadrants III and IV and you have completed your Fibonacci Quilt! Voi-la!

Incidentally, I have a friend who is a jeweler. His wife is a math teacher. He has created a beautiful pendant of the Golden Spiral. You can view his pendant on his etsy.com page.

A must-view Tedtalk summary video of the Fibonacci numbers can be found below.