Fibonacci numbers and an area puzzle: Connecting geometry and algebra in the mathematics classroom

School Science and Mathematics, Mar 1997 by Mary M Sullivan, Regina M Panasuk

A mathematical puzzle that asks about "missing " area leads to an exploration of the Fibonacci sequence as well as genuine inquiry in plane geometry connected to algebra. This article discusses the inquiry, the concepts, the solution, and an extension that deepens all students ' understanding of connections between algebra and geometry.

The National Council of Teachers of Mathematics Curriculum and Evaluation Standards (1989) encourages teachers to create opportunities for students to make connections within mathematics and across disciplines. The Fibonacci numbers, which have attracted the attention of professional and amateur mathematicians (Boles & Newman,1987; Gardner,1982; Hoggatt, 1969; Jacobs, 1994; Pappas, 1989; Steen, 1994), are simple in their definition and rich in their relationships within mathematics and connections to other fields. Because of their richness, teachers have little difficulty in incorporating them appropriately into their curriculum.

High school students' first introduction to the Fibonacci number sequence 1, 1, 2, 3, 5, 8, . . ., y, x, y x may take place in a science class because the sequence appears frequently in nature or in an art class because of its relation to the Golden Ratio. The students' introduction to the sequence in a mathematics class may be as an illustration of a recursive function, and the teacher may connect it to science and art. The Fibonacci connections that are discussed in this article are within mathematics, concepts from algebra and plane geometry, and start with the following puzzle. Cut an 8 x 8 square as indicated in Figure la and reassemble the pieces as shown in the rectangle in Figure lb. Compare the areas of the rectangular and square. What do you find? How do you explain your discovery?'

Although "solving" the puzzle and exploring Fibonacci number sequence properties is interesting, the mathematical richness lies in delving into some deceptively simple concepts behind the puzzle: collinearity, slope, and similarity. The puzzle problem prompts an investigation of these concepts, provides an opportunity to determine an appropriate mathematical model, and offers practical applications for slope and similarity. Because the mathematical models that lead to solutions are not immediately apparent, we assert that the search for an appropriate model is inquiry itself. When students are engaged in inquiry, they are constructing their mathematical framework.

When solving the puzzle problem, students have difficulties in expressing the area difference that occurs because they are guided by the visual representation (see Figure lb), deceived by it, and make an erroneous assumption (Wallace & West, 1992) that the points on the rectangle's "diagonal" lie on a straight line. Thus, the essence of the problem rests on concepts that are not readily apparent in its statement. The concepts of betweenness and collinearity, which provide the mathematical model, are not well defined, nor understood, in most secondary mathematics classrooms. Jurgensen, Brown, and Jurgensen (1988) offer the following definition: "Collinear points are points all in one line" (p. 2). Add to this definition: if three points, O, A, and B, are on a line, then one of the points is between the other two. In terms of the slope concept, slope OA = slope AB. Because the points on the "diagonal" of the rectangle appear visually to lie on a straight line, students assume that they do. Thus, an obstacle of this problem, like many geometric fallacies (for example, all triangles are isosceles), arises from this erroneous assumption.

What If

The question "What if.. .?" encourages the students to embark on a discovery exploration. In this instance, what if students were to cut the 8 x 8 square as suggested in Figure 3? What would the reconstructed figure look like? Students are prompted to continue their exploration of squares whose side length is a Fibonacci number but the segments into which the side is cut are not Fibonacci numbers. Using a 13 x 13 square and cutting sides into segment pairs (4 and 9), (3 and 10), and (6 and 7), students conjecture about patterns they observe in the reconstructed figures.

Students themselves suggest a natural extension: "What if" we were to use different-size squares whose sides are not Fibonacci numbers and vary the size of the segment pieces (see Figure 4)? What patterns exist among the reconstructed figures? The students discover that the interior of some constructed rectangles contains an open space with a shape that looks like a parallelogram, whereas others contain some overlapping area in its interior.

A Fibonacci Property

In the course of the exploration, some students discover that squares whose side lengths are Fibonacci numbers, when cut into segments with lengths of the two preceding Fibonacci numbers (as in Figure 1), form rectangles with the property that the area difference between the square and the formed rectangle is 1 or -1. Table 1 illustrates the property.