Polyominoes, Polygons, and Polyhedra: Discovering Geometry through Exploration

Have you ever watched a baby learn a new skill, like walking, crawling, or even rolling over? The child does not master the skill on the first try, but they are determined. They fall down, perhaps bump their head, but they keep trying. They readjust what they are doing, figure out how to create momentum and balance, and finally succeed. Even then they may stumble, but they persist. In the process, they grow stronger and can apply these skills to later learning, like running, climbing the cabinets, and reaching the cookie jar.

Throughout our lives, we grow by challenging ourselves to move beyond our comfort zone. We learn to read and write, play games and sports, cook, make music, or create art. Our brains are muscles that develop as we attempt new challenges. The nature of mathematics involves struggling with a problem, trying different strategies, taking breaks but persisting, and learning from failed attempts. In the explorations of this section, you may find that some of the problems are easy, but others require looking at the problem in new ways and trying different strategies. Accept the challenge, try out different ideas, and don’t worry about making mistakes. Allow these attempts to enable growth and eventual success.

To give you the experience of confronting the challenge and the exhilaration of conquering it, the answer to the last task in this exploration will not be provided here. Searching for solutions or examples will not help you grow mathematically. Instead, consider other ways in which you might attack the problem. Dare to try something new!

The above exploration starts off relatively easy. Once you understood how to create and measure shapes on the geoboard, you probably found triangles with areas of \(\frac{1}{2}, 1, \frac{3}{2}, 2, 3\text{,}\) and \(\frac{9}{2}\text{.}\) In fact, you might have found multiple triangles with these areas. Yet the remaining areas may seem impossible. If you are still in this situation, think about why these same six areas keep occurring. What do these triangles have in common? What other types of triangles exist on the GeoBoard?

As you find these new triangles, another question emerges: How does one compute the area of these triangles? To use the area formula, \(A=\frac{1}{2}bh\text{,}\) one usually uses the values of \(b\) and \(h\text{.}\) How does one find \(b\) and \(h\) with certainty? Or is it necessary to find the base and height of the triangle? Might there be other ways of finding the area in the current setting?

There are many lessons to be learned here that go beyond measuring area. Mathematics is not about doing fast calculations with provided formulas or following steps. Instead, it requires persistence, creative thinking, and exploring multiple strategies. Sometimes it is helpful to walk away from a problem for a little while and to return to it with a different perspective. It helps to share ideas with others even when you do not know where they will lead.

Did you discover a triangle that was not obvious? Did you figure out a way to compute the area of these triangles? Did you help others understand? Be proud of your discoveries and contributions. This is evidence that you, yes you, are a mathematician!

Each student brings their own individual strengths to the class. Some are great at remembering and applying formulas while others are great at drawing sketches. Some have a talent for recognizing patterns or building on others’ ideas while other students are adept at explaining results and connections. In fact, you may find yourself contributing in different ways on different activities. The important thing is to believe in yourself and be open to trying new things. Unfortunately, most people believe the myth that they can’t do math. They spend too much energy comparing themselves to others. Math is not about being perfect or fast. Instead, mathematics is about exploring, creating, experimenting, conjecturing, applying, and justifying.

If today was not your day to shine, perhaps you were hesitant to consider alternative approaches or to pursue your ideas. This particular topic might not have engaged you in the same way as it did others. Or maybe it was just an off day. There will be more opportunities to overcome challenges throughout this book. As long as you are open to learning and exploring your own ideas, your mathematical ability and confidence will grow. After all, the course has just started, and you are just adapting to this book and its style, your instructor, and your classmates. By the way, your brain may hurt a little as you confront and conquer mathematical challenges. Like every other muscle in your body, a little soreness means that you are growing in strength. Success comes from overcoming the struggle, from remembering and building on your accomplishments and missteps.

As we have noticed in this section, not all segments are horizontal or vertical. To determine the lengths of these slanted segments on a grid, the Pythagorean Theorem 2.5.2 will be helpful. The Pythagorean Theorem allows us to find the lengths of the sides of right triangles using the fact that \(a^2+b^2=c^2\) where \(c\) is the length of the hypotenuse and \(a\) and \(b\) are the lengths of the legs of the triangle. A more in-depth exploration of the Pythagorean Theorem will be pursued in Section 2.5.

Applying the Pythagorean Theorem requires facility with the algebraic operation of squaring and its inverse operation, the square root function. A review of how to work with squares and square roots is provided in A.4.1 in the Backmatter.