In my capstone class for future secondary math teachers, I ask my students to come up with ideas for *engaging* their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Sydney Araujo. Her topic, from Algebra: the quadratic formula.

D4. What are the contributions of various cultures to this topic?

The quadratic formula can be traced all the way back to the Ancient Egyptians. The ancient Egyptians knew how to calculate the area of different shapes but did not know how to calculate the length of the sides of a shape. Moving forward, it is speculated that the Babylonians developed the completing the square method to solve problems involving areas. The Babylonians used a more similar number system to the one we use today. Instead, they used hexagesimal which made addition and multiplication easier. We can also see a similar method used by the Chinese around the same time. Pythagoras and Euclid were some of the first to attempt to find a more general formula to solve quadratic equations, both using a geometric approach. They’re ideas differ slightly, Pythagoras observed that the value of a square root is not always an integer but he refused to allow for proportions that were not rational. Whereas Euclid proposed that irrational square roots are also possible. At the time, the ancient Greeks did not use the same number system that we use, so it was impossible to calculate square roots by hand. It wasn’t until the Indian mathematician, Brahmagupta, who came up with the solution to the quadratic formula. This is because Indian mathematics used the decimal system as well as zero which had a massive advantage over the Egyptians and Greeks. Brahmagupta was the one that recognized that there are two roots in the solution to the quadratic equation and described the quadratic formula.

A2. How could you as a teacher create an activity or project that involves your topic?

One of my fondest high school memories is from my junior year physics class. It was the famous Punkin’ Chunkin’ project. Students were put in groups and asked to build a trebuchet or catapult that could launch a pumpkin across a field. The only requirement was for the device to work, the distance was just fun extra credit. For this project we had to predict the pumpkins trajectory using different variables like the pumpkin’s weight, force, momentum, etc. However, by the time we were juniors, we had either taken Algebra 2 or were currently in it. So, our physics and algebra teacher were working together so that by the time this project came around we were working on quadratic equations in algebra. As the shape of the trajectory of a pumpkin was a parabola. Because of this experience, I can create an activity or even a similar project with the physics teacher. This way students see the different applications of quadratic equations and have a tangible real world math experience.

B2. How does this topic extend what your students should have learned in previous courses?

As the quadratic formula is taught in Algebra 1, students have only seen linear equations prior to that point. Students recognize that when they are solving these equations, they are looking for one solution, no solution, or infinitely many solutions. The one solution being a singular ordered pair and then they are done. What students then must extend on when they reach quadratic equations, and the quadratic formula is that they’re now looking for two separate solutions. So, at this point they know how to solve for x and understand inverses which is important when it comes to quadratic equations. During the solving process of a quadratic equation, students may have to take the square root of both sides of the equation which will give you a plus or minus sign in front of the square root. Which makes the connection on why there are two solutions to a quadratic equation and the quadratic formula, because a parabola has two roots.

__Works Cited:__

Brahambhatt, Rupendra. “Quadratic Formula: What, Why, and How It Changed Mathematics.” *Interesting Engineering*, Interesting Engineering, 16 July 2021, interestingengineering.com/quadratic-formula-what-why-and-how-it-changed-mathematics.