George Lucas Educational Foundation

# A Real Learning Curve: Catapults Demonstrate the Quadratic Equation

Algebra students use their math skills to control the arc of launched objects.
By Lisa Morehouse
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### Launch Lessons:

Ninth-grade algebra students Dontae and Gerry built catapults to test the quadratic equation in real life.

Credit: Courtesy of John Dahle

When I heard my friend and former colleague Emmanuel Medina's Algebra I students had built catapults to test their knowledge of quadratic equations and are planning to demonstrate their handiwork, I had to see it for myself.

So, on a mild spring evening, I make my way to Hayward, California, to stand on the darkened field of the Impact Academy of Arts and Technology, a small, brand-new charter high school where Medina's students are ready to exhibit their creations.

I'm treated to a beautiful sight: The night sky is lit up with streaking arcs -- parabolas, to the math initiated -- as the freshmen shoot glow sticks from their catapults. But even more beautiful is the sight of students deeply engaged in learning -- and teaching.

### "Parabolas Are Everywhere"

Before the catapult launch, the students start the evening by sharing their work with and explaining quadratic equations and parabolas to more than 200 parents and community members. One team -- Brianna, Steven, and Cody -- describe the catapult project to me.

Steven describes himself as a once-struggling math student, but he points to the hands-on work as an inspiration. "I wrote down quadratic equations before we built the catapult, but until I built and launched it, I didn't really understand," he says.

Brianna chimes in, "Parabolas are everywhere -- in science, technology, and when you shoot a catapult." She then patiently tries to sweep the cobwebs from the math corners of my brain. "When you launch an object out of a catapult, it follows a parabolic path," she explains.

Medina started this project like an old-fashioned math instructor, teaching quadratic equations and parabolas from the California state standards through examples and practice questions. Knowing that this and related subjects make up about a quarter of the math content on California's standardized assessments, Medina used released test questions in this first phase.

Then he introduced the catapult idea by connecting it to other disciplines, thereby piquing his students' interest. "Back in the day, the purpose of studying the quadratic equation was to study the effective use of weaponry," he tells me. "People used catapults up until World War I -- which the kids were studying in their history class -- when it was a tool for trench warfare."

Medina told students they'd be designing and building their own catapults but deliberately gave them few directions. "I knew if they were left to their own devices, they'd surprise me with great designs," he explains. He simply instructed them to find images online by searching for the words catapult, trebuchet, and design. Student teams then decided on the layouts for their catapults, got Medina's approval, and made blueprints with a list of needed materials.

Using \$300 in school funds, Medina purchased the materials (mostly PVC pipe, 2 x 4s, plywood, and screws) and borrowed power tools (for which students received safety training). Students used their blueprints to measure out their designs on the materials, and Medina cut the wood for each group using a circular saw.

### "Textbooks Are Great, But . . ."

The students performed the initial tests of their catapults by launching solid Styrofoam eggs, then collected the data. Medina explained to them that they needed to record two pieces of information to find the equation of the parabolas their eggs traced: hang time (the amount of time it was in the air) and distance traveled.

Brianna describes how they followed Medina's instructions to then find the vertex, or highest point, of each parabolic flight. First, she explains, they timed their egg's flight and divided the time by two to find when the object was at the vertex. Then, by measuring the distance the egg traveled along the x-axis -- from the catapult-launch site to the landing spot -- and dividing that distance in half, they identified where the vertex occurred.

Medina taught the students how to use that information to figure the height of the vertex, which allowed them to graph the exact path of the projectile. They could then experiment with controlling where their eggs landed.

"If you know where the egg is going, you can plan to hit targets at a variety of heights and distances based on that one equation," Medina says. Students got so adept at finding this information that they were able to launch their eggs into garbage cans placed at various distances. Meanwhile, to keep students developing their on-paper math skills, Medina gave them related word problems to work on in class.

Using the results of the tests, the students worked to improve their catapults. "We started out with basic weights," Cody says, describing one of his group's posttest innovations. "The catapult was working, but I thought, 'What about tension?'" The group removed the weights they were using and replaced them with bungee cords, which gave their launches more distance, height, and hang time.

Cody says students learned the most about parabolas during this process of perfecting their catapults. "Textbooks are great, but they don't give you the visual," he adds. "You have to go through a process of how parabolas are created, take the data, plug it into a quadratic equation, and analyze it."

Ho Nguyen, a mathematics-curriculum specialist for the nearby West Contra Costa Unified School District, who came to the exhibition, believes that the project has allowed students to gain insight into the quadratic equation. "I think this is a great way to motivate students," he says. "Since it is a fun project, it becomes relevant to them, which makes the teaching and learning much, much easier."

### "Making That Perfect Parabola"

As night falls on the evening of the exhibition, the crowd walks up a small hill to the school's field, where thirteen of the most consistent catapults sit at the ready. When Medina calls out, "Okay, set it up. Let's fire!" students attach thick glow sticks to their catapults' arms, ready to launch them at targets marked by glowing hula hoops on the field.

Cody and Steven hook up bungee cords, launch their glow stick, and watch the bright light fly 40 feet high in a parabolic arc, prompting giggles from little brothers and sisters who run across the field, gathering the launched sticks and the targets.

Afterward, the students talk about their sense of satisfaction at creating something and about what they have learned in the process. "I'm proud to say, 'Hey, I built that!'" says Brianna. "It took multiple steps. It fell apart a couple of times. We misplaced some of our materials. But the end result is, it launches."

"Creating a catapult was not only fun, it was the best learning experience I've ever had," adds Cody. "With a textbook, it's in one ear and out the other. But with the catapult, because it's hands-on, it has stuck with me. It was the process -- playing with it, trying to get to the goal, to make that perfect parabola."

###### Lisa Morehouse, a former teacher, is now a public-radio journalist and an education consultant.

Anonymous (not verified)

Hello,

If you are interested in projectiles for middle schoolers, I suggest using:

slingshots using small rubber bands and twigs
launching cotton balls
plastic spoon catapults
simple seesaw catapults
compressed air rocket launchers (kids' toys)
Nerf products

Good luck!

Anonymous (not verified)

Hey there,

I feel your pain. There are plenty of good things launched into the air... You could launch food and supplies to an island in class to rescue action figures.

I hope you keep up the fight with your administrators. Learning is a lot of fun for kids and adults when we have freedom to explore. There were minor injuries in class, but please tell your administrator that our test scores were not. Impact was a first year school and we out-performed every large school in the city in algebra.

Good luck to you. If you ever decide that you need more space that what you currently have, give me a holler.

Manny

PS

Anonymous (not verified)

Find a Boy Scouts of America Webelos Handbook. Instructions for building a catapult out of a shoebox, plastic spoon, two pencils and two rubberbands are in the Engineering section. Works great with marshmallows ... although I suppose if handled incorrectly you could put an eye out with one, so don't say the idea came from me!

Anonymous (not verified)

I taught middle school students for aprox 5 years before moving up to the high school level. They are more than capable of using power tools with the proper training! Simple power tools like jig saws, drills and screw guns are basic tools everyone should know how to use :) Middle school is a great place to learn how to operate these.

Anonymous (not verified)

This is an amazing project. And it's cool to see that the school did it in there math class. At our school were doing it in our physic's class and it really is a cool project to do.

Anonymous (not verified)

My son did a very homemade catapult for his science fair project. He's in 5th grade. He used a rat trap for the spring mechanism, attached a piece of wood to the arm of the trap, and screwed a plunger head on to the wood. Obviously the results are limited to the scope of the project, but it was fun, age appropriate, and a successful catapult.

Anonymous (not verified)

Your activity is very inspiring! I'm hoping this will work well with my students also.

I have a question about implementing this lesson because I am planning to use the same idea for my algebra class. How can you find the maximum height using only distance and time?

Anonymous (not verified)

I would like to try this with my Algebra 2 students this year...always looking for something different. I have done the ball bounce project and sequential bounces. I think this project will get my students thinking as to who can launch the furthest and launching into an object. I would love to have the lesson plans to this project as soon as possible.

Anonymous (not verified)

I would like to try this with my Algebra 2 students this year...always looking for something different. I have done the ball bounce project and sequential bounces. I think this project will get my students thinking as to who can launch the furthest and launching into an object. I would love to have the lesson plans to this project as soon as possible.