By Susan Zielinski, posted August 1, 2016 —

To tackle this deceptively simple problem, students need to solve a system of equations, use the quadratic formula, the equation for objects in free-fall, and the distance formula. It’s a meaty enrichment lesson suitable for the middle of algebra 2 and beyond. The complete lesson write-up, including extensions and solutions, can be found at http://tinyurl.com/queensreward. This problem can be solved in one class period or can be enhanced by including the extension questions and digital production.

The Problem

Students must write and solve a system of three equations and three unknowns that can be reduced quickly to two variables. The variables are the height of the cavern (h), the time for the ball to drop (t_down), and the time for the sound to travel back up (t_up). Refer to the linked documents for details.

Equation 1: Time equation: t_total = t_down + t_up = 6 sec

Equation 2: Free-fall equation for the cannonball on its way down: h(t) = (–g/2)t² + v₀t + h₀

Equation 3: Distance equation for the sound on its way up: distance = (speed of sound)(t)

The Questions

The handout guides students toward the solution through a series of questions: estimating the speed of sound, estimating the depth, solving for the time until the splash, then finally solving for the depth. The optional extension questions include graphing quadratics, terminal velocity research, parametrics, speed of sound, history, Newton’s Laws of Motion, and Excel modeling.

The Challenges

I’ve been surprised at the challenges this “messy” problem provides for my students, whether they are in algebra 2 or calculus, including the following:

• Students need a surprisingly long time to understand what actually happens during the 6 seconds. They must realize that two totally separate equations and physical laws govern the ball and sound.
• Although students have likely worked with both the distance and free-fall equations previously, they may not recognize when each applies. For instance, students often want the ball to fall linearly rather than accelerating due to gravity.
• At some point, students realize (perhaps with coaching) that they need to define their own variables to represent the different time quantities, preferably choosing descriptive names like t_up.
• Students must choose a frame of reference by deciding if the top or bottom of the cavern represents h = 0. Either way works as long as they are consistent.
• Estimation is helpful in deciding if a final answer is reasonable and a vital skill to develop in students. One of the questions asks students to solve a simplified problem to estimate the depth before tackling the full problem.
• Accuracy is important, so students should not round early. Showing students how to use their calculator without rounding the interim calculations is helpful.
• I usually require an all-digital product that requires Microsoft Word’s Equation Editor, Design Science’s MathType, or the Google Docs equation editor. Most students have never used these tools before and are impressed at the beautiful math they can easily create. Expect to take most of a class period to demonstrate and practice this skill.

Above all, it’s vital to avoid helping too much and instead let students debate the approach and solution, offering only hints and only when necessary. Be patient, and your students will enjoy the process and be satisfied when they solve the problem and help the mathematician “win.” I always enjoy the way my students collaboratively bring together many algebra 2 topics to collectively earn the Queen’s Reward.

Susan Zielinski, szielinski@sps.edu, teaches mathematics at St. Paul’s School in Concord, New Hampshire. She is currently a doctoral student at Northeastern University, researching collaboration and power sharing in the classroom, algebraic misconceptions, and engaging assessments. Before her teaching career, she served in the Air Force, worked as an industrial engineer, and homeschooled her children. In her free time, she loves to travel, spend time in the woods, and ballroom dance.