Thinking Outside the Book

I wanted to make math “real” for my students, to help them find meaning behind the variables and the coefficients when graphing f(x)= ax2+bx+c, so I decided to move the lesson from paper to the physical world. Nearly all the students own a digital camera with video capability. I asked them to throw a ball straight up, take a video of its flight, and use the corresponding data to study different forms of the quadratic equation.

Lesson description

I prepared a large scale marked in feet and inches with thick lines that I taped to the wall. One student stood in front of the scale and practiced tossing a ball straight up and then letting it hit the floor. When the student was able to consistently make a good toss, another student took a small digital camera and captured the toss on video. Students then watched the video frame by frame and recorded the height of the ball at each frame on an Excel spreadsheet. The students then calculated the length of each frame by dividing the length of the video segment by the number of frames during the segment. The frames on my camera were .04 seconds apart. So 0, .04, .08, .12, etc., were the elements in the first column for the independent variable, and the height in feet was recorded in the second column for the dependent variable. Once the data was entered, the students created a scatterplot and produced three models for the data: a quadratic regression equation (y = ax2 + bx + c), an equation in vertex form (y = a(x – h)2 + k), and an equation in factored form (y = a(x – z1)(x – z2)). The students were then asked to analyze the coefficients of each model.

Subject area

This lesson was designed for Algebra 2 students but could be modified for Algebra 1 students or extended to the Pre-Calculus level by including a study on the instantaneous rate of change.

Curriculum standards

For the Algebra 2 version, the lesson meets the following North Carolina Standard Course of Study goals:

• Use quadratic functions and inequalities to model and solve problems; justify results.
  a. Solve using tables, graphs and algebraic properties.
  b. Interpret the constants and coefficients in the context of the problem.
• Create and use best-fit mathematical models of linear, exponential and quadratic functions to solve problems involving sets of data.
  a. Interpret the constants, coefficients and bases in the context of the data.
  b. Check the model for goodness of fit and use the model, where appropriate, to draw conclusions or make predictions.

Resources

Students will need access to a digital camera that can capture video, computers with Excel software, a ball of some sort and materials to make a scale to measure the ball’s height.

Grading Rubric

The grading rubric for this lesson gives 20 points for each of the following tasks:

• Collecting the data
• Plotting the data
• Analyzing the quadratic regression equation
• Creating the vertex form model
• Creating the factored form model