The goal of this lesson is to help students use their understanding of Galileo's equations of motion, trigonometry and simulations to investigate the accuracy of an alternative mathematical model for range. This lesson addresses the HSF-BF.A.1a standard because it asks students to create an explicit expression for the range of a projectile when the time of flight is unknown. It aligns with the NGSS Practices (SP5) of Using Mathematical and Computational Thinking for Science, Developing and Using Models (SP2) and Planning and Carrying Out Investigations (SP3) because students will investigate the accuracy of an alternative equation for a projectile's range using a simulation. This lesson also is aligned to the NGSS Cross Cutting Idea of Models because students extend their current mathematical model for a projectile's range to one that is independent of time.
Using a Mindmap, within this lesson students construct an explanation of the equations of motion highlighted during the lessons from the introduction to projectile motion portion of our curriculum. Students then use their understanding of mathematical knowledge of trigonometry and the constraints of a projectile with known initial velocity and launch angle but unknown time of flight to construct an explicit mathematical model for the range of a projectile. Finally, students work in teams of two to four to apply their new equations to generate testable data and assess the accuracy of their data using a simulation. Within this lesson, I ask students to focus on stretching their prior algebra and physics knowledge to extend a simple model for the range to a more explicit expression for the range of a projectile in terms of initial velocity and launch angle.
The alternative expression for the range of a projectile is
R = vi^2 * sin (2 * theta)/g
R = vx * t.
Both are equivalent, but the first equation helps prevent rounding errors because students only have to perform a single calculation instead of three. I assess student understanding throughout the lesson using informal check-ins and grade each student's work at the end of the school day using the school-wide rubric.
This portion of the lesson follows a routine to communicate things students need to be proficient in by the end of the semester and the goals of the lesson to students. I summarize the key ideas through the bell-ringer activity to the class and take attendance while students transfer the projected information into their notebooks.
Today's additional piece of information is a BIG IDEA which states that the horizontal displacement depends on both initial velocity and launch angle for the range of a projectile. Later on within this lesson I ask students to use a simulation to test their calculations of range using both mathematical expressions.
During this section, I write the phrase Angry Bird Physics and circle it on the interactive whiteboard at the front of the room and ask students to create a Mind Map in their notebooks of key ideas, diagrams, descriptions and equations that come to mind when they hear the phrase. This activity asks them to identify information from previous lessons that we use to understand physics content in this lesson. The summative assessment for this course asks students to determine the effect of initial velocity and launch angle on the range of a projectile. I ask students to complete this mind map because I want to see each student's current understanding of range at this point in the semester.
I spend 5 minutes calling on students from around the room and write down key ideas and equations based on their input and show step by step how the equation for range can be manipulated to determine the range in terms of initial velocity and angle of release. During this section of the lesson I use direct instruction because I am showing students new material that extends their current level of understanding. I present students with the idea that the equation for range that we learned during past lessons on projectile motion can re-written to be dependent on initial velocity and angle of release rather than on initial horizontal velocity and time of flight. Students spend the last ten minutes of this portion of the lesson transferring this information into their lab notebooks.
During the next twenty minutes, I project a data table on the interactive whiteboard at the front of the room for students to complete. I distribute a handout with the table and ask students to solve for range both ways using the provided material. Students have to take screenshots of the obtaining the green Score splash screen in the simulation and have to compare the percentage error between their calculated range and the simulation range using both math models for the horizontal range of a projectile. I make sure to ask clarifying questions as I circulate in order to push students into supplying evidence to support their claims about the range of an Angry Bird.
After the discussion is over I circulate and students spend the next twenty minutes using a simulation to test values of the horizontal range of a projectile. Students use both mathematical expressions for range: one that depends on time and the other does not depend on time. I remind students of the digital resources available to them and then ask for resource managers to gather the materials provided at the front of the room where the equipment is kept. The digital resources include Chromebooks and the PhET simulation. Students work in groups of 2-4 to complete their data tables. Some groups split the work up where students take turns calculating columns of the data tables. Other groups work through the table together by calculating the values independently and comparing their solutions after each team member has completed the table. I check in with groups around the room to see if they have any questions or concerns as they investigate range and fill in the data. At the end of this section of the lesson, students clean the stations and return materials.
Students have two data tables one with the time-dependent mathematical expression for a projectile’s range and the other with the time independent mathematical expression for a projectile’s range. Students use the simulation to determine the accepted value for the range given a certain initial velocity and range. Students then calculate two sets of percent error: One between the accepted value for range from the simulation and their calculated value for range that depends on time; The other between the accepted value for range from the simulation and their calculated value for range that does not depend on time. Students check their answers, but the simulation has a set of acceptable answers with 2 standard deviations from the “bullseye” solution that students know as "the accepted value" for a projectile's range. I include examples of student work here that shows students' applications of both types of equations and a generated data table for the range of a set Angry Bird launches.
The closing activity is a writing prompt for a routine called headlines with which students are familiar. I ask students to create, in their notebook, a headline that illustrates the most important and the most challenging portions of the lesson. I also ask students to be prepared to explain the reasoning behind each part of their headline. The headlines routine asks students to summarize their thoughts in 25 words or less. This type of closure activity asks students to highlight connections to their previous understanding and key ideas within the lesson and also works to make student thinking visible regarding the underlying reasons behind their understanding. Click here to see an example of student work for the closing question.