The goal of this lesson is to help students use trigonometric identities to determine mathematical expressions for the horizontal and vertical components of a projectile's motion. This lesson addresses the HSA-REI.A.1 and HSG-SRT.C.8 standards because it asks students to break an initial velocity vector into its component parts and to use trigonometric ratios to solve for each component in a step-by-step manner. It aligns with the NGSS Practices of Using Mathematical Reasoning (SP5) and Constructing Explanations (SP6) for Science because students will use mathematical logic to create summaries to explain factors that are related to a projectile's motion. This lesson also is aligned to the NGSS Cross Cutting Idea of Patterns because students must recognize that the horizontal and vertical components of a projectile's velocity are independent of each other.
Within this lesson students will begin constructing an explanation of the independent motions that describe a projectile's motion using a mnemonic device of their choice. Students will then use their understanding of mathematical models from trigonometry and the mnemonic devices from the beginning of the lesson to complete a trig practice worksheet. Finally students work in teams of 2-4 to complete a set of practice problems that connect trigonometry to an object's motion.The summative assessment for this course is called a performance based assessment where students present information to a committee which consists of external evaluators made up of teachers and students. In order for a student to convince a committee that their understanding of a topic meets or exceeds the standard of proficiency the student needs to gather credible information and use multiple ways to represent the information in an easily accessible manner. Within this lesson I ask students to focus on stretching their prior trigonometry knowledge to address the velocity components of an Angry Bird . I assess student understanding throughout the lesson using informal check-ins, and will assess each student's work at the end of the school day.
This part of the lesson begins with this routine. Today's piece of additional piece of information is a Big Idea which states that trigonometric ratios can be used to find expressions for the horizontal and vertical components of a projectile's velocity. The objective of the Bell-Ringer is to give students a clear understanding of the focus of today's lesson. In this lesson I want students to get ready to leverage information gathered from their understanding of trigonometric functions to construct mathematical models for the vertical and horizontal components of a projectile's velocity.
I follow up the bell-ringer with a turn and talk routine found on this set of slides. The Turn and Talk routine consists of:
During the turn and talk I circulate and note that a few students confuse the horizontal and vertical components of velocity as they discuss their "noticings or wonderings" with their peers. This could be an artifact of our investigation as to why 45 degrees yields a maximum range. With this in mind, I emphasize the independent nature of horizontal and vertical velocity vectors and the connection to trigonometric functions within the turn and talk portion of this lesson.
During this section, I write the phrase Trigonometric Ratios and circle it on the interactive whiteboard at the front of the room. I 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 mathematics that we will use to understand physics content later in the lesson. After five minutes elapse I ask for volunteers and compose a mind map on the interactive whiteboard at the front of the room. I want students to understand how important identifying information from mathematics is to understanding patterns within physics content.
The goal of this section of the lesson is to use both mind mapping and mnemonic devices as ways to connect mathematical models from trigonometry and physics and help drive home the enduring understanding that a projectile's horizontal and vertical velocity components are independent of each other. Mnemonic devices are a common strategy for across curricula and grade level at our school to help students access information in a non-linear manner. A mnemonic device asks students to use represent physics content using an easy to recall acronym or phrase. In this section of the lesson, I present students with an extension to the mnemonic to the more traditional SOH CAH TOA, called NV NJ NY and distribute an activity for students to practice skills from trigonometry.
While students use calculators to solve the practice problems and discuss the Trig Practice Activity with their table mates, I circulate and address any questions students may have. This part of the lesson focuses on appropriately using a scientific calculators and applying trigonometry to analyze a set of practice problems. Students work through the problems, as shown here.
At the end of this section I ask for volunteers from each lab table to provide a summary of where they are because I want students to give me a feedback and to provide candid reflections on the ease of the task and the connections between the information they collected and concepts they already know. This helps me see where they are in terms of their mathematics skills and their ability to make connections between trigonometry and physics. During the next section students are given a chance to apply these skills to the motion of an angry bird following a parabolic trajectory with a known initial velocity and launch angle.
I lead a Whole Class Discussion for five minutes about the connections between trig ratios and the components of vectors. Within this discussion, I introduce the idea that vectors that make an acute angle to the horizontal axis can be broken into horizontal and vertical components that can be modeled as the unknown sides of a right angle triangle. During the next twenty minutes I project a set of Example Problems on the interactive whiteboard at the front of the room. I make sure to ask clarifying questions as I circulate in order to push students into supplying evidence to support their claims about the vector components of an Angry Bird's velocity.
After the discussion is over I circulate and students spend the next twenty minutes creating a summary that includes a problem, the physics of the problem, a step-by-step solution and tips for peers on how to analyze similar problems in the future. Students may either solve the provided Independent Practice Problem or create a similar one in their groups to work on. 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. Once we have discussed the requirements for the summaries and all of the materials have been distributed to each group, students begin working in groups of 2-4 at a location of their choice to summarize the connections between trigonometric identities and the vector components of an Angry Bird's velocity. I circulate the classroom with my classwork assessment clipboard and make notes of groupings and answer clarifying questions for students.
I include examples of student work here and here that show a student summary and solution to an example problem. At the end of this section, I pause and ask students to return the materials they used during this section to the front resource station. A resource manager returns each material to a bin or labeled drawer so that they are readily available the next time the materials are needed.
Throughout this lesson I give students multiple opportunities to listen to one another and to control the process of their learning. With this in mind, I include a closing activity for this lesson as I do in others so that students are assessed on their current level of understanding on the connections between trigonometry and a projectile's velocity.
The closure asks students to identify and describe their personal level of understanding of key ideas within the lesson and also works to make student thinking visible regarding the underlying reasons behind their understanding. I check student responses to this closure to determine whether students are proficient in the understanding of the connection between trigonometric ratios and the independent horizontal and vertical components of a projectile's velocity at the beginning of the next lesson. During the next lesson I highlight this connection and the tools from this lesson that are useful for illustrating this enduring understanding. Click here to see an example of student work for the closing activity.