Catapults and Parabolic Motion (Investigative Lesson)

Name: Charelle Smith

Title of Lesson: Connection between Quadratics and Projectile Motion

Date of Lesson:

Length of Lesson: Two days

Description of Class: Algebra I

Source of Lesson: Combination of previous lesson plans and http://galileo.phys.virginia.edu/classes/109N/more_stuff/Applets/ProjectileMotion/jarapplet.html

TEKS addressed: §111.32

(b) Foundations for functions: knowledge and skills and performance descriptions.

(1) The student understands that a function represents a dependence of one quantity on another and can be described in a variety of ways. Following are performance descriptions.

(B) The student gathers and records data, or uses data sets, to determine functional (systematic) relationships between quantities.

(D) The student represents relationships among quantities using concrete models, tables, graphs, diagrams, verbal descriptions, equations, and inequalities.

(E) The student interprets and makes inferences from functional relationships.

(2) The student uses the properties and attributes of functions. Following are performance descriptions.

(C) The student interprets situations in terms of given graphs or creates situations that fit given graphs.

(D) In solving problems, the student collects and organizes data, makes and interprets scatterplots, and models, predicts, and makes decisions and critical judgments.

(d) Quadratic and other nonlinear functions: knowledge and skills and performance descriptions.

(1) The student understands that the graphs of quadratic functions are affected by the parameters of the function and can interpret and describe the effects of changes in the parameters of quadratic functions. Following are performance descriptions.

(B) The student investigates, describes, and predicts the effects of changes in a on the graph of y = ax².

(C) The student investigates, describes, and predicts the effects of changes in c on the graph of y = x² + c.

The Lesson:
I. Overview : The students will further explore quadratic functions and projectile motion in this lesson. They will be allowed to use hands on tactics to see how the motion of an object thrown connects to graphs, and furthermore to the equations that form these graphs.

II. Performance or learner outcomes: After this lesson, the students will be able to formulate a quadratic equation that represents that parabolic path of an object and use the quadratic function to position a catapult to hit given targets.

III. Resources, materials and supplies needed

-Meter sticks (2 to 3 per group) and/or measuring tape (1 per group)

Graphing calculators

Calculator/overhead transparency connection

-Poster size Post-it^© graph paper

Three different colored markers

3 Tennis Balls for each group

IV. Supplementary materials, handouts

5 E Organization

Engagement: Teacher has 3 balls, a measuring stick, and asks for 3 volunteers. The teacher will have one of the students throw one of the tennis balls, one student measure the distance that the ball lands, and one student trace the path of the ball as it is being thrown. Evaluate: What is the predicted shape that the thrown ball will make? What is the mathematical name for this shape? Does the force behind throwing the ball effect how far it will go?	The students will watch as the teacher performs the engagement. They will answer the questions that the teacher asks and also question the importance of the assignment. A curve, a straight line, etc. Parabola Yes. The harder the ball is thrown, the further it will go and vice versa.
Exploration: The students will break up into groups of four. Before doing anything, each group will hang a piece of butcher paper on the wall behind where the group will be exploring. Each student will take on one of the following roles: (a) throwing the ball, (b) measuring the distance, (c) tracing the path of the ball, and (d) the note taker. Each group will have 3 balls that they will throw. Student A will throw the first ball outwards so that each group member can see the path of the ball. While the ball is being thrown, Student C will trace the path that the ball is taking on the sheet of butcher paper. Student B will measure how far the ball was thrown, and Student D will write down the measurements. The groups will repeat the process 3 times. Evaluation: Teacher will walk around the room to make sure that the students understand what they are doing. The teacher will help the students whenever any questions are asked.	The students break into their groups and perform the engagement by themselves. The students will ask the teacher for any help needed if they are having problems.
Explanation: The teacher will tell the students that they are going to use the picture of the flight of the three different balls to make a graph on the x, y coordinate plane with the point (0, 0) as the starting point of launch of the ball. The teacher will then have the students generate an equation for the parabolas the different balls made using their calculators. Here the students will have to remember the general form of the quadratic formula and use it to aid their findings. Evaluation: What is the equation for the quadratic formula? What is this derived from? Why is this necessary for this project? How can you measure how far the ball will be thrown?	Students will take their drawings and make a graph of the path that each ball takes. Using their calculators, the students will find the equations for their different graphs. X= [-b(+/-)√(b^2-4ac)]/[2a] The formula for parabolas: y=ax^2+bx+c It will help find certain points. Solving for the Zeros of the Equation.
Extend/Elaborate: The teacher will have the students go to the following website: http://galileo.phys.virginia.edu/classes/109N/more_stuff/Applets/ProjectileMotion/jarapplet.html and have the students will use the applet to further explore how projectile motion ties into the project of building catapults. Evaluation: The teacher will wrap up the lesson by asking students what was learned and by having the students complete their journal with ideas from the lesson that will help in the overall project.	The students will go to the website and explore projectile motion in conjunction with the quadratic formula and catapults. The students will relay what they learned and then write an entry in their journal that ties this lesson to the project.