ALEX Lesson Plan

     

JUMP!!! An Exploration into Parametric Equations

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  This lesson provided by:  
Author:Matthew Massey
System: Madison County
School: Buckhorn High School
The event this resource created for:GEMS
  General Lesson Information  
Lesson Plan ID: 23795

Title:

JUMP!!! An Exploration into Parametric Equations

Overview/Annotation:

Students will use vectors and parametric equations to determine the velocity that they should jump out of the window of a burning building in order to land safely into the rescue net. They will work in small groups and utilize graphing calculators. This interdisciplinary lesson includes the subjects of Physics and math.
This lesson plan was created as a result of the Girls Engaged in Math and Science, GEMS Project funded by the Malone Family Foundation.

 Associated Standards and Objectives 
Content Standard(s):
Mathematics
MA2015 (2016)
Grade: 9-12
Precalculus
28 ) Utilize parametric equations by graphing and by converting to rectangular form. (Alabama)

a. Solve application-based problems involving parametric equations. (Alabama)

b. Solve applied problems that include sequences with recurrence relations. (Alabama)

Local/National Standards:

 

Primary Learning Objective(s):

Students will use parametric equations to determine the position of launched objects at any point of time. Students will also utilize technology to analyze the path of the object to be evaluated when the object will hit the ground.

Additional Learning Objective(s):

Students will determine the horizontal and vertical component vectors of a launched object.

 Preparation Information 

Total Duration:

91 to 120 Minutes

Materials and Resources:

A ball (tennis, baseball, etc), class set of graphing calculators or computers that have Internet access to use graphing calculator from the Internet, video recorder, video player, Digital Projector

Technology Resources Needed:

Computers with internet connection

Background/Preparation:

Students should be efficient using the formula D = rt to solve distance, rate, and time problems. Students should also be familiar with the vertical position function P(t) = -16t^2 + tv + h.

  Procedures/Activities: 
1.)Initiate a whole class experiment with your students taking turns throwing a ball (tennis ball, baseball, etc) from the top of stadium bleachers, or comparable situation, onto a target 15 feet away from the structure.

2.)Stategically place students into groups of 3 with different learning styles to allow for peer mentoring and tutoring. Distribute attached questionnaire. Have each group collaborate to answer the questions.

QUESTION 1: This may be an appropriate time to discuss that an object launched horizontally falls at the same rate as an object dropped. Answer: 1.37 seconds. This is determined by using the vertical position function (previously discussed.) This can be determined by solving for 't' when 4 in the final height, initial velocity is 0, and beginning height is 34. Students will need a calculator to solve the quadratic 4=16t^2+0t+34. Different methods of solving this can be discussed (square root property, quadratic formula, finding roots, etc).

QUESTION 2: Using D=rt, when D is 15ft and t is 1.37sec, students should determine r to be 10.9ft/sec.

PREDICTION: This prompt should spark a debate among groups and is the gateway to using parametric equations to follow the path of objects, in this case, our jumper.

3.)Through directed questioning, guide students to see that two equations are needed to model the path of the jumper: one to measure the distance away from the building, another to measure the distance off the ground. Now distribute the second questionnaire with the general form of the two parametric equations to follow the paths of objects launched on this planet. A connection can be made to Earth Science on the gravitational pulls of other planets and our moon.

4.)Students will collaborate to answer the first question.
1. x=5.45ft y=27.44ft
2. To follow the path of the object, students will need to change modes of their graphing utility from function to parametric mode. Students can use the trace feature or other methods to determine if they will land in the net. Students who do not have graphing calculators can access one at the website linked below.
EXPLORATION: By substituting 15 for x and 4 for y, students will notice that two variables are missing: t & v. Students can solve this system of equations by substitution.
(Graphing Calculator)
This Illuminations website has a functioning graphing calculator.

5.)Show students the video of the ball tosses and facilitate a whole group discussion on adjustments that would need to be made for each throw to hit the target.

6.)Close the lesson by having the students create their own story in which parametric equations must be used to achieve a happy (or sad) ending for homework.


Attachments:
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  Assessment  

Assessment Strategies

Informally assess students by closely monitoring their participation in the small groups. Formally assess each groups answers to the second questionnaire (attached) for accuracy. The weight of the grades can vary.

Acceleration:

Further topics that could be explored would be creating horizontal and vertical component vectors for the jumper. Students could then determine the horizontal and vertical velocity of the jumper by using the Pythagorean Theorem. They could further explore component vectors in the interctive game car storm chaser

Intervention:

Exercises involving basic distance, rate, and time problems could be given students who need extra preparation before or after the lesson


View the Special Education resources for instructional guidance in providing modifications and adaptations for students with significant cognitive disabilities who qualify for the Alabama Alternate Assessment.