3, 2, 1 Launch


Author:

Kevin Metzger

Unit Title:

3, 2, 1 Launch

Grade:

10

Subject:

Math (Geometry/Algebrea II)

Estimated Duration:

14 Days, 48 minute class each day

Unit Activities:

Background Knowledge: 

Students will have to have prior knowledge of basic functions. This obviously only consists of linear functions and transformations of linear functions. Students will apply that knowledge to learn about transformations of second degree functions. Finally, students will need to have a basic knowledge of Microsoft Excel. We will be using that program to create scatterplots and trendlines to find a best fit quadratic function.

Date:

July 2013

The Big Idea (including global relevance)
Students will be given a scenario in which they will need to develop a solution to the scenario. A hostage situation has taken place!  A local prison has been taken over by three prisoners. They have locked down most of the prison, but there is one access that guards can enter and exit without being seen. However, one of the guards was injured and need emergency antibiotics. The only way the guards can get the antibiotics is if they are sent over a 300 foot wall. It is the students job to design a trebuchet that can launch the antibiotics at least 300 feet in the air. This means the trebuchet will have to at least launch at most 40 feet linearly (as to not launch straight in the air, the projectile will have to move in a linear fashion as well). They will be restricted by the following constraints:
  1. The students will have to use a scale factor 1/20 the actual size of the scenario.
  2. Students will be located 2 feet away from the prison wall.  This means students will need to reach maximum height after 2 feet of travel (theoretically 40 feet).
  3. The students will be given pvc-pipe kits that will give them the essentials for a trebuchet (throwing arm, sling, and a way to create a base).  However, the overall structure they create will be using their own imagination and the materials given.
  4. The throwing arm will have two holes drilled in the structure at 6 inches and 7.5 inches.
Once students get the minimum distance and height, they will make ten trials with their perfected launcher. Using the data points, they will use Microsoft Excel to determine the quadratic equation that best fits their data. Using this equation, they will theoretically compute the maximum height their projectile will reach. Once they compute the maximum height, they will test their theory to see if it holds true. Students will refine their design to launch the theoretical maximum.
The Essential Question
  1. What do you need to create a quadratic equation and find the maximum of that equation?
  2. What is a trebuchet? 
Justification for Selection of Content

Projectile motion is a very large topic often discussed in most physics classes. However, its applications are seen in pre-calculus and calculus as well. With that being said, projectile motion applies to a variety of real-world situations. For example, the military always uses projectile motion when shooting a variety of weapons. Also, NASA considers projectile motion when launching things into space.

The Challenge

Students need to develop a trebuchet that launches an object at least 2 feet in distance and 15 feet in height. They will need to test their trebuchet, find the best-fit quadratic representation, and then find the theoretical maximum height of their trebuchet. Once this is complete, they need to test to see if their object theoretically reaches this height.

The Hook

I will show the students a video of a trebuchet in action. There are many videos of “pumpkin-chunkin” or trebuchets launching very heavy objects, such as pianos.


Teacher's Guiding Questions
  1. What motion does the projectile follow throughout its course of flight?
  2. What do you need in order to maximize the distance of the trebuchet?
  3. What important piece of information do you need in order to find the maximum height of a quadratic?

ACS (Real world applications; career connections; societal impact)

Applications:  Students will be considering the motion of a projectile through its course of flight.  This closely resembles what the military does when analyzing the flight of a weapon or missile.  Also, this closely resembles what NASA considers when analyzing the flight of a rocket into space

Career Connections:  Students will see military applications.  Also, students can correlate this to any type of projectile motion.  This could mean aerospace engineering, and mechanical engineering.  

Societal Impact:  How could the maximum height of a projectile affect a society?  If a projectile reaches a certain height then it reaches a certain linear distance.  This could have a major impact especially in missile launches.

Engineering Design Process (EDP)

Students will use the engineering design process when considering how to create their trebuchet with the materials they are given.  Also, using the challenge based learning framework, students will complete the following activities throughout the lesson:

  1. Use the engineering design process to develop a trebuchet and launch a projectile.
  2. Apply the data they receive from their launches to quadratic equations.
  3. Analyze the quadratic equations
  4. Determine the maximum height and test this value using their trebuchets. 
Unit Academic Standard

F.IF.4 For a function that models a relationship between two quantities,

interpret key features of graphs and tables in terms of the quantities,

and sketch graphs showing key features given a verbal description

of the relationship. Key features include: intercepts; intervals where

the function is increasing, decreasing, positive, or negative; relative

maximums and minimums; symmetries; end behavior; and periodicity.


F.IF.7 Graph functions expressed symbolically and show key features

of the graph, by hand in simple cases and using technology for more

complicated cases.★

a. Graph linear and quadratic functions and show intercepts,

maxima, and minima.

b. Graph square root, cube root, and piecewise-defined functions,

including step functions and absolute value functions.


F.BF.1 Write a function that describes a relationship between two

quantities.★

a. Determine an explicit expression, a recursive process, or steps for

calculation from a context.

b. Combine standard function types using arithmetic operations.

For example, build a function that models the temperature of a

cooling body by adding a constant function to a decaying exponential,

and relate these functions to the model.


F.BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k,

k f(x), f(kx), and f(x + k) for specific values of k (both positive and

negative); find the value of k given the graphs. Experiment with

cases and illustrate an explanation of the effects on the graph using

technology. Include recognizing even and odd functions from their

graphs and algebraic expressions for them.

Tentative Schedule

Day 1:  Introduction to quadratics

            Graphing Quadratics:  Basic transformations

Day 2:  Graphs and Transformations

Day 3:  Activity 1 Worksheet A:  Students will use what they learned in class to complete graphs and transformations of quadratic functions.

Day 4:  Activity 1 Worksheet B:  Students will be introduced to the idea of the project through the Smart Board Notebook presentation.  They will then be asked to complete the Engineering Design Process (worksheet B).  This is where they can develop guiding questions and a possible model for their trebuchet launcher.

Day 5:  Activity 1 Worksheet B:  After the introduction, there may need to be another day to complete Worksheet B.

Day 6:  Discussion of standard form and properties of quadratic equations (Smart Board Notebook presentation).

Day 7:  Discussion of converting vertex to standard form as well as how to create graphs from standard form equations.

Day 8:  Activity 2 Worksheet C:  Practice converting from standard form to vertex form.

Day 9:  Activity 2 Worksheet D:  Students will build their trebuchets

Day 10:  Activity 2 Worksheet D:  Students will build their trebuchets.

Day 11:  Creating quadratic equations given a data set.  Students will use graphing calculators, and Microsoft Excel.

Day 12:  Activity 3 Worksheet E:  Practice creating equations given a data set.

Day 13:  Activity 3 Worksheet F:  Creating a quadratic equation given the data they obtain from several launches of their trebuchet.

Day 14:  Activity 4 Worksheet G:  Students will convert their standard form equation to a vertex form and find the maximum height their trebuchet will reach.  They will then refine their design and see if they can reach the theoretical maximum.

Day 15:  Extra day just in case they need extra time to refine their design.

Where the CBL and EDP appear in the Unit

- Activity 1 Worksheet B represents the EDP

Students will use this design plan to fabricate an actual trebuchet.  They will then use the trebuchet to make several launches.  After each launch they will record the maximum height of their projectile.  Using this data they will create a quadratic equation that best represents their data, theoretically find the maximum height, then refine their design.

- Activity 2 Worksheet D represents the CBL

Misconceptions

1. Students will confuse the transformations of quadratic functions.  They often confuse which part of the equation forces the parent function to move left, right, up, or down.

2. Students may not understand why a quadratic function needs at least three points to develop a solid relationship between input and output values.

Additional Resources

www.pearsonsuccess.net

Pre-Unit Assessment Instrument
Post-Unit Assessment Instrument
Results: Evidence of Growth in Student Learning

After teaching the Unit, present the evidence below that growth in learning was measured through one of the instruments identified above.  Show results of assessment data that prove growth in learning occurred.


Results
Reflection
  1. The students were able to relate quadratics with trebuchets.  Also, they were able to use their data to create a quadratic equation that appropriately described the path of their projectile.
  2. I selected the content for this unit because projectiles accurately describe quadratics.  Also, students were able to see a real life example of where they would use the mathematics.
  3. The data indicates that students have never seen quadratics before.  They were unable to answer any question on the pre-assessment.  However, given the same assessment after the unit, they were able to answer significantly more questions.
  4. The purpose of selecting this unit was met.  Students were able to relate quadratics to projectile motion.
  5. I would make students keep an engineering notebook in order to organize their changes.