The Sky is Falling


Author:

Thomas Haas

Unit Title:

The Sky is Falling

Grade:

10

Subject:

Algebra II

Estimated Duration:

11 days (50 min classes)

Unit Activities:

Activity 1: Effects on Linear Equations

Activity 2: Effects on Quadratic Equations

Activity 3: Linear Data and Line of Best Fit

Activity 4: Data and Best Fit Curve

Activity 5: Analyze the Drop

Activity 6: Challenge

Background Knowledge: 

Students must already understand how to graph points, what a linear and quadratic equation looks like, as well as what a linear graph and quadratic graph look like. They must have functional knowledge of solving a system of equations. Basic knowledge of a graphing calculator would be very beneficial as well.

Date:

July 2014

The Big Idea (including global relevance)

Stunts in movies.

Action movies are fun to watch because the stunts in them are so thrilling and dangerous looking. 

The Essential Question

How can we create a thrilling stunt that is also very safe for all those involved?

Justification for Selection of Content

This unit will review linear and quadratic families of functions and how their equations effect the change in the graphs. Students have shown in the past that they lack the capabilities to look at two equations or graphs and explain the changes that will occur from one equation/graph to the other. This concept is needed for students to explain in detail, the changes in real world graphs by changing different parameters. Analyzing a quadratic and linear equation at the same time and manipulating one or the other to have them intersect at a different place will deepen their understanding of solving different kinds of equations. It will also show them how they can change real world parameters to get a different desired result.

The Challenge

Using a tennis ball and CO2 car, model a thrilling stunt of a steel ball falling to the earth that barely misses hitting a car that’s speeding by.

The Hook

Fast and Furious© train scene

In this thrilling action scene, the two guys are drag racing their cars when a train is coming. What all had to be known by the people that developed this stunt to ensure that the actors would be completely safe from getting hit by the oncoming train?

Teacher's Guiding Questions
  1. How do we find the equation of the CO2 car?
  2. How can we change the equation of the car to determine the maximum distance the car can be from the drop zone?
  3. How long does it take the falling object to reach the ground?
  4. How do I ensure that I solve for the whole car passing the drop zone?
  5. What kind of equation will the CO2 car form? (Quadratic or Linear)
ACS (Real world applications; career connections; societal impact)

Real World Applications: Action movies are filmed all over the world and the actors/actresses need to be kept safe during any stunts that would be used.

Career Connections: The most obvious connection would be a stunt designer. However, all engineers need to be able to model a situation to predict an outcome.

Societal Impact: Movies generate income for the towns they are filmed in.

Engineering Design Process (EDP)

All steps of the EDP will be incorporated in lesson 3 activity 2. Students will identify and define the challenge. Then they will research methods to generate equations for the car and the falling ball. They will make a detailed drawing of how they will film the car and the ball being dropped to create equations. After generating an equation, they will model the stunt on a graph with both equations plotted. To minimize human error, a person should be placed midway between the ball being dropped and the C02 car being released to say go. 

Unit Academic Standard

A-CED.1 - Create equations and inequalities in one variable and use them to solve problems.

*A-REI.2 - Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

A-REI.4 – Solve quadratic equations in one variable

A-REI.7 - Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.

A-REI.11 - Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations.

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.

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

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.

S-ID.6 - . Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

a.       Fit a function to the data; use functions fitted to data to solve problems in the context of the data.


*  is for the students that are more advanced to solve the equation algebraically as well as graphically. 

Unit Activities

Lesson 1: Effects on parent graphs (3 days)

Students will study how parent graphs for linear and quadratic functions change as specific parts of the equation are changed. They will develop correct terminology to describe how different parts of the equation effect the graph.

Activity 1: Pretest.

Study the effects of how changes in the linear equation effect the graph.

Activity 2: Study the effects of how changes in the quadratic equation effect the graph

Lesson 2: Best Fit (2 days)

Students will plot data into their graphing calculators and find the line of best fit and best fit curve to describe the data. They will also look at the r2 number to determine the strength of the correlation.

Activity 3: Find the line of best fit

Activity 4: Find the best fit curve

Lesson 3: Challenge (5 days)

Students will analyze the video of a golf ball dropping to determine the equation for the ball in free fall. Then the class will discuss the Big Idea. The groups will come up with some essential questions from the big idea to guide toward creating an exciting stunt that could go into an action film. Students will be given the challenge and then begin their research and development of the solution to test.

Activity 5: Analyze dropping a golf ball

Activity 6: Issue Challenge

Where the CBL and EDP appear in the Unit

The CBL and EDP both appear completely in Activity 6

Misconceptions

Students will often mistake the drop of a golf ball to be linear since the ball drops so fast and the drop isn’t from too far of a distance to make more of the quadratic curve. Students can also mistake a negative r value as a bad correlation because it is negative.

Pre-Unit Assessment Instrument
Post-Unit Assessment Instrument
How to Make This a Hierarchical Unit

If this lesson was to be taught at a lower grade level, they would be advised to substitute the falling tennis ball and CO2 car for two constant velocity toy buggies. They could still plot data and draw a line of best fit free hand instead of using the graphing calculator. The goal would be to find where the two line intersected and have one of the cars change the distance slightly closer so they could pass by before the other car got to them.

Reflection

1) Did the students find a solution or solutions that resulted in concrete meaningful action for the Unit’s Challenge?

Yes, three of the eleven groups were completely successful (car missed getting hit by less than one meter). An additional five groups were somewhat successful (ball landed less than one meter before the car or less than three meters after the car).

2) Why did you select this content for the Unit?

This content lead students to thinking about manipulating the graph of a quadratic as well as using real data to generate equations using the graphing calculator and the r2 values.

3) What does the data indicate about growth in student learning?

The data indicated that students understood more about the content after the challenge than before the challenge. However, a closer view of the final tests indicated that the students still struggled to take information they needed to solve as a group to build toward a final solution for the challenge. The challenge needs to be modified to ensure that the students take more time to consider what each person in the group is doing to help solve the challenge so they would be able to handle that work on their own when asked to do so.

4) Was the purpose for selecting the Unit met?  If yes, provide student learning related evidence.  If not, provide possible reasons.

The entire purpose was not met. The entire unit was very involved and rigorous that even some of the more enthusiastic students were feeling depleted with the actual math it would take before they could test their model. More entrance points for the unit need to be developed so that students remain engaged and differentiation is more effective.

5) What would you change if you re-taught this Unit?

I am actively changing this unit. The unit started with attaching a safety device onto the CO2 car in order to save a person from falling debris. However, that would take too much time and take away from the focus on generating the equations using the R2 values and manipulating the graphs of the quadratics. I will be providing some students with video of the falling tennis ball in order to guide them toward how to create the equation for the CO2 car. The more advanced students will be required to show algebraic work when they solved for how back the car needs to be placed. However, lower achieving students will need to communicate the steps it would take to manipulate a graphing calculator to solve for that information.