Title III Technology Literacy Challenge Grant

Learning Unit

LU Title: Parabolas in our Lives	Author(s): Scott Ranani
Grade Level: Algebra 1, Algebra 2, Course 2	School : Oneida High School
Topic/Subject Area: Quadratic Equations/ Mathematics	Address: 560 Seneca St. Oneida, NY 13421
Email: sranani@oneida-high.moric.org	Phone/Fax: 315-363-6901

OVERVIEW

This three week unit introduces students to quadratic equations by having the students find the equation of the axis of symmetry and the coordinates of the vertex. Students graph quadratic functions, compare different parabolas, and approximate the roots of quadratic functions. Students will also be able to solve quadratic equations using the quadratic formula.

CONTENT KNOWLEDGE

Declarative

Procedural

Identify a quadratic function written in standard form.

Identify the vertex on a parabola.

Identify the axis of symmetry.

Know the formula for the axis of symmetry.

Define roots of a quadratic equation.

Identify the minimum or maximum in a parabola.

Know the quadratic formula.

Use a graphing calculator to graph a quadratic function and find the coordinates of its vertex.

Find the equation of the axis of symmetry.

Graph and interpret quadratic functions

Use estimation to find roots of quadratic equations.

Solve quadratic equations by using the quadratic formula.

ESSENTIAL QUESTIONS

How can a real situation in your life be represented by a equadratic equation?

How do certain professionals in the work force use quadratic functions?

Explain the construction of objects with a parabolic shape.

CONNECTIONS TO NYS LEARNING STANDARDS
List Standard # and Key Idea #: Write out related Performance Indicator(s) or Benchmark(s)

INITIATING ACTIVITY

Students will view segments of the video "Home Alone 2." Students will identify parabolic motion (falling objects) in certain scenes and measure the times of the falls and approximate the distances of the falls. They will also try to conclude if the distances approximated are realistic.

LEARNING EXPERIENCES
In chronological order including acquisition experiences and extending/refining
experiences for all stated declarative and procedural knowledge.

Declarative Knowledge

What declarative knowledge should students be in the process of acquiring & integrating? As a result of the unit, the student will know or understand…

What experiences or activities will be used to help students acquire & integrate this knowledge?

What strategies will be used to help students construct meaning, organize and/or store the knowledge?

Describe what will be done.

Identify a quadratic function written in standard form.

Identify the vertex on a parabola.

Identify the axis of symmetry.

Know the formula for the axis of symmetry.

Define roots of a quadratic equation.

Identify the minimum or maximum in a parabola

Know the quadratic formula.

Bulletin Board

Lecture

Notes

KWL*

3 minute pause*

Inquiry*

Reciprocal teaching*

On a KWL chart, we will generate the Know and the Want related to quadratic equations. We will fill in the Learn as we complete each experience.

Several times throughout the lecture, I will stop and ask the students to identify various vocabulary words used in the lesson.

I will provide the student with various examples of different types of parabolas. Students should make hypotheses about the vertex, axis of symmetry, roots, and the minimum or maximum. A discussion will follow.

A student leader will summarize each lesson, and will ask questions based on the lesson. The student leader will try to clarify the confusing points.

Procedural Knowledge

What procedural knowledge will students be in the process of acquiring & integrating? As a result of this unit, students will be able to:

What will be done to help students construct models, shape & internalize the knowledge?

Describe what will be done.

Use a graphing calculator to graph a quadratic function and find the coordinates of its vertex.

Find the equation of the axis of symmetry.

Graph and interpret quadratic functions

Use estimation to find roots of quadratic equations.

Solve quadratic equations by using the quadratic formula.

Think-aloud*

Similar to … different from…

Set of written steps

Practice with variation

Student think-aloud*

Practice with variation

Similar to… and different from…

Set of written steps

Practice with variation

Think-aloud*

Practice with variation

Common errors and pitfalls

Use a think-aloud process to demonstrate how to use the graphing calculator to graph a quadratic function and find its vertex.

Students will recall graphing linear functions. Using a Venn Diagram students will compare and contrast the similarities and differences in the process of graphing lines and parabolas.

We will create a set of written steps in order to graph a quadratic function and find the coordinates of the vertex.

Working in groups, students will graph various quadratic equations using the graphing calculator. They will also find the coordinates of the vertex.

Students will demonstrate how to graph quadratic equations and find the coordinate of the vertex.

Using the students’ parabolas, we will observe that parabolas are symmetrical and a line of symmetry exists. In groups, students will try to figure out the equation for the line of symmetry and generalize a rule or formula. After a class discussion students will be given the formula.

Working in groups, students will receive a variety of quadratic equations. They should find the axis of symmetry and the coordinates of the vertex of the parabola.

Students will graph several quadratic equations. The student should discuss the differences and similarities between parabolas. Students should be able to identify whether a parabola has one root, two roots, or no real roots.

We will construct a set of written steps in order to graph quadratic functions.

Students will be given a variety of quadratic equations to solve by graphing. They should use the graphing calculator and ZOOM IN on the zero. They should also use the ROOT feature on the CALC menu. Student should approximate the value of the roots.

Student will be given a quadratic equation. Students will observe a model use of the quadratic formula.

They will practice several examples.

Students will be made aware of common errors of computation.

Extending and Refining

What knowledge will students be extending and refining? Specifically, they will be extending and refining their understanding of…	What reasoning process will they be using?	Describe what will be done.
Graphing quadratic equations and linear equations	Comparing	Students have learned the process of graphing linear equations. Students will extend that process in graphing quadratic equations. Student will compare and contrast the similarities and interpret graphs in different situations.
Axis of Symmetry	Inductive Reasoning	After the students have seen several graphs of quadratic equations, students will induce the formula for finding the equation of the axis of symmetry.

Planning Guide

Unit: Quadratic Equations

Step 1

Step 2

Step 3

What knowledge will students be using meaningfully? Specifically, they will be demonstrating their understanding and ability to…

What reasoning process will they be using?

Describe what will be done.

The concept of quadratic equations and parabolas

Parabolas are found in a variety of areas.

[ ] Decision Making
(selecting from seemingly equal alternatives or examining the decisions of others)
[ ] Problem Solving
(seeking to achieve a goal by overcoming constraints or limiting conditions)
[ ] Invention
(creating something to meet a need or improve on a situation)
[ ] Experimental Inquiry
(generating an explanation for a phenomenon and testing the explanation)
[ ] Investigation
(resolving confusions or contradictions related to a historical event, a hypothetical past or future event, or to the defining characteristics of something)
[x] Systems Analysis
(analyzing the parts of a system and how they interact)
[ ] Other:

The reflective surface of headlights is designed in the shape of a parabola so that light is directed in a straight line. The light originates from the focus and is reflected outward in a line parallel to the axis of symmetry.

Using graph paper, draw a model of a headlight shining straight ahead. Use colored pencils to draw several paths of light.

Use tracing paper to trace your model. Rotate the tracing slightly (approximately 1 degree) clockwise about the focus. What happens to the direction of the light rays? What happens when you rotate the tracing one-degree counterclockwise?

This is what happens when headlights shift out of alignment. An oncoming driver’s visibility is reduced by 25% if a headlight is aimed one degree too high. Visibility is reduced 50% if the headlight is aimed one degree too low.

Student should name some other kinds of lights that use parabolic reflectors. Why is the parabola necessary?

CULMINATING PERFORMANCE
Include rubric(s)

1. Students will work in groups of six, which will be divided up in two teams of three. At the top of a blank piece of paper, each team member describes one feature of a parabola.

One student writes about the location of the vertex.

Another student writes about the location of the line of symmetry.

The third student writes about either the location of the y intercept, or the number or location of the x-intercept.

Each team passes its three pieces of paper to the other team. Below the description of the feature, both teams write at least three equations whose graphs have the feature described. When both teams have finished, check each other’s work and correct any errors. (This acts as a review not a formal assessment.)

2. Students will be given the formulas for the height versus time of the three objects listed below.

arrow h = -16t² + 112t + 0

baseball h = -16t² + 64t + 0

football h = -16t² + 64t + 75

Graph the three equations using a graphing calculator. How are the graphs similar? How are they different? What causes the difference?

3. If a baseball player throws a ball straight up in the air at 64 feet per second, the equation for the throw will be y = 64x - 16x². Since we are not interested in the negative values for x (negative time), adjust the range of the x values to x_min = 0 seconds and x_max= 10 seconds. Set the range of y values to y_min= 0 feet and y_max = 100 feet.

Graph the equation on your calculator and answer the following questions: (Note: you may need to move your cursor around and "zoom" as needed to answer some of these questions.)

1. What is the maximum height reached by the baseball?

2. How many seconds does it take to reach this maximum height?

3. How long does it take for the ball to reach the ground?

Now suppose a stronger player steps up and pitches a ball straight up with a greater speed than the first, namely 70 feet per second.

4. Write the new equation for the stronger player.

5. What is the maximum height of this throw?

6. How long does this throw stay in the air?

7. Describe the effect of changing the upward speed on a) the graph, b) the maximum height, c) the time of flight. What do you think would happen for a slower upward throw, say 50 feet per second?

8. On the moon, the gravity is roughly 1/6 that on Earth. The equation for the first player throwing the ball on the moon is y = 64x -2.67x². How high could the astronaut throw the ball on the moon?

4. The Jones family has a garden that measures 15 feet by 30 feet. They want to increase the area of the garden by 250 square feet. If they increase the length and the width by the same amount, what would amount, what would be the new dimension of the garden?

Molly and Jeff are in driver’s education class. Molly says, "My driver’s education book contains this formula for the distance a car travels once the driver has decided to stop the car. The formula is d=0.05s² + s where d is the stopping distance in feet and s is the speed in miles per hour of a car." Jeff explains that his father told him to use this rule of thumb, "Leave one car length between your car and the car in front for every 10 miles per hour of speed." (One car length represents 15 ft.) How fast should you go if the car in front of you is 4 car lengths in front of you? Is Jeff’s father’s rule of thumb accurate?

Rubric

1	2	3	4
Misses key points.	Explanation is unclear.	Good solid response with clear explanation.	A complete response with a detailed explanation.
No diagram or sketch.	Inappropriate or unclear diagram.	Clear diagram or sketch.	Includes clear diagrams or sketches.
Major math errors or serious flaws in reasoning.	May be some serious math errors or flaws in reasoning.	No major math errors or serious flaws in reasoning.	No significant math errors.
No response	Response shows some understanding of the problem.	Shows substantial understanding of the problem, ideas, and processes.	Shows complete understanding of the questions, mathematical ideas, and processes.
Does not meet the requirements.	Does not meet the requirements.	Meets the requirements of the problem.	Goes beyond the requirements of the problem.
No counterexamples	No counterexamples	No counterexamples	Includes counter examples.

PRE-REQUISITE SKILLS

Linear Relationships

Coordinate Geometry

Apply Formulas

Basic Graphing Calculator Skills

UNIT SCHEDULE/TIME PLAN

3 weeks

TECHNOLOGY USE

Graphing calculator

Computer

Resources

*Dimensions of Learning

Cord Applied Mathematics