Project AMP
Dr. Antonio Quesada – Director, Project AMP
Abstract
Grade/Subject: 9-12/Algebra
Time Frame: 3 – 40 minute periods
Strand:
Algebra
Topic:
Applying Quadratic Equations to Find Maxima – Using
Numerical, Graphical, and Algebraic approaches
Objectives:
This lesson will allow a student and instructor to use the table and
graphing capabilities of a graphing calculator and the algebraic
method to find the extrema of quadratic equations in context.
Materials:
Graphing calculator, overhead display, handout (included)
Authors:
Steve Donaldson
Prerequisite: Standard form of a quadratic equation
Zeros of a function
Area of a rectangle
Graphing calculator features: table and graphing
Concept
1. A parabola has either a maximum or a minimum; 2. For every maximum or
minimum y value, there is a corresponding x value; 3. Numerical, graphical, and
algebraic are all valid methods of solving a problem.
Learning Objectives
1. Students will construct and interpret tables and graphs as they relate to realworld mathematics (Measurement strand); 2. Students will develop graphical
techniques of solution for problem situation involving functions (Algebra strand)
Project/Task
Students will be guided through a given problem like finding area of a rectangle
or other similar problems. They will engage in individual and group work using
three methods of solution (numerical, graphical, algebraic). As a final assessment
the students will be given an opportunity to solve a quiz problem giving the
answer and a written description of their process of solution.
Assessment
Informal observation and feedback (individual and group)
Demonstration of each method by student
Pencil and paper
Project AMP
Dr. Antonio Quesada – Director, Project AMP
Learning Strategies
This lesson is designed for an Algebra II class following the topics of parabolas
and writing a quadratic equation in standard form. Approximate time: 90 minutes
(2 periods). Give the students the first example on the overhead (problem #1 on
the enclosed assignment sheet) and have them draw a picture and write an
equation. They should then be given time to attempt each type of solution
(numeric – table – and graphical) with the instructor going over each part after 5
or so minutes. Following the second type, the students may be shown or
requested to write the equation in standard form (depending on their ability) and
can draw the answer from it (algebraic). The students can be given more
problems to work on in small groups (2-3 students) allowing time to observe and
address weakness and acknowledge strengths.
Classroom and Information Management
This lesson is intended for rooms where students may view and overhead display
and be able to move into groups (individual desks are ideal). Students should be
at similar ability levels and instructor should be competent with demonstrating the
examples. Allow ample time for students to attempt problems on their own.
Sharing
Students are asked to find area and share methods of solution with classmates in
discussion and in small groups.
Results
Tables on the graphing calculator; graphs on the graphing calculator; evidence of
knowledge on paper both mathematical and narrative.
Tools and Resources
Overhead and screen, Graphing calculator with overhead display, Worksheet with
sample problems, Quiz
Do & How
The graphing calculator will be used to graph a table of values for a quadratic
equation. The graphing calculator will be used to graph the function and trace,
zoom, and/or maximum functions will be used.
Project AMP
Dr. Antonio Quesada – Director, Project AMP
Name
The following problems involve a quadratic relationship. In each problem, you
should draw a picture, set up an equation, and answer the question.
1. A landowner desires to build a rectangular fence to keep his land free from wild
animals. He has 200 feet of fencing with which to build this perimeter fence. What
is the maximum area his fence can contain?
a. Draw a picture and set up an equation.
b. Using the table feature of your calculator, find the dimensions where the
maximum area occurs, and give that maximum area.
c. Using the graph feature, verify the dimension where the maximum area occurs.
d. Put the equation in standard form. What is the vertex of the parabola?
e. How does the vertex in d. relate to the answers from b. and c.?
Solve the following problems in groups.
2. A farmer wishes to enclose a pasture to prevent his cows from running off. If he has
enough fence to form a rectangular perimeter of 540 years, what is the maximum area
of the pasture he could enclose, and what are its dimensions?
a. Draw a picture and set up an equation.
b. Solve using the table.
c. Verify using a graph.
d. Verify by putting the equations into standard form.
3. A rancher has been having problems with sheep running off, wolves wandering onto
his pastures, and various other critters causing problems. He decides to put up a
rectangular fence. Along one side of the pasture flows a river forming a natural
boundary. Knowing he only needs to construct three more sides of the rectangle,
what would the dimensions be that give him maximum area using 437 meters of fence
(nearest tenth of a meter)?
a. Draw a picture and set up an equation.
Project AMP
Dr. Antonio Quesada – Director, Project AMP
(#3 cont’d.)
b. Solve this problem using a graph.
c. Verify algebraically.
d. Verify using a table.
4. Susie has a stand where she sells T-shirts. She has on hand 129 shirts. For each $1
increase in price, she loses 4 sales. At what value should she set the price to get
maximum income and what is that income?
a. Set up an equation.
b. Solve using a graph.
c. Verify using a table.
d. Verify using standard form.
5. A street vendor in Cleveland sells a certain sports team’s jerseys for $20. During any
given hour, he can expect to sell 14 jerseys at that price. For every $1 that he lowers
the price, he can expect to sell 1 more jersey in an hour. By how much should he
decrease the price to ensure maximum income per hour?
a. Set up the equation.
b. Solve by using standard form.
c. Verify using a table.
d. Verify using a graph.
e. What amount will he pull in per hour?
6. Make your own word problem. Trade with someone in your group to have them
solve it. Make sure you have them verify with another method.
Project AMP
Dr. Antonio Quesada – Director, Project AMP
Quiz
Name
For the following problems, solve each using a graph, a table, or algebraic standard
form. Do not use the same method for both. After solving, describe how you solved
it.
1. Mark recently bought a rectangular lot that he wishes to enclose with electric fence.
He has enough material to put up 220 yards of fence. What is the maximum area that
he can enclose?
a. Solve.
b. What method did you use?
c. Describe in detail how you solved this problem.
2. Mary inherited a popcorn box company from her grandfather. The boxes are sold by
the carton. Due to increased competition for every $1/carton increase in price, she
loses 12 carton sales. If she can keep in stock 1000 cartons per day, at what price
should she set a carton to ensure maximum income for that day?
a. Solve (remember to use a different method than in #1). What is the maximum
income that she could realize? At what price?
b. What method did you use?
c. Describe in detail how you solved this problem.