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 real-world 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

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.


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.

(#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.


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.