To Deal Or Not To Deal 

Designed by: Barbara Roach       School: Keenan High 

Grade Level: 8-10      Subject: Algebra One

Competency Goal /Objective(s): Use quadratic equations to model practical situations. (Al111C) 


Overview: This lesson provides a practical application of quadratic equations. The lesson includes exploratory activities (individual and collaborative-pairs), teacher-led discussion and whole-class discussion and brainstorming.

Students will study equivalent expressions of quadratic relationships through an exploration of area of rectangles. Students will investigate how increasing one dimension of a rectangle and decreasing the other dimension affects the area of the resulting rectangle and will use the knowledge gained to move from general statements to specific equations. Ultimately, students will develop quadratic equations that model a real-life situation.

Focus/Essential Questions:  

1. If you transform a rectangle by increasing or decreasing the length of the sides of the rectangle, what effect does the transformation have on the new figure?

2. How do you solve practical area problems by using quadratic expressions?

Time Frame: One 90-minute class period.

Resources:
Health Algebra One, Health Publishing Company, 1997
Calculators
Land Deal, What's the Big Deal? And Is It a Deal? Transparencies
Worksheets - What's the Big Deal? and Is It A Deal?
Ticket to leave rubric



Assessment 

Informal - Observation, questioning, investigation responses 
Ticket to Leave (summarizer) - Graded after class by teacher 

Instructional Activities: 

Activator: 
Teacher will launch lesson by displaying the Land Trade Proposal of Richland One Mall Corporation transparency and reading the proposal to the class. 

Teacher will ask: Is the land trade proposal a fair trade? 

(NOTE: Solicit reaction, not computations.) 

Teacher will say: To determine if this transaction is a fair trade, we are going to perform an investigation. 


Demonstration with Discussion: 

Teacher will draw a square and a rectangle on the overhead or board. The square has sides of 3 meters and the rectangle's length is 2 meters longer than the square and the width is 2 meters shorter than the square. 

Ask: What are the areas of the two plots of land? 

• The square has area of 9 sq. m. 

• The rectangle has area of 5 sq. m. 

• The perimeters of the figures are the same, 12 meters. 
  
  

Small Group Activity: 

In collaborative pairs, using the What's the Big Deal Worksheet, students will complete and record information for values 1 through 7. (NOTE: The table will help students to see patterns that will tell them more about the Richland One Mall Land Deal as well as similar deals.) 

Once all pairs have finished, the teacher will display the What's the Big Deal? Transparency. Groups will share answers and selected students will enter data on transparency. 

The teacher will ask: What patterns do you see in the table? 

• The area of each square is the square of a side. 

• The area of the square increases in the patterns 7, 9, 11, 13, ... 

• The dimensions of the rectangle each increase by 1 from one entry to another. 

• The area of the rectangle increases in the pattern 7, 9,11,13, ... 

• The area of the rectangle is always 4 less than the area 
  of the square. 
  
  

The teacher will demonstrate another example with a square with length of 8. 

Ask: Are the patterns the same? 

Students will add data to table. 

Individual Practice: 

Working independently, students will find the values corresponding to a square of sides of 100 meters. Teacher will circulate and give assistance, if needed. 

Small Group Activity: 

Working in pairs, students will discuss and answer question B. Each pair will report their answer and explanation to the class. 
Ask: Can you find a general expression for each column? Allow pairs to discuss and answer question C. 

(NOTE: Translating specific into general is difficult for some students. If so, work through the process with students. 

Some students may write n² - 4 for the area of the new rectangle. If so, test the two expressions,  (n + 2)(n - 2) and n² - 4, to see whether they are the same value for a given value of n.) 

Allow pairs to finish. 

Ask: What is a general expression for the area of the square? (n² ) In general, how does the area of the rectangle compare to the area of the square? (The area of the rectangle is 4 square meters less than the area of the square.) Using n² , how would you write an expression for the area of the rectangle? (n² - 4) 

Teacher will write expressions on the overhead or board. 

Ask: Are the expressions, (n +2)(n - 2) and n² - 4 equivalent? 
(Yes) Discuss. 

Individual Practice: 

Students will complete Is it a Deal? Worksheet. 

Teacher will monitor and give assistance to students as they complete the worksheet. 


Class Discussion: 

Teacher will display Is it a Deal? Transparency, solicit answers from students, and enter data on transparency. 

Ask: How do the equations of this activity compare to those of What's the Big Deal? Worksheet. Suppose you trade a square piece of land for a rectangular lot that is 6 meters longer on one side and 6 meters shorter on the other side than a square lot. Without completing a table, how will the area of the new rectangular lot compare to the area of the original square lot? 

Summarizer - Ticket to Leave 

Written response: Is the land deal proposal of Richland One Mall Corporation fair? Students should explain and use examples and/or illustrations. Display Land Trade Proposal of Richland One Mall Corporation transparency and allow students to write their responses. (Students will turn in responses at end of class and teacher will score later using the rubric.) 

Extension: Will the land deal be fair if the original square is changed by only one dimension? Defend your answer. 



8) Check all that apply: 

Name _______________________________   Date _____________ 

What's the Big Deal? 

You are the owner of XYZ Properties and own a square piece of land that you would like to trade for a rectangular lot. The length of a side of the square lot is x. The length of the rectangular lot is 2 meters longer and the width is 2 meters shorter than a side of the square lot. 

A. Complete the table below. 

B. For side lengths (3-7), how does the area of the rectangular lots compare with the areas of the square lots? For which side lengths, if any, is this a fair trade? 



C. Assume the side length of the square lot was x meters. For each column in the table, write expressions for the values in each column in terms of x. For example, the expression for the area is x². 

Name ________________________________    Date _____________ 

Is it a Deal? 

A square has sides of length y feet. A new rectangle is created by increasing one dimension by 4 feet and decreasing the other dimension by 4 feet. 

1. Make a table showing the area of the square and the area of the new rectangle for whole number y values from 5 to 10. 









2. Compare the area of the squares with the areas of the rectangles. What do you notice? 




3. Write an equation for the area of the square and an equation for the area of the new rectangle? 



4. Compare the equations of this activity with the equations for the areas with fixed perimeters for the activity, What's the Big Deal? How are the equations similar? How are they different? 

Rubric for TICKET TO LEAVE