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**Overview:**

This lesson will provide students with an introduction to solving equations and inequalities numerically (using a table), graphically, and algebraically.

**Time Allotment:**

One 50-minute class period

**Subject Matter:**

Linear equations and inequalities

**Learning Objectives:**

Students will be able to:

- Write a profit equation for a simple business.
- Solve equations numerically, graphically, and algebraically.
- Solve inequalities numerically, graphically, and algebraically.

**Standards:**

*Principles and Standards for School Mathematics*, National Council of Teachers of Mathematics (NCTM), 2000:

NCTM Algebra Standard for Grades 6-8

http://standards.nctm.org/document/chapter6/alg.htm

NCTM Algebra Standard for Grades 9-12

http://standards.nctm.org/document/chapter7/alg.htm

**Supplies:**

Teachers will need the following:

- Copies of the Coach’s Questions handout

Students will need the following:

- Graphing calculator
- Graph paper
- Large sheets of poster paper

**Steps**

**Introductory Activity:**

**1.** Explain that the class has been asked to help the football coach determine the practicality of selling hot dogs to raise funds to buy new jerseys.

**2.** Explain that the coach needs $450 for start-up costs and plans to sell the hot dogs for $0.50 each. The start-up costs include the purchase of 2,500 hot dogs, buns and condiments, and wages for the vendors.

**3.** Ask the class to generate a list of questions that the coach might ask himself to determine how well his plan will work. Record the students’ questions. Possible questions include: How many hot dogs will he have to sell before he makes a profit? How many hot dogs can reasonably be sold at the football game? How much does the coach need to make to pay for the jerseys? Is $0.50 a reasonable price to charge per hot dog?

**4.** Discuss the student questions, and save them for use at the end of the lesson.

**Learning Activities:**

**1.** Tell students that for now the class will focus on answering the question of how many hot dogs the coach needs to sell in order to make a profit.

**2.** Solicit student thoughts about the number of hot dogs he needs to sell to break even.

**3.** Ask the class to devise a table that would produce information about the problem. Students should state that the table should show the number of hot dogs sold, the revenue made selling them, and the profit earned by selling them.

**4.** Have students create the table, starting with 100 hot dogs sold and increasing by increments of 100. The table should continue until students find the break-even point. The table should look like the one shown below:

**5.**Ask students to write formulas that can be used to find the values for both the revenue and the profit.

**6.**Discuss students’ formulas for each of the columns. Examples may include:

- Revenue = 0.50 x number of hot dogs sold
- Profit = revenue – start-up costs.

Elicit that the formulas can also be written as: R = 0.50H and P = 0.50H – 450.

**7.** Inform students that to use a graphing calculator, the formulas must be written using y and x for variables. This means the profit formula would be written in the form y = 0.50x – 450. Discuss the meaning of the 0.50 and the 450 in the equation.

**8.** Have the whole class create an electronic table using the calculator. To do this, type the equation into the “y =” area of the calculator. Then, set the table on the calculator to match the table the students made on their papers.

**9.**Students should note that the table they made using the calculator matches the table they made on paper. They should also notice that the calculator’s table continues indefinitely. Tell them they just solved this problem using a numeric method that looks at tables of values. Note that the break-even point occurs when 900 hot dogs are sold.

**10.**Students will now find the break-even point using a graph. Discuss possible values that they should select for the graph’s scale. Possibilities include: xmin = 0, xmax = 2500, xscl = 100, ymin = -500, ymax = 800, and yscl = 100.

**11.** Ask students to determine which point on the graph represents the break-even point. They should recognize that this is the point where the line crosses the x-axis; it is located at (900, 0), and it represents the fact that 900 hot dogs sold will bring a profit of $0.00. Tell them they just solved the problem using a graphical method.

**12.**Students will now find the break-even point using algebra. Elicit from the class that if they want to find the break-even point, then they should let P = 0, and then solve the equation for H. For example:

- P = 0.50H – 450
- 0 = 0.50H – 450
- 450 = 0.50H
- 900 = H

**13.**Point out that the three different methods of solving this problem all produced the same solution. Ask students which method they preferred and why.

**14.**Now, hand out the activity sheet that lists the coach’s questions for the class. The students should work together in groups, answer the questions and prepare a presentation for one of the questions. The students will need to answer all of the questions by solving the problems using tables, graphs, and algebra. Two of the questions involve solving an inequality, so students can see that these three methods of solving equations also work for solving inequalities.

**15.**Have students present their findings to the class.

**Culminating Activity/Assessment**

For homework, ask students to answer the questions the class generated at the beginning of the lesson. They should also find the break-even point for the coach if he lowered his start-up costs by $100 and charged $1.00 for each hot dog.

The questions below dealing with linear equations and inequalities have been selected from various state and national assessments. Although the lesson above may not fully equip students to answer all such test questions successfully, students who participate in active lessons like this one will eventually develop the conceptual understanding needed to succeed on these and other state assessment questions.

- Taken from the California High School Exit Examination (Spring 2002):

In the inequality 2x + $10,000 $70,000, x represents the salary of an employee in a school district. Which phrase most accurately describes the employee’s salary?

**A. At least $30,000 (correct answer)**

B. At most $30,000

C. Less than $30,000

D. More than $30,000 - Taken from the Virginia Standards of Learning (2002):

What is the solution of 2x + 3 x – 5?

A. x -8/3

**B. x -8 (correct answer)**C. x -2/3

D. x -2 - Taken from the North Carolina End-of-Course Test, Algebra 1 (October 2003):

The drama club is selling tickets to a play for $10 each. The cost to rent the theater and costumes is $500. In addition, the printers are charging $1 per ticket to print the tickets. How many tickets must the drama club sell to make a profit?

A. 54

B. 55

**C. 56 (correct answer)**

D. 57 - Taken from the Virginia Standards of Learning (2002):Which is a zero of the function f(x) = 3x – 21?

A. -21

B. -7

C. 0

**D. 7 (correct answer)** - Taken from the Virginia Standards of Learning (2002):

What is the solution to

?

A. d = -243

**B. d = -45 (correct answer)**C. d = -3

D. d = -5/9

**Teacher Commentary:**

This is the beginning of the year, and students have had very little experience with inequalities. After reviewing students’ work, I saw that I have to take time to review inequalities. I noticed that one student confused the “greater than” sign with the “less than” sign. I also observed that the students need to work on graphing inequalities. I will address both of these issues in a future lesson.