Insights Into Algebra 1: Teaching for Learning
Linear Functions and Inequalities Lesson Plan 2: Hot Dog Sales – Solving Linear Equations and Inequalities
This lesson will provide students with an introduction to solving equations and inequalities numerically (using a table), graphically, and algebraically.
One 50-minute class period
Linear equations and inequalities
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.
Principles and Standards for School Mathematics, National Council of Teachers of Mathematics (NCTM), 2000:
NCTM Algebra Standard for Grades 6-8
NCTM Algebra Standard for Grades 9-12
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
Teachers Activities and Assignment
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.
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.
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.
Related Standardized Test Questions
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?
C. 56 (correct answer)
- Taken from the Virginia Standards of Learning (2002):Which is a zero of the function f(x) = 3x – 21?
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
Student Work: Hot Dog Sales
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.
Workshop 1 Variables and Patterns of Change
In Part I, Janel Green introduces a swimming pool problem as a context to help her students understand and make connections between words and symbols as used in algebraic situations. In Part II, Jenny Novak's students work with manipulatives and algebra to develop an understanding of the equivalence transformations used to solve linear equations.
Workshop 2 Linear Functions and Inequalities
In Part I, Tom Reardon uses a phone bill to help his students deepen their understanding of linear functions and how to apply them. In Part II, Janel Green's hot dog vending scheme is a vehicle to help her students learn how to solve linear equations and inequalities using three methods: tables, graphs, and algebra.
Workshop 3 Systems of Equations and Inequalities
In Part I, Jenny Novak's students compare the speed at which they write with their right hands with the speed at which they write with their left hands. This activity enables them to explore the different types of solutions possible in systems of linear equations, and the meaning of the solutions. In Part II, Patricia Valdez's students model a real-world business situation using systems of linear inequalities.
Workshop 5 Properties
In Part I, Tom Reardon's students come to understand the process of factoring quadratic expressions by using algebra tiles, graphing, and symbolic manipulation. In Part II, Sarah Wallick's students conduct coin-tossing and die-rolling experiments and use the data to write basic recursive equations and compare them to explicit equations.
Workshop 6 Exponential Functions
In Part I, Orlando Pajon uses a population growth simulation to introduce students to exponential growth and develop the conceptual understanding underlying the principles of exponential functions. In Part II, a scenario from Alice in Wonderland helps Mike Melville's students develop a definition of a negative exponent and understand the reasoning behind the division property of exponents with like bases.
Workshop 7 Direct and Inverse Variation
In Part I, Peggy Lynn's students simulate oil spills on land and investigate the relationship between the volume and the area of the spill to develop an understanding of direct variation. In Part II, they develop the concept of inverse variation by examining the relationship of the depth and surface area of a constant volume of water that is transferred to cylinders of different sizes.
Workshop 8 Mathematical Modeling
This workshop presents two capstone lessons that demonstrate mathematical modeling activities in Algebra 1. In both lessons, the students first build a physical model and use it to collect data and then generate a mathematical model of the situation they've explored. In Part I, Sarah Wallick's students use a pulley system to explore the effects of one rotating object on another and develop the concept of transmission factor. In Part II, Orlando Pajon's students conduct a series of experiments, determine the pattern by which each set of data changes over time, and model each set of data with a linear function or an exponential function.