Linear Functions and Inequalities

Explanation

Linear functions model a wide variety of real-world situations, including predicting the cost of a telephone call that lasts a given amount of time, the profit of a hot dog stand, and the amount of tax paid for a given income. Linear functions arise when there is a constant rate of change. Students should solve problems in which they use tables, graphs, words, and symbolic expressions to represent and examine linear functions and linear patterns of change.

Linear functions can be written in three different forms, as shown in the table below. Each form provides different information about the function.

Students should be able to recognize a linear function in a table of values, a graph, or in algebraic form. When studying linear functions graphically, students should understand that the slope of the line represents a constant rate of change for the function, and that the y-intercept is the point where the graph crosses the y-axis and often represents the initial condition or starting point for the function. Through practical experience solving linear function problems in context, students will develop an understanding of the concepts and real-world meanings of the slopes and y-intercepts of lines.

Other examples:

• The linear function F = 1.8C + 32 can be used to convert temperatures between Celsius and Fahrenheit.
• If a utility company charges a fixed monthly rate plus a constant rate for each unit of power consumed, a linear function will show the monthly cost of power. If the fixed rate is $25, and the cost for each unit of power is $0.02, the linear function is C = 0.02P + 25.
• The linear function I = 400C + 1,500 yields the total monthly income of a car salesman who makes a monthly base salary of $1,500 and receives $400 dollars for each car sold.

A linear function, whose graph is a line, can be written in the form y = mx + b, where m and b are constants and m 0. As with any function, students can represent a linear function as a table, an equation, or a graph.

Linear functions are fundamental to the study of mathematics. Students can transfer many of the important concepts learned through the study of linear functions to the understanding of other functions. According to the National Council of Teachers of Mathematics (NCTM):

It is essential that [students] become comfortable in relating symbolic expressions containing variables to verbal, tabular, and graphical representations of numerical and quantitative relationships. Students should develop an initial understanding of several different meanings and uses of variables through representing quantities in a variety of problem situations.

(NCTM, Principles and Standards for School Mathematics,
2000, p. 223)

Students should be able to move fluently between the different representations of linear functions and, given a description of a situation, should be able to produce a table, equation, and graph. Likewise, when given one representation of a linear function, students should be able to produce the others. To do this, they need ample opportunity to explore situations involving linear functions in all representations. For example, in the video for Workshop 2, Part I, Tom Reardon started with a verbal description and a table of values. The students then produced an equation and a graph to further describe the situation. They discussed the meaning of the constants in the equation and how those constants affected both the table and the graph.

See what Diane Briars has to say about the important aspects of linear functions that Tom Reardon applied in his lesson:

When the students were talking about the slope and they said that 24 cents was the slope, he didn’t stop there. He asked students to look for any pattern they saw in the table and they could see that there was a 24-cent difference between each of the y-values in the table. So he was connecting the slope back to the incremental change in cost, which was a really nice connection. The other thing that made this a rich problem is that he was able to ask students to make the connections and put meaning to those numbers: what does slope mean in this particular situation, what does the intercept mean in this situation? He was able to go back and say, let’s make it real, and let’s make it concrete.

Linear equations and inequalities can help solve a wide range of problems, including predicting the cost of a phone call that takes a given number of minutes and predicting the number of hot dog sales necessary to make a profit of $100, or a profit of at least $250.

Other examples:

• To find the Celsius equivalent of 77 degrees Fahrenheit, solve the linear equation 77 = 1.8C + 32.
• If a utility company plan uses the formula C = 0.02P + 25 to calculate monthly charges, a family can determine how many units of power they can consume to pay no more than $50 per month by solving 0.02P + 25 50.
• If a car salesman makes a monthly base salary of $1,500 and receives an additional $400 dollars for each car sold, the equation 400C + 1500 4,700 gives the number of cars that he must sell to earn a monthly income of at least $4,700.

In many classrooms, students learn only the algebraic manipulations necessary to solve problems. It is important for them to know that they can also solve linear equations and inequalities using tables and graphs. Below, note the three different ways of solving the equation 0.24x + 0.85 = 6.13. In this problem, x represents the length (in minutes) of a phone call that costs a total of $6.13. To use the table and graph method, let y = 0.24x + 0.85 where y represents the total cost of a call lasting x minutes.

Conclusion: A 22-minute phone call costs $6.13.

Solving Algebraically
Students solve the equation by choosing equivalence transformations to apply to both sides of the equation. First, they subtract 0.85 from both sides of the equation and simplify the results. Then, they divide both sides by 0.24 and simplify the results. It is important to note that this process only works for linear equations. When students study other functions, they must learn the algebraic techniques that allow them to produce the solution to that particular function.

Solving With Tables
Students enter the equation y = 0.24x + 0.85 into a graphing calculator. Looking at the table of values, they can identify the x-value that produces the desired y-value of 6.13. That occurs when the x-value is 22. Essentially, the table is a quick and efficient guess-and-check method of solution that works for any type of function. To refine guesses, students can adjust the increments in the table.

Solving With Graphs
Students enter the equation y = 0.24x + 0.85 into the calculator. Looking at the graph of the line, they can identify 22 as the x-value that produces the desired y-value of 6.13. One way to find the desired value is to use the calculator’s trace function to follow the line to the desired point. Like the table, the graph is a quick and efficient guess-and-check method of solution that works for any type of function. To refine guesses, students can also graph y = 6.13 and find the x-value where the two lines intersect.

While students should understand and make connections between all three solution methods, the tabular and graphic methods are particularly helpful to students who find symbolic manipulation difficult. (This will be explored fully in the “Rule of Four” section of Workshop 5.)

A linear equation is an equation that can be written in the form Ax + By = C and whose graph is a straight line. An inequality is an open sentence that contains the symbol , , >, or <.

The following definitions come from a mathematics dictionary:

Linear equation or expression: An algebraic equation or expression which is of the first degree in its variable (or variables); i.e., its highest degree term in the variable (or variables) is of the first degree. The equations x + 2 = 0 and x + y + 3 = 0 are linear.

Inequality: A statement that one quantity is less than (or greater than) another. If a is less than b, their relation is denoted symbolically by a < b; the relation a greater than b is written a > b. Inequalities have many important properties … An inequality which is not true for all values of the variables involved is a conditional inequality; e.g., (x + 2) > 3 is a conditional inequality, because it is true only for x [values] greater than 1.

(Source: James, Robert C. and Glenn James. Mathematics Dictionary [4th edition]. New York: Van Nostrand Reinhold, 1976.)

Solving linear equations and inequalities is a major focus in algebra. Students should be proficient in solving equations and inequalities algebraically, as well as with tables and graphs. According to the National Council of Teachers of Mathematics (NCTM):

Algebra … should provide students with insights into mathematical abstraction and structure. Students should develop an understanding of the algebraic properties that govern that manipulation of symbols in expressions, equations, and inequalities. They should become fluent in performing such manipulations by appropriate means – mentally, by hand, or by machine – to solve equations and inequalities, to generate equivalent forms of expressions or functions, or to prove general results.

(NCTM, Principles and Standards for School Mathematics, 2000, p. 297)

NCTM further states that students need a solid understanding of the order of operations and the distributive, associative, and commutative properties in order to become fluent with symbol manipulation (PSSM, 2000, p. 227). See the Workshop 2 video for examples of how two algebra teachers promote this in their classrooms.

Read Fran Curcio’s thoughts about the important aspects of solving equations and inequalities that Janel Green applies in her lesson in the video for Workshop 2, Part II:

Mathematics is such a wonderful subject that allows students to explore expressions and relationships in a variety of ways. In some cases, it may be appropriate to express a relationship in a graph, and there are questions that can be answered using information in graphical form. There are different questions that can be answered easily by using the relationships expressed in a table. The generalization of particular patterns in relationships expressed in algebraic form also allows for different questions to be answered. Having these tools at their fingertips and the ability to interpret these three representations gives students power in applying their mathematical content knowledge.

The links below are to pages within stable sites and are current as of the date of publication of this workshop. Due to the ever-changing nature of the Web, it is possible that some links may change. Should you reach a non-working link, we recommend entering a couple words from its description into the site’s search function, or into a Web-based search engine.

Related Standards:

NCTM Middle Grades Algebra Standard
http://standards.nctm.org/document/chapter6/alg.htm
This Web page describes what students should know and be able to do algebraically in grades 6-8, and offers suggestions for the type of classroom activities necessary to develop conceptual understanding.

NCTM High School Algebra Standard
http://standards.nctm.org/document/chapter7/alg.htm
This Web page describes what students should know and be able to do algebraically in grades 9-12, and offers suggestions for the type of classroom activities necessary to develop conceptual understanding.

Worthwhile Mathematical Tasks:

A Research Companion to Principles and Standards for School Mathematics.Reston, VA: National Council of Teachers of Mathematics, 2003.

Assessment Standards for School Mathematics. Reston, VA: National Council of Teachers of Mathematics, 1995.

High School Mathematics at Work: Essays and Examples for the Education of All Students. Washington, DC: National Academy Press, 1998. This book was produced by the Mathematical Sciences Education Board under the direction of the National Research Council. It is available for free online at http://www.nap.edu/.

Professional Standards for Teaching Mathematics. Reston, VA: National Council of Teachers of Mathematics, 1991.

Appropriate Use of Technology:

Carlson, Scott. “Back to Basics: A Bridge Too Far” From Pi in the Sky; issue 5. Pacific Institute for the Mathematical Sciences. Pi in the Sky is available online at http://www.pims.math.ca/pi/. This article on the use of calculators states that “the intelligent and appropriate use of calculators does not create … gaps, but mitigates them.”

Pomerantz, Heidi and Bert Waits. “The Role of Calculators in Math Education: Dispelling the Myths.” Dallas, TX: Urban Systemic Initiative/Comprehensive Partnership for Mathematics and Science Achievement, 1997. Available online in the Resources section of http://education.ti.com/. This article, on the Texas Instruments Web site, addresses the appropriate use of calculators and the benefit of having more time to explore challenging and interesting mathematics.

Handheld Graphing Technology at the Secondary Level: Research Findings and Implications for Classroom Practice. Dallas, TX: Texas Instruments, 2000. Available online in the Resources section of http://education.ti.com/. Conducted by an independent team of educational researchers, this study examines the effect of handheld graphing technology on students, teachers, and the content of secondary mathematics.

“Setting the Record Straight About Changes in Mathematics Education: Commonsense Facts to Clear the Air.” California Mathematics Council Web site, May 2001. Available online.

The article states: “In light of the accessibility, speed, and accuracy of calculators, NCTM advocates a mathematics curriculum that balances an appropriate use of calculators with other educational concerns.”

SMART Board
The SMART Board is a commercial product offered by SMART Technologies, Inc.

Linear Equations and Inequalities:

Friel, Susan, Sid Rachlin and Dot Doyle (eds). Navigating Through Algebra in Grades 6-8. Reston, VA: National Council of Teachers of Mathematics, 2001.
This book provides classroom activities that support the NCTM’s Algebra Standard.

Math Forum
http://mathforum.org/
This site contains problems involving solving linear inequalities.

National Council of Teachers of Mathematics (NCTM)
http://www.nctm.org
The NCTM Web site contains E-Example 7.5, “Exploring Linear Functions,”
a java applet that lets students explore linear functions and allows the linking of multiple representations of mathematical concepts. Additionally, E-Example 6.2.1, “Learning about Rate of Change in Linear Functions,” is an applet with interactive graphs to help students understand the concept of slope as a rate of change.

Purple Math
http://www.purplemath.com
Purple Math is a very well-organized site with lessons on solving linear equations and solving linear inequalities.