# Absolute Value

Absolute Value (Pedagogical Content Knowledge Project)

by Michael Sarris (UCIrvine, August 2009)

Absolute Value Powerpoint at https://eee.uci.edu/09m/12364/pckwikiproject/Absolute+Value+PCK+Powerpoint.ppt

## Cognition and Learning Background: Absolute Value

### Methods of Teaching

The typical method of teaching absolute value involves memorization of the formal definition and application of the case-by-case method and the properties of inequalities in order to solve problems (McLauren, 1985, p. 91).

Typical instruction with absolute value begins with the definition used in conjunction with algebraic representation in simple equality equations (such as $|x| = 4$ or $|2x + 3| = 7$). While more textbooks are emphasizing conception of absolute value as distance and using number lines as a tool to understanding, the more common approach is procedural solution using algebra and representation of the answers algebraically and on a number line. The prolonged connection of absolute value with distance or number line representation tends to fall by the wayside in favor of procedural fluency as a goal.

### California State Standards Involving Absolute Value

• 2.5 - Understand the meaning of the absolute value of a number; interpret the absolute value as the distance of the number from zero on a number line; and determine the absolute value of real numbers.
• Algebra I
• 3.0 - Students solve equations and inequalities involving absolute values.
• 25.3 - Given a specific algebraic statement involving linear, quadratic, or absolute value expressions or equations or inequalities, students determine whether the statement is true sometimes, always, or never.
• Algebra II
• 1.0 - Students solve equations and inequalities involving absolute value.

### California State Standards Glossary Definition of Absolute Value

• A number’s distance from zero on the number line. The absolute value of -4 is 4; the absolute value of 4 is 4.

## Cognitive Obstacles and Common Misconceptions

### Basic misconceptions

• Absolute value is "always positive" meaning that the solution of an absolute value problem can never be negative
• Absolute value is "always positive" and so any solutions must be made positive
• Absolute value problems only have one solution
• Absolute value problems have two solutions which are always mirrors (i.e. 4 and -4)

Note that these misconceptions are related but distinct. There is a difference between a negative solution being impossible (i.e. not to be considered) and having only positive solutions, which may involve considering a negative solution but subsequently making it positive.

### Cognitive Obstacles with the Definition of Absolute Value

The definition of absolute value is:

$\left | x \right | = x \text { if } x \ge\ 0\,\!$
$\left | x \right | = -x \text { if } x < 0\,\!$

Sink (1979) found that students ignore the conditions (the if statements above). Students have been trained since elementary grades to treat the absolute value of a quantity as a nonnegative value. Students will often say that absolute value means "the answer is always positive". However, as they tend to ignore the condition statements $x \ge\ 0\,\!$ and $x < 0\,\!$, students become confused with the second sentence of the definition. The idea that $\left | x \right | = -x \,\!$ contradicts their previous understanding that the absolute value would be nonnegative.

### Cognitive Obstacles with Inequalities

Absolute value is often found in conjunction with inequalities, leading to all of the cognitive obstacles that are associated with linear equality and inequality equations. Because absolute value in conjunction with inequalities typically leads to compound inequalities, the number of intertwining cognitive obstacles can quickly increase.

This section can be removed and expanded upon when the Inequalities page is developed.

• Students have difficulty reading inequalities correctly
• As with equalities, students are confused by variables on the right side of the inequality
• $5 > x$ is more cognitively demanding than $x < 5$
• Compound inequalities necessitate the ability to use variables on the right sight of the inequality sign
• Students do not understand that inequalities can not be read left to right and right to left in the same way
• Students are used to equality equations, which can be read in any order
• Students read $6 = x$ as "6 is equal to x" and "x is equal to 6"
• Students likewise read $6 > x$ as "6 is greater than x" and "x is greater than 6" even though this is incorrect

• Conjunction and disjunction of compound inequalities are not well understood
• Compound inequalities can be written as one statement or as two statements joined by a keyword
• The singular statement guarantees that one part of the inequality must be correctly read right to left (see above)
• The joined statements require selection of the correct keyword ("and" or "or")
• The keyword may or may not be present in the presentation of the problem
• Sometimes it is given in the problem or implied by the section of the book being studied (see below)
• Sometimes it must be inferred from the behavior or the graph
• Students are encouraged to think of problems as "and" or "or" inequalities
• Students do not often understand which is the correct keyword and why
• Textbooks often divide these problems artificially into two sections
• Students are led to believe they are separate kinds of problems
• The cases of overlapping inequalities or all real number solutions are not always presented as possibilities
• Students are trained to classify the problems but do not understand the underlying meaning or purpose of the classification
• Compound inequalities can be represented with a number line
• Early problems often require usage of the number line
• Textbooks typically drop the requirement for the solution to be represented as a number line with the more complicated problems
• Students are taught to observe the basic characteristics without understanding what they truly mean

• Students have difficulty with the various kinds of notation
• Inequality notation ($x < 6\,\!$)
• Used to formulate problems and solutions
• Compound inequalities can be represented with inequality notation, but have their own cognitive obstacles (see above)
• Inequality notation requires the use of a variable
• Interval notation ($\left [ -3, 5 \right ]$)is typically introduced earlier in the school year
• Interval notation is used only for representing solutions to problems written in other notations
• Interval notation does not make use of variables
• Compound inequalities can be represented with interval notation, but require special symbols
• The union and intersection symbols ($\cup$ and $\cap$) are typically taught during set theory and hastily reviewed (and then forgotten) when teaching interval notation
• The union and intersection symbols are another way of representing "and" and "or"
• Set notation ($\left \{ x | x \in \Re, x > 5 \right \}$
• Used to formulate problems and solutions
• Less frequently used to describe inequalities before Algebra II
• Number Line Graph notation
• Can be used as ways to understand and conceptualize problems
• Used mainly to describe solutions; usage of the number line as a tool for solving the problem diminishes after early examples
• Have two styles that relate to interval notation

### Cognitive Obstacles with Absolute Value Inequalities

• Students are instructed to solve problems using the case-by-case method
• There is a "positive" case and a "negative" case
• The "shortcut" for the negative case is to reverse the inequality sign and the sign of the constant on the side opposite the variable
• The reason for this "shortcut" is often poorly explained
• The quantity inside the absolute value sign can be positive or negative
• If this quantity is negative, the resultant equation can be multiplied or divided on both sides by -1
• Even if this process is explained, it is often not related back to distance or another way of making sense of the operation
• Since students do not understand the procedure, they will often forget to reverse one or both signs

This illustrates a typical example of the case-by-case solution method. The procedure skips right to the "shortcut", where the inequality and the right hand side constant have their signs reversed without explanation. Since the coefficient of the variable is negative, the solution obtained by dividing both sides by -2 requires the inequality to be reversed again. Students will mistakenly think this has already been taken care of since they have already flipped the inequality sign once.

• Textbooks sometimes instruct that answers to absolute value inequalities will be "or" if the original problem is > or $\ge$ and "and" if the original problem is < or $\le$
• This is an additional procedural step to memorize or forget
• Sometimes the mnemonics "greatOR than" or "less thAND" are used
• This is only true for absolute value inequalities -- it does not transfer to compound inequalities as a whole
• Students are often only given this procedural step and mnemonic and do not understand the underlying concepts
• Furthermore, students will neglect to check that their answer makes sense and will default to the what the mnemonic or shortcut indicates

Dean (1985) pointed out that students will assume that the solution set always contains only alternating intervals. Multiplicity and the sign of factors can change the solution intervals even when the critical points remain the same.

## Pedagogical Tools and Strategies

### Ahuja - absolute value as distance

Ahuja (1976) suggests considering absolute value in terms of distance, making the definition of absolute value:

For real numbers $x\,\!$ and $y\,\!$, consider $\left | x - y\right |\,\!$ as the distance between $A\,\!$ and $B\,\!$, where $A\,\!$ and $B\,\!$ are the representations of the numbers $x\,\!$ and $y\,\!$, respectively, on the real line.
Example: Solve $\left | x - 5\right | = 2\,\!$
Since the distance of $x\,\!$ from 5 is 2, $x\,\!$ can be either 7 or 3.

Example: Solve $\left | x + 7\right | = 3\,\!$
$\left | x + 7\right | = \left | x - (-7)\right |\,\!$. This equation says that the distance of $x\,\!$ from -7 is 3. The answer would be either -4 or -10.

### Arcidiacono - piecewise functions illustrate absolute value algebraically and graphically

Arcidiacono (1983) recommends using piecewise functions to reinforce the algebraic and graphical representations of the absolute value function together from the beginning of instruction. His three stage approach to absolute value is:

1. Derive the algebraic definition of absolute value by studying graphs
2. Graphing absolute value functions using student-generated tables and exercises
3. Use graphs of absolute value functions to solve word problems

### Ballowe - use square roots to rewrite absolute value problems

Ballowe (1998) suggests using the definition $\left | x \right | = \sqrt{x^2}\,\!$ to rewrite the problem without absolute value signs.

Example: $\left | 3x - 4\right | = \left | 6 - 2x\right |\,\!$
$\sqrt{(3x - 4)^2} = \sqrt{(6 - 2x)^2}\,\!$ rewrite using the definition
$(3x - 4)^2 = (6 - 2x)^2\,\!$ square both sides

### Brumfiel - teach all five definitions of absolute value

Brumfiel (1980) recommends teaching five definitions of absolute value and the investigation of one problem with all five definitions. The definitions are:

1. Let $x\,\!$ be any real number. On a coordinate line let $X\,\!$ be the point whose coordinate is $x\,\!$. Then $\left | x \right |\,\!$ is the undirected distance between $X\,\!$ and the origin.
2. Let $r\,\!$ be any real number. Choose any numbers $x\,\!$ and $y\,\!$ so that $x - y = r\,\!$. Let $X\,\!$ and $Y\,\!$ be the points whose coordinates are $x\,\!$ and $y\,\!$. Then $\left | x - y \right |\,\!$ is the undirected distance between the points $X\,\!$ and $Y\,\!$.
3. $\left | x \right |\,\!$ is the "larger" of the numbers $x\,\!$, $-x\,\!$. We write this briefly as $\left | x \right |\,\!$ = Max {$x\,\!$, $-x\,\!$}, that is, $\left | x \right |\,\!$ is the maximum element of the set that consists of $x\,\!$ and $-x\,\!$. Of course we agree that Max {$a\,\!$,$a\,\!$} = $a\,\!$. This takes care of the case $\left | 0 \right |\,\!$.
4. $\left | x \right | = \sqrt{x^2}\,\!$
5. $\text { if } x \ge\ 0,\text{ then } \left | x \right | = x \,\!$
$\text { if } x < 0, \text{ then } \left | x \right | = -x \,\!$

Note that Definition 2 is essentially the same as Ahuja's(1976) recommendation.
Note that Ballowe (1998) expanded upon the ideas in Definition 4
Note that in Definition 5, Brumfiel (1980) has ordered the definition such that the condition comes first in each statement, addressing Sink's (1979) concern that the condition would be ignored by the student.

### Friedlander & Hadas - combine and spiral

Friedlander & Hadas (1988) combined Sink (1979), McLaurin (1985), and Brumfiel's (1980) approaches into a spiral sequence

1. Distance on the number line
1. Exercises using the number line
2. Tables combining verbal descriptions with algebraic representations
3. Tables combining open sentences, verbal descriptions, graphs, and solution sets
2. Coordinate system graphics
1. Graphing piecewise functions
2. Graphical solutions of absolute value sentences in two-dimensional coordinate systems

### McLauren - evaluate the critical points

McLauren (1985) recommends solving absolute value inequalities as equalities, plotting the results on a number line, listing the intervals defined by the critical points, and testing each interval to see if it is a solution. This technique can be extended to quadratic and rational inequalities.

### Sink - rooting out definition misconceptions

The teacher must show with examples how substituting values according to the conditions causes the result to be nonnegative.

Example
$\left | x \right | = -x \text { if } x < 0\,\!$
$\text { if } x = -4 \text { then } -x = -(-4) = 4\,\!$
Example
$\left | x - 6 \right | = -(x - 6) \text { if } x < 6\,\!$
$\text { if } x = 4 \text { then } -(x - 6) = -(-2) = 2\,\!$

### Stallings-Roberts - the Absolute Value Scale (AVS)

The Absolute Value Scale (AVS) (Stallings-Roberts, 1991) is a manipulative tool that students can create themselves that helps to visually associate absolute value problems with number lines and the concept of absolute value as distance.

• Students can make the AVS from a sheet of ruled notebook paper, creased lengthwise, creating two one-inch by eleven-inch strips that are placed next to one another. The bottom strip represents the number line. The top strip represents the distance scale that corresponds to the number line.
• AVS can be used to model algebraic problems
• AVS can be used to understand how to rewrite some equalities and inequalities in terms of absolute value.

### Wagster - solve by intervals

Wagster (1986) recommends a way similar to McLauren's (1985). First, the zeros of each expression in an absolute value inequality are marked on a number line. Then, in each interval on the number line, the equations are solved for that interval only. If the solution is within the interval, it is a solution of the inequality.

### Wei - absolute value and ellipses/hyperbolas

Wei(2005) suggested comparing the geometrical meaning of specific absolute value equations to ellipses and hyperbolas.

• The geometrical meaning of $|x - a| + |x - b| = c$ is "finding all points on a number line, the sum of whose distances from two points a and b is equal to a constant c.
• The definition of an ellipse is "the set of all points in a plane, the sum of whose distances from two fixed points, called foci, is a constant"
• The x coordinates of the two vertices of an ellipse are $\frac{(a + b) + c}{2}$ or $\frac{(a + b) c c}{2}$
• The solutions of the absolute value equation $|x - a| + |x - b| = c$ are $\frac{(a + b)}{2} + \frac{c}{2}$ or $\frac{(a + b)}{2} - \frac{c}{2}$

• Likewise, the geometrical meaning of $|x - a| - |x - b| = c$ corresponds to the intersection points of a hyperbola with two foci at a and b on the x-axis

## Curricula and Technological Resources

### Technological Activities

Investigating Absolute Value Equations with the Graphing Calculator (Horak, 1994)

Students can use graphing calculators to plot absolute value equations with two expressions. Students can examine equations with one, two, or no solutions and use the graphing calculator to find precise intersections.

### Websites - Interactive

The following links are tools to help visualize absolute value graphs, both on the number line and in the coordinate plane. A indicates the best tools for learning. Many of these are simple tools that allow users to see how a graph of an absolute value equation behaves. The better tools show different aspects of absolute value or show how intersecting graphs produce solutions to the equations being graphed.

### Websites - Games

The following links are to games that utilize absolute value in some way. None of the games are involved or teach the concepts. They could, however, be used to reinforce lessons or provide extra practice with the basic concepts of absolute value. These are all drill-and-practice type activities and don't explore the conceptual side of absolute value like the tools above.

## Annotated References

Ahuja, M. (1976). An approach to absolute value problems. Mathematics Teacher, 69(7), 594-596.

• Understanding absolute value in terms of distance and number lines.

Arcidiacono, M. J. (1983). A visual approach to absolute value. Mathematics Teacher, 76(3), 197-201.

• Using piecewise functions algebraically and graphically to understand absolute value.

Ballowe, J. M. (1988). Teaching difficult problems involving absolute value signs. Mathematics Teacher, 81(5), 197-201.

• Using squares and square roots to rewrite absolute value problems.

Brumfiel, C. (1980). Teaching the absolute value function. Mathematics Teacher, 73(1), 24-30.

• The five definitions of absolute value that should be taught together.

Dean, E. (1985). Reader reflections: Inequalities I. Mathematics Teacher, 78(8), 590.

• Response to McLauren pointing out that students will always think solutions contain alternating intervals.

Friedlander, A. & Hadas, N. (1988). Teaching absolute value spirally. In The Ideas of Algebra, K-12, eds. A. Coxford & A. Shulte, pp. 212-220. Reston, VA: National Council of Teachers of Mathematics.

• Combination of Brumfiel, Sink, and McLauren's ideas organized into a spiral sequence.

Horak, V. M. (1994). Investigating absolute-value equations with the graphing calculator. Mathematics Teacher, 87(1), 9-11.

• Strategies and ideas for using a graphing calculator to graph and solve sets of absolute value equations by showing intersections graphically.

McLauren, S. C. (1985). A unified way to teach the solution of inequalities. Mathematics Teacher, 78(2), 91-95.

• Strategy for solving absolute value inequalities by solving equations as equalities, plotting critical points on the number line, and testing each interval defined by a critical point.

Sink, S. C. (1979). Understanding absolute value. Mathematics Teacher 72(3), 191-195.

• Explained misconception in definition of absolute value (position of conditions) and how to overcome it.

Stallings-Roberts, V. (1991). An ABSOLUTE-ly VALUE-able manipulative. Mathematics Teacher, 83(1), 303-307.

• Description of AVS dual-number line manipulative tool to help solve absolute value problems.

Wagster, L. W. (1986). Using number lines to solve difficult absolute-value problems. Mathematics Teacher, 79(4), 260-263.

• Solving absolute value inequalities by plotting zeros on a number line and solving the equations in each interval

Wei, S. (2005). Solving absolute value equations algebraically and geometrically. Mathematics Teacher, 99(1), pp. 72-74

• Understanding absolute value in terms of ellipses and hyperbolas.