ONeill_Janelle_RedesigningAbsoluteValueMethodology

ABSOLUTE VALUE EQUATIONS AND INEQUALITIES O’Neill, 1
Redesigning the Methodology for Teaching
Absolute Value Equations and Inequalities
With a Multi-Representational Approach
Janelle O’Neill
Hunter College

Introduction
In a very basic sense, if the notion of absolute value is internalized as “the number which is
always positive, even if it’s negative” students will encounter difficulties transitioning from
numerical absolute values to functional absolute values (Almog & Ilany, 2012; Karp &
Marcantonio, 2010). Karp and Marcantonio (2010) regard absolute value as a complex, difficult
subject and cite that an inability to work with absolute value can often hamper comprehension of
more advanced concepts. As students’ progress to absolute value equations and inequalities,
structural similarities between equations and inequalities contribute to a competition between
intuitive beliefs and formal acquired knowledge about the solving process (Tsamir & Almog, 2001;
Tsamir & Bazzini, 2004). Too often in a classroom setting, students are only given a simple
algorithm for solving absolute value equations and inequalities with little details or reference to
definitions (Karp & Marcantonio, 2010).
Researchers agree on common issues students have when solving absolute value equations
and inequalities. First, students can fail to see the absolute value notation as a relevant operator. The
symbol is often disregard, omit or ignore and the equation or inequality is solved as a linear
function (Almog & Ilany, 2012; Karp & Marcantonio, 2010). In another situation, students
attributed the absolute value notation to the x variable only so the original question | |
became | | (Karp & Marcantonio, 2010). Another error comes from students’ tendency to
overgeneralize knowledge of solving equations to solving inequalities (Almog & Ilany, 2012;
Schreiber & Tsamir, 2012; Tsamir & Almog, 2001; Tsamir & Bazzini, 2004). During an interview,
a student replied “I know I solved correctly since I used methods that I have already successfully
used many times before when solving equations” (Tsamir & Almog, 2001, p. 520). Major
generalizations comes in the form of solving an inequality as if it were an equation and failing to
change the direction of the inequality when dividing or multiplying by a negative number. Almog

and Ilany (2012) also refer to issues students have when misinterpreting quadratic and rational
inequalities as equations.
Controversy also arises as students come to conclusions about their results. In instances
where appears as part of the solution, students may reject this case because it contradicts the
assertion that absolute value is non-negative (Karp & Marcantonio, 2010). During interviews,
Almog and Ilany (2012) learned that “students are confused by the fact that the result of an absolute
value task must be non-negative, while the value of the expression inside the absolute value sign
can be negative” (p. 358). Students also exhibited symmetry errors such as when solving | |
they noted that the number 2 was a solution so concluded that the number –2 must also be a
solution without fully reasoning through the problem (Karp & Marcantonio, 2010). Student
misconceptions include the notion that “absolute value is always equal to the number and to its
opposite” (Almog & Ilany, 2012, p. 358). In a study by Almog and Ilany (2012) a percentage of
participants believed the solution to the absolute value inequality could only contain integers. Other
students misunderstood when ‘all real numbers’ or ‘no solution’ was appropriate to use. Schreiber
and Tsamir (2012) also found that students confused ‘all real numbers’ and ‘no solution’ when
solving quadratic inequalities as well. In a study by Tsamir and Bazzini (2004) students believed the
solution to an inequality must be an inequality and rejected the possibility of a single-value solution.
Lastly, many students possess a misunderstanding of the logical connectors and and or,
which cause the students to switch them around or use them interchangeably (Almog & Ilany, 2012;
Schreiber & Tsamir, 2012; Tsamir & Almog, 2001). Almog and Ilany (2012) also cite that students
wrote a final inequality excluding any logical connector, or would write a logical connector in the
final answer contrary to the logical connector used to solve the problem. “We can see that some
students do not fully understand the meaning of the logical connectors or and and” (p. 353).

To minimize common issues and misconceptions students have when solving absolute value
equations and inequalities, this study will focus on a multi-representational approach combining a
graphical method with an algebraic method. From their findings, Tsamir and Almog (2001) suggest
that a heavier emphasis on a graphical strategy could provide students with visual images of
solutions to help facilitate interpretation of the results. Additionally, Tsamir and Reshef (2006)
recommend introducing the graphical method first because students can benefit from its visual
representation.
Arcidiacono (1983) affirms that “a visual approach to absolute value intuitively illustrates
how a problem can be analyzed by breaking it down into parts” (p. 197). A graphical approach
involves students interpreting a graph of an absolute value function as it compares to some
condition on y in the coordinate plane. Students visualize an absolute value equation as a system of
equations and look to calculate the intersection point or points, or where the function lies above or
below a given condition for an inequality (Dreyfus & Eisenburg, 1985). Arcidiacono (1983)
correlates a graphical representation of | | with its algebraic representation as a piece-wise
function { from the very beginning, which gets students to see these two forms as
complementary rather than competitors. Dreyfus and Eisenburg (1985) warn that jumping right into
an algebraic case approach can be inefficient and take a considerable amount of work just to obtain
the empty set as the solution, which was not possible to foresee from the beginning but which a
graphical approach could immediately illustrate.
However, in a study by Almog and Ilany (2012) none of the students used graphs to solve
the given tasks, which may suggest students do not inherently think of the graphical approach as a
viable option or students have little exposure to the graphical method. Tsamir and Almog (2001)
had a similar result because an algebraic manipulation approach was the most prevalent amongst
students in their study. Interestingly, the use of only algebraic manipulations also yielded the

highest rate of incorrect responses (Tsamir & Almog, 2001). Researchers cite there is a need for
fundamental and diverse graphing skills if students are to fully benefit from a graphical approach
(Arcidiacono, 1983; Dreyfus & Eisenberg, 1985). However, with the invention of the graphing
calculator, graphing requirements and proficiency definitions have changed. With the technological
advances calculators have gone through, a graphical approach could improve conceptual
understanding and the interpretation of results (Dreyfus & Eisenberg, 1985; Horak, 1994; Kiser,
1990; Tsamir & Almog, 2001). In a study by Kiser (1990), two groups of students started with no
significant difference between pretest scores, then after a presentation on absolute value inequalities
with one of the groups using computer-enhanced instruction, scores on a posttest reveal a
statistically significant increase in scores for the computer-enhanced group. Still instructors must be
aware that complications will arise if students are unfamiliar with the graphing calculator
capabilities at no fault to their understanding of equations and inequalities (Piez & Voxman, 1997;
Tsamir & Reshef, 2006).
Currently, as students’ transition from linear equations and inequalities, to absolute value
equations and inequalities, to quadratic equations and inequalities to rational equations and
inequalities, etc., each problem can be taught using a different approach (McLaurin, 1985).
Considering a graphical approach fused with the algebra can provide students with a transferable
method in which they can successfully solve an equation or inequality for any function. A strong
understanding of the graphical approach can also lead to a clear conceptual understanding of more
advanced function equations or inequalities, such as a system with a compound absolute value
inequality, two absolute values functions, or with an absolute value and a linear function
(Arcidiacono, 1983). A graphical approach can also help answers such as ‘all real numbers’ or ‘no
solution’ make more sense to students.

Methodology
The lesson on absolute value equations and inequalities was implemented with 80 students
enrolled in an Algebra 2 & Trigonometry course. Students received a packet consisting of eight
absolute value examples, four equations and four inequalities. Students were given a double class
period to complete the graphical solution upper portion of the worksheets. The graphical solution
required the students to create a system of equations by treating each side of the equal sign as a
separate function; usually the left side of the equation as and the right side of the equation as .
This method was consistent with Horak’s (1994) methodology and Dreyfus and Eisenberg’s (1985)
procedure of using ( ) and ( ) for function names instead. Students then entered the equations
into their graphing calculator to create a sketch of the system, adjusting the window for a better
picture if necessary. The equation examples then asked the students to estimate the number of
solutions, with the understanding that the solution to a system is a point or points of intersection.
Both the equation and inequality examples had students calculate the intersection point or points by
using a program imbedded in the graphing calculator which students’ studied during the unit on
linear equations (see Appendix I). Since the intersection program in the graphing calculator returns
both the x and y coordinates of the point of intersection, a discussion ensued about which value
would be the solution to the original problem. Students agreed that since the original equation or
inequality only had an x variable, then the x-coordinate would be the correct choice for part of the
final answer.
The next class involved connecting the graphical solutions with an algebraic method for
absolute value equations. Visually, the first two equation examples involve an absolute value
function intersecting a constant horizontal function at two distinct locations. Students were advised
to look at the absolute value graph as two separate pieces; a line with a negative slope and a line
with a positive slope. Students separated the absolute value function into two separate cases:

| | became ( ) and ( )
Consequently, the algebraic approach had a clear connection with the graphical approach. Rather
than turning the 12 into a negative and positive quantity, students can visually see the change in
sign as it relates to a negative and positive sloped line. This method was consistent with
Arcidiacono’s (1983) methodology. A discussion with the students emphasized that the terms inside
the absolute value must be included in parenthesis to reflect a change, if necessary, in both the slope
of the line and the y-intercept, if the line was to continue and intersect with the y-axis. Though the
plus sign in front of the positive slope line is not necessary, it emphasizes the two cases that
students will consider. With the distributive property and standard algebraic methods to solve linear
equations, students calculated the same x-values algebraically that were achieved with the graphical
approach the previous lesson. The second equation example included numerical quantities outside
of the absolute value. Students were again instructed to focus on the linear nature of the graphical
sketch and evaluate the absolute value with two separate cases:
This example with outside influences gave students a different perspective on the absolute value
equation, but still showcased the same negative slope line and positive slope line analysis.
The last two absolute value equations were specifically chosen to introduce the students to
extraneous solutions. In the New York State Regents curriculum for Integrated Algebra, students
learn about solving linear-quadratic systems and the possibility of there being two, one or no
solutions depending on the intersection points. When now transitioning to a constant-absolute value
system, students should be able to see a correlation. Graphically, students can see that the equation
| | only has one point of intersection, and many students only wanted to consider
the positive slope line algebraically. However, a discussion followed about whether students would
remember to check the solutions graphically before jumping right into the algebra, and many

students agreed that they would probably go right into the two case method. Therefore, the analysis
began by creating two cases:
After the students algebraically solved each case, they knew only one of the solutions was correct.
A discussion arose about where the other solution could have possibly come from and what it meant
to the problem. Students returned to the graphical representation to make sense of the problem.
They concluded that since the two cases removed the absolute value notation and turned the
problem into two separate lines, if the negative slope line was to continue it would eventually
intersect the second part of the system. Since the absolute value is restricted and does not continue,
the other solution is said to be extraneous. This was a new vocabulary word for many students, but
gave them valuable reason to always remember to check their answers whether numerically with
substitution or graphically in the calculator in the future. Students should discover that the
extraneous value of x appears because the extension of one of the absolute value branches would
intersect with the constant horizontal function (Horak, 1994). The last equation | | was
treated very similarly to the previous problem. Students knew there would be no solution because
graphically there were no intersections, but many were curious as to what numerical answers would
be achieved algebraically; a few students in every class were even able to conclude that both values
would be extraneous before finishing the algebraic approach.
The analysis of absolute value inequalities took two separate class periods. The first period
was devoted to completing the graphical approach by determining the inequality notation that would
satisfy the original problem graphically (see Appendix I). In the first two inequalities, students were
asked to identify the location on the sketch where the absolute value was greater than the horizontal
line:
| | and | |

Visually, students concluded that the two separate portions of the absolute value above the line
showed where it was greater than the horizontal. This method was consistent with Dreyfus and
Eisenberg’s (1985) methodology for analyzing whether the function ( ) is lower than or higher
than the function ( ). Students were instructed to highlight these separate portions on the sketch.
The next process of writing the inequality notation took some time because students had to take a
two-dimensional sketch and focus only on the possible x-values. This difficulty is consistent with
the findings of Piez and Voxman (1997) concerning reading a one-dimensional solution from a two-
dimensional graph. However, relating this process to finding the domain proved helpful because
students studies domain, along with inequality notation, in the first quarter. Also, transitioning the
two highlighted pieces to the x-axis also proved helpful for students when creating the inequality
notation in terms of x values only. This method was consistent with Dreyfus and Eisenberg’s (1985)
methodology since “the graphical method thus leads naturally to a representation of the solution set
on the x-axis which avoids the somewhat cumbersome set notation” (p. 655). Students identified
that the one-dimensional solutions were positioned outside the intersection points towards the
arrows, very similar to the two-dimensional highlighted sketch. Again, students agreed that since
the original inequality only had an x variable, then the x values would be the correct choice for the
final inequality. At this point the logical connector or was introduced to the students. Students were
told that since the two solutions were disjointed the final answer is either “here or over there”; the
instructor would point to the two disconnected pieces of the graph at this time. Conversely, the last
two inequalities were examples of the logical connector and. Students were asked to identify the
location on the sketch where the absolute value’s position fell in relation to the horizontal line:
| | and | |
Visually, these two examples were slightly different. In the first, students concluded that the
connected portion of the absolute value below the line showed where it was less than the horizontal.

In the second, students determined that the connected portion of the absolute value above the line
showed where it was greater than the horizontal. However, each graph illustrated the joint nature of
the logical connector and. Students were instructed to highlight these separate portions on the
sketch. To emphasize the creation of a compound inequality for the and inequality, the instructor
held a rubber band between two fingers and a student volunteer stretch the rubber band either below
or above the instructors fingers to create the same shape highlighted in the sketch. Slowly the
student returned the rubber band to the center and students were able to see the two-dimensional
figure become a one-dimensional number line. From here students were comfortable concluding
that the solutions to the last two inequality examples lied between the two intersection points.
There was a fluid transition from the graphical approach to the algebraic approach for
absolute value inequalities. Based on interviews conducted by Almog & Ilany (2012), “it is
recommended to examine an approach that strengthens students’ understanding of the solution of
the equation | | where is positive, negative or 0, and then to discuss the solution of
inequalities in a similar way” (p. 362). Students began by separating the absolute value inequality
into two separate cases just like the algebraic process with absolute value equations:
As a result, the algebraic approach still has a clear connection with the graphical approach. Rather
than turning the 12 into a positive and negative quantity, and demanding that students change the
direction of the inequality, students can visually see the change in sign as it relates to a negative and
positive sloped line and its location above the horizontal line at 12. As students used the distribution
property and other algebraic manipulations an important note had to be considered when dividing,
or multiplying, by a negative number. Many students remembered that the direction of the
inequality had to switch. This approach has the direction change of the inequality come naturally
with the solving process and makes more sense rather than starting off by telling students that the

direction of the inequality must switch. When students end with two separate inequalities, a
discussion resulted as to whether this represented an or or and scenario. Some students were quick
to state it was an or inequality because the answers were separate and it is what the graphical
approach revealed, but the instructor stressed that when faced with the solving process on their own
students must supplement the algebraic inequality solutions with a number line to provide
conclusive proof. Student’s familiarity with creating numbers lines from previous courses proved
useful, and for the first example the number line substantiated the or claim. The second inequality
example was very similar to the first:
This example provided students with two different situations to consider. First, without the
graphical solution coupled with the algebraic approach, the negative 2 on the right side of the
inequality could have tempted students to believe there was ‘no solution’ because that was the result
when there was a negative number in one of the equation examples. However, students could verify
graphically that there was a solution and therefore suppress any inclination to automatically believe
there will be ‘no solution’ if a negative number appears as part of the original problem. Also, this
was the first and only example done in class where the x values of the intersection points were
symmetrical. Students were able to recognize that the absolute value sketch was symmetric across
the y-axis which led to symmetrical x values. The instructor emphasized that symmetrical solutions
were possible but only in special circumstances so students should not expect to find one
intersection and negate it to find the other intersection.
The last two inequalities were chosen to help students recognize that creating a number line
after finding the inequality solutions algebraically was essential for concluding which logical
connector was represented. By this point students easily separated the inequality into two cases:

A discussion followed about getting the two separate inequalities to be represented by the logical
connector and and written as a compound inequality. Students attested that without the graphical
solution above they would have been tempted to put the two inequalities together with the logical
connector or. The importance of creating a number line before coming to a conclusion about the
logical connector was again emphasized. The very last example was chosen to make students aware
that the direction of the inequality in the original problem did not always dictate the logical
connector that would result. At least one student in every class asked if the inequality faced the
absolute value was it an or and if it faced away from the absolute value was it an and. The instructor
underlined the importance that students must be very careful when coming to any absolute
conclusions and applying these intuition in the solving process. When students separated the last
example:
| | became ( ) along with ( )
The and outcome was almost counterintuitive for the students that originally believed an or
inequality should have resulted, but the number line along with the graphical depiction convinced
them otherwise and helped solidify that inequalities must be accompanied by some graphical
representation to substantiate the final chosen logical connector.
After much discussion and practice solving absolute value equations and inequalities both
graphically and algebraically, students were given a homework assignment consisting of twelve
questions from previous New York State Math B and Algebra 2 & Trigonometry Regents exams
(See Appendix II). The following results section describes some common errors and issues that
arose when student work was examined.
Results
The issues and misconceptions shown by students were consistent with the literature. Some
students did not have a clear understanding of the concept of absolute value. For the equation

| | , students separated it into 2 cases such as ( ) and ( ) such
as to first make everything inside the absolute value have a positive coefficient, and then separate
the equation into a line with a negative slope and a line with a positive slope. Students also found
the ‘reverse order’ task of question 7 quiet challenging, just as Tsamir and Bazzini (2004) warn.
Students chose option 2, | | , as a final answer and could not recognize ‘all real
numbers’ as the solution set to this choice. In other questions, some students created a compound
inequality for instances where the absolute value would represent an and inequality, however this
was not part of the methodology previously discussed. It is assumed students received assistance
from another person or textbook outside of the current course. The compound inequality approach is
of course fine, but it is hoped that students are able to create a graphical sketch of the absolute value
inequality in their mind or on paper to solidify that an and inequality will result.
Concerning errors in student’s final results, for the question: solve algebraically for all
values of x: | | , students solved , as if ignoring the absolute value symbol,
received as a solution and negative it to give as the other solution. A few students
considered the negative case first and then negated their answer which led them to
as their final solution. These students exhibited a misunderstanding of symmetry as it relates to
absolute value. Also, for the question: Solve algebraically for the negative value of x: | |
, students again disregarded the absolute value notation as an irrelevant operator, found
as one of the solutions then concluded must be the negative solution. This
misunderstanding was addressed with a look at the graphical solution to these questions.
Surprisingly consistent with studies by Almog and Ilany (2012) and Tsamir and Almog (2001) was
many students’ reluctance to use the graphical method to solve short answer or multiple choice
questions. It could definitely be attributed to a lack of use and familiarity with the potential the
graphing calculator has to make solving easier. Students just jumped right into the algebra because

that was what they were most accustomed. Unfortunately, it was also noticed that even though
students had experience with the same graphing calculators since freshmen year, their level of
competence was very low and many features had remained unused.
Other errors in selecting a final answer choice combined confusion with the concept of
absolute value with intuitive notions about the final solution of an absolute value. When asked what
type of number the solution set of | | contains, many students chose “both positive and
negative real numbers” because they attributed this choice to the idea that the quantity inside the
absolute value can be positive or negative. Students did not recognize the difference between
‘solutions’ to the inequality and the ‘expression’ inside the absolute value. Other students chose ‘no
real numbers’ because their final inequality included all negative values and fractions; this was
interesting because imaginary number had not yet been studied by the students.
The most prevalent issue revolved around the misunderstanding of the logical connectors
and and or. When asked: Which graph represents the solution set of | | , first an algebraic
manipulation was most common and second students mixed up the choice of logical connectors.
Some students even chose the number line graph with closed circles rather than open which
constitutes a concern for an understanding of inequality notation in general. Other students
combined confusion with intuitive notions about the final solution of an absolute value with a
misunderstanding of the logical connectors. For the question: what is the solution set of the
inequality | | , students chose option { | } without realizing the inequality
statement does not make sense. Students recognized that the direction of the inequalities was
correct, but disregarded the idea of having to decide whether the statement should be an and or an
or.
Even after focusing on a multi-representational approach combining a graphical method
with an algebraic method to minimize common issues students have when solving absolute value

equations and inequalities, errors arose due to misconceptions, intuitive conflict or other reasons. In
a study by Karp & Marcantonio (2010) shows no evidence that students actually anticipate the
solutions to problems based on previously solved problems. It was apparent that some students were
very new to receiving a multi-representational approach and there was little cohesive connection
between similar notions represented in different ways. As mentioned in the literature, one would
have thought that “after discovering that the horizontal straight line has two points of
intersection with a graph of the function | |, students can recode this information in algebraic
terms and anticipate the fact that not only does the equation | | have two solutions, but so
does, say, the similar equation | | ” (Karp & Marcantonio, 2010, p. 49). Since a graphical
approach seemed very new to many students, it will not be abandoned but emphasized as an
additional tool for students to solve, and possible anticipate solutions to, various other functions.
Discussion and Conclusion
Sandra C. McLaurin (1985) entitled her article “A Unified Way to Teach the Solution of
Inequalities”, and after almost 30 years mathematics curriculums are still in need of a method to
connect equations and inequalities of different functions so students can make sense of the solving
process. The National Council of Teachers of Mathematics (2010) urges reasoning and sense
making to be at the forefront of classroom lessons every day. With substantiation from the
literature, the method previously discussed intimately connects what can be an abstract algebraic
process with a more concrete graphical representation to give students a vantage point on the
outcome of an equation or inequality. Arcidiacono (1983), Dreyfus and Eisenburg (1985) and
Horak (1994) all cite that a graphical representation can help absolute value equations and
inequalities make more sense to students. “In the words of one of our students: The graphical
method makes sense to use” (Dreyfus & Eisenberg, 1985, p. 662). Additionally Arcidiacono (1983)

and Dreyfus and Eisenburg (1985) discuss the benefits of a graphical representation for absolute
value functions even before students had access to computer-enhanced instruction. With the
invention and evolution of graphing calculator capabilities, creating a function illustration and
visualizing the intersection of a system of equations is even more accessible to students of all
different abilities. “Those educators involved in computer instruction should be encouraged that the
microcomputer can be an effective tool to enhance classroom instruction and can be more effective
than traditional presentations for certain groups of students” (Kiser, 1990, p. 95). A graphical
approach, especially with graphing calculator technology, can even be an effective tool and could
lead to the greatest dividend for average or weaker students (Dreyfus & Eisenberg, 1985). Solving
problems that involve geometric representation can be easier for some students than solving ones
that involve manipulating variables (Karp & Marcantonio, 2010).
Ultimately, this graphical model can be transferred to quadratic equations and inequalities,
cubic equations and inequalities, exponential equations and inequalities, rational equations and
inequalities, and every other function explored in algebra. Tsamir & Almog (2001) found that
students usually had correct solutions when using a graphical representation for rational and
quadratic inequalities. If a student has the ability to create a graphical representation of a problem,
this is a tool they can continue to utilize in higher level mathematics classes like trigonometry and
calculus. Dreyfus and Eisenberg (1985) believe students achieve an objective with the graphical
approach which is at least as important as the procedural skills involved in algebraically solving
equations. In the philosophy section of the Advanced Placement Calculus course description, the
College Board dictates that “the courses emphasize a multi-representational approach to calculus,
with concepts, results, and problems being expressed graphically, numerically, analytically, and
verbally” (p. 5). This “rule of four”, as it is commonly referred to among educators, should be a
staple in all mathematics curriculum from elementary school through middle school and high

school. Piez and Voxman (1997) believe that “students gain a more thorough understanding of a
function if it is explored using numerical, graphical and analytical methods” (p. 164). Piez and
Voxman (1997) insist that students should be strongly encouraged, if not required, to work with
multi-representations.
Mathematics educators must continue to look for opportunities to create a unified approach
to teaching mathematics so students can see graphical and algebraic methods as complements to
each other and begin to make connections over various function types. Looking at all mathematical
concepts from different perspectives will help to guide students towards a conceptual understanding
and give them the problem solving skills necessary to confidently confront any obstacle in today’s
society.

References
Almog, N., & Ilany, B. (2012). Absolute Value Inequalities: High School Students' Solutions and
Misconceptions. Educational Studies In Mathematics, 81(3), 347-364.
Arcidiacono, M.J. (1983). A Visual Approach to Absolute Value. The Mathematics Teacher, 76(3),
197-201.
Dreyfus, T. & Eisenberg, T. (1985). A Graphical Approach to Solving Inequalities. School Science
and Mathematics, 85(8), 651-662.
Horak, V.M. (1994). Absolute-Value Equations with the Graphing Calculator. The Mathematics
Teacher, 87(1), 9-11.
Karp, A., & Marcantonio, N. (2010). “The Number Which Is Always Positive, Even If It's
Negative” (On Studying the Concept of Absolute Value). Investigations In Mathematics
Learning, 2(3), 43-68.
Kiser, L. (1990). Interaction of Spatial Visualization with Computer-Enhanced and Traditional
Presentations of Linear Absolute-Value Inequalities. Journal of Computers in Mathematics
and Science Teaching, 10(1), 85-96.
McLaurin, S.C. (1985). A Unified Way to Teach the Solution of Inequalities. The Mathematics
Teacher, 78(2), 91-95.
National Council of Teachers of Mathematics. (2010, April 9). Focus in High School Mathematics:
Reasoning and Sense Making in Algebra. Retrieved from:
http://www.nctm.org/catalog/product.aspx?id=13524
Piez, C.M., & Voxman, M.H. (1997). Multiple representations – Using different perspectives to
form a clearer picture. The Mathematics Teacher, 90(2), 164-166.

Schreiber, I., & Tsamir, P. (2012). Different Approaches to Errors in Classroom Discussions: The
Case of Algebraic Inequalities. Investigations in Mathematics Learning, 5(1), 1-20.
The College Board. (2012). Calculus: Calculus AB Calculus BC Course Description. Retrieved
from: http://apcentral.collegeboard.com/apc/public/repository/ap-calculus-course-
description.pdf
Tsamir, P. P., & Almog, N. N. (2001). Students' strategies and difficulties: the case of algebraic
inequalities. International Journal Of Mathematical Education In Science & Technology,
32(4), 513-524.
Tsamir, P., & Bazzini, L. (2004). Consistencies and inconsistencies in student’s solutions to
algebraic ‘single-value’ inequalities. International Journal of Mathematical Education In
Science & Technology, 35(6), 793-812.
Tsamir, P., & Reshef, M. (2006). Students’ Preferences When Solving Quadratic Inequalities.
Focus on Learning Problems in Mathematics, 28(1), 37-50.

Appendix I - Unit 1: Solving Absolute Value Equations and Inequalities
1. Solve Graphically:
| |
System: Sketch:
{
Estimate the Number of Solutions: Calculate the Intersection(s):
Solve Algebraically:
| |

Solution(s):
| |
System: Sketch:
{
| |

Solution(s):
| |
System: Sketch:
{
Calculate the Intersection(s): Inequality Notation:
Solve Algebraically and Graph the Solution on a Number Line:
| |

Solution(s):
| |
System: Sketch:
{
| |

Solution(s):
Appendix II – Regents Prep Unit 1: Solving Absolute Value Equations and Inequalities
1. The graph to the right represents f x( ) .
Which graph best represents f x( ) ?
2. Which equation is represented by the accompanying graph?
(1) y x  3 (3) y x  3 1

(2) 1)3( 2
 xy (4) y x  3 1
3. What is the solution set for the equation ?
(1) (3)
(2) (4)
4. Solve algebraically for all values of x: | |
5. Solve algebraically for the negative value of x: | |
6. Which graph represents the solution set of 2 1 7x   ?
7. The solution set of which inequality is represented by the accompanying graph?
(1) 72 x
(2) 72  x
(3) 72 x
(4) 72  x
8. Which graph represents the solution set for the expression 2 3 7x   ?

9. What is the solution of the inequality x  3 5?
(1)   8 2x
(2) x x  8 2or
(3)   2 8x
(4) x x  2 8or
10. The solution of | |
(1) x < -1 or x > 4
(2) x > -1
(3) -1 < x < 4
(4) x < 4
11. What is the solution set of the inequality 3 2 4 x ?
(1) }
2
1
2
7
|{  xx
(2) }
2
7
2
1
|{  xorxx
(3) }
2
7
2
1
|{  xx
(4) }
2
1
2
7
|{  xorxx

12. The solution set of 3 2 1x +  contains
(1) only negative real numbers
(2) only positive real numbers
(3) both positive and negative real numbers
(4) no real numbers

ONeill_Janelle_RedesigningAbsoluteValueMethodology

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