“Making Sense of Extraneous Solutions” (Zelkowski, 2013)

In the article, “Making Sense of Extraneous Solutions,” Zelkowski (2013) proposes two problems that were initially created to enhance the TPACK development of preservice teachers may also be used to deepen secondary students’ understanding of extraneous solutions.

1- Comment on these problems and discuss how your students might respond.

2- Additionally, comment on Zelkowski’s assertion that the current generation of teachers are likely to feel comfortable incorporating similar problems in their teaching in half the time of previous generations of teachers.

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Making Sense of Extraneous Solutions

Article in Mathematics Teacher · February 2013

DOI: 10.5951/mathteacher.106.6.0452

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Jeremy Zelkowski

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MAKING

EXTRANEOUS

Do you always have to check your answers

“A

lways check your answers when

solving any radical equation.” While

observing a class as a mentor, I overheard this response from a student

teacher to a student’s question, “Do

we need to check our answers to this [cube root]

equation?” Good teachers will always emphasize

checking work after solving a problem, but this particular question referred to the possible introduction of an extraneous root. The solution of a radical

equation involving only cube roots, where checking

for extraneous algebraic solutions is not required,

was not discussed by the student teacher; there was

no discussion or examination of function domain.

The underlying reasoning and sense making of

extraneous solutions did not occur at any point in

the lesson. As a result, I began to develop a technology-based lesson using the TI-Nspire™ CAS.

Principles and Standards for School Mathematics

(NCTM 2000) states, “Technology is essential in

teaching and learning mathematics; it influences the

mathematics that is taught and enhances students’

learning” (p. 11). The lesson in which I observed

the teacher-student exchange quoted above took

place in a second-year algebra classroom. It focused

solely on procedures and used technology simply as

a computational tool for checking answers to equations containing square and cube roots. The focus

on reasoning and sense making with technology in

the lesson presented here will enable students to do

more than just carry out procedures; they will be

able to understand the procedures and know how

452 MatheMatics teacher | Vol. 106, No. 6 • February 2013

Copyright © 2013 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.

This material may not be copied or distributed electronically or in any other format without written permission from NCTM.

SENSE OF

SOLUTIONS

when solving a radical equation?

Jeremy s. Zelkowski

the procedures might be used in different situations

while interpreting the results (NCTM 2009).

The lesson includes three key elements of reasoning and sense making with functions: (1) using

multiple representations of functions as a means of

demonstrating mathematical flexibility in problem

solving; (2) modeling by using families of functions;

and (3) analyzing the effects of parameters (NCTM

2010). According to the Programme for International

Student Assessment (PISA 2007), American students

lag behind international students in their ability to

analyze, reason, communicate, solve, and interpret a

variety of mathematical problems. The following lesson

allows teachers to use a pedagogical approach to teaching mathematics through problem solving (see Schoen

2003) and also address these PISA areas of concern.

By exploring the following two problems, students will improve their ability to understand and

solve problems involving radicals:

Problem 1: Determine an equation with a single

radical that, when solved algebraically, will

yield one unique real solution and two extraneous solutions.

Problem 2: Without working out this problem

with paper and pencil, predict how many real

and extraneous solutions would be obtained

when solving 5x + 3 – 2 = 2x + 3 algebraically.

For each problem, explain your reasoning in

support of your responses.

Vol. 106, No. 6 • February 2013 | MatheMatics teacher 453

(a)

Fig. 2 Graphing the negation of the radical function

reveals the extraneous algebraic solution.

(b)

(c)

Fig. 1 the power of the graphical solutions (a and c) and

the cas solution (b) to equations 1 and 2 adds to student

understanding and establishes new connections.

When students demonstrate the algebraic solutions on the TI-Nspire CAS, the handheld tool gives

this warning: Operation might introduce false solutions, a signal for students to explore further.

In figure 1c, the graph indicates that equation

2 has only one real solution, x = 2; thus, the other

algebraic solution (x = –1) must be extraneous.

Students should explore how the graph can be used

to explain the extraneous solution, which is visible

when –f1(x) is graphed and an intersection point

occurs at (–1, –1) (see fig. 2). Hence, the x = –1

extraneous solution now appears graphically.

Students must learn that squaring both sides

of the equation, even one as simple as x = 2, will

introduce extraneous roots. They need to realize

that squaring to undo the square-root function

means that both cases of the radical function need

to be considered. For example, in the equation

±5x + 2 = x, both

(

( x + 2)

)

2

(

= x 2 and − ( x + 2)

)

2

= x2

result in x + 2 = x2 → x2 – x – 2 = 0 .

SOLVING PROBLEM 1

At the start of the lesson, the teacher and students

can work through skill-focused problems similar to

equations 1 and 2, as needed.

Equation 1: 5x – 2 = 5

Equation 2: 5x + 2 = x

Then students can begin to make connections

between solution methods by using paper and

pencil or a computer algebra system or by creating

a graph. Alternatively, the teacher may elect to

begin the TI-Nspire lesson through discovery. The

CAS eliminates computational errors and allows

students to focus on content and connections

between multiple representations without losing

the opportunity to demonstrate problem-solving

ability and algebra skills (see fig. 1 for graphical

and CAS solutions).

454 MatheMatics teacher | Vol. 106, No. 6 • February 2013

EXAMINING FAMILIES OF EQUATIONS

DYNAMICALLY

The interactive geometry capability of the TINspire (with or without CAS) can take the lesson

to a higher cognitive level. As figure 3 shows,

students are able to manipulate the linear function f1(x) dynamically. They can manipulate the

y-intercept and slope independently. To manipulate the y-intercept, students move the cursor

arrow near the y-intercept (see the crosshairs near

the origin in fig. 3a) and grab the line (center of

touchpad). To change the slope, students move the

cursor arrow near the ends of the line (see fig. 3b,

lower right) and grab.

The parameters a, b, and c of the radical function can be changed incrementally. Figure 3 shows

a student’s work in shifting the radical function,

f2(x), to the right three units, thus eliminating the

(a)

Fig. 4 This student initially attempted to solve problem 1

by graphing a radical and a quadratic function. The

parabola is tangent to the radical function and intersects

the negation of the radical function twice.

(b)

Fig. 3 A student changes parameters to transform both

the radical and the linear function. Going from (a) to (b),

the slope of the line and the x-intercept of the radical

function have been adjusted.

intersection with the linear function. The student

also changed the slope of f1(x) from 1 to –0.5.

These transformations produce two functions that,

when set equal to each other, will produce an equation with two extraneous algebraic solutions. The

dynamic capability empowers students to build

their own mathematical understanding as well as to

exercise their reasoning and sense making.

However, the possibility of complex solutions of

equations from the family of functions generated

by f1(x) and f2(x) in figure 3 can easily be overlooked by students. Teachers need to be ready to

incorporate guiding questions such as these: What

additional possibilities exist for solution sets that

we have not yet examined graphically? What are

the graphical representations for each combination

of solutions? Are there possibilities that we may be

overlooking? What would happen if f1(x), f2(x),

and –f2(x) do not intersect at all?

After discussing the complex solution possibilities, students should begin to think about the

necessary conditions that will add a third possible

solution, as required by problem 1 of the lesson.

The solution sets that can be generated by the family of functions explored in figure 3 have only two

algebraic solutions (real, extraneous, or complex).

This will help differentiate the differences between

Fig. 5 A student’s algebraic solution to the proposed

graphical solution seen in figure 4 uses CAS.

extraneous and complex solutions—which students

sometimes think are the same thing.

When I have used this lesson, most students

gravitate toward sketching a graphical solution for

problem 1. At times, students use only polynomial

functions and try a linear function composed with

the basic radical function. This approach limits the

initial examination.

Without an equation, students have trouble

making a start on this problem. Trial and error or

guess and check usually do not yield a solution.

One student proposed a possible graphical solution

(see fig. 4), an equation that would involve the

radical function previously explored and a quadratic function. This choice shows great graphical

intuition, but this solution needs further examination. The intersection near or on the y-axis must be

at a point of tangency. This proposed solution leads

to one piece of mathematical understanding needed

for a solution.

Vol. 106, No. 6 • February 2013 | Mathematics Teacher 455

Fig. 6 Students’ only recourse to solving an equation such

as 2.3 sin x = 5x — 1 is to use a graph or a CAS.

Students can then explore this proposed solution

by solving an equation (see the CAS solution shown

in fig. 5). By using the CAS to solve this proposed

equation algebraically, students realize that four

answers result: x ≈ –1.86, –0.25, 0, and 2.11. The

point (in fig. 4) near (0, 2) is worthy of discussion

because it ended up not being a point of tangency.

The two functions intersect twice (at x = 0 and at

x ≈ –0.25) in the region near the positive y-axis.

Students need to make two important discoveries in the lesson to be able to solve problem 1. First,

they must realize that an equation with a single

cube-root radical will not generate any extraneous

solutions; only an even-root radical will. (A fuller

discussion of this concept is available at the website

provided at the end of the article.) Second, students

must discover that after they square both sides of

the equation, the result must generate three solutions. In other words, squaring the desired equation

must produce a cubic polynomial.

Once students make the mathematical connections needed, they can use the TI-Nspire capabilities (CAS, parameterization, transformation of

linear functions) to generate a correct solution to

problem 1. For example, 6×3 – 4x = x – 0.5 yields

one solution, but there are an infinite number of

answers involving any radical with an even index.

At this point in the lesson, teachers could ask questions such as these:

• What would happen if we try to generate a solution that includes a trigonometric or a transcendental function?

• What would happen if we try to generate an

equation with only integer or rational solutions?

One student used the TI-Nspire to generate a

fairly convincing graphical argument to a trigonometric-transcendental question (see fig. 6).

Some students have tried to generate the point of

tangency seen in figure 4 or integer-only solutions

by reversing the problem—that is, by starting with

456 Mathematics Teacher | Vol. 106, No. 6 • February 2013

(a)

(b)

Fig. 7 A student uses reversibility to generate a solution to

problem 1 (a) and then verifies the solution graphically (b).

a factored form of the equation and working backward algebraically. For example, one student started

with x • x • (x + 2) • (x – 3) = 0, which has a double

root at x = 0 and could be a solution to the graphical

representation seen in figure 4. The student first

removed parentheses and then ended up with the

equation 7×3 + 6×2 = –x2 (see fig. 7). An examination of the graphical representation made clear that

the x2 should be on the right side of the equation and

should be negative. Although this double root is not

a point of tangency as defined in geometry or calculus, the student found a correct solution to problem 1

with integer-only algebraic solutions.

This lesson thus helps develop students’ ability

to use mathematical flexibility and reversibility in

problem solving (Krutetskii 1976; see also Rachlin

1985). Making connections between algebraic and

graphical solutions also demonstrates the use of

mathematical flexibility in problem solving. The

family of equations of the form a5x + b + c =

mx + B cannot be used to solve problem 1. Deep

understanding of the mathematical connections

between algebraic and graphical solutions from

this family of functions presents an environment

conducive to reasoning and sense making. Both are

needed to move to mathematical problems requiring higher levels of cognitive effort.

(a)

Fig. 9 The graphical representation presents four

solutions to problem 2, three of which are extraneous.

(b)

Fig. 8 Students can solve problem 2 using a graph (a);

the other branch of the sideways parabola shows an

extraneous root (b).

SOLVING PROBLEM 2

The beauty of this problem is that it allows for

different approaches and different solutions. The

previous class discussions, leading questions, and

equations explore reflecting the simple square-root

function over the x-axis, thus creating a parabola

opening to the right or left with its vertex on the

x-axis. The lesson does not guide students through

the examination of a square-root function that

crosses the x-axis, as is the case with problem 2,

or one that lies completely above the x-axis. If we

plot the left side of the equation in problem 2 or

visualize y = 1x shifted to the left three units and

down two units, we see that this radical function

crosses the x-axis (see fig. 8). Student responses

to this problem have generally fit into one of two

approaches.

Most students’ first response includes shifting

the radical and the linear functions up two units.

Essentially, this is what we would expect or want

students to do first algebraically—add 2 to both

sides of the equation first, then square, and so on.

Thus, students accurately predict one real solution

and one extraneous solution.

The second type of response generally comes

from creating the parabola seen in figure 8b. I

want students to examine and understand—

algebraically and graphically—what will happen if

we square both sides of the equation first. Doing so

is a common first step by some students when they

see a square root sign. Mathematically, squaring

first is not as efficient. Only a few students have

ever submitted an answer such as that shown in

figure 9. This solution accurately depicts squaring both sides first and eventually solving a quartic

polynomial equation that will generate a real solution and three extraneous solutions.

Students must consider the four cases of the

radical equation ± [±5x + 3 – 2] = 2x + 3: (1) 5x + 3

– 2 = 2x + 3; (2) –5x + 3 – 2 = 2x + 3; (3) –[5x + 3]

– 2] = 2x + 3; and (4) –[–5x + 3 – 2] = 2x + 3. Cases

1 and 3 yield the equation –45x + 3 = 4×2 + 11x + 2,

and cases 2 and 4 yield the equation 45x + 3 = 4×2

+ 11x + 2. We now solve ±45x + 3 = 4×2 + 11x + 2,

which both produce the quartic polynomial equation 16(x + 3) = 16×4 + 88×3 + 137×2 + 44x + 4, with

solutions at x = –2.75, –2, (–341 – 3)/8 ≈ –1.1754,

and (341 – 3)/8 ≈ 0.4254.

In figure 9, the highlighted point depicts the

only real solution at x = −2 and represents case 1,

f1(x) = f2(x). The three extraneous solutions are

depicted with the dotted functions intersecting the

linear function. Case 2, f3(x) = f2(x), yields x =

−2.75. Case 3, f5(x) = f2(x), yields x ≈ −1.1754.

Case 4, f4(x) = f2(x), yields x ≈ 0.4254.

By not providing a direct path or direction to

solve problem 2, I am able to assess students on a

higher cognitive level (Stein et al. 2000).Thus, I do

not provide an in-depth exploration of this problem

type during the actual lesson. This approach also

serves as a way to create discussion about the

mathematics of this lesson after the assignment

has been collected.

DEVELOPING TPACK

This lesson was originally planned for secondary

school mathematics preservice teachers and their

Vol. 106, No. 6 • February 2013 | Mathematics Teacher 457

development of Technological Pedagogical and Content Knowledge (TPACK; see Mishra and Koehler

2006), mathematical reversibility, flexibility, and

reasoning and sense making.

It has generally been acknowledged that it

takes three to five years of training in technology

for teachers to be comfortable and knowledgeable

enough to use it regularly in teaching (Dwyer,

Ringstaff, and Sandholtz 1991; Means and Olson

1994). That was true for a different generation of

teachers. In today’s technologically driven world,

three to five years is too long. In theory, because

today’s generation of secondary school mathematics teachers grew up with technology in their hands

often and early, this time should be cut in half (see

Leatham 2007; Norton, McRobbie, and Cooper

2000; Zelkowski 2011).

This lesson is just one that can help in the development of TPACK as well as content knowledge.

Moreover, this approach to a topic traditionally

taught through rote procedure can be extremely

engaging in the high school classroom.

REFERENCES

Dwyer, David C., Cathy Ringstaff, and Judy H. Sandholtz. 1991. “Changes in Teachers’ Beliefs and

Practices in Technology-Rich Classrooms.” Educational Leadership 48 (8): 45–52.

Krutetskii, Vadim A. 1976. The Psychology of Mathematical Abilities in School Children. Trans. from the Russian by Joan Tell; edited by Jeremy Kilpatrick and

Izaak Wirszup. Chicago: University of Chicago Press.

Leatham, Keith R. 2007. “Pre-service Secondary

Mathematics Teachers’ Beliefs about the Nature of

Technology in the Classroom.” Canadian Journal

of Science, Mathematics, and Technology Education 7

(2/3): 183–207.

Means, Barbara, and Kerry Olson. 1994. “Tomorrow’s

Schools: Technology and Reform in Partnership.” In

Technology and Education Reform, edited by Barbara

Means, pp. 191–222. San Francisco: Jossey-Bass.

Mishra, Punya, and Matthew J. Koehler. 2006. “Technological, Pedagogical, and Content Knowledge: A

Framework for Integrating Technology in Teacher

Knowledge.” Teachers College Record 108 (6):

1017–54.

National Council of Teachers of Mathematics

(NCTM). 2000. Principles and Standards for School

Mathematics. Reston, VA: NCTM.

———. 2009. Focus in High School Mathematics: Reasoning and Sense Making. Reston, VA: NCTM.

———. 2010. Focus in High School Mathematics: Reasoning and Sense Making in Algebra. Reston, VA: NCTM.

Norton, Stephen, Campbell J. McRobbie, and Tom J.

Cooper. 2000. “Exploring Secondary Mathematics Teachers’ Reasons for Not Using Computers

in Their Teaching: Five Case Studies.” Journal of

458 Mathematics Teacher | Vol. 106, No. 6 • February 2013

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Research on Computing in Education 33 (1): 87–109.

Programme for International Student Assessment

(PISA). 2007. PISA 2006: Science Competencies

for Tomorrow’s World. Paris: Organisation for

Economic Co-operation and Development. http://

www.pisa.oecd.org/dataoecd/30/1739703267.pdf.

Rachlin, Sidney L. 1985. “The Development of Problem-Solving Processes in a Heterogeneous EighthGrade Algebra Class.” Paper presented at the annual

meeting of the North American Chapter of the International Group for the Psychology of Mathematics

Education, October 21–24, Columbus, OH.

Schoen, Harold L, ed. 2003. Teaching Mathematics

through Problem Solving: Grades 6–12. Reston, VA:

National Council of Teachers of Mathematics.

Stein, Mary K., Margaret S. Smith, Marjorie A. Henningsen, and Edward A. Silver. 2000. Implementing Standards-Based Mathematics Instruction: A

Casebook for Professional Development. New York:

Teachers College Press.

Zelkowski, Jeremy. 2011. “Developing Secondary

Mathematics Preservice Teachers’ Technological

Pedagogical and Content Knowledge (TPACK):

Influencing Positive Growth.” In Research Highlights in Technology and Teacher Education: 2011,

edited by Cleborne D. Maddux, David Gibson,

Bernie Dodge, Carl Owen, Punya Mishra, and

Matthew Koehler, pp. 31–38. Chesapeake, VA:

Society for Information Technology and Teacher

Education.

Editor’s note: The student activity sheet, teacher

notes, and TI-Nspire lesson file can be downloaded

from the author’s website: http://sites.google.com

/site/jszelkowski/Home/mtextraneous.

JEREMY S. ZELKOWSKI, jzelkowski@

bamaed.ua.edu, is an assistant professor of secondary mathematics

education at the University of Alabama in Tuscaloosa. He focuses on preparing

preservice mathematics teachers to incorporate

technology effectively into instruction when

they enter the teaching profession.