Making Sense of Extraneous Solutions Discussion

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"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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See discussions, stats, and author profiles for this publication at: Making Sense of Extraneous Solutions Article in Mathematics Teacher · February 2013 DOI: 10.5951/mathteacher.106.6.0452 CITATION READS 1 243 1 author: Jeremy Zelkowski University of Alabama 19 PUBLICATIONS 62 CITATIONS SEE PROFILE Some of the authors of this publication are also working on these related projects: Mathematics (Secondary) Teacher Education View project Using Technology in the Teaching of Mathematics - Practice View project All content following this page was uploaded by Jeremy Zelkowski on 20 January 2016. The user has requested enhancement of the downloaded file. 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. 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, 6x3 – 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 7x3 + 6x2 = –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 = 4x2 + 11x + 2, and cases 2 and 4 yield the equation 45x + 3 = 4x2 + 11x + 2. We now solve ±45x + 3 = 4x2 + 11x + 2, which both produce the quartic polynomial equation 16(x + 3) = 16x4 + 88x3 + 137x2 + 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 View publication stats 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:// Rachlin, Sidney L. 1985. “The Development of Problem-Solving Processes in a Heterogeneous EighthGrade Algebra Class.” Paper presented at the annual meeting ...
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When teachers are demonstrating mathematical flexibility in solving problems, it is
required they explain in the best way possible since students may lag behind during the lesson.
To avoid this, I think each teacher should be willing to foc...

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