230 MatheMatics teacher | Vol. 104, No. 3 • October 2010

deeperDelving

“Delving Deeper” offers a forum for classroom teachers to share the mathemat-ics from their own work with the journal’s readership; it appears in every issue of Mathematics Teacher. Manuscripts for the department should be submitted via http://mt.msubmit.net. For more background information on the department and guidelines for submitting a manuscript, visit http://www.nctm.org/publications/content.aspx?id=10440#delving.

Edited by J. Kevin Colligan, jkcolligan@verizon.netRABA Center of SRA International, Columbia, MD 20145

Dan Kalman, kalman@american.eduAmerican University, Washington, DC 20016

Virginia Stallings, vstalli@american.eduAmerican University, Washington, DC 20016

Jeffrey Wanko, wankojj@muohio.eduMiami University, Oxford, OH 45056

New Life for an Old Topic: Completing the Square Using Technology

mathematics experience at all levels. As students progress through their school mathematics experi-ence, their ability to see the same mathematical structure in seemingly different settings should increase” (p. 64).

A recent technology-oriented investigation led us to surprising connections among quadratics, mathematical envelopes, tangent lines, and tangent parabolas. In particular, our work with the TI-Nspire computer algebra system (CAS) enabled our students to generate conjectures and test hypotheses in ways not possible with pencil and paper alone while providing new insights into tra-ditional, skill-oriented topics.

COMPleTing THe SQUARe WiTH TileS: An OveRvieWAt a local teaching conference focusing on the use of interactive whiteboards (IWB), our interest was piqued by the speaker’s discussion of a seemingly routine topic—completing the square. The presen-tation began innocently enough, with the speaker demonstrating the drawing capabilities of IWB soft-ware. As he represented algebraic expressions with virtual Algebra Tiles (Bruner 1966; Dienes 1960; ETA/Cuisenaire 2008; Picciotto 2008), the speaker used tiles of three different dimensions—with areas of 1, x, and x2 square units (Corn 2004). The shapes (see fig. 1) were familiar to many in the audience.

The speaker then combined tiles to represent more complicated algebraic expressions. For

steve Phelps and Michael todd edwards

Mathematics teaching has always been a curi-ous blend of the old and the new. As the use of technology becomes more common-

place in school classrooms, this blend becomes even more pronounced. When teachers and students revisit traditional topics using technology, they are afforded opportunities to connect mathematical ideas in powerful, previously unimagined ways. The National Council of Teachers of Mathematics (NCTM) captures the importance of connections clearly in its Principles and Standards for School Mathematics (2000): “The notion that mathemati-cal ideas are connected should permeate the school

Copyright © 2010 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.

Vol. 104, No. 3 • October 2010 | MatheMatics teacher 231

instance, the trinomial x2 + 6x + 1 was represented as shown in figure 2.

By manipulating the tiles within the IWB environ-ment, the presenter demonstrated several examples of completing the square, such as the one illustrated in figure 3. The arrangement of tiles in figure 3a sug-gests that 8 units are needed to complete the square for the expression x2 + 6x + 1 (Corn 2004). The expression that results from this addition—namely, x2 + 6x + 9—is a perfect square trinomial, expressible as the square of a binomial: (x + 3)2 (see fig. 3b).

The speaker shared several more examples of this sort, each time completing the square by split-ting the x tiles into two piles with the same number of pieces—placing one pile in a horizontal “stack” below the x2 piece and the other pile in a vertical “stack” to the right of the x2 piece. While discussing pedagogical advantages and limitations of such an approach, the presenter made an off-hand comment that intrigued us: He noted that tiles were not help-ful when completing the square with trinomials con-taining odd linear coefficients (such as x2 + 5x + 1) because odd numbers of x tiles cannot be split into two piles with the same number of pieces.

Although no one questioned this claim, the authors sat through the remainder of the session wondering whether this statement were true. At the conclusion of the presentation, we sat in the conference center lobby, constructing squares with presentation software on our laptops. Rather than splitting the x blocks into two equal piles, we added blocks of various dimensions onto our initial con-struction to complete the square. Within five min-utes, we had generated a family of expressions that could be added to x2 + 5x + 1 to complete squares.

Two examples of completed squares are shown in figure 4, one completed by adding x + 8 and the other completed by adding 3x + 15.

We now explore unexpected mathematical con-nections fostered by our new interpretation of com-pleting a square, in particular the following:

• Thefamilyoflinearfunctionsthatcomplete

squares for various trinomials (i.e., square com-pleters) and the relationship of these functions to the original (i.e., seed) trinomial

• Connectionsbetweensquare-completerfami-lies and the traditional completing-the-square algorithm

Technology plays a central role throughout our exploration, informed by concrete examples. For instance, IWB and virtual Algebra Tiles motivated our initial questions about alternative conceptions of completing the square. Subsequent explorations involving more sophisticated mathematical

Fig. 1 the three basic shapes of algebra tiles may be used

to represent algebra terms.

Fig. 2 this arrangement of tiles represents the algebraic

expression x2 + 6x + 1.

Fig. 3 adding tiles to (a) completes the square (b).

(a) (b)

Fig. 4 there are many ways to complete a square begun with x2 + 5x + 1; two are

shown here.

(a) (b)

232 MatheMatics teacher | Vol. 104, No. 3 • October 2010

simulation and data analysis are fostered using TI-Nspire CAS and dynamic geometry software.

FAMILIES OF SQUARE COMPLETERSConsider our previous example involving x2 + 5x + 1. Students are typically taught that exactly one expression will complete the square for such a trinomial when in fact any one of an entire family of linear expressions may be added to gen-erate perfect square trinomials. Teachers (and their students) typically add only a constant term when given the choice because adding additional variables complicates algebraic expressions when solving equations. Restricting one’s attention to square completers with no variables, although ped-agogically justifiable, ignores rich and unexpected mathematical connections. By exploring the fam-

ily of expressions that, when added to the original expression, create a perfect square polynomial, we provide students with opportunities to reconsider traditional content using inquiry-oriented, technol-ogy-rich perspectives.

For instance, figure 4 illustrates that adding x + 8 to x2 + 5x + 1 generates the perfect square tri-nomial x2 + 6x + 9 = (x + 3)2. Likewise, adding 3x + 15 yields x2 + 8x + 16 = (x + 4)2. In fact, an infinite number of different expressions may be added to x2 + 5x + 1 to generate perfect square trinomials. We define the nth square completer of p(x) as the expression one adds to p(x) to yield the perfect square trinomial (x + n)2. Table 1 illustrates family members for p(x) = x2 + 5x + 1 for various n.

Note that the second differences from entries in the second column are constant, suggesting a qua-dratic relationship between n and the nth square completer. Later we discuss a CAS-assisted deriva-tion of a general formula for the nth square com-pleter for arbitrary p(x).

INITIAL INVESTIGATIONS WITH CASTo gain a more thorough understanding of the family of square completers, we constructed a multipage TI-Nspire document to generate fam-ily members for a given trinomial p(x) automati-cally. On the first page of the document, students manipulated sliders to change the coefficients of the seed trinomial: ax2 + bx + c. On the second page, algebraic expressions for various family members were generated within a CAS-enhanced spread-sheet. Screen shots from the first two pages of the TI-Nspire document are highlighted in figure 5.

As figure 5a suggests, coefficients of the seed trinomial (a, b, and c) controlled by the sliders in a TI-Nspire Graphs & Geometry window were linked to a CAS-enhanced spreadsheet, as shown in figure 5b. Algebraic expressions for square com-pleters were calculated in column D of the spread-sheet and graphed back in the Graphs & Geometry window. In figure 5a, the envelope formed by family members appeared to be quadratic. To explore this conjecture more rigorously, students constructed intersection points among consecutive

Fig. 5 Graphical representations (a) and tabular representations (b) can be linked.

Table 1 Various Family Members for Seed Polynomial p(x) = x2 + 5x + 1

nnth

Square Completer Sum Factored Form

3 x + 8 (x2 + 5x + 1) + (x + 8) = x2 + 6x + 9 (x + 3)2

4 3x + 15 (x2 + 5x + 1) + (3x + 15) = x2 + 8x + 16 (x + 4)2

5 5x + 24 (x2 + 5x + 1) + (5x + 24) = x2 + 10x + 25 (x + 5)2

6 7x + 35 (x2 + 5x + 1) + (7x + 35) = x2 + 12x + 36 (x + 6)2

100 195x + 9999 (x2 + 5x + 1) + (195x + 9999) (x + 100)2

(a) (b)

Fig. 6 intersection points are located (a), and a quadratic regression is performed (b).

(a) (b)

Vol. 104, No. 3 • October 2010 | MatheMatics teacher 233

family members and then generated a quadratic function to fit these ordered pairs (see fig. 6).

The white dots in figure 6a depict intersections of nth and (n + 1)st square completer graphs. The bold curve was generated by fitting a quadratic function to these intersections. As students manipu-lated values of a, b, and c with sliders, the envelope and associated quadratic regression were updated dynamically. This feature enabled students to look for connections between various seed polynomials and corresponding envelopes of square completers such as those provided in table 2. Because of the amount of time required to generate such graphs by hand, such an exploration would have been highly impractical for our students without technology.

Clearly, the entries in table 2 suggested a rela-tionship between the equations of quadratic enve-lopes and seed polynomials—namely, seed p(x) = x2 + bx + c appeared to have envelope –x2 – bx – c + 0.25. Although we were not certain whether this conjecture held in general, the cases warranted further investigation.

COnneCTiOnS WiTH TAngenT CURveSBecause the sign of the coefficients of the seed and the envelope appeared to be opposites, we modi-fied our TI-Nspire document to graph the opposites of square completers of p(x), defining qn(x) as the opposite of the nth square completer for a given p(x). Explicitly, qn(x) = p(x) – (x + n)2. The modified TI-Nspire document simultaneously graphed qn(x) for various n and p(x). A sample graph for p(x) = x2 + 5x + 1 and qn(x) with –20 < n < 20 is shown in figure 7.

By inspection, students conjectured that the graph of qn(x) was tangent to p(x) for all n. To determine whether such a hypothesis were true, we first con-structed a formula for qn(x) given p(x) = x2 + bx + c:

qn(x) = p(x) – (x + n)2 by definition = x2 + bx + c – (x2 + 2nx + n2) by substitution = (b – 2n)x + (c – n2) algebraic simplification

Several students performed similar calcula-tions in a step-by-step fashion using CAS. Such an approach is highlighted in figure 8.

Insight into the general formula qn(x) was pro-vided through analysis of a concrete representation

Table 2 Seed Polynomials and Corresponding Quadratic Envelope Equations

Seed Polynomial

Envelope of Square Completers

x2 + 5x + 1 –x2 – 5x – 0.75

x2 + 5x + 2 –x2 – 5x – 1.75

x2 + 5x + 3 –x2 – 5x – 2.75

x2 + 6x + 1 –x2 – 6x – 0.75

x2 + 6x + 2 –x2 – 6x – 1.75

x2 + 6x + 3 –x2 – 6x – 2.75

x2 + 9x + 9 –x2 – 9x – 8.75Fig. 7 the graph of p(x) and qn(x) with –20 < n < 20 led

students to a conjecture.

Fig. 8 students can calculate qn(x) using the ti-Nspire cas.

Fig. 9 students construct a representation of qn(x) =

(x2 + bx + c) – (x + n)2 using algebra tiles.

234 MATHEMATICS TEACHER | Vol. 104, No. 3 • October 2010

of (x – n)2 – (x2 + bx + c) with Algebra Tiles (see fig. 9). The gray and white regions—that is, (2n – b)x + n2 – c—represent the nth square completer of x2 + bx + c—that is, – qn(x). With qn(x) defined as –(2n – b)x – (n2 – c) for arbitrary p(x) = x2 + bx + c, students used CAS to verify that the graph of qn(x) was tangent to p(x) for all n.

In using CAS to verify that each square-com-pleter family member was tangent, we first defined p(x) as the general trinomial x2 + bx + c and qn(x) as –(2n – b)x – (n2 – c). Solving the equation p(x) = qn(x), we determined intersection points of x2 + bx + c and qn(x), the nth member of the square completers. Solutions to the equation, as calculated with TI-Nspire CAS, are shown in figure 10.

From figure 10, we saw that there was exactly one solution to the equation—namely, x = –n. This result strongly suggested that every member of the square-completer family qn(x) was tangent to p(x) at x = –n. Establishing with certainty that qn(x) was tangent to p(x) required little more than elementary calculus. Noticing that the slope of qn(x) is –(2n – b),students noted that this is precisely the value of p′(x) = 2x + b evaluated at x = –n. Hence, they cor-rectly concluded that qn(x) and the tangent line to

p(x) at x = –n were parallel. Because qn(x) and p(x) have only one point in common, students concluded that qn(x) was the tangent line to p(x) at x = –n.

GRAPHICAL INTERPRETATION OF SQUARE COMPLETERSHow do p(x), qn(x), and the completed square (x + n)2 fit together graphically? As an example, we asked students to consider p(x) = x2 + 5x + 1 and the spe-cific square completers q-1(x) and q2(x) (see fig. 11). In figure 11a, the vertex of the completed square is below the point of tangency; in figure 11b, the ver-tex is above the point of tangency. Square completers q-1(x) and q2(x) are members of qn(x). Students were able to recognize that the plotted functions were tangent to p(x) at x = 1 and x = –2, respectively. Moreover, they noted that each member of qn(x) led to a completed square trinomial whose vertex was located on the x-axis directly above (or below) the point of tangency. By completing the square in this manner—that is, by adding on the appropriate Algebra Tiles—we effectively translated p(x) so that the vertex was vertically aligned with the point of tangency and on the x-axis.

COMPLETING THE SQUARE REVISITEDAt this juncture, it was instructive to connect our new observation back to more traditional notions of completing the square—that is, taking half the linear coefficient, squaring, and adding. In the traditionally understood method of completing the square, students add nothing but unit blocks to the initial trinomial; they do not add x blocks. Algebraically, this approach implies that qn(x) = k, a constant function. Hence, when we complete the square in the conventional sense, the coefficient of the linear term of qn(x) is zero—in other words, –(2n – b) = 0, implying that n = b/2. We know that qn(x) is tangent to p(x) at x = –n, which implies that, in traditional completing-the-square situa-tions, qn(x) is a horizontal line tangent to p(x) at the vertex of the trinomial.

This observation led us to a novel interpretation of completing the square. Geometrically speaking, by completing the square, one translates p(x) vertically so that its vertex lies on the x-axis. We encouraged students to see that completing the square for p(x) = x2 + 5x + 1 involves steps similar to the following:

Step 1:

Step 2:

p x x x

p x x

( )p x( )p x

( )p x( )p x

= +x x= +x x

+ −+ −+ −+ −+ −+ −+ −+ −+ − =

2= +2= +x x= +x x2x x= +x x2+ −2+ −

5 1x x5 1x x +5 1+

2 5.2 5.+ −2 5+ −.+ −.2 5.+ −. 1 2 222 22

2

5 12 25 12 25 12 22 52 22 5 1

5 25 525 52 6

+ +2 2+ +2 25 1+ +5 12 25 12 2+ +2 25 12 2( )2 2)2 2+ −2 2+ −2 2+ −2 2+ −2 22 5+ −2 52 22 52 2+ −2 22 52 2+ −2 2+ −2 22 22 2+ −+ −2 2+ −2 22 2+ −2 2

+ −+ −+ −+ −+ −+ −2 2+ −2 22 2+ −2 2

2 2+ −2 22 2+ −2 2

+ =5 2+ =5 25 5+ =5 5+ +5 5+ +5 5

x5 1x5 15 1+ +5 1x5 1+ +5 12 25 12 2+ +2 25 12 2x2 25 12 2+ +2 25 12 2

p x 5 5x x5 525 52x x25 52 + +x x+ +5 5+ +5 5x x5 5+ +5 5x x

2 5.2 52 5+ −2 5.2 5+ −2 5

( )p x( )p x . .6. .6+ +. .+ +. .5 2. .5 25 5. .5 5+ =. .+ =5 2+ =5 2. .5 2+ =5 25 5+ =5 5. .5 5+ =5 5x x. .x x5 5x x5 5. .5 5x x5 5+ +x x+ +. .+ +x x+ +5 5+ +5 5x x5 5+ +5 5. .5 5+ +5 5x x5 5+ +5 5Step 3: 252252

5 25 2 5

2

2Step 4:

Step 5:

p x 5 2x5 2

p x x

( )p x( )p x . .5 2. .5 2. .5 2. .5 25 2. .5 25 2x5 2. .5 2x5 2

( )p x( )p x

+ =5 2+ =5 25 2+ =5 2. .+ =. .5 2. .5 2+ =5 2. .5 25 2. .5 2+ =5 2. .5 25 2+5 25 2. .5 2+5 2. .5 2(5 2(5 25 2. .5 2(5 2. .5 2 )= +x= +x= + . ... ..5 5. .5 5. .5 5. .5 5. .25

2(= +(= + )5 5)5 5. .5 5. .). .5 5. .5 5−5 5

Fig. 10 Points of intersection for p(x) and qn(x) were

determined using the CAS.

Fig. 11 Note the location of the vertex of the completed square trinomial in relation

to the points of tangency of the square completers in both (a) and (b).

(a) (b)

Vol. 104, No. 3 • October 2010 | MatheMatics teacher 235

In step 4 of the calculations above, the seed function p(x) is translated vertically +5.25 units. In particular, the vertex of p(x) is translated from (–2.5, –5.25) to (–2.5, 0). We represented this idea graphically by plotting p(x) and q2.5(x) simultaneously with the TI-Nspire CAS (see fig. 12). Recalling previous discussions of the square completers q-1(x) and q2(x), students viewed the traditional method of completing the square as a special case of a much larger complet-ing-the-square theory.

FOUR SUggeSTiOnS FOR FURTHeR ReSeARCHThere are many fruitful paths to explore when we combine the concrete visual clues of Algebra Tiles with the rich technology applications found in the TI-Nspire CAS. The following four investiga-tions offer interested readers a glimpse of the ways a technologically enhanced Algebra Tiles view of polynomials can lead students to previously unforeseen connections. These investigations are readily accessible to students in first-year algebra through precalculus.

1. Quadratics with Leading Coefficients Other Than 1Consider p(x) = 2x2 + 5x + 2. The traditional algo-rithm for this case would have students first divide through by 2 (the leading coefficient) to make the leading coefficient 1 and then proceed as the tra-ditional algorithm dictates—that is, take half the linear coefficient, square it, and add. Instead, we look for opportunities to add on tiles to complete a square. As we have established, many combinations of Algebra Tiles would suffice to complete a square; for instance, adding 2x2 + 3x + 2 would work, as would adding on 2x2 + 11x + 14. Graphing p(x) along with the family of the opposites of all possible square completers—that is, qn(x)—produces the graph in figure 13.

2. Completing the Square for Linear EquationsBegin with a single x block and 3 unit blocks and model x + 3. What Algebra Tiles would one need to add to complete the square? One could add x2 + 3x + 1 or even x2 + 9x + 22. Either way, if one exam-ines the family of square completers, the result is a family of parabolas tangent to the line y = x + 3 (see fig. 14).

Fig. 13 changing the coefficient of the leading term to a number other than 1 yields

similar results.

(a) (b)

Fig. 12 the traditional completing-the-square algorithm

can be given a graphical interpretation.

Fig. 14 square completers for a linear function form a

family of tangent parabolas.

Fig. 15 3D algebra Blocks (a) can be used to model completion of a cube; the graph

(b) shows p(x) and some cube completers.

(a) (b)

236 MatheMatics teacher | Vol. 104, No. 3 • October 2010

and their manipulation more concretely. Moreover, the use of technology afforded both

us and our students opportunities to connect math-ematical ideas in powerful, previously unimagined ways. The process of completing the square is transformed from a series of algebraic steps to be memorized, executed, and quickly forgotten to a geometric process that reveals the vertex of a parab-ola through geometric translation.

BiBliOgRAPHYBrown, S. I., and M. I. Walter. The Art of Problem Pos-

ing. 3rd ed. Hillsdale, NJ: Lawrence Erlbaum Asso-ciates, 2004.

Bruner, Jerome S. Toward a Theory of Instruction. Cambridge, MA: Belknap Press, Harvard Univer-sity Press, 1966.

Corn, J. “Completing the Square: Virtual Algebra Tiles, Smart Boards, and SMG.” Presentation at the Teachers Teaching with Technology Regional Con-ference, Cleveland, OH, Nov. 13, 2004.

Dienes, Zoltan P. Building Up Mathematics. London: Anchor Press, Hutchinson Educational, 1960.

ETA Cuisenaire. Algebra Tiles. Vernon Hills, IL: ETA Cuisenaire, 2008. www.etacuisenaire.com/.

National Council of Teachers of Mathematics (NCTM). Principles and Standards for School Math-ematics. Reston, VA: NCTM, 2000.

Picciotto, Henri. “Factoring x ^ 2 + bx + c.” 3D Alge-bra Lab Gear. 2008. http://www.picciotto.org/math-ed/manipulatives/factoring.html.

3. Completing CubesSimilarly, 3D Algebra Blocks can be used to model p(x) = x3 + 2x2 + x (see fig. 15). What would be required to complete the cube? As before, we could add x2 + 2x + 1, or 4x2 + 11x + 8, or 7x2 + 26x + 27, to name just a few cube completers. Graphing the opposite of these yields a family of parabolas in which some are tangent although each member intersects the cubic p(x) at only one point. Further investigation will yield connections between these cube completers and Taylor polynomials.

4. Squaring the CircleExperienced Algebra Tiles users will recognize both x and y blocks; many also are comfortable with var-ious methods of modeling negative integers. These methods can be used to model quadratics that rep-resent conics. For instance, consider the collection of x2 + y2 – 4 tiles used to model the circle x2 + y2 = 4. In general, to complete the square (x + y + n)2, one could add 2xy + 2x + 2y + 5, or 2xy + 4x + 4y + 8, or 2xy + 16x + 16y + 68. Graphing the circle and the family of square completers for the circle produces a family of hyperbolas, some of which appear to be tangent to the circle (see fig. 16).

SUMMARYWe have explored alternative interpretations of completing the square along with the mathematical implications of such interpretations. Throughout our investigations, technology—specifically, Alge-bra Tiles—has played a central role.

From generating initial questions with inter-active whiteboards to constructing and testing hypotheses with students using TI-Nspire CAS and dynamic geometry software, technology has made various facets of this investigation accessible to sec-ondary school students. In particular, the tools pro-vide students with a means of considering abstract mathematical objects such as symbolic polynomials

STEVE PHELPS, sphelps@ madeiracityschools.org, teaches geometry at Madeira High School in Cincinnati, Ohio. MICHAEL TODD EDWARDS, edwardm2@muohio.edu, is an associate professor in the Department of Teacher Education at Miami University in Oxford, Ohio.

Both are interested in the use of technology in teaching and learning mathematics, with particular emphasis on computer algebra systems, dynamic geometry software, and pencasting technology.

Fig. 16 completing the square for the equation of a circle

produces graphs like this one.