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The Illinois Mathematics Teacher

EditorsDaniel JordanColumbia College Chicago623 S Michigan Ave, Suite 500Chicago, IL 60605

Christopher ShawColumbia College Chicago623 S Michigan Ave, Suite 500Chicago, IL 60605

Reviewers

Kris AdamsEdna BazikSusan Beal

Carol BensonPatty Bruzek

Dane CampBill Carroll

Mike CartonChip Day

Lesley EbelTodd Edwards

Linda Fosnaugh

Dianna GalanteLinda Hankey

Pat HerringtonAlan Holverson

John JohnsonRobin Levine-Wissing

George MarinoKaren Meyer

Jackie MurawskaNicolette Norris

Jacqueline PalmquistJames Pelech

Randy PippenSue Pippen

Aurelia SkibaAndrzej Sokolowski

Clare StaudacherJoe Stickles

Robert UrbainNicole Wessman-Enzinger

Darlene WhitkanackPeter WilesLara Willox

ICTM Governing Board

OfficersPresident: Robert Mann Secretary: Lannette Jennings

President Elect: George Reese Treasurer: Rich WyllieBoard Chair: Kara Leaman

Directors

Early Childhood

Carly MoralesJennie Winters

Grades 5–8

Eric BrightAnita Reid

Grades 9–12

Martin FunkKara Leaman

Community College/University

Craig CullenPeter Wiles

At-Large

Zachary HerrmannJackie MurawskaAdam Poetzel

The Illinois Mathematics Teacher is the official journal of the Illinois Council of Teachers of Mathematics andis devoted to providing ideas and information to support the professional development of teachers at all levels of thecurriculum (K-16). Current issues and select back-issues are available online at http://ictm.org/imt.html.

This work is licensed under the Creative Commons Attribution 3.0 United States License. To view a copy of this license,visit http://creativecommons.org/licenses/by/3.0/us/.

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Table of Contents

Editors’ Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2014 ICTM Awards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Susan Brown

Articles

Constructivist Approach to Algebraically Expressing a Line . . . . . . . . . . . . . . . . . . . . . . 4Andrzej Sokolowski

Teaching Tricks Can Be Tricky . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Linda Fosnaugh & Patrick Mitchell

Cracking the Codes of Number Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Lara Willox & Sarah C. Newman

iPad Fun: Exploring Slope and Functions Using iPads with the TI-NSpire App 21Ann Wheeler & Brandi Falley

Puzzle

“Follow the Directions,” a crossword puzzle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Christopher Shaw

Editors’ Note

Welcome to volume 62 of Illinois MathematicsTeacher, the journal of the Illinois Council ofTeachers of Mathematics. This marks our first fullissue as editors of the journal, and we are excitedto continue the tradition of quality scholarshipthat this journal represents.

During our transition into the editorship, wereceived invaluable help from many sources fromacross ICTM. In particular, we would like tothank Marilyn Hasty and Tammy Voepel, whoserved for a long time as editors of the journal andprovided us with valuable resources and insight.We would also like to mention Don Porzio,who provided us with administrative guidance asformer president of the ICTM, Ann Hanson, whorecommended us for the task and continues tosupport us in many ways, and Don Beaty, whomanages the online platform for the journal.

In this issue

Each year at the annual meeting of the ICTM,winners of the ICTM awards are invited to presentremarks to the audience in attendance. Amongthe awardees this year was Susan Brown, whosubmitted her inspirational remarks to the jour-nal for publication. Andrzej Sokolowski presentsan inquiry-based lesson for exploring linear geom-etry in “Constructivist Approach to AlgebraicallyExpressing a Line.” Linda Fosnaugh and PatrickMitchell argue for grounding mathematical proce-dures in conceptual understanding in “TeachingTricks Can Be Tricky.” In “Cracking the Codesof Number Relationships,” Lara Willox and SarahC. Newman present the results of their work de-veloping number sense in early mathematics class-rooms. In “iPad Fun: Exploring Slope and Func-tions Using iPads with the TI-Nspire App,” AnnWheeler and Brandi Falley present a classroomactivity in which students use iPad software toanalyze the slopes of real objects. The final con-

tribution to this issue is a mathematics-themedcrossword puzzle composed by Christopher Shaw.

Finally, as of this issue, the journal willbe published exclusively online through theICTM website. New articles will be publishedas they are accepted on the journal websiteand periodically collected into complete issues.With this transition, several other functionshave also moved online. If you would liketo join ICTM, we encourage you to visit theICTM website at http://ictm.org. To submitan article or volunteer to be a reviewer forthe IMT, please go to the journal website athttp://www.ictm.org/journal, where you willfind a detailed set of guidelines for submission tothe journal. You may also send an email to theeditors at [email protected].

Thank you for reading, and we look forwardto working with you.

Daniel Jordan & Christopher Shaw

Illinois Mathematics Teacher 1

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mailto:[email protected]

2014 ICTM Awards

From the editors: The editors congratulatethe winners of the 2014 ICTM awards, whichwere presented at the 64th Annual Meeting of theIllinois Council of Teachers of Mathematics. Theawardees are listed below. We also present theremarks delivered by Susan Brown, the winnerof the 2014 T.E. Rine Secondary MathematicsTeaching Award, who submitted the text of heracceptance speech to the IMT. Information aboutthe awards and nomination instructions can befound at http://ictm.org/ictmawards/.

Max Beberman Mathematics Educator Award

Alan ZollmanNorthern Illinois University, DeKalb

Fred Flener Award

Paul J. KarafiolWalter Payton College Prep High School, Chicago

T.E. Rine Secondary Mathematics Teaching Award

Susan BrownYork High School, Elmhurst

Distinguished Life Achievement in Mathematics

Angela AndrewsNational Louis University - Retired

Illinois Promising New Teacher of Mathematics

Esther SongNiles West High School, Niles

Lee Yunker Mathematics Leadership Award

David SpanglerMcGraw-Hill Education, Chicago

Elementary Mathematics Teaching Award

Denise BrownCarruthers Elementary School, Murphysboro

Middle School Mathematics Teaching Award

Annie ForestFreedom Middle School, Berwyn

A Thought About Teaching (for MathTeachers)

Remarks by Susan Brown

The first time I taught geometry was manyyears ago. The textbook was very theoretical,a holdover from the New Math era. Every timewe encountered a new relationship, students wereasked to determine whether it was an equivalencerelationship, and if it was, they had to prove it. Ihad never heard the term before, and had to learnthat an equivalence relationship is one that isreflexive, symmetric, and transitive. For instance,early in the year we studied congruence, and weproved the following:

Reflexive Property:

∠A ' ∠A.

Symmetric Property:

If ∠A ' ∠B, then ∠B ' ∠A.

Transitive Property:

If ∠A ' ∠B and ∠B ' ∠C, then ∠A ' ∠C.

My students did not find this to be very interest-ing, and I admit that I did not either. (The onlyintriguing situation was perpendicularity. Unlikecongruence and similarity, it is not an equivalencerelationship.) But years afterward, this experi-ence led me to a thought that I did find interest-ing: Teaching is an equivalence relationship. Letme convince you that this is true.

Reflexive Property: Person A teaches person A.As beginning teachers quickly find out, you mustteach yourself an enormous body of knowledge inorder to do your job. You must enlarge whatyou know about mathematics as you struggle toanswer questions like “Why does it work thisway?” You must formulate alternate explanationsto things that you took for granted as a student,

2 Illinois Mathematics Teacher

http://ictm.org/ictmawards/

2014 ICTM Awards

or learned on the surface and never reallypondered. Beyond the subject matter, much ofwhat you must teach yourself is about learningand about kids. How do you do these things?Of course, collaboration, professional reading, andfurther coursework play a role. But they havelimited impact unless you also watch yourself andwhat unfolds in your classroom, and think deeplyabout what you observe.

Symmetric Property: If person A teaches personB, then person B teaches person A. Perhaps thebest teachers you have are your own students.If I have any advice for new teachers, it is toalways, always, always listen to your students.Their thinking is the end product of what youdo. Their words and actions are the clues thatlet you discover what they know. Their questionsand comments, however odd and puzzling, helpyou to understand what’s really going on. Andthey can have wonderful insights and ideas thatyou never even considered.

Transitive Property: If person A teaches personB, and person B teaches person C, then person Ateaches person C. I can still hear the voice of myown geometry teacher in what I say to students.I hope that I have been able to capture someof her enthusiasm and passion for mathematics,and to pass them on to the students and teacherswith whom I work. But that is not all. Throughmy teacher there is a chain that leads backwardto Euclid and his predecessors. That chain alsomoves forward. My students will someday playa role in the early mathematical education oftheir own children. Some of my students havegone on to become math teachers themselves, andwill influence thousands of students in the future.What we do as teachers does not end with us.

Q.E.D.

Susan Brown

MATHEMATICS DEPARTMENT

YORK HIGH SCHOOL

355 W ST. CHARLES RD.

ELMHURST, IL 60126

E-mail : [email protected]


Constructivist Approach to Algebraically Expressing a Line

Andrzej Sokolowski

Abstract

Although sketching and building linear functions surfaces in many mathematics classes, taking directmeasurements leading to function formulation is not often exercised. This paper—written in the form ofa lesson accompanied by commentaries—is to fill in the gap. Guided by constructivist learning theory,the lesson places the students in the roles of scientists who integrate their mathematical theoreticalknowledge with their skills of measuring to produce algebraic representations of straight lines. The unitdoes not involve complex equipment, except rulers that are readily available in any math classroom.Despite its simplicity, this lesson generates an engaging, discovery type learning environment thatinvolves not only the students but also the teacher.

Keywords: mathematics, education, constructivism, slope

Published online by Illinois Mathematics Teacher on October 7, 2014.

1. Introduction

Several studies (e.g., Stanton & Moore-Russo,2012; Lobato & Siebert, 2002) have revealedthat the concept of slope, despite its multipleapplications, appears to students as an abstractmathematical entity. It is hypothesized thatthis difficulty might be rooted in presenting theslope as a ratio of dimensionless quantities, whichdiminishes its physical interpretation and weakensits relevance to students’ prior experiences. Thispaper presents a lesson that is designed to enrichthe meaning of slope—and consequently themeaning of linear functions—by applying directmeasurement.

The lesson is rooted in the constructivisttheory of knowledge acquisition. Constructivistsclaim that learners construct their own knowledgebased on received impulses (Von Glasersfeld,1995). One of the opportunities to apply thislearning theory is to place learners in realisticsettings that resonate with their prior experiencesand let them construct the knowledge.

The lesson presented here will reflect theconstructivist approach. It can serve as aunit summarizing the process of building linearfunctions. It can be conducted in any mathclass provided students are familiar with the

concepts of slope and the algebraic formulationof linear functions using slope-point or slope-intercept form.

2. Lesson Description

2.1. Lesson Outline

The commentary provided in the body ofthe manuscript is based on a lesson conductedwith a group of precalculus students in a ruralEast Texas high school. During the lesson, thestudents—guided by the teacher—unfolded thephysical meanings of all parameters and processesneeded to express a given line in a symbolicalgebraic form. A metric ruler was used toquantify the needed parameters, such as slope andintercepts. Measurement, which is an elementof scientific inquiry, has received significantattention in the newly developed Common CoreState Standards (Porter et al., 2011). Thus,including this element in the lesson enhanced notonly the constructivist approach but also reflectednew mathematics teaching recommendations. Inaddition, the students used graphing calculatorsto verify the derived mathematical models.

Since measuring is intertwined with thealgebraic meanings of these processes in thislesson, it was hypothesized that a unified



view of expressing a physical object—a line—algebraically would emerge in students’ mindsas a result. It was also expected that, duringthis process, the mathematical tools used toexpress the slope of a line in mathematical formwould become tangible and more meaningful tostudents.

The lesson made use of guided discovery asthe method of instruction. Thus, the teacherwas available to guide the students through theprocess of transferring a sketched line into itsalgebraic form.

2.2. Using Measurement to Connect withStudents’ Prior Experiences

With the blackboard cleaned of any coordinateaxes, the teacher opened the lesson by drawinga line on the board and positing the followingquestion: how can one construct an algebraicequation of the line? The students were puzzled atfirst because no axes or grid lines were provided—i.e., they experienced a dissonance that prompteda discussion. The teacher directed students’attention to a general structure of a linearfunction, y − y1 = m(x − x1), and asked howto find the parameters that are needed to usethe structure. Students suggested that selectingany two points from the line was necessary. Theteacher selected two points and labeled themon the line (see figure 1) and then asked forthe next step needed to find the line’s equation.The students were again in doubt because, eventhough the points were labeled, the coordinateswere still not quantified, i.e., the line was placed inan environment without associated mathematicalrepresentations that the students were used tohaving provided for them.

The teacher asked what was needed toestablish these points’ numerical coordinates.The students realized that a frame of referencecalled the “x-y coordinate axes” was missing. Theteacher then asked where exactly the axes shouldbe drawn. The students suggested locating themnear the line. The teacher drew, or asked astudent to draw, the coordinate axes on the boardin a standard vertical and horizontal fashion, asdepicted in figure 2.

A

B

Figure 1: Line with two points identified.

x

y

A

B

Figure 2: Line with coordinate axes.


Andrzej Sokolowski

Since neither the scaling of the axes nor gridlines were provided, the students experienced yetanother dissonance resulting in their inabilityto continue with the process of the linequantification. Some students suggested “makingup a scale” on the x and y axes. The teacher askedif there was a more precise method of finding exactnumerical magnitudes of the coordinates. Thestudents had difficulty associating the coordinateswith their measurable physical lengths. In orderto make the transition, the teacher referred toa meter-stick and said that since the line waspositioned with reference to the established x-y axes, certain quantities such as lengths couldbe measured in order to quantify the inclinationknown as slope and the line’s exact locationwith reference to the axes. This was a pivotalelement of the lesson and apparently “an eyeopener” for some students. They realized that thecommonly given coordinates represent, in fact,physical distances of points from an establishedreference, called the origin in mathematics.

A short discussion of what measuring systemto use—initiated by the teacher—also tookplace. Students realized that whether they usedcentimeters or inches, the equation of the line,although expressed in different units, shouldresemble the same position and inclination as theline sketched on the board.

In order to fully experience this idea, thestudents were further guided through the processof deciding what segments to measure and how tomeasure the segments (see figures 3 and 4). Theteacher asked some students to participate andmeasure the necessary lengths.

The students decided to quantify the verticalintercept first. Its magnitude as measured fromthe origin was 20 cm. They concluded that thenext step was obtaining a numerical value for theslope, which was possible in two ways—either bymeasuring the coordinates of the points or bymeasuring the lengths of the run and rise betweenthe points. Most of the students suggestedmeasuring the run and rise between the points.The teacher sparked conversation by asking ifmeasuring the length of the segment AB would besignificant in the process of slope quantification.

x

y

A

B

Figure 3: Measuring the vertical intercept.

x

y

A

B

Figure 4: Slope quantification.

Most students rejected this idea. Yet, as a righttriangle was “formed,” this question generated afurther inquiry about the differences between theconcepts of slope and hypotenuse. Not all of thestudents understood that these two fundamentalconcepts were different. After a short discussionconsensus was reached that the segment AB,when representing the slope, was a ratio, while, asa hypotenuse of a triangle, it represented a length.

2.3. Integrating the Measurements into SymbolicLine Representation

The inclination of the line was determined bythe ratio of the vertical and horizontal distancesbetween the points, i.e.,

slope =vertical displacement

horizontal displacement=

∆y

∆x.



In the example, the slope took the value of 43 cm50 cm =

0.86cmcm . By applying the slope-intercept form,

the students concluded that the equation of theline was y = 0.86cm

cmx + 20 cm, or more simply,y = 0.86x + 20. The students were then asked touse a graphing calculator to verify that the graphon the board resembled the function equation.A discussion about selecting a calculator viewwindow that would provide a full view of theline and also reflect the blackboard dimensionsensued. This discussion helped students to realizethat the view window that would match theblackboard was determined by the horizontaland vertical distances of the established x-yaxes from the edges of the blackboard. Acomment supporting the use of negative valuesfor the horizontal distance to the vertical axis onthe left (Xmin) and for the distance below thehorizontal axis (Ymin) completed the discussion.The teacher concluded that the students’ termnegative distance is not being used as in it isin physics, but instead a negative position isimplied. This comment was especially of interestto students who were concurrently taken a physicscourse. It was also interesting to note thatthe position of the line was already determinedwith reference to the x-y axes, thus no furtherconsideration of the line was needed to have itgenerated by the graphing technology. Typingthe equation into the graphing calculator wassufficient. Students found it rewarding that thetechnology provided a line similar to that shownon the blackboard.

3. Interpreting the Slope

Just as the tangible tasks of measuringlengths can enhance slope quantification andhelp students to construct meaning, linking thetasks to applications and function monotonicitycan expand student understanding of the slope’ssignificance.

3.1. Interpretation of the Units of Slope

It was important to note that in the process ofalgebraically expressing the line, the units of thelengths that constituted the slope were cancelled

out, and therefore only the slope magnitude wasused. This is the special case of dimensionlesslinear function representations, which are usedwidely in mathematics classrooms. Although thealternative of linking slope’s geometric meaningto its interpretation when the axes representdifferent quantities was not highlighted duringthis lesson, the following questions could inspiresuch an extension. For instance, could aspeed of 5m

s be perceived as ratio of lengths inwhich m

s is viewed in a manner similar to the

geometric representation[

5 cm1 cm

]ms ? Would this

representation lead the students to correct slopeinterpretations while simultaneously helping themretain the slope geometrical meaning? It seemsthat this idea is worth of further research.

3.2. Interpretation of the Slope’s Sign

The lesson also provided opportunities forhighlighting the association between the sign ofthe slope and function increase or decrease. Fromthis point, we use the general slope definition∆y∆x

, rather than the less formal riserun . In the

analyzed example, the value was ∆y∆x

= 0.86.Solving for ∆y, we have ∆y = 0.86∆x. Thisstatement indicates that for every increase inthe horizontal distance by, for instance, 1 cm,the vertical height of the line increases by 0.86cm (∆y = 0.86(1 cm) = 0.86 cm), and thispositive slope causes an increase of the functionvalues. Respectively, if one obtains the slopeto be, for example, −2, then this will beinterpreted to mean that for every increase ofx by one unit, the y coordinate of the functiondecreases by 2 units according to ∆y = −2∆x.The slope appears as an indicator of how thefunction’s y-coordinates change. Consequently,this interpretation will enhance the calculusinterpretation of the derivative (often called theslope function by students), which says that ifthe derivative of f(x) is negative on a giveninterval (and therefore the instantaneous slopesare negative), the values of the function willdecrease on the interval and vice versa (e.g., seeStewart, 2007).


Andrzej Sokolowski

3.3. Example Combining Both Interpretations

When mathematizing the slope, the studentsrealized that the formula rise

run constitutes aratio of two lengths—the horizontal separationbetween the selected points and their verticalseparation. Seeing the units of lengths in the slopequantification helped initiate students’ transitionto thinking in other contexts. The teacherposed the following question: suppose that theslope of a line representing water temperaturemeasured in degrees Fahrenheit versus time oversome time interval is 9. Then suppose that thewater temperature was labeled on the vertical axisand the time of heating the water, in minutes,was labeled on the horizontal axis. What isthe geometric and scientific interpretation of thisslope? The students realized that a positive slopeindicated a ratio of a positive rise over positive run(it was suggested that run was always consideredpositive). In a scientific interpretation, it meantthat the water temperature was increasing by 9◦Ffor every minute of heating.

3.4. Suggestions for Independent StudentPractice

Providing students with other contexts inwhich the slope can be calculated will broadenthe idea and enhance its relevance. The followingare some suggested extensions.

1. Use simulations (see, for example, theBalancing Act simulation on the PhETwebsite, http://phet.colorado.edu/en/

simulation/balancing-act) to copy andpaste a few snapshots and have students cal-culate the slopes and define the correspond-ing function equations.

2. Provide students with various triangles cutout from a piece of cardboard. Have thestudents trace the triangles, determine thecoordinates, and then construct the math-ematical equations of the selected inclina-tions.

3. Supply students with fixed linear graphsrepresenting various quantities and ask thestudents to determine function parametersfrom them.

4. Reflections

This lesson arose from the author’s search tocontextualize mathematical ideas and create learn-ing environments according to the contemporaryconstructivist learning theory. As a novice tryinga simple approach, the lesson generated severalreflections. Despite being conducted in precalcu-lus classes—where linear functions play a rathermarginal role—it turned out to be an engagingand meaningful learning experience for the stu-dents and for the teacher. As mathematics con-cepts are usually presented to students deduc-tively by providing fixed formulas and practic-ing their applications, this lesson brought in el-ements of investigation that will be meaningfulin the process of function formulation. The stu-dents were surprised, for example that x-y coor-dinates can be physically measured. Their ex-perience deepened when they observed—on theirgraphing calculators—a line that resembled theone originally sketched on the board by the teacher.This gave them an evident proof that abstractmath concepts are tangible and applicable in re-ality. By using measurement, the students seemedto find meaning and became encouraged to con-struct the function on their own.

The question remains: what should be theappropriate grade level to introduce such lesson?It is hypothesized that the most appropriategrade level is when students are introduced tothe idea of linear functions for the first time. Itis further hypothesized that such a lesson willprovide students with a reference for sketchinglinear functions in their further math courses.The author would like to encourage readers totrying to conduct this lesson and share theirreflections.

References

Lobato, J., & Siebert, D. (2002). Quantitative reasoningin a reconceived view of transfer. The Journal of Math-ematical Behavior , 21 , 87–116.

Porter, A., McMaken, J., Hwang, J., & Yang, R. (2011).Assessing the common core standards opportunities forimproving measures of instruction. Educational Re-searcher , 40 , 186–188.


http://phet.colorado.edu/en/simulation/balancing-act

http://phet.colorado.edu/en/simulation/balancing-act


Stanton, M., & Moore-Russo, D. (2012). Conceptualiza-tions of slope: A review of state standards. School Sci-ence and Mathematics, 112 , 270–277.

Stewart, J. (2007). Single variable calculus: Concepts andcontexts. Cengage Learning.

Von Glasersfeld, E. (1995). A constructivist approach toteaching. In L. P. Steffe, & J. E. Gale (Eds.), Construc-tivism in education (pp. 3–15). Hillsdale, NJ: LawrenceErlbaum.

Andrzej Sokolowski

MAGNOLIA WEST HIGH SCHOOL

42202 FM 1774 RD.

MAGNOLIA, TX 77354



Teaching Tricks Can Be Tricky

Linda Fosnaugh∗, Patrick Mitchell

Abstract

In this article, a concrete model is used to demonstrate that integer multiplication can be generalizedto introduce binomial multiplication, which provides a background for factoring quadratic polynomials.The authors advocate conceptual understanding over rote memorization of procedures and present theshortfalls of the bottoms-up technique.

Keywords: mathematics, education, conceptual understanding, procedural fluency, factoringquadratic polynomials

Published online by Illinois Mathematics Teacher on September 29, 2014.

Procedures and algorithms abound in themathematics classroom. Throughout their schoolyears, children are taught to manipulate numbers.While drill and practice are important in amath class, an understanding of the underlyingconcepts is equally important if the student is toretain information and use this knowledge to learnnew concepts. According to the National Councilof Teachers of Mathematics, a mathematicsstudent must learn with understanding (NCTM,2000). The student must be able to seethe connection between new ideas and priorknowledge of mathematics and should not viewmathematics as a series of unrelated topics.Conceptual understanding of mathematics leadsto proficiency. Rote memorization of factsand algorithms without understanding leads toconfusion as to when to apply a procedure and aninability to solve new problems. These sentimentsare reinforced by the National Research Council(NRC) in their report, Adding It Up: HelpingChildren Learn Mathematics (NRC, 2001).

The term mathematical proficiency was intro-duced by the NRC to describe what it meansfor a student to learn mathematics successfully(NRC, 2001). This term consists of five interwo-ven strands: conceptual understanding, procedu-ral fluency, strategic competence, adaptive rea-soning, and productive disposition. We wish to

∗Corresponding author

focus on two of these strands, namely concep-tual understanding and procedural fluency. Asdescribed by the NRC, conceptual understandingis the comprehension of mathematical concepts,operations, and relations, while procedural flu-ency refers to the skill in carrying out proceduresflexibly, accurately, efficiently and appropriately(NRC, 2001). We will begin our discussion withthe development of integer multiplication, bino-mial multiplication, and factoring quadratic poly-nomials. We will then demonstrate a “trick” forfactoring quadratic polynomials that yields thecorrect answer but does not develop conceptualunderstanding.

1. Conceptual Development of Multiplica-tion

A student who has a conceptual understand-ing of a mathematical concept is able to learnnew ideas by connecting them to the previouslylearned concept. For example, teaching multipli-cation of binomials using algebra tiles connects tothe way that multiplication of two digit numbersis taught using base ten blocks. Consider the vi-sualization of 22×13 using base ten blocks shownin figure 1.

The representation helps a student visualize



Figure 1: Base ten block representation of 22 × 13

that 22 × 13 = (20 + 2)(10 + 3). That is,

20 + 210 + 33 × 2 + 3 × 20 + 10 × 2 + 10 × 20

This concept carries over to multiplication ofbinomials. The product (x+2)(x+3) is naturallyset up using algebra tiles, as illustrated in figure2.

The student can use and extend theirknowledge of integer multiplication to obtain

(x + 2)(x + 3) = x2 + 3x + 2x + 6.

A student who has an understanding of multi-plication of binomials with positive constants canthen tackle signed values. Such a student knowsthat

(x + 3)(x− 4) = x2 − 4x + 3x− 12

= x2 − x− 12,

and knows where the constant term comes from.Students can then extend their knowledge to in-clude all binomials and can multiply an expression

1

1

1

x

x

x

x

x2

1

1

1

x

1

1

1

x

x 1 1

Figure 2: Algebra tile representation of (x + 2)(x + 3)

such as (3x + 1)(2x− 5).

(3x + 1)(2x− 5) = 6x2 − 15x + 2x− 5

= 6x2 − 13x− 5.

Notice that the middle term, −13x, is theresult of adding 3x × (−5) and 1 × 2x, and thatthe factors 3 × (−5) and 1 × 2 are also factors of6× (−5), the product of the leading and constantterms of the binomial.

2. Factoring Quadratic Polynomials

The NRC advocates teaching conceptualunderstanding along with drill and practicein order to develop procedural fluency (NRC,2001). Conceptual understanding helps studentsremember the procedures practiced in the drill.In addition, the NRC states that students needto know when and how to use the procedures(NRC, 2001). In this section, we illustrate howa student with a conceptual understanding ofmultiplication of binomials would be preparedto factor a quadratic function by reversing theprocess.

A student who has developed an understand-ing of multiplication and factoring and whounderstands the multiplication procedure knowsthat

(ax + b)(cx + d) = acx2 + adx + bcx + bd.

Conceptual and procedural understanding ofinteger multiplication will help the student findthe correct values for a, b, c, and d. If that studentis faced with factoring the polynomial 6x2+7x−5,then the student understands that the term ac iseither 6 × 1 or 3 × 2 (the two factorizations of 6)and the term bd is either (−1) × 5 or (−5) × 1(the factorizations of −5). The student has alsolearned that the middle term, 7x, is the result ofadding two terms whose coefficients are factors of6 × 5, or 30. In this case, the coefficients are −3and 10.

Using the above information, we obtain

6x2 + 7x− 5 = 6x2 − 3x + 10x− 5.


Linda Fosnaugh, Patrick Mitchell

Notice that the first two terms, 6x2 and −3x, con-tain the common factor 3x, and that the last twoterms, 10x and −5, contain the common factor 5.Using the technique known as factoring by group-ing, the student can factor out the common termsfrom each pair. That is, 6x2 − 3x = 3x(2x − 1)and 10x − 5 = 5(2x − 1). The factorization of6x2 + 7x− 5 is the result of combining these pairsof terms. Thus, factoring by grouping yields

6x2 + 7x− 5 = (6x2 − 3x) + (10x− 5)

= 3x(2x− 1) + 5(2x− 1)

= (3x + 5)(2x− 1).

This approach to factoring follows naturallyfrom previous knowledge of binomial multiplica-tion and can be useful in generating new insightsabout factoring higher order polynomials.

3. A Factoring “Trick”

The following method is a popular algorithmfor factoring. It is presented under various namessuch as “bottoms-up factoring” and “slide anddivide.” Although the method works and is basedon substitution, at first glance the individualsteps do not appear to be mathematicallyjustified. This method focuses on a proceduralalgorithm in lieu of conceptual understanding.Implementation of this method is as follows.Suppose we want to factor the trinomial

6x2 + 7x− 5.

Step 1 Multiply the leading coefficient andconstant and replace the constant term withthe result. Replace the leading coefficientwith 1.

x2 + 7x− 30

Step 2 Factor as you usually would with aleading coefficient of 1.

(x + 10)(x− 3)

Step 3 Divide the constants by the originalleading coefficient of 6.(

x +10

6

)(x− 3

6

)

Step 4 Simplify rational numbers.(x +

5

3

)(x− 1

2

)Step 5 “Bottoms up” by multiplying each

binomial by the denominator of theconstant term.

(3x + 5)(2x− 1)

Some steps in the above procedure are of greatconcern. In the first step, the leading coefficient isremoved and multiplied by the constant to forma new trinomial. The third step involves dividingeach of the constant terms of the binomials bythe original leading coefficient. Step five allowsus to move the denominator to the position ofleading coefficient. Are these steps “legal”? Ifso, why? Can we use the same method whenwe encounter other trinomials in another context?In general, these steps change the expressionand hence change the problem. Where did thismethod come from and why does it seem to work?

4. The Trick Revealed

The bottoms-up technique for factoringa quadratic polynomial involves an implicitsubstitution of variables, which is not appropriatefor a beginning algebra student who is learningpolynomial factorization. The bottoms-upmethod relies on a student’s ability to factorexpressions of the form

x2 + bx + c.

The next logical step is to be able to factorexpressions of the form

ax2 + bx + c.

Often in mathematics we change the probleminto one we are able to solve and then makethe appropriate adjustments. In this case weassume that we know how to factor a quadraticpolynomial with leading coefficient 1. To changethe general quadratic into a form that we can



handle, we perform a change of variable by settingx = z

a. Thus we have the following:

a(za

)2

+ b(za

)+ c,

orz2

a+

bz

a+ c.

Now multiply the expression by a. (We mustremember that we did this.)

z2 + bz + ac

Note that we have a quadratic polynomialwith leading coefficient 1, and thus we canfactor it. After factoring the above expression,substitute z = ax. Recall that we multiplied theexpression by a previously; thus we need to undothat process and divide the expression by a.

Example: Factor the expression 6x2 +7x−5.

Substitute x = z6:

6(z

6

)+ 7

(z6

)− 5

Simplify:z2

6+

7z

6− 5

Multiply by 6:

z2 + 7z − 30

Factor:(z + 10)(z − 3)

In the bottoms-up method we divided theconstant term by 6, reduced the fraction and thenmoved the denominator to the leading coefficient.In reality, we are just making the followingsubstitution.

Substitute z = 6x:

(6x + 10)(6x− 3)

Factor GCD:

2(3x + 5)3(2x− 1)

Divide by 6:

(3x + 5)(2x− 1)

5. Conclusion

The bottoms-up (or slide and divide) methodpresents a shortcut that hides the underlyingmathematics. For example, step 1 tells us tomultiply the leading coefficient by the constantterm, put that result as the new constant term,and then replace the leading coefficient with 1. Itis only when we introduce the change of variablesthat we see why this series of moves makes sense.

Another point to make is that the variablex in the original problem is not the same asthe variable x in step 1 and step 2; however, insteps 3 and 4 it returns to its original meaning.This is performed without any notification to thestudent. This sort of shortcut is better suited fora magic trick than for a mathematical procedure.It is very important in algebra to realize that,although variables can represent any number,in the context of an algebraic expression orprocedure they are fixed. The final step neglectsto mention that we are actually multiplying theexpression by 6, that is, by the original leadingcoefficient. The bottoms-up factoring methodis mathematically correct, but its justification isabove the level of the beginning algebra student.Hence, the method itself appears to work bymagic.

Manipulatives and mnemonics that highlightessential steps in mathematical procedures areoften used to get our students to reachmathematical proficiency. Our goal as teachersis not to simply find a method that works butrather to establish a conceptual understandingof mathematics. When applying these varioustechniques, we must ensure that the underlyingmathematics is transparent and transportable.The Factoring Quadratic Polynomials sectionserves as a good example of a techniquethat is both transparent and transportable.The bottoms-up technique falls short in bothcategories.

References

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


Linda Fosnaugh, Patrick Mitchell

National Research Council (NRC) (2001). Adding It Up:Helping Children Learn Mathematics. Kilpatrick, J.,Swafford, J., & Findell, B. (Eds.). Mathematics Learn-ing Study Committee, Center for Education, Division ofBehavioral and Social Sciences and Education. Wash-ington, DC: National Academies Press.

Linda Fosnaugh

DEPARTMENT OF MATHEMATICS

MIDWESTERN STATE UNIVERSITY

3410 TAFT BOULEVARD

WICHITA FALLS, TX 76310


Patrick Mitchell

DEPARTMENT OF MATHEMATICS

MIDWESTERN STATE UNIVERSITY

3410 TAFT BOULEVARD

WICHITA FALLS, TX 76310


Cracking the Codes of Number Relationships

Lara M. Willox∗, Sarah C. Newman

Abstract

First-grade students investigated number relationships though a series of hands-on activity stationsand assessments, gaining experience that altered their perception of numbers.

Keywords: mathematics, education, numeracy, decomposition of numbers, number sense

Published online by Illinois Mathematics Teacher on October 6, 2014.

1. Overview

All numbers are related. Large numbersmay be built through mathematical operationson smaller numbers, which in turn may be builtfrom even smaller numbers. Additionally, thereare multiple combinations of smaller numbersthat comprise larger ones. According to Vande Walle et al., understanding this part-part-whole relationship of numbers is crucial todeveloping number sense. They assert that, “Toconceptualize a number as being made up of twoor more parts is the most important relationshipthat can be developed about numbers. Forexample, 7 can be thought of as a set of 3 anda set of 4 or a set of 2 and a set of 5” (Van deWalle et al., 2007). To an adult who has had yearsof schooling, the idea that seven can be composedin different ways is likely evident, but to a childonly beginning to develop number sense, it canseem as new and complex as a foreign language.

Number sense refers to an understandingof the relationships between numbers. Formerdistrict mathematics coordinator for AlbuquerquePublic Schools, Hilde Howden, offers the followingdefinition of number sense:

Number sense can be described asgood intuition about numbers andtheir relationships. It developsgradually as a result of exploringnumbers, visualizing them in a variety


of contexts, and relating them in waysthat are not limited by traditionalalgorithms (Howden, 1989).

This article explores the impact of activitiesusing manipulatives on the overall number senseof a class of first-graders. In addition toproviding students with a fun and inquiry-basedlearning environment, the number sense activitiesalso help students to meet state standardsand expectations. There are eight overarchingexpectations for first-grade students outlinedin the Common Core Georgia PerformanceStandards (CCGPS). According to the standards,students will be expected to, “1) make sense ofproblems and persevere in solving them, 2) reasonabstractly and quantitatively, 3) construct viablearguments and critique the reasoning of others,4) model with mathematics, 5) use appropriatetools strategically, 6) attend to precision, 7) lookfor and make use of structure, and 8) look forand express regularity in repeated reasoning”(Georgia Department of Education, 2014). Theseeight expectations need to be met consistently aseach standard is addressed.

Many students learn the traditionalalgorithms for performing basic operations inelementary school, but they may not understandwhat they are doing or why. They may learn tocount, skip-count, add, subtract, even multiplyand divide, all without fully comprehending thetasks they are completing. This occurs becausestudents can be taught to calculate the correctanswer without understanding the process ofobtaining that answer. A student may have


Lara M. Willox, Sarah C. Newman

memorized, for example, that 6 + 4 = 10, withoutcomprehending the quantities that 6, 4, and 10represent in this problem. In his 2011 book, ShaiSimonson explains what happens when studentsmemorize mathematics without understanding:

Memorizing mathematics withoutcomprehension is often harmful. Ifyou memorize a poem that you don’tunderstand, there is still the chancethat the flow of the words may havean effect on you. When you memorizedates of historical events, at least youknow the chronological order of thoseevents, even if you may not know theirsignificance. When you memorizemathematics without understanding,you delude yourself into thinking thatyou know something, when in factyou do not. This delusion compoundsthe lack of understanding; your abilityto apply the knowledge, generalizeit, or even question its truth iscompromised. (Simonson, 2011)

Simonson’s sentiments concerning learningmathematics without comprehension are sharedby Kathy Richardson, a nationally renownedmathematics educator who says of suchmemorization processes, “When children learn toadd, subtract, multiply, and divide as a set ofprocedures, but do not also understand what’shappening to the numbers structurally, they maybe able to get the correct answers, but theirlearning is limited” (Richardson, 2004). Tocombat such limited learning and ensure students’success in mastering math concepts, educatorsmust encourage students to explore how numbersare structured and how they function in varioussettings by teaching them how concepts are linkedto procedures.

Experimenting with manipulatives is anexcellent way for students to begin constructingsuch number sense in the primary grades.Manipulatives are simply any concrete objectsthat students can use to attain knowledge ofsubject matter through hands-on interaction.

When used appropriately, manipulatives are toolsthat help students to do their job–learning–moreeffectively, and to meet the expectations of stategrade-level standards.

2. Context

In a first-grade classroom at a Title I school inGeorgia, a teacher and student teacher ponderedwhich topic they should choose for a collaborativeaction research project with a local university.Their assignment was to select a content areain which they believed their students needed toimprove. After determining the subject, theyworked collaboratively with an assistant professorand a student researcher at the university todevelop and execute an intervention plan forgrowth in the selected content area. At thebeginning of the year, the teacher observed thatmost of her class was unable to fluently composeand decompose numbers to ten1, a skill thatshould be mastered by the end of kindergarten,according to Common Core standards. Afterobserving the students’ work, it was evident tothe teacher that the children needed remediationin the area of number sense before they couldtackle more complex concepts. To address thisneed, the teachers decided to select number senseas the area for the research project.

3. Necessary Tools

The teacher and student teacher begancollaborating with the research team to finda math intervention for their class. Duringthe initial research phase, the team developedan interest in the work of Kathy Richardsonand decided to try her methods and materials.The primary resource they used for selectingand implementing number sense activity centerswas Richardson’s 1999 book, Developing NumberSense: Counting, Comparing, and Pattern, and

1Common Core Standard MCCK.OA.3: “Decomposenumbers less than or equal to 10 into pairs in more thanone way, e.g., by using objects or drawings, and record eachdecomposition by a drawing or equation (e.g., 5 = 2 + 3and 5 = 4 + 1).”



the manipulatives associated with it. Themanipulatives included color tiles, wooden cubes,number cubes, and Unifixr cubes. Theyalso included what Richardson refers to as“collections,” which are simply groups ofordinary, everyday objects such as paper clips,beans, and rocks (Richardson, 1999). Theprimary assessment utilized was the AMC(Assessing Math Concepts) Anywhere HidingAssessmentr, which the teacher supplementedwith informal assessments of her own. Oncethe necessary tools had been selected, thestudents, teachers, and researchers were preparedto embark on an eight month investigation ofnumber relationships.

4. How Many Are Hiding?

The Hiding Assessmentr is designed toassess each student’s individual understanding ofcombinations that comprise the numbers threethrough ten. The teacher begins by asking astudent to select a specified number of objects; forthis example we use three. The objects should beidentical. Once the student has counted the threerequested objects, the teacher initiates a dialogue,which typically resembles the following:

teacher: “How many color tiles do youhave?”

student: “Three.”teacher: “That’s exactly right. There are

three color tiles. We are going to playa game with these color tiles. I amgoing to hide some in my hand andshow you the rest. I want you to tellme how many I am hiding.”

After this initial conversation, the teacherwill hide various subsets of the group of objectsand show the student the remaining membersuntil the student has solved all combinations forthat number. The process continues through theremaining numbers until the student has masteredeach of the numbers from three through ten,or until the student has reached a place wherehe or she struggles to denumerate the hidden

subset accurately or automatically. The numberthat the student cannot complete becomes hisor her instructional level. The numbers thatthe student will practice independently (throughthe use of hands-on activities, such as the onesmentioned in the next section) include the numberof the instructional level, one number below itand one number above it. So, if a studentknows all of the combinations of four, but onlysome of the combinations of five, then five isthat student’s instructional level. The range ofnumbers that student should practice would befour through six. The student has confidencewith parts of four, is ready to learn parts offive, and can practice parts of six for a challenge.Students should be initially assessed to determinetheir starting instructional level, and they shouldbe reassessed once they have worked on theirinstructional level for a specified period of time.This allows students to progress to the next leveland enables teachers to document their growthand plan future instruction more effectively.

5. Investigating Relationships

Once the teacher and student teacherhad discovered and recorded each student’sinstructional level, the students could begin theirinvestigation of the numbers three through ten.The students needed to develop the concreteevidence that only self-exploration could provide.To aid students in their investigation, the teacherof this classroom provided a number of hands-on activity stations taken from Richardson’sbook (Richardson, 1999). Students used theactivity stations in twenty minute incrementsseveral times per week. Though game-likein construction, each of the various stationsrequired students to perform mathematical tasksusing manipulatives and to record their results.Because different quantities may be used for eachactivity, the students worked independently andin small groups to investigate different concepts,multiple times, until they were satisfied that theyhad collected enough evidence to support the ideathat all numbers are indeed related.

While several different activity centers were



used, the children had two favorite stationsthat they visited consistently when investigatingtheir numbers: Number Trains and NumberCombination Stories. At the Number Trainstation, students used the recording sheet thatcorresponded to their instructional number, twosets of different colored Unifixr cubes (each setcontaining the instructional number of cubes),and crayons matching the colors of the Unifixr

cubes. Using the different colored cubes, studentscomposed their instructional number in a varietyof ways and recorded those combinations ontheir sheet. For example, suppose a student’sinstructional level is seven. This student wouldneed the number train sheet for seven, sevenUnifixr cubes of one color (blue, for instance),seven Unifixr cubes of another color (red, forinstance), a blue crayon and a red crayon. As thestudent manipulates the Unifixr cubes, he or shenotices that seven reds plus zero blues is seven,six blues and one red is seven, one red and sixblues is seven, and so on. The student colorsthe blocks on his or her recording sheet to matchthe combinations that he or she made with theUnifixr cubes. This process continues until thestudent has illustrated all possible combinationsfor the instructional number. (See figures 1 and2.)

Coming up with as many combinations aspossible is a critical part of a child’s learningprocess. While someone with developed numbersense would see that four reds plus three bluesis the same as three blues plus four reds, ayoung student developing number sense seesthese as two different statements. Even though4 + 3 has the same value as 3 + 4, educatorscannot assume that a student already knowsthis information. Working with manipulatives,students will discover that these statements areequivalent. They will also notice that regardlessof which number they put first and which numberthey put last (3 or 4 in this case) they still get thesame answer, 7.

The Number Train station allowed studentsto create mathematical statements and to makesense of them. They used their reasoning skillsand manipulatives to determine the quantities

Figure 1: A student uses a number train sheet forcombinations of ten, working with red and yellow Unifixr

cubes.

Figure 2: A sample of a student’s finished work.



Figure 3: A student writes a number combination storyfor the number six using a corral template.

behind number symbols and represent them asobjects. Students created number train charts,which served as models to help them explain whatthey did during their center time, justify theirconclusions, and remember what they learned.

Students also enjoyed the NumberCombination Stories center. In addition toteaching mathematical concepts and introducingstudents to relevant word problems, this stationencourages writing and creative thought. At theNumber Combinations Stories station, studentsbegan by selecting a background setting such as agarden or an ocean. Next, they select a number ofobjects equivalent to their instructional number.Once students have completed these steps, theygenerate math stories based on problems createdusing parts of their individual number, writing thestories on a recording sheet as they go along. (Seefigure 3.)

With the introduction of the Common CoreState Standards and the increasingly popular ideaof “writing across the curriculum,” students arenow writing in every subject, including math,validating the need for centers such as thisone. Students may use addition or subtractionproblems for this activity, but they must representtheir work with pictures or manipulatives, andwith the traditional algorithm. This allowsthe teacher to determine whether or not theinvestigations were conducted thoroughly and

correctly.

6. Examining the Impact

After eight months of investigations, thestudents saw conclusive evidence that numbersare indeed related, and can be created from a hostof different combinations. Throughout the periodof these investigations, the teacher and studentteacher carefully tracked the progress of thestudents. Their data shows that the majority ofthe students increased their instructional numberby at least two levels. The data does notshow that all of the students mastered theability to work fluently with combinations toten, but it does show substantial improvement.Samples of student work from centers andindependent practice sheets, as well as increasedunderstanding demonstrated in class numbertalks, provide evidence that the students arelaying solid foundations upon which to buildfurther mathematical knowledge. Though all ofthe students in this classroom participated inthe Hiding Assessmentr, the graph only showsthe data for the fifteen who remained in thisclassroom throughout the entire school year.Students who joined the class in the middle ofthe year or changed schools during the course ofthe school year did not have the opportunity totake the Hiding Assessmentr three different timesas did their peers, providing inconclusive dataconcerning their progress.

The teacher and student teacher of thisclassroom were thrilled by the difference thatthe activities made in helping their students tobecome effective “number detectives.” At thebeginning of the year, the number sense of manyof their students was limited to the numbers3 through 6. Even though most students didnot master every combination for the numbersthree through ten, they made significant gainsin their understandings of how numbers work asevidenced by our data, which is shown in figure 4.

7. Conclusion

The students were not the only oneschallenged by the investigation of number



Figure 4: Graph showing student improvement over thecourse of the activity from September 2012 to April 2013.

relationships. Previously, the teacher had taughtmathematics through direct instruction with amethodical approach to computation, focusing onstrategies for finding the correct answer, ratherthan fostering a deep understanding of numbersand their relationships. Relinquishing absolutecontrol of math activities and allowing studentsto investigate mathematics independently proveddifficult for the teacher and student teacher.Instead of being “dictators” during mathinstruction, they became “facilitators.” At first,it was hard for them to stand back and watchthe students choose their own activities, ignoringthe overwhelming instinct to tell them whatthey should work on. As they let the studentstake responsibility for their learning, however,the teachers were astonished to see studentsgravitating towards the activities that were rightfor them, working on the concepts they needed topractice. Having personally experienced successwith the stations, both the teacher and studentteacher are convinced that the ideas presented inRichardson’s book, when properly implemented,wield the power to instill a deep sense of numberrelationships in young children.

After the conclusion of our study and the startof the new school year, the assistant professorconducted a follow-up interview with the teacher.She is still using the activity centers in herclassroom to guide her new students in obtaining

number sense so that they will have a solidfoundation on which to build the knowledge ofother standard mathematic ideas and principles.As for her former students, they are excellingin mathematics. The second grade teachersreport that they are extremely impressed by thenumber sense exhibited by the first-graders whoparticipated in the number sense building activitystations. They have seen such a difference thatthey have requested that the first-grade teacherscontinue to use the number sense activity stationsin their classrooms so that their future studentswill be ready for second grade math when theymove up.

References

Common Core State Standards Initiative and others(2010). Common Core State Standards for Mathemat-ics. Washington, DC: National Governors AssociationCenter for Best Practices and the Council of Chief StateSchool Officers.

Georgia Department of Education (2014). CCGPSGrade Level Overview: 1st Grade, Mathematics.https://www.georgiastandards.org/Common-Core/

Common%20Core%20Frameworks/CCGPS_Math_1_

GradeLevelOverview.pdf. Accessed August 2014.Howden, H. (1989). Teaching number sense. Arithmetic

Teacher , 36 , 6–11.Richardson, K. (1999). Developing Number Concepts,

Book 1: Counting, Comparing and Pattern. NJ: Pear-son Education, Inc.

Richardson, K. (2004). Going beyond the right answer.Teaching Pre K-8 , 34 , 52–53.

Simonson, S. (2011). Rediscovering Mathematics: You Dothe Math. MAA.

Van de Walle, J. A., Karp, K. S., Bay-Williams, J. M.,Wray, J. A., & Rigelman, N. R. M. (2007). Elementaryand middle school mathematics: Teaching developmen-tally . Pearson/Allyn and Bacon.

Lara M. Willox

COLLEGE OF EDUCATION

UNIVERSITY OF WEST GEORGIA

1601 MAPLE ST.

CARROLLTON, GA 30118


Sarah C. Newman

COLLEGE OF EDUCATION

UNIVERSITY OF WEST GEORGIA

1601 MAPLE ST.

CARROLLTON, GA 30118



https://www.georgiastandards.org/Common-Core/Common%20Core%20Frameworks/CCGPS_Math_1_GradeLevelOverview.pdf



iPad Fun: Exploring Slope and Functions Using iPads with the

TI-Nspire App

Ann Wheeler∗, Brandi Falley

Abstract

In this article, we discuss two iPad-driven projects that can be used in an algebra classroom to teachslope and families of functions. The activities are hands-on and allow students to see the connectionsof certain careers to mathematics and the beauty of mathematics around them.

Keywords: mathematics, education, slope, function, technology, tablets

Published online by Illinois Mathematics Teacher on December 17, 2014.

1. Overview

Today’s students thrive in engaging class-rooms, where technology abounds and often isvital to the classroom culture. Calculators,computers, and even tablets, such as iPads,have transformed the mathematics classroominto a virtual learning environment of endlesspossibilities. In addition, national organizations,such as the National Council of Teachers ofMathematics (NCTM), see the importance oftechnology by listing it as one of the six principlesfor mathematics that is integral to a mathematicsteacher’s instruction (NCTM, 2000).

Besides NCTM’s endorsement of technology,other researchers have also studied the usefulnessof such devices in education. Eichenlaub et al.argued that the iPad, though not as integral toacademic life as a computer, can be a powerfultool in aiding collaboration, encouraging organi-zation, and assisting learning, regardless of field orlevel of academic achievement (Eichenlaub et al.,2011). While many features make the iPad auseful portable computing device, Johnston andStoll contend that the iPad’s custom applications(apps) provide the leading benefit because they al-low manipulation and direct interaction with con-tent (Johnston & Stoll, 2011).

Not only are the uses of technology importantin the classroom, but the relevance of mathemat-


ics to student learning is key to many students’interests in the lessons. Slope and functions, themathematical concepts our lessons address, aretwo concepts that teachers can make applicableto students. For example, one way we relatedslope to students was through civil engineering:we used the concept of grade or slope to con-struct various structures, including roads, park-ing lots, staircases, and wheelchair ramps. Wediscussed how the steepness of each of these struc-tures influenced their slopes: for example, a steepstaircase would have a slope with a greater mag-nitude than a fairly flat one. For functions, weexplored with students the shapes of curves theycould see around the school, such as in paintingsand architecture. Depending on the object, itsgeneral shape might resemble a parabola, an ab-solute value curve, or something else entirely.

Utilizing these mathematical ideas of slope andfunctions, coupled with iPad technology and theTI-Nspire calculator, we created two projects foran honors college algebra class during fall 2013.Both activities involved students exploring theirsurroundings by searching for mathematics in thereal world. In the first project, “InvestigationUsing Slope,” students investigated the mathe-matics behind the steepness of wheelchair rampsand staircases. For the second project, “Find thatCurve!”, students found curves from various fam-ilies of functions. Not only are these two projectsinteractive and engaging, they also address Com-


Ann Wheeler, Brandi Falley

mon Core State Standards.1 Sample student workis provided throughout the paper, and both activ-ities are given with TI-Nspire instructions in theAppendices.

2. Project #1: Investigation Using Slope

As a culminating activity for the unit, stu-dents utilized iPads with the TI-Nspire App forProject #1: Investigation Using Slope to see howmathematics was a part of their everyday lives(see Appendix A for the complete project, in-cluding instructions for the TI-Nspire App). Pre-viously, students learned about the definition ofslope and how to calculate its value, given twopoints or a graph, using the slope formula. Forthis project, the students reinforced their knowl-edge about the concept of slope and how to deter-mine its value in various applications. Studentswere organized into groups of two or three, andeach group was given the following materials:

1. the Investigation Using Slope handout,

2. an iPad with the TI-Nspire App,

3. a level, and

4. a tape measure.

Students then walked around campus, searchingfor two wheelchair ramps and two staircases. Eachramp or staircase needed to be visible enough sothat a picture could be taken of its steepness froma level position, as well as accessible enough sothat students could measure its steepness with atape measure. Before students left the classroom,the teacher gave instructions on how to take atwo-dimensional looking photo so that no depthwas showing (see figures 1 and 2).

In addition, the teacher demonstrated how touse a level so that students could be confidentthey were taking pictures that were perpendicularto the ground. By taking the extra time to showstudents how to successfully take photographswith the iPad, the teacher ensured that studentsproduced fairly accurate slope calculations.

Once students completed the picture-takingphase of the assignment, they returned to the

1CCSS 8.EE.B.5, resp. CCSS HSF-IF.C.7a/b

Figure 1: Photo of a staircase with exposed depth showing.

Figure 2: Photo of a staircase, correctly taken with noexposed depth showing.


iPad Fun: Exploring Slope and Functions Using iPads with the TI-Nspire App

Figure 3: Wheelchair ramp with best fit line.

Figure 4: Staircase with best fit line.

classroom to calculate the slope of each wheelchairramp and staircase using the TI-Nspire App ontheir iPads. One at a time, students added theirphotos to the app and created best fit lines tomatch the slope of each ramp and staircase. Theycould easily change the slope of their best fitlines by selecting the line with their finger and/orstylus and shifting the line to fit the steepness ofthe ramp or staircase. Figures 3 and 4, producedby students in class, show the result of using theTI-Nspire App to calculate the best fit lines for awheelchair ramp and a staircase.

Besides calculating slope with their iPads, wefelt it was important for students to also be ableto check to see whether their measurements weresimilar if they completed the measurements witha tape measure. Sixty percent of the studentcalculations agreed to within one tenth of an inch,

with the majority of measurement differencesbeing attributed to difficulties in measuring theslopes accurately.

An additional portion of the assignment re-quired students to compare their measurementsto building regulations, which helps students seethe mathematics that has to go into structures likestaircases and wheelchair ramps we often take forgranted. As stated in the activity, both structuresmust meet certain slope requirements. For exam-ple, according to the American Disability Associ-ation, wheelchair ramps should have a slope ratio(height to length) of no more than 1:12 (Com-forth, 2009). Most of the students found that thewheelchair ramps around campus had a slope ra-tio of roughly 1:12, confirming that our campusramps met the requirements. Two groups had oneset of measurements that were slightly higher thanthe standard but attributed their calculation dif-ferences to difficulties in being able to accuratelymeasure the ramp. For the staircase portion ofthe assignment, all students found all staircasesto meet the standard.

3. Project #2: Find That Curve!

For Find that Curve!, students had alreadylearned about the definition of a function, prop-erties of different functions, and transformationsof functions. Similar to the first project, stu-dents went outside and engaged in mathematics(see Appendix B for the complete project, includ-ing instructions for the TI-Nspire App). Studentswere once again organized into groups of two orthree with each group given a Find that Curve!handout and an iPad with the TI-Nspire App.We gave students descriptions of specific charac-teristics of curves we wanted them to find acrosscampus. These curves included quadratic, abso-lute value, square root, and cubic curves.

After students took pictures of the curves withtheir iPads, they returned to the classroom todetermine equations whose graphs approximatedthe curves in their pictures. Similar to Project#1, students added each photo to their TI-NspireApp and plotted curves, where adjustments couldbe made to their lines by using their finger and/or


Ann Wheeler, Brandi Falley

Figure 5: Absolute value curve.

Figure 6: Square root curve.

stylus. Below are some examples of their work forvarious curves.

4. Student feedback

Students completed a survey after the projectswere completed to provide feedback about theprojects. All students indicated that they enjoyedthe projects; there were multiple comments madeabout the positive experiences using iPads andreal life applications. One student commented:

I liked becoming aware of the detailput into building slope for stairwaysand ramps. I also liked using an iPadas a multi-functional tool to take thepicture and find the slope.

Another student remarked on the usefulness of theactivity to her knowledge of class concepts:

Going out and finding these graphshelped me remember what graphs looklike because I had to apply what welearned in class to know what to lookfor.

Figure 7: Parabolic curve with a reflection over the x-axis.

One unexpected observation from a student ad-dressed the effectiveness of the collaborative na-ture of the project:

This might not be pertinent, but Ifound that having this as an honors-specific course is fantastic! Groupprojects usually end with me doing allthe work, but not here. It was truly aunique experience.

Students also mentioned some areas for improve-ment in the future. Multiple students commentedthat certain curves in the second project were dif-ficult to find, such as the square root function.Some felt that giving examples of student workwould be useful so that they had a clearer ideaabout what they should be looking for aroundthe university. Even though students struggled tofind certain curves, they rose to the challenge andfound them without assistance from the teacher.

Along with student feedback, we noted a fewlimitations along the way. For Investigating Us-ing Slope, one essential requirement was to takepictures that looked two-dimensional, where therewas no depth to the staircase and/or ramp. Thishelped in determining an accurate slope via theTI-Nspire App. However, achieving a perfect two-dimensional image took some practice, which wasdifficult in the short time allotted for the outdoorinvestigation portion of the project. Another con-cern was that the interface caused the iPad slopes



to vary from the actual slopes. Students createdbest fit lines with their finger and/or stylus, whichdid not allow for great precision in the measure-ments. In Project #2, Find That Curve!, themain confounding factor was that pictures weretaken from varying perspectives (i.e., taking a pic-ture from below), causing distortion in the imagesof the curves.

5. Conclusion

Technology in the classroom is always evolv-ing, and we, as educators, need to be comfort-able and stay up-to-date with current trends andadvances. As we started working with tablets inour classroom, we found that the iPads were user-friendly, and that the students loved to use them.Using iPads with the TI-Nspire App to teach stu-dents about slope and families of functions can bea fun and exciting experience for both the teacherand students. With the projects described in thisarticle, we found that the classroom became amore meaningful environment, where students ac-tually saw the mathematics come alive. Studentshad to search outside the textbook to find theslopes and curves they were studying. We hope inthe future to potentially create more iPad-basedlessons to engage students in other areas of math-ematics, such as gathering real world data andperforming statistical measures.

References

Comforth, E. (2009). Constructing wheelchair rampsto ADA specifications. URL: http://www.

disabled-world.com/assistivedevices/ramps/

ada-specs.php. Accessed Dec. 1, 2014.Common Core State Standards Initiative and others

(2010). Common Core State Standards for Mathemat-ics. Washington, DC: National Governors AssociationCenter for Best Practices and the Council of Chief StateSchool Officers.

Eichenlaub, N., Gabel, L., Jakubek, D., McCarthy, G., &Wang, W. (2011). Project iPad: Investigating tabletintegration in learning and libraries at Ryerson Univer-sity. Computers in Libraries, 31 .

Johnston, H. B., & Stoll, C. J. (2011). It’s the pedagogy,stupid: Lessons from an ipad lending program. eLearnMagazine, May 2011 . URL: http://elearnmag.acm.org/featured.cfm?aid=1999656. Accessed Dec. 1,2014.

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

Stairway Manufacturers Association (2006). Vi-sual interpretation of the international residen-tial code: 2006 stair building code. URL: http:

//www.stairways.org/Resources/Documents/2006%

20Stair%20IRC%20SCREEN%20web%20download.pdf.Accessed Dec. 1, 2014.

Ann Wheeler

MATHEMATICS & COMPUTER SCIENCE DEPARTMENT

TEXAS WOMAN’S UNIVERSITY

PO BOX 425886

DENTON, TX 76204-5886


Brandi Falley

MATHEMATICS & COMPUTER SCIENCE DEPARTMENT

TEXAS WOMAN’S UNIVERSITY

PO BOX 425886

DENTON, TX 76204-5886



http://www.disabled-world.com/assistivedevices/ramps/ada-specs.php



http://elearnmag.acm.org/featured.cfm?aid=1999656

http://elearnmag.acm.org/featured.cfm?aid=1999656

http://www.stairways.org/Resources/Documents/2006%20Stair%20IRC%20SCREEN%20web%20download.pdf



Brandi Falley, Ann Wheeler

Appendix AProject #1: Investigation Using Slope

You will be using the iPad and the TI-Nspire App to investigate the slope of stairsand wheelchair ramps across campus. Take pictures of at least two wheelchair rampsand two staircases, making sure you hold the iPad level and perpendicular to theramp/staircase. (If you have done this correctly, your level should have the bubblein the middle, and the stairs/wheelchair ramp should look two-dimensional.) Afteryou take a picture, use a tape measure to calculate the height and depth of eachramp/staircase. Record your results in the below table.

Object - rampor staircase

Locationon campus

Slope - iPadcalculation

Slope - tapemeasure

calculation

1. Were your iPad and tape measure calculations similar? If not, explain why youthink there were discrepancies.

2. According to the American Disabilities Association (ADA)1, a wheelchair rampshould have a slope ratio of no more than 1:12. Do the wheelchair ramps youmeasured meet this standard? Explain.

3. According to the International Residential Code (IRC)2, a staircase’s verticalheight should be no more than 196 mm with a horizontal length of at least 254mm. Do the staircases you measured meet this standard? Explain.

1Source: http://www.disabledworld.com/assistivedevices/ramps/ada-specs.php2Source: http://www.stairways.org/Resources/Documents/2006%20Stair%20IRC%20SCREEN%

20web%20download.pdf


http://www.disabledworld.com/assistivedevices/ramps/ada-specs.php




Calculating Slope using the iPad with the TI-Nspire App

Directions:For calculating the slope of a ramp or staircase using your iPad with the TI-NspireApp (version 3.10.0), follow the below directions:

1. Press the TI-Nspire App icon.

2. Press the + sign in the upper left hand corner of your screen.

3. Select “Graphs” from the dropdown menu.

4. Select the camera icon in the upper right hand corner of the screen.

5. Select “Add Photo” from the dropdown menu. (Your selected photo shouldnow be on your screen.)

6. Your cursor should be at the f1(x):= line. Make an educated guess as to whatyou believe is the slope of the ramp or staircase picture you have as a backdrop.(Note: You will need to write an equation of the line, not just the slope, or youwill get an error message.) Once you have written an equation, press enter. Ablue line with your equation should appear on your screen.

7. Depending on the accuracy of your equation, you may need to adjust your lineon the screen. To do so, select the line with your finger and move it to thedesired location. (Note: Selecting the line near the origin will result in shiftingthe y-intercept, whereas selecting the line away from the origin will change theslope.)

8. Once you have the desired equation, record your slope in the table.

9. Email each picture to the instructor.


Brandi Falley, Ann Wheeler

Appendix BProject #2: Find That Curve!

Using your iPad with the TI-Nspire App, take pictures of objects around campus thatshow the types of curves listed below. Then, determine an equation of best fit for thecurve in each picture. No “double-dipping” please - each type of curve should have adifferent object/picture associated with it.

1. A parabolic curve with a positive “a” value, where the formula for a parabolais of the form y = a(x− h)2 + k

2. A parabolic curve with a reflection over the x-axis

3. A parabolic curve with a vertical stretch

4. An absolute value curve

5. A square root curve

6. A cubic curve reflected through the origin

After you’ve completed the tasks, take screen captures of your pictures with theoverlaid lines of best fit, and send them to the instructor via email.



Determining the Line or Curve of Best Fit using the iPad with the TI-Nspire App

Directions:For determining the line or curve of best fit using your iPad with the TI-Nspire App(version 3.10.0), follow the below directions:

1. Press the TI-Nspire App icon.

2. Press the + sign in the upper left hand corner of your screen.

3. Select “Graphs” from the dropdown menu.

4. Select the camera icon in the upper right hand corner of the screen.

5. Select “Add Photo” from the dropdown menu. (Your selected photo shouldnow be on your screen.)

6. Your cursor should be at the f1(x):= line. Make an educated guess as to whatyou believe is the the equation of the line or curve that you have as a backdrop.Once you have written an equation, press enter. A blue line or curve with yourequation should appear on your screen.

7. Depending on the accuracy of your equation, you may need to adjust your curveon the screen. To do so, select the curve with your finger and move it to thedesired location. (Note: Selecting points near the origin will result in shiftingthe y-intercept, whereas selecting the points away from the origin will changethe slope of a line, or the width and orientation of a parabola.)

8. Once you have the desired equation, record your equation on your handoutunder the appropriate equation description.

9. Email each picture to the instructor.


Follow the Directionscrossword puzzle by Christopher Shaw
