VMT, Vol. 44, No. 2, Spring 2018

Virginia Council of Teachers of Mathematics |www.vctm.orgVIRGINIA MATHEMATICSVol. 44, No. 2 TEACHERSpring 2018 Moving Mountains with Mathematics!Virginia Mathematics Teacher vol. 44, no. 2

Editorial StaffDr. Agida Manizade Dr. Jean Mistele Ms. Alexandra Largen Ms. Cameron Leo Editor-in-Chief Associate Editor Assistant Editor Assistant Editor Radford University Radford UniversityRadford University [email protected] Radford University [email protected] Printed by Wordsprint Blacksburg, 2200 Kraft Drive, Suite 2050 Blacksburg, Virginia 24060Virginia Council of Teachers of Mathematics Many Thanks to our Reviewers for Spring 2018 President: Jamey Lovin Caleb Adams, Radford University Past President: Cathy Shelton Eric Choate, Radford University Secretary: Kim Bender Laura Jacobsen, Radford UniversityMembership Chair: Ruth Harbin-Miles Betti Kreye, Virginia Tech Treasurer: Virginia Lewis Virginia Lewis, Longwood University Webmaster: Ian Shenk Jamey Lovin, Virginia Beach City Public SchoolsNCTM Representative: Lisa Hall John McGee, Radford UniversityElementary Representatives: Meghann Cope; Vicki Bohidar Laura Moss, Radford UniversityMiddle School Representatives: Melanie Pruett; Skip Tyler Andy Norton, Virginia Tech Secondary Representatives: Pat Gabriel; Lynn Reed Ryan Smith, Radford UniversityMath Specialist Representative: Spencer Jamieson Kateri Thunder, James Madison University 2 Year College: Joe Joyner Skip Tyler, Henrico County Public Schools4 Year College: Ann Wallace; Courtney Baker Katy Ulrich, Virginia TechVMT Editor in Chief: Dr. Agida Manizade Jay Wilkins, Virginia Tech VMT Associate Editor: Dr. Jean Mistele Denise Wilkinson, Virginia Wesleyan UniversityVirginia MathemaTthicissisTsueeachhadera v15o%l. 4a4c,cenpota. n2ce rate Special thanks to all reviewers! We truly appreciate your time and service. The high-quality journal is only possible because of your dedication and 2 hard work.

Virginia Mathematics Teacher vol. 44, no. 2 Table of Contents: Featured Awards ................................................... 5 Note from the Editor.............................................. 6 Message from the President .................................. 7 Note from the VDOE ............................................. 8 Organization Membership Information................. 9 Assessing Fractions Formatively in the First Grade................................................................... 10 Problem Solving Opportunities for Students....... 17 Grant Opportunities ............................................ 21 Busting Blockbusters ........................................... 21 HEXA Challenge Spring 2018............................. 22 Understanding Preservice Elementary Teachers’ Perceptions .......................................................... 24 Presidential Awards for Excellence .................... 28 Good Reads ......................................................... 29 Math Jokes........................................................... 29 Call for Manuscripts ........................................... 30 Key to the Fall 2017 Puzzlemaker....................... 31 How to Share an Island: Divergent Thinking and Creativity ...................................................... 32 Solutions to Fall 2017 HEXA Challenge............. 38 Technology Review.............................................. 41 Motion with LEGOs and Dynamic Geometry ..... 43 Upcoming Math Competitions............................. 48 The Puzzlemaker.................................................. 51 Conferences of Interest........................................ 52 3

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Featured Awards 5 Congratulations to the 2017 State-Level Finalists for the Presidential Awards for Excellence in Mathematics and Science Teaching: William Daly, Albemarle High, Albemarle County Public Schools Blythe Samuels, Powhatan High, Powhatan County Public Schools Elisa Tedone, James River High, Chesterfield County Public Schools See page 28 for more details about these awards. Congratulations to our HEXA Challenge Winner: Craig West! He teaches Calculus, Analysis, Statistics, Discrete Mathematics, and Geometry at East Rockingham High School, Elkton, VA. See page 23 for a picture of our winner. Try our new HEXA Challenge problems featured in this issue!Virginia Mathematics Teacher vol. 44, no. 2

Contributors to Our Success Story Dr. Agida Manizade Radford University This year, the Virginia Mathematics Teach- matics teacher at Blacksburg Middle School iner was awarded the National Council of Teachers Blacksburg, VA, plans to be a math teacher in Bos-of Mathematics’ 2018 Publication Award for ton, MA next year. He was an Assistant Editor forOutstanding Journal. Without the continued sup- the journal. All five contributed to our 2016 issuesport from the mathematics community, this would of the journal.not be possible. Our editorial board plays a tremen-dous role in keeping the Virginia Mathematics One of the main contributors for the 2017Teacher as a high-quality source of relevant infor- issues of the journal is our current Associate Edi-mation for the community. We strive to strike a tor, Dr. Jean Mistele, Associate Professor in thebalance between providing information related to Department of Mathematics and Statistics at RU,mathematical content and helpful pedagogical who joined the journal in Spring 2017. Our currentpractices in a mathematics classroom. The journal Assistant Editors, Ms. Cameron Leo, a graduateincludes practitioner-oriented articles, mathematics student of Mathematics Education at RU, as well ascompetitions, puzzles, jokes, and other information Ms. Alexandra Largen, an undergraduate studentspecifically designed for mathematics teachers at of Computer Science, have been tremendous con-all levels. tributors in developing and designing recent issues of the journal. Last, but not least, Section Editors, Every member of the editorial board has Mr. Christophe Hirel, who provides critical re-contributed to our success, as the award was given views and information on cutting-edge educationalbased on the quality of the journal publications technologies, and Dr. Betti Kreye, who providesfor two consecutive years. The Virginia Mathe- suggestions and reviews of current professional lit-matics Teacher has been published since 1974. erature for our audience, have been contributing toTwo and a half years ago, I was given the privilege the journal for the past two and a half years.to serve as Editor-In-Chief. It’s been a great pleas-ure to work with an extremely talented and devoted I would like to express my deepest gratitudegroup of professionals. This group includes our to all of the aforementioned individuals, as well asfirst Associate Editor, Dr. Rayya Younes, a for- the dozens of reviewers who have helped us pro-mer Radford University (RU) Assistant Professor duce this wonderful journal. On behalf of the edito-of Mathematics Education, currently teaching at rial team, I would also like to thank the Artis Col-the University of Balamand in Lebanon. Our first lege of Science and Technology and the Mathe-Assistant Editor, Ms. Michelle Graham, a former matics Education program in the School ofRU graduate student of Education, is currently Teacher Education and Leadership at RU, asteaching at Blacksburg High School in Blacksburg, well as the Virginia Council of Teachers ofVA. Mr. Brian Pratt, our second Assistant Editor, Mathematics for their continued support.a former RU undergraduate student of Biology, iscurrently working as a medical scribe at Lewis- Agida ManizadeGale Medical Practices. He is applying to medical Professor, RUschools this year. Dr. Margie Mason, our second Editor-In-Chief, VMT JournalAssociate Editor, is a Professor of Mathematics [email protected] at The College of William and Mary in www.vctm.org/VMTWilliamsburg, VA. Mr. Liam Downey, a formerRU graduate student of Education, who is a mathe-Virginia Mathematics Teacher vol. 44, no. 2 6

Message from the President Jamey LovinVirginia Council of Teachers of Mathematics As I write my last message as President of nered with the VDOE to present the Ideas for Im-the VCTM Executive Board, I am filled with grati- plementing the 2016 Virginia Standards of Learn-tude and privilege to have worked with such talent- ing Focus Conference on October 6, 2017 at theed board members and to have represented the Central Virginia Community College Campus inamazing teachers of the Commonwealth of Virgin- Lynchburg, VA for select schools in several under-ia. We have much to celebrate! served school districts in Regions 5 and 6. The pur- pose of funding this outreach was to fulfill our mis-The Virginia Mathematics Teacher. At its No- sion to promote the improvement of mathematicsvember 2017 meeting, the NCTM Membership and education, to provide leadership in the professionalAffiliate Relations Committee (MARC) reviewed development of teachers, and facilitate cooperationand selected the Virginia Council of Teachers of among organizations at the local, state, and nation-Mathematics’ Virginia Mathematics Teacher jour- al levels.nal for the 2018 Publication Award for outstandingjournal. Congratulations to our editorial board: Dr. VCTM 2018 Annual Conference at RadfordAgida Manizade, Dr. Jean Mistele, Dr. Christophe University. Our annual conference was held atHirel, Dr. Betti Kreye, Ms. Cameron Leo, and Ms. Radford University on March 9-10. The conferenceAlexandra Largen. committee and conference planners did a wonder- ful job of soliciting quality presentations and insti-VCTM Awards and Grants. Our awards and tuting several platform changes. We are in the pro-grants committees have been working tirelessly to cess of evaluating feedback, but preliminary resultsrevamp our application and award procedures. A say: BEST EVER!big step in that direction was introducing an onlineapplication process for many of the awards. Many Although your board is grateful for all weof this years’ awardees credited the streamlined have been able to accomplish, we continue to lookonline application process as a reason they had pur- for ways in which VCTM can continue to servesued their award. math educators in Virginia. Please do not hesitate to let us know how we can better serve you!Website. Over the past two years, our webmasterhas been renovating and refreshing our website. Hecontinues to investigate ways to keep our membersinformed via this engaging platform.Affiliate Support. Members of the Executive Jamey Lovin, VCTM PresidentBoard have been attending local affiliate events to [email protected] support from, and information about,VCTM. These efforts have been appreciated by lo-cal affiliates, and invitations to future events con-tinue to arrive in our emails!Professional Development Outreach CommitteeFacilitates First Outreach Effort. VCTM part-Virginia Mathematics Teacher vol. 44, no. 2 7

Note from the VirginiaDepartment of Education Tina Mazzacane Virginia Board of Education’s Regulations proach to state accountability.Establishing Standards for Accrediting Public The revised SOA implements the “ProfileSchools in Virginia ensure that an effective educa-tional program is established and maintained in of a Virginia Graduate” which includes knowledge,Virginia’s public schools. In November 2017, the skills, attributes, and experiences that prepare stu-Board of Education approved revisions to the dents for college and the workplace. These includeStandards of Accreditation (SOA) that include academic content knowledge, workplace skills,changes in school accreditation standards and grad- community and civic responsibility, and careeruation requirements aligned to the Profile of a Vir- planning. The Profile of a Graduate also embedsginia Graduate. competencies that ensures students develop the “Five C’s”: The revised accreditation standards provide • Critical thinkinga more comprehensive view of school quality by • Creative thinkingincluding multiple school-quality indicators, not • Communicationjust overall achievement on state tests. Included in • Collaborationthese school-quality indicators are overall profi- • Citizenshipciency and growth in mathematics at the elemen- These competencies may also strengthen the devel-tary and middle school levels, and mathematics opment of mathematical understanding, and highlyachievement gaps among student groups. Other in- correlate with the Virginia Mathematics Processdicators to evaluate high schools will include ab- Goals for Students (problem solving, communica-senteeism; dropout rate; graduation and comple- tion, reasoning, connections, and representations).tion; and college, career, and civic readiness. The Venn diagram below outlines some of the rela-School accreditation ratings for the 2018-2019 tionships that may exist between the Five C’s andschool year will be the first to reflect this new ap- mathematical thinking:Virginia Mathematics Teacher vol. 44, no. 2 8

New graduation requirements are included the implementation of the 2016 Standards of Learn-in the 2017 revision to the SOA and take effect ing Mathematics assessments has been adjusted, inwith students entering the ninth grade in the fall of anticipation of fewer students testing in the fall of2018. The number of standard credits for a Stand- 2018:ard Diploma and Advanced Studies diploma stay • Fall 2018, End-of-Course (Algebra I, Geome-the same but the number of verified credits, earnedby passing a course with an associated end-of- try, and Algebra II) will measure the 2009course assessment (e.g., Algebra I, Geometry, or Mathematics Standards of Learning and in-Algebra II) reduces to only one in mathematics for clude field test items measuring the 2016 Math-both diplomas. ematics Standards of Learning. • Spring 2019 (Grades 3-8 and End-of-Course) Effective fall 2018, all students (regardless Standards of Learning assessments will meas-of when they enter ninth grade) will be exempt ure the 2016 Mathematics Standards of Learn-from additional end-of-course SOL tests in a sub- ing.ject once they have earned the required number ofverified credits as part of the 2017 revisions to the Mathematics educators wishing to learnSOA. For example, a tenth grader taking Algebra II more about the 2017 SOA revisions can visit theduring the 2018-2019 school year who has earned VDOE Standards of Accreditation webpage attwo verified credits in mathematics toward an Ad- http://www.doe.virginia.gov/boe/accreditation/vanced Studies diploma would be exempt from tak- index.shtml.ing the Algebra II SOL assessment. However, fed-eral requirements under the Every Student Suc- Tina Mazzacaneceeds Act (ESSA) may require additional end-of- Mathematics Coordinatorcourse testing in the subject area regardless of the Virginia Department of Educationexemption under the revised SOA. Federal law re-quires the testing of students in mathematics atleast once during high school (grades 9-12). Due to the SOA change in exemption fromadditional end-of-course SOL tests outlined above, Organization Membership Information 9 National Council of Teachers of Mathematics Membership Options: Individual One-Year Membership: $96/year, full membership Individual One-Year Membership, plus research journal: $124/year Base Student E-Membership: $48/year Student E-Membership plus online research journal: $62/year Current National Council of Teachers of Mathematics Membership: 60,000+ Members Virginia Council of Teachers of Mathematics Membership Options: $20 Student Membership (For Full-Time College Students) $20 Individual One-Year Membership $20 Institutional One-Year Membership $39 Individual Two-Year Membership $57 Individual Three-Year Membership $500 Individual Lifetime Membership Current Virginia Council of Teachers of Mathematics Membership: 1,800+ MembersVirginia Mathematics Teacher vol. 44, no. 2

Assessing Fractions Formatively in the First Grade Ann H. Wallace, Rich Busi, and Jenna Rinker An audible sigh is heard in the classroom as planning. Although assessments may take manythe first-grade students listen to their teacher say, forms (performance, diagnostic, etc.) when used“We are going to continue learning about fractions formatively, research has shown they improve stu-today.” Students groan and whisper to each other. dent learning (Black & Wiliam, 2006). UsingAs she overhears their complaints, she is discour- knowledge of the curriculum and formative assess-aged by their attitude but determined to provide a ments, teachers can engage in the following keymeaningful experience for her students. The previ- processes:ous day she gave a pre-test with both area and setmodel problems and used the results to make her 1. Identify learners’ status;subsequent plans. The pre-assessment indicated her 2. Identify learning objectives; andstudents were more successful using the area model 3. Devise a plan (Wiliam & Thompson, 2007).than the set model to represent and describe frac-tions; therefore, she focused the following days’ In order to achieve these key processes, thelessons on the set model for fractions. National Council of Teachers of Mathematics (2015) recommends 5 formative assessment strate- The authors of Principles to Actions: En- gies be leveraged as teachers plan and implementsuring Mathematical Success for All (NCTM, instruction. These strategies are:2014) call on educators to use assessments as anongoing teaching tool to help promote student un- 1. Observation – These give the teacher anderstanding and to inform their mathematics les- overall glimpse of student understanding. They cansons. The process of assessing student learning re- happen formally or be used spontaneously during aquires teachers to look beyond merely correct and lesson. When used formally they should be inten-incorrect responses that students have given to dif- tionally connected to the actual planning and im-ferent questions and assessment items (Brendefur, plementation of the lesson.Strother, Rich & Appleton, 2016). According to theNational Council of Teachers of Mathematics 2. Interviews – Interviews are more interac-(2015), formative assessment refers to a wide vari- tive than observations. They require asking ques-ety of methods that teachers use to conduct in- tions and carrying on dialogues with individuals orprocess evaluations of student comprehension, small groups of students in an attempt to better un-learning needs, and academic progress during a les- derstand where they are in relation to the learningson, unit, or course. Formative assessments help goals. Interviews allow teachers to dig deeper intoteachers identify concepts that students are strug- a students’ understanding following an observation.gling to understand, skills they are having difficultyacquiring, or learning standards they have not yet 3. Show Me Activities – A deliberately cho-achieved so that adjustments can be made to les- sen task used to elicit the level of understandingsons, instructional techniques, and academic sup- sought during the observation or interview. Theseport. Ideally, formative assessments blend into tasks may include written student work, verbal re-classroom instruction without disrupting instruc- sponses to questions, and/or explanations of com-tional time (Heritage, 2010). pleted work. Assessments are necessary and important 4. Hinge Questions – A carefully designedparts of a student’s education, as well as teachers’ question within the lesson that enables the teacher to determine whether the class is ready to move on.Virginia Mathematics Teacher vol. 44, no. 2 The design of the question draws out the errors or weak points that may have been noted during the observations and/or interviews. 10

5. Exit Tasks – A hinge question that is Figure 2. Children using counters.placed at the end of a lesson, where students applywhat they have learned, helps the teacher plan fu- garoo. She wanted the students to use the chips toture instruction (Fennell, Kobett, & Wray, 2015). represent the animals in the story. Although these terms may be easily de- The first competitive event for the campersfined, it is not always clear what these strategies is tug-of-war, and the campers count off into twolook like and in what order they should occur with- equal teams. Mrs. Rinker asks her students to showin a normal teaching context. The purpose of this the two teams with their counters. The studentsarticle is to unpack these strategies within the con- separate their counters into two groups with sixtext of a first-grade lesson sequence on fractions. counters in each group. She asks, ‘How many campers are on each team?’ The students reply, We collaborated with a first-grade teacher ‘Six’. She follows with, ‘Six team members repre-from Virginia’s Shenandoah Valley to capture howshe uses these strategies within instruction. The sent of the campers.’state of Virginia addresses both area and set models The next event in the story is a swimmingof fractions in the first grade (Mathematics Stand-ards of Learning for Virginia Public Schools relay race. There are three lanes in the pool. Using(2009) 1.3 – The student will identify the parts of a a model of a three-lane pool, Mrs. Rinker asks herset or region that represent fractions for halves, students to show how the 12 campers (counters)thirds, and fourths and write the fractions). This ar- can be divided in a way that represents an equalticle focuses on how she addressed her students’ number for each lane in the pool (Figure 3). Theunderstanding of the set model of fractions. students divide their chips into three equal groups.Day 1: Set Model Fraction Lesson. She adds, “Each team represents of the campers. Mrs. Rinker asks her first-grade students to How many campers are on each team?” The stu-place 12 counters on their desks. She wants her stu- dents state there are four campers in each lane.dents to experience what it means to separate a Mrs. Rinker states, “One-third of the campersgroup of counters into subsets prior to introducing equals four campers”.them to the set model for fractions. She opens thebook Jump, Kangaroo, Jump! (Murphy, 1999) and The next event is a canoe competition onbegins to read the story (Figure 1) about 12 animal the lake. She reads, “There are four canoes. If therecampers participating in field day. As she introduc-es the animals, she asks the students to show themwith their counters (Figure 2): 1 kookaburra, 1emu, 2 platypuses, 3 koalas, 4 dingoes, and 1 kan-Figure 1. Reading to children. Figure 3. Students’ models.Virginia Mathematics Teacher vol. 44, no. 2 11

is an equal number of campers in each canoe, how groups I need to make? (She hands the mark-many campers are in each canoe?” Using a model er to Sam who approaches the whiteboardwith four canoes, the students show three campers and circles 2 triangles to create one groupin each canoe to represent the 12 total campers be- and two more triangles to create a seconding divided equally into four groups (Figure 3). group.) Mrs. Rinker: (Pointing to the fraction) HowMrs. Rinker states, “Three campers represent of did you know to circle groups of two?the 12 campers.” Sam: The parts have to be equal. Mrs. Rinker: What about those triangles Finally, Mrs. Rinker asks her students if (pointing to the remaining four triangles)?they can use their counters to determine how many (Sam circles the remaining triangles bystudents are on a team if there are six teams with an grouping them into two more groups of two.)equal number of campers on each team. The stu- Now how many groups do we have (Figuredents separate their counters into six groups with 4)?two campers (counters) in each group. She further Sam: Four. (Teacher circles the four in theasks, ‘Two campers represent what fraction of all denominator of the fraction in this exampleof the campers?’ The students struggled to deter- and reminds the class that is the number of equal groups we need.)mine that two campers represent of the campers. Mrs. Rinker: Now, how many of those She next uses the context of the story to groups do we want? Mariah: Three.help the students equally divide a set into groups. Mrs. Rinker hands Mariah the marker andStanding at the white board, Mrs. Rinker formally asks her to shade in the three. Mariah shadesintroduces the set model as a way to show fraction the first three triangles (Figure 5a).meaning. Her goal for having the students manipu-late the counters was to help them interpret the de- Figure 4. Groups.nominator of a fraction (referred to as the part ofthe number ‘below the line’) as the number of The teacher realizes Mariah is seeing the ‘3’ as rep-equal groups into which a number of objects could resenting three individual triangles rather than threebe divided. The numerator of a fraction (called the of the groups they created. Based on this observa-part of the number ‘above the line’) tells how many tion, she asks another question.groups to select. Mrs. Rinker: What does the part of the num- Mrs. Rinker: Who can tell me a fraction us- ber above the line mean? ing halves or fourths? Mariah: It tells us the number of groups we Sam: Three-fourths. want. Mrs. Rinker: Let’s imagine that the triangles Mrs. Rinker: (Writes on the board.) Now are eight of our campers from the story. Now we need to know what this fraction means. pretend that they have to form teams of two. Together, the numerals above and below the If three of the teams are girls and one team is line make up one number called a fraction. boys, can you shade all the girls? Mariah The part of the number below the line (pointing to the 4) tells us how many equal 12 groups we need to make. The part of the number above the line (pointing to the 3) tells us how many of those groups we want to have. (She draws eight triangles on the board and asks her students to represent the trian- gles with their counters.) Mrs. Rinker: Let’s pretend these triangles are eight of the campers from our story. When looking at , can you show me how manyVirginia Mathematics Teacher vol. 44, no. 2

Figure 5a. Mariah’s strategy 1. Figure 6. Hinge question.Figure 5b. Mariah’s strategy 2. might look like. Some students are able to divide their eight counters into four groups of two and shades an additional three triangles to fill the first three groups (Figure 5b). represent the with yellow counters (Figure 6). Mrs. Rinker: How is that different from shad- ing three triangles? Those students were given two more fractions ( Mariah: We needed six.The teacher relates the drawing and shading direct- and ) and asked to show a partner how theyly back to the numeric representation by asking would represent them with chips as they had justthe students three questions: 1. Where do you see the 4? Students reply, ‘We done for . However, some students did not pass the hinge question (Figure 6) and would have gone have four groups.’ unnoticed if the whole group answers were accept- 2. Where do you see the 3? Students reply, ‘We ed as a reflection of each individual’s understand- ing. She calls the four students who produced in- have three groups that are just girls.’ correct representations with the counters to the 3. How many triangles did we shade to show back table. [Note: All of the students who incor- rectly represented the fraction had similar incorrect of our set? Students reply, ‘Six triangles representations. If they had not, each grouping of represent the six girl campers.’ similar mistakes would have been called back sepa- The class answers these questions and seem rately.] After conducting an interview with theseconfident. However, to make sure they understood students, which consisted of asking the students tothis meaning of fractions, she poses the following share what they did with their counters as well ashinge question: ‘Take out eight red counters and why they did it, she determines that they interpret-place them on your desk. If these are eight camp- ed the ‘4’ as the number of counters to be turneders, can you show me of the campers by flipping over instead of the number of groups to divide thethose chips over to the yellow side?’ counters. Mrs. Rinker concludes that this requires She observes while the students work and small group instruction to resolve.makes anecdotal notes as the students are solvingthe problem. She uses this second round of obser- While the small group was collecting theirvation data to determine which students need addi- counters, Mrs. Rinker quickly passes around thetional assistance as well as what that assistance following exit task to the twelve students who did pass the hinge question and had also completed theVirginia Mathematics Teacher vol. 44, no. 2 other two fraction problems they had been given. “I have 12 flowers in my garden. of the flowers have red petals. How many flowers have red pet- als? Use pictures, numbers and words to solve.” She was confident, based on her observations of them using the 2-color counters that they could solve the problem. Small Group Instruction. 13

For the four students who needed additional Figures 8a (top), 8b (center), and 8c (bottom). Representa-help, she provides assistance by giving each stu- tions.dent four 2-color counters. She chooses a show meproblem where the total number of counters also groups needed additional, but different, instruction.represents the part of the number ‘below the line’, Group 2 viewed the denominator as the number ofso that they would only have to create groups of flowers that needed to be colored (Figure 8a).one. She tells the students that their four counters Group 3 needed re-teaching based on various otherrepresent the four dingoes from the story. issues (ie. drew 12 flowers and colored 1; drew eight flowers and colored four; etc.) (Figures 8b Mrs. Rinker: If of the dingoes are wearing and 8c).glasses, how many are wearing glasses? Can you Day 2: Set Model Fraction Lesson.show that with your counters? With her assistance,they are able to divide the counters into four equal Group 1 was successful on the previousgroups and turn one over as she reminds them of day’s exit task, and therefore are given a slightlythe meaning of the numbers above and below the more challenging problem to complete (Figure 9)line (Figure 7a). along with 4 crayons. This problem extends the students’ knowledge of equal groups because they Next, she provides each student with four now must determine and write the fraction repre-more counters and revisits the original set that in-volved 8 triangles. She writes the fraction andasks the students what the part of the number be-low the line represents. One shares it is the numberof ‘teams’ like they had talked about with the class.Mrs. Rinker asks if they could show her the teams.She further asks them to model the (girls) withtheir counters. All four are able to represent theby flipping over six counters and demonstratingtheir knowledge of the fraction (Figure 7b).Assessing Student Work for the Next Day’s Les-son. Mrs. Rinker collects the flower problemfrom the students (the four students in the smallgroup subsequently complete it later in the day)and uses this exit task to create three differentiatedgroups for the next day’s lesson. Of the groups,Group 1 ‘got it’ by solving the problem correctlyand explaining their thinking, but the other twoFigures 7a (left) and 7b (right). Student’s knowledge of frac- Figure 9. More challenging task.tions. 14Virginia Mathematics Teacher vol. 44, no. 2

Figures 10 (left) and 11 (right). Using multilink cubes. Figures 12a (left) and 12b (right). Representing fractions.senting the set they create. Previously, they mod- 1.eled the fraction using the parts above and below Group 3 Instruction.the line. Now they have to reverse their thinking inorder to apply their understanding. Group 3 is then called to the carpet. Mrs.Group 2 Instruction. Rinker works with this group using the same coun- ters and similar story-based problems as the day Group 2 did not create equal groups based before. Their representations of the flowers on theon the part of the number below the line on the pre- exit task showed numerous misconceptions such asvious day’s exit task; instead, they colored the dividing the total number into two parts regardlessnumber of flowers equal to the number below the of the fraction, only coloring one object in the setline. Mrs. Rinker gives each student in this group (possibly because all the fractions on the exit taskeight blue multi-link cubes. She writes the fraction were unit fractions, meaning they had a numerator of one), and drawing an incorrect number of flow- and asks the group what the parts of a number ers to start the problem. Using the characters fromabove and below the line represent. A student the story and the scenario of dividing the campersshares that below the line represents how many into equal teams over again from the day beforegroups. She then asks the students to use the eight proved successful. The students in this group weremultilink cubes to show her the two groups (Figure able to divide a given number of chips equally and10). relate the number of groups to the number below the line of a fraction after being exposed to it a sec- Mrs. Rinker: Now can you trade in some of ond day. They were also able to turn over the ap- propriate number of counters based on the number your blue cubes for pink cubes so that of above the line of a fraction. the cubes you have right now become pink? Conclusion. She explains to them that trading the color The original set model fraction lesson fromcubes represents the shading they did on the board day one was only successful for one group of stu-yesterday. The students each physically trade four dents. By collecting the information about the stu-of their blue cubes for four pink cubes showing one dents’ learning (through observations, hinge ques-of the two groups to be pink (Figure 11). This re- tions, show me activities, interviews, and exitlates back to her original hinge question that asks tasks) Mrs. Rinker was able to regroup and differ-students to separate the counters into groups based entiate her instruction for the needs of each group,on the part of the number below the line. She then which made her assessments truly formative. Shecollects the cubes and redistributes six blue cubes initially used observation to reveal her students’ conceptions and misconceptions about the set mod-to each student. Now, she asks them to show . el of fractions. Her observations revealed that someShe watches to see if they would separate their cu- students viewed the numerator as a whole numberbes into three groups before trading blue cubes for as opposed to a part of a whole group. She asks apink cubes to represent the fraction (Figures 12a & hinge question and further observes her students tob). As she observes each student demonstrate the determine if the students can solve a similar setthree groups of blue cubes and trade one group forpink cubes, she sends them back to their desks to 15work on the same garden problem given to GroupVirginia Mathematics Teacher vol. 44, no. 2

model problem without her help. She uses inter- Murphy, S. (1999). Jump, kangaroo, jump! Newviews and show me activities in small groups with York, NY: Scholastic.the students who exhibited a misunderstanding fol-lowing the lesson. She uses an exit ticket to plan the National Council of Teachers of Mathematicsnext days’ instruction. Ultimately, most of the stu- (NCTM). (2014). Principles to actions: Ensur-dents were able to interpret the number of equal ing mathematical success for all. Reston, VA:groups into which a set of objects could be divided NCTM.as well as how many of those groups to select. National Council of Teachers of Mathematics Our experience shows that using the 5 (NCTM). (2015). Annual perspectives in math-formative assessment strategies, as recommended ematics education: Assessment to enhanceby NCTM, is a helpful way to improve teachers’ teaching and learning. Reston, VA: NCTM.ability to gather and interpret evidence of studentthinking, and to use that evidence to enrich mathe- Virginia Board of Education. (2009). Mathematicsmatics instruction. We, as teachers, can incorporate standards of learning for Virginia publicformative assessment by using strategies illustrated schools. Richmond, VA: Commonwealth ofhere to inform instruction based on the needs of Virginia, Department of Education. Re-students. The formative assessment strategies we il- trieved from http://www.doe.virginia.gov/lustrated can be used to develop instructional inter- testing/sol/standards_docs/mathematics/ 2009/ventions that are more appropriate for each individ- stds_math.pdfual student. We have modeled each of the strategiesand we feel these strategies can be adapted to any Wiliam, D., & Thompson, M. (2007). Integratingmathematics lesson regardless of grade, topic, or assessment with instruction: What will it take topopulation. make it work? In C.A. Dwyer (Ed.), The future of assessment: Shaping teaching and learningReferences (pp. 53-82). Mahwah, NJ: Lawrence ErlbaumBlack, P., & Wiliam, D. (2006). Assessment and Associates. classroom learning. Assessment in Education, 5 *The electronic version of VMT will include the (July), 7-74. link to the authors’ reflection on the lesson present-Brendefur, J.L., Strother, S., Rich, K., & Appleton, ed here and describe some strategies this teacher S. (2016). Assessing student understanding: A could use to further improve the quality of experi- framework for testing and teaching. Teaching ences for her students. Children Mathematics, 23(3), 174-181.Council of Chief State School Officers (CCSSO). Ann H. Wallace (2010). Common Core State Standards. James Madison University Retrieved from http://corestandards.org [email protected], F., Kobett, B., & Wray, J. (2015). Class- room-based formative assessments: Guiding Rich Busi teaching and learning. In C. Suurtamm & A R. James Madison University McDuffie (Eds.), Annual perspectives in mathe- [email protected] matics education: Assessment to enhance teaching and learning, (pp. 51-62). Reston, Jenna Rinker VA: National Council of Teachers of Mathe- Frederick County Public Schools matics. [email protected], M. (2010). Formative assessment and next-generation assessment systems: Are we 16 losing an opportunity? Washington, DC: Coun- cil of Chief State School Officers.Virginia Mathematics Teacher vol. 44, no. 2

Problem-Solving Opportunities for Students Danny Cline, Mike Coco, and Kevin Peterson Teachers, regardless of discipline, want es for high impact problem solving experiencestheir students to think critically and become strong- where students have an opportunity to think criti-er problem solvers. Hence, we must ask the ques- cally, fail, and try again with gentle guidance whention, how does one teach problem solving? Is it needed? While exercises give them the practiceenough to simply assign many exercises or give nu- needed to become proficient at using a particularmerous exams? Perhaps it is sufficient to deliver mathematical tool, the right problem will encour-multiple lectures on problem solving techniques or age inquiry and exploration. In this paper, we giveto solve hundreds of example problems on the a few examples that inspire our students to solveboard. Although each of these ideas might help, real problems on their own.problem solving is not a spectator sport. We must The Hall of Lights.supply our students with an opportunity to experi-ence the problem-solving cycle of inquiry, explora- The first problem we exhibit is the simplest.tion, conjecture, and proof. We believe that the rig- This is a version of the popular locker problem.orous and unambiguous nature of mathematicsmakes it a very safe place to practice and learn The Statement: Imagine a long corridor. At-problem solving and navigate the path from inquiry tached to the corridor ceiling are lights, with eachto discovery. one operated by a pull cord. Assume that there are 32,000 lights in the hallway and they are all off. Unfortunately, in mathematics, most stu- Imagine further that at the entrance to the corridor,dents believe that there is an algorithm that must be there is a line of 32,000 people. The first person inlearned to solve each “type” of problem. This the line walks down the entire corridor and pullsshould be no surprise as they are told throughout each of the chains, turning on every light. The sec-their mathematical career, “Here is how you solve ond person comes right behind, and pulls the chainlinear equations,” “Here is how you solve quadratic on every second light, thereby turning off lights 2,equations,” “Here is how you find relative extre- 4, 6, 8, and so on. Now, the third person comesma,” or “Here is how you solve a first order linear along and pulls the cord of every third light. Thatdifferential equation.” If our goal, as teachers, is to is, lights numbered 3, 6, 9, 12, 15, etcetera. Now itteach critical thinking and problem solving, should is becoming difficult to keep track of which lightsnot we make the distinction between exercises and are on and which lights are off. Continue in thisproblems? Should not we make space in our cours- way until, finally, the 32,000th person walks down the corridor and pulls the chain of the 32,000thVirginia Mathematics Teacher vol. 44, no. 2 light. Which lights are on (or off)? The Exploration: The interesting thing about this problem is that almost no one has a gut feeling for what the answer will be, yet our intui- tion on how to proceed is perfect and almost uni- versal. Simply play the same game with fewer than 32,000 lights to see if a pattern emerges, and one quickly does. The columns of the table below are labeled with the number of each light. The ith row represents the status of each light (0 for off and 1 for on) after the ith person passes through the hall. 17

Change how the lights work: In this prob- lem each light cycles through 3 settings when its chain is pulled: off, dim, bright. As in the original problem, the first person in the line walks down the entire corridor and pulls each of the chains, turning every light to dim. The second person comes right behind, and pulls the chain on every second light and so on until the 32,000th person walks down the corridor and pulls the chain of the 32,000th light. Which lights are dim or bright or off? Change the corridor: This time there are n lights, each one operated with a pull cord. Howev- The Conjecture: The last row shows that af- er, in this problem, the corridor is circular. Imagineter the 16th person passes through, the only lightsthat are on are 1, 4, 9, and 16. The obvious (and there is only one cord puller and they begin pullingcorrect) conjecture is that only lights whose num-bers are perfect squares will be on. the chain of the light numbered 1 and pull every The proof requires the student to recognize third light’s chain. For what numbers, n, will it betwo important facts: possible to turn all of the lights on? 1. A light will be on if it is switched an odd number of times, and off if it is switched an It should be obvious that each of these mod- even number of times. ifications has many more modifications, making it 2. A light’s chain is only pulled by someone whose number is a factor of the light’s a very interesting problem to explore. number. Once they make these connections, it does Trees on Fire.not take them long to realize that only the numbers This is an interesting application in cellularwith an odd number of factors will be on and thatonly the perfect squares have an odd number of automata. A cellular automaton is a lattice of cellsfactors. that can take on different ‘colors.’ The ‘color’ of The Inquiry: At the end of each problem,students are asked to list interesting questions that each cell changes through a number of discretethey would like to explore further. The following isa list of a few new questions: time steps according to a set of rules based on the Change the switching rules: The first per- ‘color’ of its neighboring cells.son in the line walks down the entire corridor andpulls each of the chains, turning on every light. The The Original Problem: There is a forestsecond person comes right behind, and pulls thechain on every light with a number that is a power consisting of an n x n grid of trees. If a tree has 2of 2, thereby turning off lights 2, 4, 8, 16, and soon. Now, the third person comes along and pulls neighbors (N, S, E, or W, not diagonally) on fire,the cord of every light with a number that is a pow-er of 3. That is, lights numbered 3, 9, 27, 81, etcet- then that tree catches fire. What is the minimalera. Continue in this way until, finally, the 32,000thperson walks down the corridor and pulls the chain number of trees (and their positions) required to in-of the 32,000th light. Which lights are on (or off)? itially be on fire to guarantee that the entire forestVirginia Mathematics Teacher vol. 44, no. 2 will catch on fire? The Exploration: As in the previous prob- lem, it seems that the obvious approach is to play the game with a particular square board and a cho- sen initial number of trees on fire. In the example, a 1 represents an ignited tree and a 0 represents a non-ignited tree. Example 1: Step 0: Step 1: Step 2: 18

Step 3 and all subsequent steps: This did not yield the desired result, which time is non-increasing. Hence, you must start withforces us to ask the following: Do we need to set a perimeter of at least 4n because the desired end-more trees on fire initially or do we simply need to ing state has a perimeter of 4n.situate the initially ignited trees differently? Unlikethe Hall of Lights problem, which has one starting The Inquiry: We could dedicate an entirestate (all lights off) and one ending state (the per- paper to generalizations of this problem. We listfect square lights on), when n = 1 this problem has one modification by changing only one detail at a2 starting states. When n = 2, there are 16 starting time.states, and when n = 4, there are 216 = 65536 start-ing states. Hence, for n > 2, this problem is better Change the shape: Interesting results comesuited for computer exploration. A short computer from simply changing the shape of the board to aprogram can run through all the possibilities in a rectangle, L-shape, symmetric cross, or asymmetricfew seconds, keeping track of those starting posi- cross while asking the same question: What is thetions that end with all the trees ablaze. The results minimal number of trees (and their positions) re-of such a program will yield several examples in quired to be on fire initially to guarantee that thewhich 4 trees initially aflame will end with all the entire forest will catch on fire if, at each step, a treetrees on fire, and no examples that start with 3 trees needs 2 neighbors (N, S, E, or W) to be on fire foron fire. that tree to catch fire? Here are two successful starting states: Change the rules: Starting with a square grid of trees, what is the minimal number of treesNote that Example 1 shows that not every initial ar- (and their positions) required to initially be on firerangement of 4 trees on fire ends the same. to guarantee that the entire forest will catch on fire if we change the rules to one of those listed below? The Conjecture: After checking what hap-pens for the first 4 or 5 values of n, we conjecture 1. A tree requires 3 or 4 neighbors (N, S, E, orthat there needs to be at least n trees initially on fire W) on fire for that tree to catch fire.in an n x n grid of trees to catch all the trees on fire. 2. A tree requires 2, 3, 4, 5, or 6 neighbors (N, The proof has two main parts: S, E, W, or diagonally) on fire for that tree 1. Show that if the n trees on (either) diagonal to catch fire. Change the question: Starting with a square are initially on fire, the entire grid will eventually catch fire. grid of trees and the original rule requiring 2 neigh- 2. No arrangement of n – 1 trees initially on fire bors (N, S, E, or W) on fire for a tree to catch fire, will end with the entire grid on fire. we ask the new question: What is the minimum The second part is clearly more complicat- number of trees required to initially be on fire toed. For a particular n, one can easily check all guarantee that the entire forest will catch on fire re- gardless of their initial positions? arrangements of n – 1 ignited trees and show Metallic Ratios.that none of them ends satisfactorily. Surprisingly,proving this result in general involves showing that The final problem requires the most mathe-the perimeter of the trees on fire as a function of matical sophistication and is the most well-known,Virginia Mathematics Teacher vol. 44, no. 2 19

yet it is the easiest to describe. This problem relates and sizes, we believe problems that lead the solver,the limit of ratios of subsequent terms of the Fibo- logically, along the cyclic path of exploration, con-nacci sequence to the golden ratio. What makes jecture, proof, and inquiry give students a positivethis problem noteworthy is the way it connects a problem solving experience. Problems like thesesequence of integers (a question of arithmetic), a can be found at virtually every mathematical levelroot of the polynomial x2 – x – 1 (a problem of al- and have the following characteristics: they are rel-gebra), and a limit of a different sequence (a prob- atively easy to understand, the solution is not obvi-lem of calculus). ous until one does some manageable exploration and calculation, the proof is appropriate for the giv- The Original problem: We define the Fibo- en course, and the problem is easy to generalize. Ifnacci sequence as follows: f1 = 1, f2 = 1 and, for all students are given regular doses of this type of sup-n > 2, define fn = fn-1 + fn-2. Now create a new se- ported challenge, they will grow into stronger andquence Fn = fn+1 / fn. Find Fn if it exists. more confident problem solvers. When done cor- rectly, it is not we who teach problem solving, but The Exploration: Students usually start by the problems themselves.using a calculator or a spreadsheet to calculate thefirst several values of Fn. It converges rather quick- Referencesly. In fact, F15 = 1.618032787. Rike, T., & Stankova, Z. (Eds.). (2015). A Decade The Conjecture: Fn = . of the Berkeley Math Circle: The American Ex- This proof has two major components: perience (Vol. 1). Providence, RI: American Mathematical Society. 1. Prove Fn exists. Tanton, J.S. (2012). Mathematics Galore! Wash- ington, D.C.: Mathematical Association of 2. Prove Fn = . America. Once one proves that the limit actually ex- Tanton, J.S. (2001). Solve This: Math Activities for Students and Clubs. Washington, D.C.: Mathe-ists (the hard part), assume Fn = L and use the matical Association of America.definition of the Fibonacci sequence to show that Vandervelde, S. (2009). Circle in a Box. Provi-L2 = L + 1. dence, RI: American Mathematical Society. Young, R. M. (1992). Excursions in Calculus: An The Inquiry: An obvious generalization of Interplay of the Continuous and the Discrete.this problem is to ask the same question about the Washington, D.C.: Mathematical Associationlimit of ratios of subsequent terms for any starting of America.sequence. We list examples only of sequencesclosely related to the Fibonacci sequence where the Danny Cline, Ph.D.ratios converge to the positive solution of xn – xn-1 - Lynchburg College… - x – 1 = 0. These solutions are often called me- [email protected] ratios. Mike Coco, Ph.D. Change the initial sequence: Lynchburg College 1. Define the Tribonacci sequence as follows: [email protected] t1 = 1, t2 = 1, t3 = 1, and for all n > 3 define Kevin Peterson, Ph.D. tn = tn-1 + tn-2 + tn-3. Now create a new se- Lynchburg College quence Tn = tn+1 / tn. Find Tn if it exists. [email protected] 2. Define the m-bonacci sequence as follows: g1 = 1, g2 = 1, …, gm = 1 and for all n > m de- 20 fine gn+m = gn-1 + gn-2 + … + gn-m+1. Now create a new sequence Mn = gn+1 / gn. Find Mn if it exists. Related question: Let rn represent the posi-tive solution of xn – xn-1 - … - x – 1 = 0. Find rnif it exists. Although problems come in many shapesVirginia Mathematics Teacher vol. 44, no. 2

Grant OpportunitiesFlanagan Innovation in Mathematics Education Grant: The Virginia Council of Teachers of Mathematics isoffering Virginia mathematics educators a one-year grant worth up to $5000. This grant is designed to sup-port educators who wish to create a meaningful, innovative project that enhances some aspect of the K-12mathematics curriculum.Karen Dee Michalowicz First Timers Grant: This grant is named for Karen Dee Michalowicz who activelypromoted and supported conference development and attendance. The purpose of this grant is to providefunding support for: VCTM members who have NOT previously attended but wish to attend a regional orannual NCTM meeting; or any Virginia teacher (including non-members) who wish to attend a VCTM Annu-al Conference or VCTM Academy for the first time. Three grants are available. One award is given for theregional or annual NCTM conference of $800 and two awards of $400 each for the state conference or acade-my.For more information about VCTM grants, please visit: vctm.org/GrantsFor information about grants through NCTM, please visit: nctm.org/Grants/Busting Blockbusters!Many Hollywood movies have scenes that In The Wizard of Oz, the scarecrow is givenseem mathematically inaccurate, if not, a Th.D. (a doctorate of Thinkology) from theimpossible. Are these scenes truly impossi- Wizard. The scarecrow then recites a com-ble, or are they more plausible than they monly used mathematical equation. Is hisseem? The goal of the contest is to provide statement at 0:55 correct? Why or why not?the best mathematical explanations for thefollowing scene. The solutions that best The video can be found at:explain a scene’s possibility or impossibilityand the different elements that help form this https://www.youtube.com/watch?v=ky7DMCHQJZYwill receive an award and the winner will befeatured in the Fall 2018 issue. Answers This issue of Busting Block Busters ismay be submitted by July 31, 2018 to contributed by Dr. Neil Sigmon, [email protected] with the subject line:Busting Blockbusters entry.Virginia Mathematics Teacher vol. 44, no. 2 21

HEXA Challenge Problems created by: Dr. Oscar TagiyevApril Challenge:The ABC is inscribed into a circle. There is a tangentline through point B. The distance from point A to thetangent line is a units. The distance from point C to thetangent line is c units. Calculate the height of the ∆ABCfrom point B to the base AC.May Challenge:A rectangular garden has a rectangular shed on it. You intendto plant two different vegetable gardens, but you want tomake sure they cover equal amounts of area. How would yousplit the garden using a straight line so that the purple area issplit equally? Find a way to generalize your answer so thatthe location of the shed does not matter.June Challenge:Seaweed Volvox represent a sphere formed with pentagon, hexagon, and septagon-shaped neighboring cells.There are 3 polygons in each vertex. Prove that there are exactly 12 more pentagons than septagons.Virginia Mathematics Teacher vol. 44, no. 2 22

Please be sure to state your Contest Alert! assumptions as you solve Virginia Mathematics Teacher is each problem. conducting a contest for educatorsAnswers to the Spring 2018 and students who can solve the great- Hexa Challenge Problems est number of problems correctly bywill be featured in the Fall 08/01/2018. The winner will receive a prize and will be featured in the next 2018 Issue of Virginia issue of the VMT. Send your solu- Mathematics Teacher. tions to [email protected] with the email subject line: Hexa ChallengeJuly Challenge: 1000000If you are to write all of the whole numbers from 1 to 1,000,000 in one line 1 2 3 4 5 6 … 999999and then add all of the digits, what is the sum of all of the digits of these numbers?August Challenge:There is an isosceles triangle ABC, where AC BC. A point E is chosen on the base AB. In triangles ACE andBCE, two circles are inscribed so that they touch CE at two points, K and H, respectively. The lengths for AE =a and BE = b are known. Find the length of KH in such a way that the obtained formula would be valid for allcases.September Challenge:The BD median of the triangle ABC is divided by an inscribed circle into three equal segments, each of lengthx. AB = c, BC = a, and AC = b. Find the ratio of any two side lengths of the triangle. Congratulations to our HEXA Challenge Winner: Craig West!He teaches Calculus, Analysis, Statistics, Discrete Mathematics, and Geometry at East Rockingham High School, Elkton, VA.Virginia Mathematics Teacher vol. 44, no. 2 23

Understanding Preservice Elementary Teachers’ PerceptionsRebecca Roark, Patricia Cummane, Brittany F. Crawford, Cindy Jong, and Molly H. Fisher In the National Research Council report, Background and Literature.Adding it Up: Helping Children Learn Mathemat- Elementary teachers have had the reputa-ics (Kilpatrick, Swafford, & Findell, 2001), the au-thors present mathematical proficiency as five in- tion of poor performance in mathematics for manyterrelated strands to include: conceptual under- years. This affects their overall attitude towards thestanding, procedural fluency, strategic competence, subject both as a whole and in teaching it. Quinnadaptive reasoning, and productive disposition. (1997) stated two decades ago, “Numerous re-Productive disposition is defined as “habitual incli- searchers have pointed out shortcomings in thenation to see mathematics as sensible, useful, and mathematical content knowledge of preservice andworthwhile, coupled with a belief in diligence and inservice elementary teachers” (p. 108). Unfortu-one’s own efficacy” (p. 116). While a variety of nately, when preservice and inservice elementarymathematical thinking skills are required for profi- school teachers portray a negative attitude towardsciency, it is clear that beliefs about one’s ability mathematics, their students can sometimes reflectand value of mathematics are equally important. this negativity in their overall mathematical perfor-The purpose of this research was to examine pre- mance in the classroom. Philipp (2007) argues that,service elementary teachers’ perceptions of the fac- “Teachers make important decisions in the mannertors responsible for teachers’ success in teaching in which they teach mathematics, and elementarymathematics. We aim to gather insight into the fol- teachers often decide how much time to devote tolowing research question: What factors do preserv- mathematics each day” (p. 309). When any teacherice elementary teachers believe to be influential in shows lack of enthusiasm or lack of interest in amathematic teachers’ success? To examine this particular subject, s/he is not as likely to make thequestion, pre and post responses from an open- subject a priority and his/her students might alsoended question on the Mathematics Experiences develop a lack of interest as a result. Additionally,and Conceptions Surveys (MECS, Jong & Hodges, when teachers do not enjoy the subject they are2015) were analyzed. Preservice teachers were teaching, students do not feel excited to learn andasked: “Imagine you walked into a classroom and do not feel like the teacher motivates them to learnsaw the ‘best’ teacher teaching mathematics. What that particular subject. Swars (2005) exhorts teach-factors do you think could be contributing to this ers to work hard and strive to the best of their abil-teacher's success?” Variations in responses could ity to be an effective teacher, then their students arebe attributed to differences in preservice teachers’ positioned to learn regardless of home life, racialeducational and classroom experiences. The pre- issues, income level, and other extraneous factors.service teachers’ responses were examined to helpinform teacher education programs to better pre- “A strong command of meaningful mathe-pare future teachers or support teachers in their matical content and a positive attitude toward themathematics instruction. We found that factors re- subject are critical attributes for educators chargedlating to teacher strategies and actions ranked with teaching mathematics to children,” (Quinn,amongst the most influential factors in perceived 1997, p. 108). If teachers do not have a deep under-teacher success, as reflected in preservice teachers’ standing of a subject, then they will not have theresponses. full capacity to explain it fully. A teacher's attitude and belief about how they teach and teaching inVirginia Mathematics Teacher vol. 44, no. 2 general is key to superior teaching (Murphy, Delli, 24

& Edwards, 2004, p. 70). Beliefs are related to alyzed frequencies across codes and themes afterone's experience that can be altered with an open all responses were coded.mind. Swars (2005) studied four preservice teach- Findings with Discussion.ers’ perceptions of effectiveness in teaching mathe-matics via interviews. In her study, three themes When studying the responses from preserv-were perceived as influential: past experiences with ice elementary teachers (n = 183), their responsesmathematics, mathematics teaching self-efficacy, were coded into thirty-one unique factors contrib-and mathematics instructional strategies. Our study uting to teacher effectiveness in teaching mathe-extended Swars’ (2005) work by examining the matics. Variation was observed between the mostperceptions of preservice teachers using a larger and least frequent factors contributing to teachersample. success between teachers’ pre and post responses.Methodology. In the pre responses, 31 out of 183 preservice teacher responses listed the teacher’s action of dif- Participants in this study were preservice el- ferentiating student instruction as the most promi-ementary school teachers enrolled in an elementary nent factor contributing to teacher success. Themathematics methods course as part of a teacher next most popular factor, described in 30 out ofeducation program across three universities in the 183 responses, was the teacher’s effort in creatingEastern United States. There were 183 participants engaging lessons where the work served a purposewho each participated in a mathematics methods and did not just consist of “busy work.” Creating acourse and completed pre- and post-versions of the positive classroom community was next, identifiedMECS (Jong & Hodges, 2015). The surveys were by 29 out of 183 preservice teachers as a factor ofadministered during the beginning and end of the teacher success. Creating a positive classroom con-semester. The surveys consisted primarily of Likert sisted of characteristics such as: respect, patience,scale items on attitudes, beliefs, and dispositions student voice and creating an open, risk-free envi-toward the teaching and learning of mathematics ronment. 21 out of 183 preservice teachers listedalong with a few open-ended questions. For this knowledge of math, including being up to date withstudy, we focused on the following open-ended research, as a contributing factor. This factor wasquestion, “Imagine you walked into a classroom followed by the teacher’s apparent enjoyment ofand saw the ‘best’ teacher teaching mathematics. mathematics: enthusiasm, positive attitude, moti-What factors do you think could be contributing to vated about subject, described by 20 out of 183this teacher's success?” Thus, the question elicited preservice teachers.preservice teacher responses about inservice teach-ers; however, only data on preservice teacher per- Many of the thirty-one factors contributingceptions were gathered. Once data were gathered, to teacher success in teaching mathematics werethe researchers used an inductive approach to cre- mentioned less frequently. The least common fac-ate codes and themes based on the data. There were tor was the teacher continuing their learningseven themes consisting of 31 different codes in through professional development opportunities.which the responses were assigned. The teacher having a positive relationship with the parents of the student was mentioned at the same Before analyzing the data, the researchers low frequency. These two aforementioned codescoded the data separately to establish interrater reli- were each listed only once in the preservice teach-ability. Once individually coded, they were com- ers’ responses. Three factors were each mentionedpared and measured for reliability to ensure all re- only twice to describe teacher success in teachingsearchers were applying the same codes to respons- mathematics. These factors were: having clear and/es with at least 80% accuracy. Codes were created or high expectations, the teacher enjoying workingand revised throughout this process. As more codes with children, as well as, encouraging studentwere added, themes were developed. After reach- thinking through verbal explanations and using dif-ing 82% accuracy, the responses were divided and ferent strategies in mathematics.individually scored by the researchers. Then we an- 25Virginia Mathematics Teacher vol. 44, no. 2

Higher variation in responses was observed cy of responses. Nevertheless, the theme teachers’between the least popular factors when comparing strategies and actions, with over ten codes, re-the pre and post responses than the most popular ceived an extremely high number of responses.factors. From the pre to post responses, 3 out of 5 Pre and Post Responses.of the most popular factors remained consistent:differentiated instruction, positive class community, Although the same participants respondedand the teacher’s knowledge of mathematics. In the to the pre- and post-surveys, their answers varied.post responses, the teacher having a positive rela- The codes were developed from the pre-survey re-tionship with/getting to know their students was sponses; thus, when coding the post-survey re-mentioned by 24 of the 183 preservice teachers. sponses, new themes emerged. In the pre-survey,Encouraging students to work together and learn the responses were more detailed as opposed to thefrom each other was also listed in 22 of 183 of the post-survey where responses became more general.preservice teacher’s responses. There was only one During the post-survey, some participants began toleast popular response, which remained consistent reflect more on the effectiveness of teaching poten-from the pre to post responses, and this was the tially based on the student instead of the teacher.teacher having a positive relationship with the par- Topics that were mentioned in the post-test but notents. This particular factor, which was mentioned the pre-test include students’ prior knowledge, so-only once in the pre responses was unmentioned in cioeconomic status (SES), and school funding.the post responses. A small selection of the thirty- These factors are beyond the control of the teach-one factors were listed at a lower frequency. Four ers, but they can play a major role in the classroom.of the thirty-one factors were listed less than three Students of a higher SES are more likely to havetimes in the preservice teachers’ responses. These the support and resources to reinforce what theyresponses include the teacher’s ability to communi- learned in the classroom, while high povertycate clearly, student success rate/ student mastery schools face challenges such as greater teacherof concepts/ student achievement, collaborating turnover rate and more novice teachers who havewith colleagues and having a positive or negative an overwhelming workload (Kitchen, 2013). Theseteaching model. higher SES students are more likely to have greaterOverall Themes. parent support. One participant mentioned the fac- tor of student parent relationship playing a positive The thirty-one factors used for coding the role in the success of the students. It may haveresponses fit into a total of seven overall themes as- been the case that experiences in the mathematicssociated with the responses. Some of the themes methods course and co-requisite field experiencewere very broad and contained several codes; these influenced changes in preservice teachers’ percep-included: teachers’ strategies and actions, teacher tions of factors involved in teaching mathematicsattitudes, and teacher experience and knowledge. effectively that are more complex.Other themes, like resources or teacher improve- Final Remarks.ment contained as little as one code and were morespecific. The remaining two themes, student ac- Our descriptive study highlights importanttions and attitudes and teacher relationships each factors that preservice teachers perceive as influ-contained three codes. The student actions and atti- encing teacher effectiveness in the mathematicstudes theme was not as frequently selected because classroom. The themes we found included those ofthe codes in it were dependent on the characteris- Swars’ (2005) study and more. The themes primar-tics of the student rather than the teacher. Even ily focused on characteristics, skills, attitudes, andthough resources only contained one code, many experiences of teachers such as the teaching mod-responses referred to the theme. However, the els they have observed, knowledge and confidencetheme teacher improvement with only two codes, in mathematics, creating a positive classroom com-received very few responses. The number of codes munity, making mathematics meaningful, and par-in a theme did not necessarily relate to the frequen- ticipating in professional development. Student characteristics, skills, and attitudes in mathematicsVirginia Mathematics Teacher vol. 44, no. 2 26

were not as common, especially in the prerespons- education reform in high-poverty communi-es. Ball, Hill, and Bass (2005) have made a clear ties. For the Learning of Mathematics, 23(3),case that mathematical knowledge for teaching is 16-22.critical to be effective and increase students’ math- Murphy, P. K., Delli, L. A. M., & Edwards, M. N.ematics achievement, which is central to teaching (2004). The good teacher and good teaching:success. However, the realities of teaching de- Comparing beliefs of second-grade students,mands require more than teachers’ mathematics preservice teachers, and inservice teachers. Theknowledge and teaching skills. Journal of Experimental Education, 72(2), 69- 92. Teacher educators can leverage learning op- Philipp, R. A. (2007). Mathematics teachers’ be-portunities around these factors to extend teachers’ liefs and affect. In F. K. Lester (Ed.), Secondperceptions to include broader and more complex handbook of research on mathematics teachingnotions of teaching that highlight student back- and learning (pp. 257-315). United States: In-ground and school contexts. Just as teachers build formation Age Publishing.on students’ prior knowledge, it is important that Quinn, R. J. (1997). Effects of mathematics meth-teacher educators do the same with preservice ods courses on the mathematical attitudes andteachers’ perceptions to strengthen their learning content knowledge of preservice teachers. Theexperience. Journal of educational research, 91(2), 108- 114.References Swars, S. L. (2005). Examining perceptions ofBall, D. L., Hill, H. C., & Bass, H. (2005). Know- mathematics teaching effectiveness among ele- mentary preservice teachers with differing lev- ing mathematics for teaching: Who knows els of mathematics teacher efficacy. Journal of mathematics well enough to teach third grade, Instructional Psychology, 32(2). and how can we decide? American Educator.Jong, C., & Hodges, T.E. (2015). Assessing atti- *For readers interested in this topic, we suggest tudes toward mathematics across teacher edu- reading the seminal pieces featured in the electron- cation contexts. Journal of Mathematics Teach- ic VMT. er Education, 18(5), 407-425.Kilpatrick, J., Swafford, J., & Findell, B. (Eds.). (2001). Adding it up: Helping children learn mathematics. National Academies Press.Kitchen, R. (2003). Getting real about mathematics *See additional authors on the next page.Rebecca Roark Patricia Cummane Brittany F. CrawfordUniversity of Kentucky Jefferson County Public Schools University of [email protected] [email protected] [email protected] Mathematics Teacher vol. 44, no. 2 27

Cindy Jong Molly H. FisherUniversity of Kentucky University of [email protected] [email protected] Department of Educationannounces finalists for nationalmathematics and science teachingawards The Virginia Department of Education Policy.(VDOE) has announced the 2017 state-level final- On October 25, 2017, the Virginia Board ofists for the Presidential Awards for Excellence inMathematics and Science Teaching (PAEMST) at Education recognized the 2017 Virginia state final-the secondary level (grades 7 – 12). The Presiden- ists in secondary mathematics. Congratulations to:tial Awards for Excellence in Mathematics and Sci- • William Daly – Albemarle High, Albemarleence Teaching (PAEMST) are the highest honorsbestowed by the United States government specifi- County Public Schoolscally for K-12 mathematics and science (including • Blythe Samuels – Powhatan High, Powhatancomputer science) teaching. Awards alternate eachyear between elementary and secondary teachers. County Public Schools • Elisa Tedona – previously at Salem Church Awards are given to mathematics and sci-ence (including computer science) teachers from Middle and now at James River High, Chester-each of the 50 states, the District of Columbia, the field County Public SchoolsCommonwealth of Puerto Rico, the Department ofDefense Education Activity schools, or the U.S. PAEMST winners are typically announcedterritories as a group (American Samoa, Guam, the and honored the year following the receipt of theCommonwealth of the Northern Mariana Islands, application. Each Presidential Awardee receives aand U.S. Virgin Islands). certificate signed by the President of the United States and a $10,000 award from NSF. Awardees The award recognizes those teachers who also are honored during recognition events thatdevelop and implement a high-quality instructional take place in Washington, D.C. These events in-program that is informed by content knowledge clude an award ceremony, professional develop-and enhances student learning. ment opportunities, and discussions with policy- makers on how to improve science, technology, Awardees serve as models for their col- engineering, and mathematics (STEM) education.leagues, inspiration to their communities, and lead-ers in the improvement of STEM (including com- Nominate a K-6 outstandingputer science) education. The National Science mathematics teacher today atFoundation administers PAEMST on behalf of TheWhite House Office of Science and Technology www.paemst.org.Virginia Mathematics Teacher vol. 44, no. 2 28

Good ReadsSection Editor: Dr. Betti KreyeIn this section, we feature mathematics literature that is appropriate for K-12 Mathematicsinstruction. If you use specific literature for your mathematics classroom and wish to shareit with the Virginia Mathematics Teacher community, please respond to the Call forManuscripts on page 30.The Formative 5: Everyday Assessment Dr. Betti Kreye Section Editor, Good ReadsTechniques for Every Classroom [email protected] The Formative 5 outlines ways to insert The Formative 5: Everyday Assessment Techniquesformative assessments into everyday teaching prac- for Every Classroomtices; it is a practical look at implementing these Francis (Skip) Fennell, Beth Kobett, and Jonathanstrategies. The book focuses on the five formative Wrayassessment strategies that include observations, in- NCTM, 2017terviews, show me (students demonstrating and ex-plaining their thinking), hinge questions, and exit Review by Dr. Betti Kreyetasks. Each of these is explained, discussed as apart of the lesson planning process, and illustratedthrough examples, vignettes, and student work.Teachers will find additional support from the in-cluded templates and ready-to-use tools. This bookwould be a valuable resource for preservice andveteran teachers of mathematics at every grade lev-el. It would also be a useful resource when coach-ing teachers to use formative assessments or as abook-study for a group of teachers searching forways to understand how to begin or to increase theeffective use of formative assessments in their ownclassrooms. Math JokesHow does a mathematician plow his fields? With a pro-tractor.After a talking sheepdog gets all the sheep in the pen, he reports back to the farmer that all 30 are accountedfor. “But I only have 28 sheep,” said the farmer. “I know,” said the sheepdog. “But I rounded them up.”47% of statistics are made up on the spot.Did you hear about the statistician who drowned crossing a river? It was three feet deep on average.What do you call a number that can’t keep still? A roamin’ numeral.Virginia Mathematics Teacher vol. 44, no. 2 29

Call for ManuscriptsAll submitted articles should be a single-spaced Word document in 12 pt. Times New Roman font with APAstyle citations and references. Please send your manuscripts or questions to Dr. Agida Manizade [email protected] with the appropriate subject line. The deadline for each call is July 15, 2018. Virginia resi-dents whose articles appear in the VMT will be granted free membership in the VCTM for one year. Teaching Dilemmas Unsolved Mathematical MysteriesWe are accepting articles that include teachers’ We are accepting articles about fascinating mathe-reflections on mathematical topics you find chal- matical problems that have not yet been solved.lenging to teach. If you have a difficult problem or The problem itself should be described simply sotopic you use for your students, describe the prob- that a middle school student could understand it. Itlem, discuss common student difficulties with it, should also include the progress that has beenand the way you approach teaching this topic. made by the mathematics community on solvingSubject line: Teaching Dilemmas this problem. Subject line: Unsolved Mathematical Mysteries Research or Practitioner ArticlesWe are accepting papers on elementary, middle, Math Girlsand secondary mathematics teaching and educa- What does your school district or institution do totion. Articles will be peer-reviewed and comments encourage girls in mathematics? Send any infor-will be shared with the author. Please provide mation or publications related to girls in mathe-articles in a word document no more than 5 pages matics education.(not including corresponding images, graphs, etc.). Subject line: Math GirlsSubject line: Research or Practitioner Article Busting Blockbusters Good Reads In this section we are accepting suggestions forDo you know of a teaching resource or literature scenes from movies for readers to analyze andladen with mathematics content? Send any litera- explain the mathematical plausibility of them.ture reviews to Dr. Betti Kreye at [email protected]. Please describe the scene and provide a timestampSubject line: Good Reads for it. Subject line: Busting Blockbusters Suggestion Technology ReviewDo you have a review of an app, website, or Reviewersonline resource? Send your critical review to If you are willing to serve as a reviewer, pleaseChristophe Hirel at [email protected]. contact us at [email protected] line: Technology ReviewVirginia Mathematics Teacher vol. 44, no. 2 30

Key to the Fall 2017Puzzlemaker ProblemVirginia Mathematics Teacher vol. 44, no. 2 31

How to Share an Island: Divergent Thinking and Creativity Nikita Patterson and Darryl CoreyIntroduction and Motivation. creative. In most instances, teachers are the ones Divergent thinking, namely producing as who shape students’ perception of and performance in mathematics. Improving mathematics perfor-many ideas as possible about a problem, is often mance and understanding are very important goalsnot a regular part of mathematics instruction in the for the education community. One way to accom-United States. Often times, it is the lack of a teach- plish this goal is to improve students’ ability toer’s confidence in his/her own ability to think “think outside the box” by engaging them in diver-‘outside the box’ for additional solution methods gent thinking while solving problems in mathemat-that prevents the introduction of such activities to ics (Corey, Unal, & Jakubowski, 2007). In order tostudents” (Corey & Unal, 2004, pg 1). Moreover, do this we, as teachers, have to learn how to thinkmost problems in mathematics yield only one cor- creatively while problem solving.rect answer, thus they are highly likely to discour- Theoretical Framework.age students from exploring or discussing diverseideas while solving problems (Kwon, Park, & Park, According to Guilford (1967), divergent2006). However, both state and national standards thinking is the mental process by which we drawfor mathematics require that we “apply and adapt a from many ideas while searching for different solu-variety of appropriate strategies” when engaging in tions to the same problem or set of situations. Inmathematical problem solving (NCTM, 2009). We this study, creativity is defined as “the process ofcan improve our own learning by becoming aware sensing gaps and disturbing missing elements;of our own thinking as we read, write, and reflect forming ideas or hypothesis concerning them; test-on our own problem solving process while seeking ing these hypotheses; and communicating the re-different solution strategies to the same problem. sults; possibly modifying and retesting the hypoth-As teachers, we can promote this awareness direct- esis again” (Torrance, 1962, p. 16). According toly by informing students about a variety of strate- Guilford and Merrifield (1960) there are six kindsgies for solving mathematics problems. This can of thinking abilities that involve creativity and di-help to construct classroom environments that fo- vergent thinking (shown in Figure 1).cus on strategic learning that is both flexible and Purpose.Figure 1. Six kinds of creative thinking abilities. 32Virginia Mathematics Teacher vol. 44, no. 2

The purpose of this on-going study is to ex- • Each brother’s portion of the island must haveamine practicing middle and secondary teachers’ an area equal to the other brother’s portion.different strategies and methods of solving givenproblems in a hybrid geometry course. The authors Teachers were given one week to work with theirwill address the following research questions: groups and submit their solutions electronically online. Analysis of the groups’ solutions revealed 1. In what ways, if any, do in-service mathe- that the three thinking abilities displayed were Flu- matics teachers demonstrate divergent ency, Flexibility, and Originality while solving this thinking and creativity skills while problem problem. solving? Fluency (F): Shively (2011) states the first 2. When teachers spend time inventing new step in problem solving is to have as many ideas as solutions, what kind of thinking abilities as possible to choose from and “play with” if the en- defined by Guilford and Merrifield (1960) deavor is to be creative. Fluency, as defined by emerge? Guilford and Merrifield (1960) is the ability to gen- erate a lot of ideas which increases the chances ofParticipants. significant responses or in the case of mathematical The participants were 25 middle and sec- problem solving, meaningful or correct solutions. Conway (1999) suggests one way to determine Flu-ondary mathematics teachers enrolled in a graduate ency while problem solving is to simply count thelevel geometry course as part of a teacher educa- number of correct solutions.tion Master’s Degree program at a regional univer-sity in the southeastern United States. The teachers Flexibility (X): Flexibility, according towere randomly divided into six problem-solving Shively (2011), is the ability to look at a problemgroups for the duration of the course. (Please check from many different angles. “The measurement ofthe electronic version of this issue of VMT at flexibility considers how many different ways ofwww.vctm.org/VMT for additional details related thinking about a problem are employed” during theto the methodology of this study.) problem solving process (Conway, 1999, p 511).The Assignment. Flexible thinkers discover new possibilities, em- ploy different interpretations, and are able to shift Divergent thinking can be assessed using from one approach to another.divergent thinking tasks. Divergent thinking tasksare intended to capture the creative quality of the Originality (O): Originality refers to solu-participants’ responses, not simply the number of tions that are unusual, infrequent, insightful, orresponses (Silva et al, 2008). In this study, teach- clever. Original thinkers produce unexpected ideas,ers were instructed to find as many different pro- first-of-a-kind solutions, and have the ability tocesses/procedures as possible when solving each think beyond the obvious (Conway, 1999; Shively,problem. In this paper, the solutions of a group of 2011; Torrence, 1972). “Originality requires theteachers are shared and discussed. greatest risk-taking and is the crux of innovation,The Problem. yet it is the most fragile dimension of creativity in school settings oriented to correct an- Two brothers are part of an expedition and they have swers” (Shively, 2011, p. 12). discovered a new island. From the sky they notice that Solutions. the island is shaped like an irregular convex quadrilat- eral. They are not sure of the actual dimensions of the This geometry problem resulted in total of 4 island, but they want to determine a way that they can strategies and 7 different methods for the entire fairly divide the island between the two of them. class. For Group A, the problem resulted in a total of 3 strategies and 5 different methods as seen be-The class determined the following to be features low.of the problem:• The island is shaped like an irregular convex Strategy 1: “Cake Cutting”: Figure 2 shows Group A’s initial analysis of the problem. “The quadrilateral. first suggested method is to divide the land by• The island dimensions are unknown. 33Virginia Mathematics Teacher vol. 44, no. 2

Figure 2. Sketch of cake-cutting strategy. points of its adjacent sides. The resulting quadrilat- eral is a parallelogram whose area is equal to halfFigure 3. Quadrilateral midpoint strategy - A. of the total area. This method showed flexibility because the group approached the problem from asimply drawing a diagonal. We found that this different perspective. Their sketch is shown in Fig-would not be fair. We are given that the island is ure 3.an irregular convex quadrilateral with unknown di-mensions; therefore we cannot be sure that this While correct, not all of the members ofmethod will produce equal parcels. In fact, an Group A were satisfied with this solution. One fe-equal division in this manner would be an unlikely male student stated, “We can give the interiorcoincidence. However, even if with an uneven divi- quadrilateral to one brother and the remaining 4sion using this method, a fair result could be triangles to the other brother. However, althoughachieved if the brothers agreed that one brother this resolves the problem of dividing the islandchoose the direction of the division, then the otherbrother claim a parcel. This is the ‘cake cutting’ evenly, it is not quite fair. One brothermethod that depends on the goodwill of the broth- would acquire all of the beachfront prop-ers, so it is not optimal.” This method demonstrates erty whereas the other brother would ac-the group showed risk-taking and offered a clever, quire none.” The idea of fairness was veryunexpected result. They gave no thought to finding important to this student and led her tothe actual answer, but rather a creative possibility. pursue another solution.This shows Originality. Another attempt at a solution from Strategy 2: Midpoints of the Sides of the Group A started with marking the mid-Quadrilateral: After much debate, the teachers in points on each of the sides of the quadrilat-Group A decided to use the dynamic software pro- eral, then the midpoints were connectedgram Geometer’s Sketchpad to divide the land by with the opposite midpoint to form lineconnecting the midpoints of each side to the mid- segments. The intersection of these seg- ments was marked. These line segments formed 4Virginia Mathematics Teacher vol. 44, no. 2 quadrilaterals within the original quadrilateral. This group found that the sum of the areas of the non- adjacent quadrilateral is equal to the sum of the re- maining quadrilaterals. This strategy is shown in Figure 4. Strategy 3: Triangle Areas: The following solution also used the midpoints of the sides of the quadrilateral, but the groups focused on their prior knowledge of triangle areas instead of the area of quadrilaterals. In an attempt to solve this problem, Group A decided to construct the line HE that joins the midpoints of side AD and side BC respectively, then the line FG that joins the midpoints of sides DC and AB. They labeled the intersection point as X. Next, they constructed the diagonals from X to each of the vertex angles to form triangles. These first steps are shown in Figure 5. They shaded the triangular areas of AXB, XCB, XCD, and ADX and used Geometer’s Sketchpad to measure the triangular areas. This next set of steps is shown in Figure 6. 34

Figure 4. Quadrilateral midpoint strategy - B. vertex would divide each of the four triangles into two equal parts (ex:Figure 5. Triangle area strategy - A. ∆CID is divided into ∆GIC and ∆DIG). If we assign half of each tri- Group A gave an alternate method of find- angle to a brother, they will both re-ing the Triangle Area. They constructed midpoints ceive the same amount of land. Weand drew a line segment between the midpoints of decided to group the eight trianglesthe opposite sides of the quadrilateral. Next they into four quadrilaterals (the quadri-constructed line segments from the vertices to the laterals are HAEI, EBFI, FCGI, andintersection of the midpoint connector lines to find GDHI) so the brothers will eachthe centroid of the quadrilateral. They described have two sections of land. Each brother will havetheir process as follows: “Disregarding the mid- opposite quadrilaterals, HAEI and FCGI are forpoint connector lines, we could see four triangles. one brother and EBFI and GDHI are for the otherWe knew that the medians we constructed between brother.” This work is shown in Figure 7.the midpoint of the hypotenuse and the opposite Strategy 4: Triangle Altitudes: This strategy did not involve the midpoints of the quadrilateralFigure 6. Final steps in triangle area strategy - A. sides, but rather focused on the diagonals of theVirginia Mathematics Teacher vol. 44, no. 2 quadrilateral, shown in Figure 8. The group found the midpoint, point I, of this diagonal segment to create equal bases. They also constructed a seg- ment from the midpoint of the diagonal to the op- posite vertices. In doing so, they constructed two triangles (with equal bases and equal heights). Therefore, GIH and JIH have the same area. Like- wise, GIF and FIJ also have the same area. They shaded this in the diagram and also measured the areas in Geometer’s sketchpad to show equality. Another group used the idea of triangle alti- tudes along with the midpoint of only two sides of the quadrilateral. “We begin solving this problem by first using Sketchpad to construct the midpoints of the two opposite sides C͞ B and A͞ D. After finding the midpoints of the two opposite sides C͞ B and A͞ D, we construct D͞ E, F͞ B, and D͞ B to form 4 triangles, ∆DCE, ∆DEB, ∆ABF and ∆DBF. Next we construct GD, the height or altitude for ∆DCE and ∆DEB. We call it h1. The next step is to construct B͞ H, the height or altitude for ∆ABF and ∆DBF. We call it h2. These heights are used to determine the areas of the respective triangles (see Figure 9). One brother will get the outer triangles; the other broth- er will get the inner quadrilateral.” This geometry problem resulted in a total of 4 strategies and 7 different methods for the entire class, showing flexibility for the class as a whole. The majority of the groups created triangles and fo- cused on triangle areas. Another popular strategy 35

Figure 7. Triangle area strategy - B. was to use the midpoints of the quadrilateral in their methods. The idea of fairness forced the classFigure 8. Triangle altitude strategy - A. to find multiple ways of dividing the land to be the “most fair” which was evidence of the fluency.Figure 9. Triangle altitude strategy - B. The “cake cutting” method showed originality.Figure 10. Clean sketch of triangle altitude strategy - B. Group A was the only group to demonstrate fluen-Virginia Mathematics Teacher vol. 44, no. 2 cy, flexibility, and originality. Upon analysis of the student work, there was no evidence of sensitivity, elaboration, or redefinition, possibly due to the type of problem and juxtaposition of the problem to the course work. It was assigned after a lesson on quadrilaterals so the students immediately con- verged to the idea of using midsegments. Conclusions. The results of this problem are just one ex- ample of the levels of divergent thinking skills that emerged during this study. The in-service teachers in the course were assigned a set of open-ended problems and asked to solve them. They were later specifically instructed to “think outside the box” and find as many solutions as possible for each problem. In-service teachers are engaging in diver- gent thinking and seeking creative ways to solve geometry problems, however, this only material- ized after they were specifically instructed to do so. Once specifically instructed to find many solutions, teachers working in groups demonstrated some are- as of the Guilford Model for divergent thinking. When teachers were not specifically instructed to seek many solutions, some groups quickly reduced to one method of solution and stopped. If we teach- ers value divergent thinking, then we can foster an environment of divergent thinkers in our class- rooms. Engaging ourselves in creative problem solving activities may be a first step. The use of technologies such as MS Excel and Geometer’s Sketchpad served as key elements in sparking divergent thinking and mathematical creativity. The technology assisted the teachers’ exploration of ideas and verification of possible so- lutions, and also helped to dispense with erroneous plans or solution methods. The technology promot- ed creativity in Group A. It was also observed that the teachers became more confident when discuss- ing their ideas online rather than in class. They de- veloped “keyboard courage” that they typically do not have in a face-to-face setting. The online envi- 36

ronment allowed a better sense of equitable footing through an Open-Ended Approach. Asia Pacificin the group and transparency amongst group mem- Education Review 2006, 7(1), 51-61.bers that helped to form a comfort level with teach- National Council of Teachers of Mathematics.ers. (2009). Principles and standards for school mathematics. Reston, VA: Author.References Shively, C. H. (2011). Grow creativity. Learning &Conway, K. (1999). Assessing open-ended prob- Leading with Technology, 38(7), 10-15. Silva, P., Winterstein, B.P., Willse, J.T., Barona, lems. Mathematics Teaching in the Middle C.M., Cram, J.T., Hess, K.I., Martinez, J.L., School, 4(8), 510-515. and Richard, C.A. (2008). Assessing creativityCorey, D., Unal, H., Jakubowski, E., An 8th grade with divergent thinking tasks: Exploring relia- Geometry Problem from Japan and American bility and validity of new subjective scoring Teachers Solutions, Learning and Teaching methods. Psychology of Aesthetics, Creativity Mathematics , Vol. 5, Issue 5, pp. 12-16, (2007) and the Arts, 2(2), 68-85.Corey, D. & Unal, H. (2004). Activating Metacog- Torrance, E. P. (1962). Guiding creative talent. nition via Distance Learning: An Exploratory Englewood Cliffs, NJ: Prentice Hall. Study of In-Service Math Teachers’ use of Treffinger, D.J., & Isaksen, S.G. (1992). Creative Metacognitive Monitoring Skills in an Online problem solving: An introduction. Sarasota, FL: Problem Solving Course. In C. Crawford et al. Center for Creative Learning. (Eds.), Proceedings of Society for Information Technology and Teacher Education Interna- Nikita Patterson tional Conference 2004 (pp. 4385-4388). Ches- Georgia State University apeake, VA: AACE. [email protected], A.J. (2001). Creativity in Education and Learning: A Guide for Teachers and Educators. Darryl L. Corey London: Kogan Page Limited. Radford UniversityGarofalo, J., Drier, H., Harper, S., Timmerman, [email protected] M.A., & Shockey, T. (2000). Promoting appro- priate uses of technology in mathematics teach- 37 er preparation. Contemporary Issues in Tech- nology and Teacher Education, 1(1). Retrieved from http://www.citejournal.org/vol1/iss1/ currentissues/mathematics/article1.htm.Guilford, J.P. (1967). The nature of human intelli- gence. New York: McGraw- Glucksberg, S., Manfredi, D. A. & McGlone, M.S. (1997). Metaphor comprehension: How metaphors cre- ate new categories. In T. B. Ward, S. N. Smith, & J. Vaid (Eds.), Creative Thought: An investi- gation of conceptual structures and processes (pp. 327-350), Washington D.C. American Psy- chological Association.Guilford, J.P. & Merrifield, P.R. (1960). The Struc- ture and Intellect Model: Its Uses and Implica- tions. Los Angeles: University of Southern Cal- ifornia.Kwon, O.N., Park, J.S., & Park, J. H. (2006). Culti- vating Divergent Thinking in MathematicsVirginia Mathematics Teacher vol. 44, no. 2

Solutions to Fall 2017 HEXA Challenge ProblemsOctober Challenge:In the triangle ABC, angle C is obtuse. The side AB = x. A circle goes through points A, B, and the point ofintersection of the heights (orthocenter) of the triangle ABC. Find the radius of such circle.SOLUTION:AE is perpendicular to BH and BF is perpendicular to AH, because BE and AF both are the heights of the tri-angle ABC . Let’s label angle FCE as α which means: FCE = α AEH = BFH = 90° AHB = 180° - αBecause of the SINE rule, = d (where dis the diameter of the triangle circumcircle). Therefore:For ∆ABC: 2R = For ∆ABH: 2R1 =R1 =R1 = R.November Challenge:There are a total N amount of nuts in all three pockets. What is the probability that each pocket contains thesame amount of nuts (N/3)?SOLUTION:We toss nuts randomly into three containers: C1, C2, and C3. The probability that a nut falls in one of thesecontainers on a single try is constant, given by: P(C1) = p1, P(C2) = p2, P(C3) = p3, where p1, p2, p3 [0, 1] and p1 + p2 + p3 = 1.Then, the distribution of the total counts in each container (Y1, Y2, and Y3) is a multinomial given by:Virginia Mathematics Teacher vol. 44, no. 2 38

P(Y1 = n1, Y2 = n2, Y3 = n3) = p1n1p2n2p3n3.The counts in each container can be equal only when n1 + n2 + n3 is a multiple of 3. Let the total count be 3m.Then the probability that the counts are equal in each container is given by: P(Y1 = m, Y2 = m, Y3 = m) = p1mp2mp3m.For the special case of equal probabilities for each container, p1 = p2 = p3 = 1/3, this simplifies to P(Y1 = m, Y2 = m, Y3 = m) = (1/3)3m.For example, with m = 1, giving a total of 3 nuts, we have P(E) = 2/9. For m = 2, giving a total of 6 nuts, wefind P(E) = 10/81.December Challenge:Everyone is aware of the murderer-waves, tsunamis, that are able to move in the ocean at a tremendous rate.They can break and turn over ships and/or vessels. Estimate the propagation rate of the tsunami wave near thecoastline, where the water depth is about 100 meters.SOLUTION:Let’s assume the velocity of the water is V, the depth of the coastal water is H, and the mass of the waterswept by the wave is m.Because of the energy laws, kinetic energy converts to potential energy and we obtain the equation: V= = ≈ 31 m/sec ≈ 70 mphJanuary Challenge:Find all the integer pairs (X; Y) that satisfy the equation X2 – 5XY + 6Y2 = 11.SOLUTION:The given equation could be rewritten as: X2 – 3XY – 2XY + 6Y2 = 11 X(X – 3Y) – 2Y(X – 3Y) = 11 (X – 3Y)(X – 2Y) = 11Since 11 is a prime number, it can be factorized in the following equations:Virginia Mathematics Teacher vol. 44, no. 2 39

When you solve the system of equations, you obtain the following integer couples: (31, 10) (-31, -10) (19, 10) (-19, -10)February Challenge:You bought a duck from a market place. Your duck lays 10 eggs, and then all the 10 eggs safely hatch. Afterthese eggs hatch you eat the duck for dinner. The newly hatched ducklings that are drakes (male ducklings)get eaten. However, you allow the female ducklings to mature and have 10 eggs each. Once they lay ten eggs,you eat the duck. The same principle continues: male ducklings become eaten as early as possible, but youallow female ducklings to mature and have 10 eggs, only then you eat the female ducklings. Unfortunately,one day only drakes (male ducklings) hatch from all eggs. You eat them all. You have counted that all togeth-er you have eaten exactly 1000 drakes. How many ducks (female) have you eaten?SOLTUION:Let’s label X as the number of ducks hatched. Therefore, the total number of drakes and ducks that havehatched will be X + 1000. On the other hand, the total number of the hatched is 10 · (1 + X). Set these equalto each other: 10 · (1 + X) = X + 1000 10 + 10X = X + 1000 9X = 990 X = 110You have eaten (1 + X) = 111 ducks.March Challenge:There is a coordinate system with the origin O, where OA = 5units, OB = 6 units, and OC = 7 units. Estimate the area of thetriangle ABC.SOLUTION:Assuming that the pyramid is filled with some gas, let’s say Pis the pressure of the gas perpendicular to the face of the pyra-mid. The solid given is actually a pyramid, a triangular four-faced solid, whose three side faces are three right triangles,each of which is perpendicular to the other triangles. The righttriangles have three sides {7, 5, 6}, {7, 6, 5}, and {6, 5, 7}, respectively. Then solve their areas accordingly.Virginia Mathematics Teacher vol. 44, no. 2 40

Area of ∆AOB will be: Area of ∆BOC will be: Area of ∆AOC will be:(1/2) · b · h = (1/2) · 6 · 5 = 15 units2 (1/2) · 6 · 7 = 21 units2 (1/2) · 5 · 7 = 17.5 units2Assuming the solid contains gas with internal pressure P units, then (P · Area of ∆AOB)2 + (P · Area of ∆BOC)2 + (P · Area of ∆AOC)2 = (P · Area of ∆ABC)2 P2 (225 + 441 + 306.25) = P2 · Area of ∆ABC2 Area of ∆ABC ≈ 31 units2Technology Review Section Editor: Christophe HirelIn this section, we feature websites, online manipulatives, and web-based applications Christophe Hirelthat are appropriate for K-12 mathematics instruction. We are looking for critical Section Editor,reviews of technologies which focus on both the benefits and limitations of using these Technology Reviewtools in a K-12 mathematics classroom. If you use a technological tool and wish to share [email protected] us, please respond to the call for manuscripts on page 30.Augmented Reality App: GeoGebraVirginia Mathematics Teacher vol. 44, no. 2 In this issue, we are featuring the GeoGebra Augmented Reality app. We often hear augmented reality and virtual reality – two concepts that can sometimes be confused. The augmented reality (AR) is a technology that superimposes a computer-generated image on a user's view of the real world, thus providing a composite view. Augmented reality enhances expe- riences by adding computer-generated enhance- ments (digital images or graphics) on top of an ex- isting reality in order to make it more meaningful through the ability to interact with it. Effective motion tracking is the key to main- taining the magic of AR. When interacting with a vir- 41

tual object as if it is in the real world, the mobile de- floor, or any flat surface, walk around them, andvice (such as laptops, smart phones, and tablets) take screenshots from different angles. The app canmust accurately characterize the position of the object display several examples of 3D math objects suchin relation to itself. AR technology is quickly com- as basic solids, Penrose triangle, Sierpinski pyra-ing into the mainstream. You may remember the mid, Football, 3D Function, Klein’s bottle, Ruledpopular game “Pokémon GO” a few years ago us- Surface, and Spiral Staircase.ing this technology. Guided activities can lead the user to dis- In contrast, virtual reality (VR) creates its cover math in the real world by observing and tak-own reality that is completely computer generated ing screenshots from different perspectives. Theand driven. Virtual Reality is usually delivered to app allows the user/student to enter his/her ownthe user through a head-mounted or hand-held con- functions. Two functions can be shown at the sametroller. This equipment connects people to the vir- time and they can be modified at any time.tual reality and allows them to control and navigatetheir actions in an environment meant to simulate The app, which was implemented 6 monthsthe real world. ago, will be further developed with more features in the near future. The app is currently available While Virtual Reality is more immersive, for installation on iPhone and on newer versionsAugmented Reality provides more freedom for the of iPad. The iPad must have support for augmenteduser because it does not require a head-mounted reality.display. The Geogebra Augmented Reality Appwas created to explore the potential of augmented In conclusion, using augmented reality,reality for learning and teaching mathematics. this app can transform the way students learn. By interacting with virtual objects as if they were in The user can place math objects on a table, their surroundings, students will be able to im- merse themselves among mathematical figures in a way they never imagined. *The editorial board thanks Dr. Stephen Jull for his collaboration and his feedback on this piece.Virginia Mathematics Teacher vol. 44, no. 2 42

Motion with LEGOs and Dynamic Geometry Zoltán Kovács A basic task of engineering is the problem have coordinates (-3.5, 1) and (3.5, -1), respective-of assembling mechanisms that make an element ly, with the horizontal rods 4 units long and themove along a prescribed path. Modern electrical vertical rod 2 units long, then the green curve mustengineering allows devices to move with incredible be a part of the geometric image of the polynomialprecision by means of microcontrollers, but many equation x6 + 12x4y2 - 212x4 + 112x3y + 12x2y4 -applications still rely on classic methods that use 44x2y2 + 2989x2 + 112xy3 - 1820xy + 4y6 - 32y4 +belts, chains, shafts or ball bearings. 64y2 = 0 (Figure 2). In this paper focused on mathematical mod- Figure 2. Watt’s linkage modeled as a GeoGebra construc-eling of mechanical engineering, a new approach is tion, also available as an online activity at https://presented that extends the classic way of learning www.geogebra.org/m/vanSAKUcanalytic geometry by using computer algebra forcalculation and a LEGO Technic compilation for We recall that an equation of a curve is anfurther experiments. We discuss some novel meth- algebraic representation of a geometrical object.ods in dynamic geometry to study motion linkages When all solutions of that equation are plotted in afrom both the computation and the experimental Cartesian coordinate system, all points of the curveviews at the same time. A classic proof is also pro- can be obtained. For example, lines are associatedvided which covers some particular cases of the ob- with linear equations of x and y such as 2x + y = 5,tained statement. but circles can be described by quadratic equations like x2 + y2 = 4, or the equation y = x2 + 1 depicts a James Watt's parallel motion linkage (1784) parabola.is an example of an element moving along a givenpath, particularly along a straight line. It is still Polynomial equations with higher degreesused in the rear axle of some car suspensions than quadratic equations are rarely discussed in(Figure 1), among others. schools, but the idea of defining an arbitrary alge- braic curve (including this one of degree 6) is not Despite Watt being very proud of his inven- difficult. By utilizing a dynamic geometry softwaretion (Bryant & Sangwin, 2008), it can be verified system, like GeoGebra to sketch this model, it isby various methods that the center of the short ver- surprisingly easy using the following instructions:tical rod moves along just on an approximately • Create points A and Cstraight line segment. The correct motion is a part • Define the segments f = 4 and g = 2of a lemniscate that can be described by a sextic • Draw a circle with center A and radius fpolynomial equation, that is, of order 6. For exam- • Attach a point M on this circleple, when the left and right blue suspension points • Draw circles with center M and radius g, andFigure 1. Watt’s linkage used in a 1998 Ford Ranger EV center C and radius f(Wikipedia, 2018b).Virginia Mathematics Teacher vol. 44, no. 2 43

• Intersect these two circles by defining point F Figure 3. Algebraic and geometric form of a mathematical• Create the midpoint T of segment MF object in GeoGebra. The input (2x + y – 5)(x2 + y2 – 4) = 0When moving M on the first circle, the trace of was entered by the user. In the Algebra Window, an expand-point T can be obtained by issuing the GeoGebra ed form of the product is shown.command Locus(T,M). Lastly, the algebraic equa-tion of the movement (eventually extended by these two, we may claim that the union indeed con-some other points) can be computed by typing Lo- tains an exact line. It can be said that the equationcusEquation(T,M). of the union contains a linear factor, even if this fact cannot be easily obtained from the expanded Here we trust that the software tool reliably form of the product (Figure 3).computes the equation, that is, the drawn curve is asubset of the algebraic curve being computed. But It can be proven in general that a polynomi-should we believe that no parts of the Watt curve al curve f(x, y) = 0 consists of a line if and only ifare completely linear at all? We can recall some f(x, y) contains a linear factor. For a general proofcomputations on elementary algebra to prove that we refer to Bézout's theorem (Bézout, 1779; Bé-no section of a parabola can be linear: indirectly, zout theorem, 2017).let us assume that the parabola y = x2 has a smalllinear segment appearing somewhere. The exten- Let us continue our investigations on Watt'ssion of this segment can be expressed as a linear linkage. It is displayed by GeoGebra directly afterfunction y = ax + b. Since the whole part of the lin- computing its implicit algebraic equation. One ben-ear segment must overlap the parabola, the equa- efit of using dynamic geometry is the immediatetion y = x2 = ax + b must be valid for infinitely availability of changing the positions of A and C inmany cases. This yields x2 - ax - b = 0 for infinitely Figure 2, and the lengths f and g, to immediatelymany cases, but this is impossible because a quad- change the displayed curve and its equation, ac-ratic equation has only a finite number of zeros cordingly. Our students, the prospective innovators(here, just 2 or even less). Clearly, this contradic- of the future, should start playing with draggingtion can be applied for other polynomial functions and changing these objects to get first-hand experi-as well, not just for the case of a parabola. ence on how the algebraic curve changes when the parameters are different from the initial setup. Af- In general, however, a polynomial curve ter a few minutes of experiments, it may turn outcannot always be written in an explicit form y=f(x). that only by modifying these parameters, no linearEven for lines, sometimes it is not possible to ex- factors will appear. (In the software tool the factor-press them in an explicit form like that. The easiest ization will be performed automatically for the Lo-example is probably the equation of the y-axis, x=0, cusEquation command and a straight line will bebut all lines parallel to this are of the same type. On immediately shown if a linear factor is present.the other hand, by using an arbitrary polynomialequation, not just of explicit forms, we can extend 44our possibilities to describe multiple curves simul-taneously. For example, the implicit equation xy =0 depicts the union of two perpendicular lines.Here we can learn that the factors of the productxy, specifically x and y, separately define both ele-ments of the union. We can think of them as lineswith equations x = 0 and y = 0, that is, both axeswill appear here at the same time when using xy=0.To generalize this idea, we may consider the equa-tion (2x + y - 5)(x2 + y2 - 4) = 0, which describes theunion of a line and a circle, namely 2x + y - 5=0and x2 + y2 – 4 = 0. When considering the union ofVirginia Mathematics Teacher vol. 44, no. 2

(See Kovács, Recio, & Vélez, 2017, for more de- Figure 5. A simple compass constructed as a LEGO Technictails.) model. It may help to visualize the experiments by straight line. The models in Figure 6 describe theutilizing a non-electronic tool as well. A handy same algebraic curve, again a sextic, that does notway may be to use a LEGO Technic set of beams have a linear factor. While the left one has a self-and connector pegs. A compilation of LEGO parts collision and can produce two disjunct curves, athat consists of 6 kinds of beams, 2 kinds of con- seemingly straight and a seemingly circular one,nector pegs, and a pen refill (Table 1) can be com- the right model can draw the whole closed curve inponents to construct not only Watt's engine but oth- one movement. For further experiments one caner linkages as well. We will see how an exactly observe the Chebyshev linkage by having a closerstraight line can be constructed with them. look on it in Kovács & Kovács (2017a), by follow- ing the steps in GeoGebra on constructing the mod-Table 1. A compilation of LEGO Technic parts and a pen el virtually. Other methods are mentioned in Bryantrefill (Kovács & Kovács, 2017b). & Sangwin (2008, p. 28-32). For beginners, it may be easier to start with Finally, we show a possible LEGO Technica construction having less components than the one model to construct an exactly straight line (Figureshown in Figure 4. The simplest one, shown in Fig- 7). Without using computer algebra, however,ure 5, produces a circle. The red bar is intended tobe a fixed part held firmly on a sheet of paper bythe left hand. The pen refill is inserted in one of theconnector pegs, and while the red part is fixed, thepen refill will leave its trace on the paper during itsmovement with the right hand. By using the components of the set, furtherlinkages can be built. Two of them (Figure 6), im-portant historically, are Chebyshev's linkage andlambda mechanism that also produce an almostFigure 4. Watt’s engine as a LEGO Technic construction. Figure 6. Chebyshev’s linkage (1854) (top) and lambdaThe pen refill is to be inserted in the connector peg pin in the mechanism (1878) (bottom), and the curves they produce.middle of the white beam. (The sketches in this paper werecreated with the LEGO Digital Designer (LDD) software 45tool, version 4.3, available at http://ldd.lego.com.)Virginia Mathematics Teacher vol. 44, no. 2

Figure 7. A model of Hart’s inversor (1874). See also Ko- midpoint of H'1H),vács & Kovács, 2017b for more details. The pen refill must • (v7 - v3)2 + (v8 - v4)2 = 82 (because the length ofbe inserted in the middle of the white bar at the rear side. H'1I is 8),proving that the pen refill moves on a straight line • 2x = 8 - v5 + v7 and 2y = -v6 + v8 (because J isis somewhat difficult. By following the steps inFigure 8, we can obtain that there is a linear factor the midpoint of H'1I),of the algebraic curve being computed by the Lo- • (v5 - 4)2 + (v6 - 0) = 42 (since the length of BH iscusEquation command in GeoGebra. This factor,(2x - 11), is a divisor of a polynomial of degree 7 4),that is a solution of an equation system of several • (v3 - v5)2 + (v4 - v6)2 = 42 (since the length ofequations describing the geometric setup. H'1H is 4, too), and Like in Figure 8, we will assume A = (0, 0), • (v7 - (8 - v5))2 + (v8 - (-v6))2 = 42 (since the lengthB = (4, 0), C = (v1, v2), H'1 = (v3, v4), H = (v5, v6), I= (v7, v8), and J = (x, y). Clearly, H' = (8 - v5, -v6), of H'I is also 4).and the following nine equations also hold:• v12 + v22 = 42 (since C is lying on a circle with These equations are easy to follow for be- center A and radius 4), ginners, too, but eliminating all variables except x• v3 + v5 = 2v1 and v4 + v6 = 2v2 (since C is the and y is a difficult task without utilizing a comput- er. Luckily, for most computer algebra systems (including GeoGebra), this equation system is man- ageable; therefore the solution of the 7th degree is quickly achievable. An online GeoGebra worksheet with the full list of the required computations can be found at https://www.geogebra.org/m/f7x2Euj2. When assembling this linkage as a LEGO model, it turns out that the remaining polynomial of degree 6 describes another motion, a non-linear one, that can be realized by pushing the linkage over its extents, changing the antiparallelogram HH'IH'1 into a par- allelogram (Kovács & Kovács, 2017b). It seems a bit surprising why the blue lineFigure 8. A GeoGebra model of Hart’s inversor, also available online at https://www.geogebra.org/m/VzvHr3nU 46Virginia Mathematics Teacher vol. 44, no. 2

Figure 9. A short elementary proof that there exists at least Let us emphasize finally that there is a bigone point on the curve drawn by Hart’s inversor that lies on difference between approximately and trulythe line 2x - 11 = 0. straight. To an engineer it can mean the difference between success and failure—and to a car driverin Figure 8 crosses the x-axis at point (5.5, 0) per- between the peaceful error-free use and the annoy-pendicularly. For the skeptic, we show that in the ing regular technical mistakes.borderline case when H'1, C, H, B (=I), J, and H'are collinear, the x-coordinate of J is indeed 5.5, In this paper, we considered some novelimplied by the factor (2x - 11). Since triangle BAC ways that extend the classic mathematical approachis isosceles and BC = CH + HB = 2 + 4 = 6, the by using computer algebra techniques for computa-midpoint K of BC creates segments BK = CK = 3 tion and a LEGO Technic compilation for furtherunits (Figure 9). Let M and L be the perpendicular experimenting. Also, we discussed methods in dy-projections of H' and J on the x-axis, respectively. namic geometry to study the topic from both theSince AB = BH' and the angles ABK and H'BM are computational and the experimental views at theequal, and the angles AKB and BMH' are equal (90 same time. Lastly, a classic proof was given whichdegrees), the triangles ABK and H'BM are congru- covered some particular cases of our statement butent, hence BK = BM = 3 units. Since J is midpoint was still incomplete for the general study. In ourof BH', L is necessarily the midpoint of BM. There- opinion, all these approaches can work together asfore AL = 5.5 units. a successful combination to connect learners to solving real life problems, including still unsolved There is a similar borderline case when all ones.points in this short proof are mirrored about the x-axis. On the other hand, another borderline case Referencescan be checked when the consecutive collinearpoints are H, C, B (=H'1), H', J, and I. In that case, Bézout, É. (1779). Théorie générale des équationsin a similar manner, it follows that the x-coordinateof J is 5.5 again. Because of symmetry, we now algébriques. Paris, France: Impr. de P.D.have four examples that J lies on the line 2x - 11 =0. To avoid a hasty conclusion, we recall that a fi- Pierres.nite number of positive examples may still be in-sufficient to be convinced since a polynomial of nth Bézout's theorem. Retrieved December 4, 2017degree may indeed have n different intersectionpoints with the x-axis. Thus, in general, we may from https://en.wikipedia.org/wiki/need a more complete study of the mathematicalbackground of the motion. For some other details Bézout's_theoremwe refer to Recio, Kovács, & Vélez (2017). Bryant, J. & Sangwin, C. (2008). How round isVirginia Mathematics Teacher vol. 44, no. 2 your circle? Princeton, N.J.: Princeton Univer- sity Press. Kempe, A. B. (1877). How to draw a straight line: A lecture on linkages. London, England: Mac- millan. Kobel, A. (2008). Automated generation of Kempe linkages for algebraic curves in a dynamic ge- ometry system. (Unpublished Bachelor’s the- sis). University of Saarbrücken, Saarbrücken, Germany. Kovács, Z. & Kovács, B. (2017a, May). No, this is not a line! Paper presented at the STEM 2017 conference, Johannes Kepler University, Linz, Austria. Kovács, Z. & Kovács, B. (2017b). A compilation of LEGO Technic parts to support learning ex- periments on linkages. Retrieved December 4, 2017 from https://arxiv.org/abs/1712.00440 47

Kovács, Z., Recio, T. & Vélez, M. P. (2017, July). Zoltán Kovács Automated reasoning tools in GeoGebra: A tu- Assistant Professor torial. Paper presented at the International Con- Linz, Austria ference on Technology in Mathematics Teach- [email protected] ings 13, Institute Français de l’education, France.Recio, T., Kovács, Z. & Vélez, M. P. (2017, July). Reasoning on linkages. Paper presented at the GeoGebra Global Gathering, Linz, Austria.The Vision Board, LLC. (2017). What is STEAM. Retrieved December 4, 2017 from https:// educationcloset.com/steam/what-is-steam/Watt’s linkage. Retrieved December 4, 2017 from https://en.wikipedia.org/wiki/Watt's_linkageUpcoming Math CompetitionsName of Organization Website Dates of Competitions Math Con http://www.mathcon.org/ May 5, 2018 Math Counts May 12-15, 2018 http://www.mathcounts.org/programs/competition-series/ competition-faq The Math League http://www.mathleague.com Month depends on grade level SUM Dog Online http://www.sumdog.com/enter_contest/ Online competitionAmerican Regions Math League USA Math Talent Search http://mathleague.org/arml.php Register through your Regional Team http://www.usamts.org/TipsFAQ/U_Tips.php Online, month-long contest Mandelbrot TBA Continental Math League http://www.mandelbrot.org/ TBA https://www.cmleague.comVirginia Mathematics Teacher vol. 44, no. 2 48

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