Virginia Council of Teachers of Mathematics |www.vctm.orgVIRGINIA MATHEMATICSVol. 44, No. 1 TEACHERFall 2017 Community of Heroes!Virginia Mathematics Teacher vol. 44, no. 1
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 2050Virginia Council of Teachers of Mathematics Many Thanks to our Reviewers for Fall 2017 President: Jamey Lovin Carrie Case, Radford University Past President: Cathy Shelton Darryl Corey, Radford University Secretary: Kim Bender Betti Kreye, Virginia TechMembership Chair: Ruth Harbin-Miles Virginia Lewis, Longwood University Treasurer: Virginia Lewis John McGee, Radford University Webmaster: Ian Shenk Laura Moss, Radford UniversityNCTM Representative: Lisa Hall Andy Norton, Virginia TechElementary Representatives: Meghann Cope; Vicki Bohidar Matthew Reames, University of VirginiaMiddle School Representatives: Melanie Pruett; Skip Tyler Padhu Seshaiyer, George Mason University Secondary Representatives: Pat Gabriel; Lynn Reed Ryan Smith, Radford UniversityMath Specialist Representative: Spencer Jamieson Maria Timmerman, Longwood University 2 Year College: Joe Joyner Katy Ulrich, Virginia Tech4 Year College: Ann Wallace; Courtney Baker Jay Wilkins, Virginia TechVMT Editor in Chief: Dr. Agida Manizade Andrew Wynn, Virginia Commonwealth University VMT Associate Editor: Dr. Jean Mistele Special thanks to all reviewers! We truly appreciate your time and service.Virginia MathemaTthicissisTsueeachhadera v2o0%l. 4a4c,cenpot.an1ce rate The high-quality journal is only possible because of your dedication and 2 hard work.
Virginia Mathematics Teacher vol. 44, no. 1 Table of Contents: Featured Awards ................................................... 5 Note from the Editor.............................................. 6 Message from the President .................................. 7 Note from the VDOE ............................................. 8 Ideas for the K-6 Classroom ................................. 9 An Example of Nature’s Mathematics: The Rainbow............................................................... 12 Opportunities for K-12 Teachers ........................ 20 MathGirls ............................................................... 21 Math Jokes........................................................... 27 Grant Opportunities ............................................ 28 Busting Blockbusters ........................................... 28 NCTM Annual Conference .................................. 29 HEXA Challenge Fall 2017................................. 30 Teaching Dilemmas ............................................. 32 Solutions to Spring HEXA Challenge.................. 36 Upcoming Math Competitions............................. 40 Technology Review.............................................. 41 Unsolved Mathematical Mysteries ...................... 43 The Shape of Ordered Pairs ................................ 44 Call for Manuscripts ........................................... 51 Good Reads ......................................................... 52 Key to the Spring 2017 Puzzlemaker................... 53 VCTM Annual Conference .................................. 54 Puzzlemaker......................................................... 57 Conferences of Interest........................................ 58 3
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Featured Awards 5 Congratulations to the 2017 Winners of the William C. Lowry Mathematics Educator Award: Elementary School Awardee: Maria Swartzentruber, Rockingham County Middle School Awardee: Matthew Reames, Burgundy Farm Country Day School High School Awardee: Jillian Marballie, Montgomery County College Awardee: Andrew Wynn, Virginia State University Math Specialists Awardee: Tash Fitzgerald, Culpeper CountyVirginia Mathematics Teacher vol. 44, no. 1
Superheroes at Work:Note from the Editor Dr. Agida Manizade In this issue, we are celebrating the contri- engage on a daily basis while dealing with pres-butions by the community of superheroes, individ- sures of standardized testing, often the lack ofuals with extraordinary skills and abilities: our resources, and many other factors, not under theirmathematics teachers. Every day, this group of control, that can affect the teaching and learningexceptional professionals engage in incredible process.activities, such as Helping young mathematicians to develop deep These superheroes also participate in ongo- ing professional development in order to continue conceptual understandings of the subject matter improving their own skills and abilities. We en- through hands-on experiences, as described in courage our readers to consider professional devel- Dana Johnson’s piece; opment (PD) opportunities discussed in this issue, Motivating students to do mathematics, regard- as well as grant opportunities that can fund addi- less of existing challenges, as explained by tional PD. Nicole Joseph’s research; Finding and analyzing great examples of math- Our mathematics teachers are phenomenal ematics in nature and the world around us, as individuals that continue their work and commit- presented by John Adam in his article about ment to excellence in teaching mathematics to our rainbows; children. We thank them for their invaluable ser- Designing high cognitive demand tasks, lessons vice in shaping future generations. and activities that promote mathematics dis- course in the classroom, similar to ideas by Agida Manizade Denise Wilkinson; and Editor in Chief, Inspiring children to see the beauty of mathe- Virginia Mathematics Teacher matics as well as developing their problem- [email protected] solving and critical thinking skills. 6 These are just a few examples of extraordi-nary activities in which our mathematics teachersVirginia Mathematics Teacher vol. 44, no. 1
Message from the President Jamey Lovin Yesterday my neighbor iate issue. Our issue? Providing services that attract told me that her son had and retain membership. How? Seeking a platform just finished his “What I that reflects the diverse needs of Virginia educa- Did This Summer” essay tors. We weren’t surprised to see many groups for the first day of school. sharing the same goal for their organization. In She mentioned he referred fact, Matt Larsen, in his July 19, 2017 NCTM Pres- to having done lots of math ident’s message, shared the following from the practice because THE very first NCTM president: “Unless we give value MATH LADY lived next received for the membership fee no one will be- door! I was pleased, not come a member a second time” - that was 1921 -only because he had been working to keep his math almost 100 years ago.skills sharp, but also because he had provided mewith the perfect topic for this message ... VCTM: In August, the Executive Board met to planWhat We Did This Summer! initiatives of “value” for educators in Virginia. We In July, NCTM hosted a Leadership Con- are very excited with the activities we have plannedference in Baltimore, Maryland. The theme was for this year and beyond. First on our planning listIntent to Impact: Addressing Access, Equity, and is the 2018 conference at Radford University. TheAdvocacy in Your Affiliate. Affiliate members theme will be Moving Mountains with Mathemat-from all over the United States discussed ways we ics. We will be inviting proposals that addresscan take action to provide a high-quality mathemat- four conference strands: teaching with the newics education for each and every learner. In a standards, access and equity, enhancing teachingpresentation from Marilyn Strutchens, we were and learning, and enhancing instruction with tech-challenged not to tell a different version of the nology. We are also in the process of revampingequity story, but to create one that was truly our our award and grant applications to make themown. The afternoon was filled with dynamic con- more accessible. We promise to keep you updatedversations that shaped our discussions for the re- about this and many more upcoming events spon-mainder of the conference. sored by VCTM and its affiliates. Check our web- site, www.vctm.org, and watch for email blasts and updates! Here’s to an awesome year!Figure 1. See https://www.storybasedstrategy.org/ Jamey Lovin, VCTM Presidentthe4box.html for information about this graphic and more [email protected], equity, and advocacy conversations. 7 Cathy Blair, Chelsea Prue, Melissa Demlein(GRCTM), Lisa LoConte-Allen (TCTM), and Iformulated an action plan to address our state Affil-Virginia Mathematics Teacher vol. 44, no. 1
Note from the Virginia Department of Education:Teaching Mathematics During the 2017-2018 Crosswalk Year Tina Mazzacane During the 2017-2018 school year, Virginia 2009 to 2016 Crosswalk (summary of revi-mathematics teachers will include the 2009 and sions) documents2016 Mathematics Standards of Learning in boththe written and taught curricula. This is known as a 2016 Mathematics SOL Video Playlist“crosswalk” year and requires curricula to blend (Overview, Vertical Progression & Support,two sets of standards, leading to full implementa- Implementation and Resources)tion of the 2016 Mathematics Standards of Learn-ing in 2018-2019. The chart below summarizes the Narrated Crosswalk Presentationsinstructional and assessment implications for im- Teachers may also find resources from theplementation of the 2016 Mathematics Standards ofLearning: 2017 Mathematics SOL Institutes useful in plan- ning instruction during the Crosswalk Year.2017-2018 School Year – Crosswalk Year 2009 Mathematics Standards of Learning and 2016 The Virginia Department of Education is currently working to revise the following resources Mathematics Standards of Learning are included in the and anticipates posting updated electronic versions written and taught curricula. on the 2016 Mathematics Standards of Learning Fall 2017 Standards of Learning Assessments measure (SOL) and Testing webpage in 2018 as the work is the 2009 Mathematics Standards of Learning, but will finalized: not include field test items measuring the 2016 Mathe- matics Standards of Learning. Updated and New Lesson Plans (Enhanced Spring 2018 Standards of Learning assessments meas- Scope and Sequence Lessons) ure the 2009 Mathematics Standards of Learning and will include field test items measuring the 2016 Mathe- Updated Vocabulary Word Wall Cards matics Standards of Learning. Virginia Board of Education Textbook Approv-2018-2019 School Year – Full-Implementation Year al List Written and taught curricula reflect the 2016 Mathemat- Instructional Video Resources for Teachers ics Standards of Learning. Educators with questions about the implementation Fall 2018 End-of-Course (Algebra I, Geometry, and of the 2016 Mathematics Standards of Learning should email [email protected]. Algebra 2) and Spring 2019 (Grades 3-8 and EOC) Standards of Learning assessments measure the 2016 Tina Mazzacane Mathematics Standards of Learning. Mathematics Coordinator Virginia Department of Education The Virginia Department of Educationprovides several resources to assist teachers as theynavigate the crosswalk year. The following re-sources are currently available on the 2016 Mathe-matics Standards of Learning (SOL) and Testingwebpage: 2016 Mathematics Standards of Learning 2016 Mathematics Standards Curriculum FrameworksVirginia Mathematics Teacher vol. 44, no. 1 8
Ideas for the K-6 Classroom:The Game of Krypto to Support Number Sense Dana T. Johnson The purpose of this article is to share a can be found through internet search terms “NCTMteaching activity that is useful and powerful for and Krypto.”instruction, motivation of students, and enrichmentin math classrooms for grades 3 – 8. It is also a Mental arithmetic is the preferred methodgreat game to teach to families. of solving the hands. Some students may be al- lowed to work with paper and pencil. For young The game ofKrypto is a card game students or those whothat consists of 56 need a more concretecards. In the deck there approach, you mayare three each of the have them write the sixnumbers 1 – 6, four numbers on a long stripeach of 7 – 10, two each of scrap paper. Thenof 11 – 17, and one they tear the numberseach of 18 – 25. Play apart, thus creating theirbegins by dealing a set own set of mini-cardsof six numbers. The for the hand. Somefirst five are combined students are more suc-in any order along with cessful when they canany of the operations +, physically rearrange the–, x, or ÷ to obtain a numbers.result equal to the sixthnumber (called the When you workobjective or target num- with young students,ber). you may want to use only the numbers 1-10. For example, suppose the numbers dealt are For primary grade stu-20, 15, 17, 3, 9, and 4. Here is a solution: 20 ÷ [(17 dents, you may allow them to solve the hand using-15) + (9 ÷ 3)] = 4. This game is similar to the fewer than all five numbers. For example, supposegame called “24,” which uses four numbers that are the numbers dealt are 4, 3, 6, 8, 1 with an objectiveprinted on a card to get the objective number 24. number of 7. Students may find solutions such as: 2-number solutions: 4 + 3 = 7 In Krypto, the purpose of the cards is to 6+1=7generate numbers for use in the game. If you do not 8-1=7have cards in your classroom and you are playing 3-number solution: 4 + 6 – 3 = 7as a whole-class activity, you may simply ask six 4-number solution: 6 ÷ 3 x 4 – 1 = 7students to choose a number between 1 and 25. 5- number solution: 6 ÷ (8 ÷ 4) + 3 + 1 = 7Write the numbers on the board and have everyone This strategy allows students to differentiate thework to find a solution. When a student finds a activity for their own level of comfort.solution, s/he calls “Krypto” and explains it to the Over the last few decades I have played thisclass. Students may also play in small groups with game with students in grades 3 to 12. No one hasa deck of cards. The game is also an excellent soli- ever cared much about scoring. The satisfactiontaire game. The National Council of Teachers of seems to be in finding a solution or seeing someoneMathematics has an online version of the game that else find one. Sometimes several students shareVirginia Mathematics Teacher vol. 44, no. 1 9
different solutions and no one seems to care who Order of operations. Ask students to wr itegets points. But if you want to score, small groups their solutions in correct notation, using rulescan give one point to the first person to solve the for order of operations or “algebraic logic.”hand correctly. I use a fun scoring method for Once a student writes a solution, others canwhole-class teams – I divide the class in half and check. For example, if one student incorrectlywrite the numbers for each hand on the board. The writes 2 + 3 x 6 – (7 + 3) = 20 then othersfirst person with a correct solution earns many should note that parentheses are requiredpoints for the team’s score – the sum of the six around 2 + 3.numbers! If they call “Krypto” but cannot producea correct solution, they have the sum of the six Commutative and Associative Properties. Innumbers SUBTRACTED from their team’s score. comparing solutions, students will see thatThis minimizes impulsive, false claims. variations in grouping and order may produce the same result. For example, one student may There are many possible benefits to playing write (3 + 2) x 1 + 4 + 5 = 14. Another maythis game in your classroom. May (1995) enthusi- claim to have a different solution: 1 x (2 + 3) +astically describes and recommends the game of 4 + 5 = 14. Students should recognize that twoKrypto in an article on motivating activities for the instances of commutative property are used –math classroom. Way (2011) describes additional one for addition and one for multiplication.benefits of games to support mathematical cogni- This can lead to a good discussion. Some stu-tive objectives, including application of math skills dents will say it is really the same solution andin a context that is meaningful to students, building others will say they are different solutions.of positive attitudes towards math, increased skilllevels, opportunities for students to participate at Multiplicative Property of Zero. If you canvarious levels of thinking, and opportunities to make a zero from two of the numbers, you canconnect with families as students share the games eliminate other numbers that you don’t need.at home. Example: 6 4 3 6 22 Lach and Sakshaug (2005) discuss their Objective number is 7.action research on games in a sixth grade class- Solution: (3 + 4) + (6 – 6) x 22 = 7room. Two of them, Muggins and 24, are similar toKrypto. After 12 weeks of playing math games the Identity elements. If you can get the objec-authors found students scored better in an assess- tive number with one or two numbers, then tryment of algebraic reasoning. to get a zero or 1 with the remaining numbers. Multiply by one or add zero. The game of Krypto does not present factsin the way flash cards do, but incorporates problem Example: 7 9 2 15 20solving and pattern searching into fact practice. Objective number is 8.Beyond the obvious practice in mental arithmeticand developing number sense, it can be an environ- Notice that 15 – 7 = 8. Can you get a 1ment for applying properties of real numbers and from the other three numbers? Yes, 20 ÷ 2 -the rules for order of operations. This game pro- 9. So the identity element for multiplicationmotes the kind of number juggling used in factor- helps with a solution ofing quadratic trinomials. (15 – 7)(20 ÷ 2 – 9) = 8 or 8 x 1 = 8. Here are some examples: Example: 2 4 5 3 7 Factoring quadratic trinomials. When we Objective number is 5. factor x2 – 8x + 12 we are looking for numbers Notice that you are given a 5 in the middle whose product is 12 and whose sum is -8. of the set of five numbers. Can you get a Once when I was teaching factoring to an 8th grade algebra class, a student blurted out, “It’s 10 easy. It’s just like Krypto!”Virginia Mathematics Teacher vol. 44, no. 1
zero from the other four numbers? One Way, J. (2011). Learning mathematics throughway is 2 [(4 + 3) – 7] = 0. The solution be- games. Series 1: Why games? Retrievedcomes from http://nrich.maths.org/2489.5 + 2 [(4 + 3) – 7] = 5 or 5 + 0 = 5. Use these five numbers to get each of theIs it always possible to find a solution using objectives listed on the left.all five numbers to get the objective? It is rare, but Use correct Order of Operations symbols! Two of them are done for you.sometimes there is no solution. Here are the only 3 6 8 25 22two examples I have encountered in the 45 years I 1= 2 = (25 – 22) ÷ 3 x (8 – 6)have been playing the game. 3= 4 = 22 ÷ (25 – 14) + 8 – 63, 23, 13, 16, 1 Objective = 20 5= 6=9, 9, 7, 16, 4 Objective = 25 7= 8= If you search the internet for a “Krypto 9=solver” you will find some sites that will check 10 =your hand for you. If it is solvable, they do it for ...you. If not, they will tell you it is unsolvable. 23 =Whenever we hit a tough hand in a classroom that 24 =no one seems to be able to solve, I write it on the 25 =corner of the board. By the next day, it alwaysseems to be solved by someone! Dana T. Johnson Retired Faculty As a challenge for middle school students College of William and Marywho seem adept at this game, I give five numbers [email protected] no objective. They write the five numbers at thetop of a piece of lined notebook paper. They write 11the numbers 1-25 down the margin on the left. Thisgives them 25 possible objectives. An example isgiven in the next column. This activity can be doneby individuals or groups. It is an excellent settingto motivate the use of order of operations notation. The game of Krypto can also be used as aclassroom management activity. I use it to fill thelast minute or two before dismissal. I tell studentsto cross their arms when they are ready to leave. Iquickly write the numbers on the board and askthem to solve the hand using only mental arithme-tic. They stay focused and quiet while thinking. If itis solved before the bell, I generate a new hand.This strategy helps make every minute count inmath class!ReferencesLach, T. and Sakshaug, L. (2005). Let’s do math: Wanna play? Mathematics Teaching in the Middle School, 11(4), 172-176.May, L. (1995). Motivating activities, Teaching PreK-8, 26(1), 26-27.Virginia Mathematics Teacher vol. 44, no. 1
An Example of Nature’s Mathematics: The Rainbow John A. AdamIntroduction. with the horizontal, just as the direction to the top It is the author’s contention that ‘nature’ is of a tree makes an angle with the direction of its shadow on the ground (in fact that angle is exactlya wonderful resource and vehicle for teaching stu- the solar altitude!). By making a large paper conedents at all levels about mathematics, be it qualita- to mimic the ‘rainbow cone’ and varying the angletively at elementary schools (shapes, circular arcs, at which students hold it, (Figure 3), they will seepolygonal patterns) or more quantitatively at mid- that if the sun is very low (i.e. close to the horizon),dle and high schools (geometrical concepts, alge- then the rainbow arc is almost a complete semicir-bra, trigonometry and calculus of a single variable). cle, whereas if the sun is too high (altitude greaterThis was the motivation for writing A Mathemati- than 42o), then the top of the rainbow is below thecal Nature Walk (as well as the somewhat more horizon and therefore not visible (unless the ob-advanced Mathematics in Nature). Within the server is on a hill or in flight; see http://realm of nature the subject of meteorological optics www.slate.com/content/dam/slate/blogs/is a particularly fascinating one; it includes the bad_astronomy/2014/09/01/study of the rainbow as well as others such as ice circular_rainbow.jpg.CROP.original-original.jpg).crystal halos and glories. Obviously there is some If the student (or anyone!) is fortunate enough tophysics involved in the explanation of these phe- see a nearly semicircular rainbow, then the anglenomena, but fortunately it is not necessary to go between the two ‘ends’ of the rainbow and theinto a lot of physical detail in order to appreciate observer – its ‘angular diameter’ – is twice 42o,the value of geometry, trigonometry and high- which is not far from a right angle!school calculus concepts used in modeling thebeautiful rainbow arcs in the sky. What about middle-school students? In the summer of 2015 I was privileged to teach a dozen For students in elementary school there is a specially selected 6th – 8th grade students in thevariety of angle-based concepts that can be ad- Virginia STEAM Academy at Old Dominion Uni-dressed when discussing rainbows. Thus, ‘solar versity. The acronym refers to Science, Technolo-altitude’ is the angle the direction to the sun makes 12Virginia Mathematics Teacher vol. 44, no. 1
gy, Engineering and Applied Mathematics. The in terms of the angles of incidence (i) and reflec-topics covered included rainbows, ice crystal halos, tion (r) respectively (see Figure 1 where the inci-water waves, glitter paths and sunbeams; addition- dent ray is refracted and reflected inside the spheri-ally, the topics ‘Guesstimation’ (i.e. back-of-the- cal drop; Figure 2 illustrates the ray path for theenvelope problems that require estimation) and secondary bow). But what is this angle? Essentially‘dimensional analysis’ (i.e. what happens as things it is the direction through which an incoming rayget bigger?) were incorporated into the week-long from the sun is ‘bent’ by its interaction with theclass. Given that the mathematical background of drop to reach the observer’s eye (the reader is re-these students included algebra, geometry and ferred to the caption for Figure 1 for more details).trigonometry, much of the material discussed inthis article was covered, and the results from the The angle of refraction inside the drop is acalculus-based topics were presented qualitatively function of the angle of incidence of the incoming(and very successfully) by engaging the students on ray. This relationship is being expressed in terms oftheir understanding of maxima and minima, and Snell's famous law of refraction, namely sini =applying those ideas in this context. nsinr, where n is the relative index of refraction (ofDoing the mathematics. water, in this case). This relative index is defined as the ratio of the speed of light in medium I (air) The primary rainbow is caused by light to the speed of light in medium II (water); note thatfrom the sun entering the observer's eye after it has n > 1; in fact n ≈ 4/3 for the rainbow, but it doesundergone one reflection and two refractions in depend slightly on wavelength (this is the phenom-myriads of raindrops. An additional internal reflec- enon of dispersion, and without it we would onlytion produces a frequently-observed secondary have bright ‘whitebows’!). The article by Austin &bow, and so forth (but tertiary and higher bows are Dunning (1991) provides a helpful summary of therarely, if ever, seen with the naked eye for reasons ‘calculus of rainbows,’ as does the even brieferdiscussed below). By adding all the contributions ‘Applied Project’ in Chapter 4 of Stewart (1998).to angular deviations of the ray from its originaldirection, the middle- or high-school student can In view of Snell’s law the high school stu-verify that for a primary bow the ray undergoes a dent should attempt to write the angle of refractiontotal deviation of D(i) radians, where r in terms of the angle of incidence i using the inverse sine function, thus: D i 2i r 2r 2i 4r i, (1) r arcsin sin i . (2) n Hence equation (1) may be rewritten as D i 2i 4 arcsin sin i . (3) n Figure 1. The path of a ray inside a spherical raindrop Figure 2. The corresponding ray path for the secondary which, along with myriads of other such drops, contributes rainbow, arising because of a second reflection within the to the formation of a primary rainbow (k = 1). The devia- raindrops. tion angle D(i) referred to in the text (see equation (1)) is the obtuse angle between the extension of the horizontal 13 ray from the sun and the extension of the ray entering the observer’s eye. Its value is approximately 138o. Its supple- ment, 42o, is the semi-angle of the ‘rainbow cone’ in Figure 3.Virginia Mathematics Teacher vol. 44, no. 1
By examining the graph of D(i) in Figure 4 the expressionit is seen that for ii [00,, π/ 2/2] (which is the only D i 2i r k 2r k 2i 2k 1 r iinterval of interest for physical reasons), the condi- = k 2i 2k 1 arcsin sin i . (9)tion for an extremum (minimum in this case) im- n plies there exists a critical angle of incidence icsuch that D'(ic) = 0. To prove this the student may Note that the result in equation (9) is modulo 2π.either use implicit differentiation of equation (1)(with subsequent use of Snell’s law) to obtain Although realistically k ≤ 2 (see below for details), with k internal reflections the corresponding result dr cos i , (4) for the critical angle of incidence that gives rise to di n cos r the minimum deviation isor directly differentiate the expression (3) and ic arccos n2 1 1/ 2 . (10) k(k 2) equate it to zero to find the critical angle ic from the resulting expression below, i.e. This result is established using exactly the same 1 cos2 i 4 1 cos2 , (5) method to arrive at equation (6). For the primary n2 i bow (k = 1) this reduces (as it should) to equationfrom which it can be found that (6) above. Additionally, equation (9) reduces to n2 1 1/2 equation (3) for k = 1 since k(k + 2) = 3. i ic arccos 3 . (6) It is an interesting trigonometric exercise to eliminate all dependence on the angle of incidenceThus, with a generic value for n of 4/3, ic ≈ 1.04 (as Kepler did in 1652) to prove from equation (8)radians, or about 59.4o for the primary bow. that As noted above this extremum is a mini- 1 4 n2 3/2 n2 3 .mum, i.e. D\"(ic) > 0, as can be shown by differenti- D ic 2 arccos (11)ating the expression (1) a second time. In fact, by noting from equation (1) that D\"(i) = –4r\"(i) and To achieve this, rewrite equation (8) asutilizing equation (4) it follows that (after some D(ic ) n2 1 1/ 2 4 n2 1/ 2 2 3 3n2 algebraic manipulation) arccos 2 arcsin A 2B d 2r n2 1 sin i 0, (7) ≡ A – 2B, (12). n3 cos3 r Then, by expanding the equation di2So, D\"(ic) > 0 as indicated. Note that in this in- sin D(ic ) sin( A 2B),stance it was not necessary to specify ic so the re- 2 sult is a global one, i.e. the concavity of the graphof D(i) does not change in [0, π / 2], the interval of it is possible to write cos[D(ic)] in terms of sin A , cos A , sin B and cos B, each of which can be foundphysical interest.Exercise for the student: Using equations (3) easily from the definitions of A and B in equationand (6) show that the minimum angle of deviation (12). The result is a rather nasty expression which(the ‘rainbow angle’) is can be reduced algebraically to equation (11).D(ic ) 2 arccos n2 1 1/2 4 arcsin 4 n2 1/ 2 . (8) Voilà! This has been generalized to higher-order 3n2 3 bows (see Adam 2008), but it would take us too far Each internal reflection adds an amount of afield to describe here; essentially the same ideasπ - 2r radians to the total deviation of the incident are involved.ray. Thus, for k internal reflections within a Some numerical values.raindrop, a term k(π - 2r) is added to the angle Thus far, we have been describing a gener-through which an incident ray is deviated, (see ic, colorless type of rainbow. For a ‘generic’ mono-Figure 2, for the secondary bow, k = 2), yielding chromatic rainbow (the ‘whitebow’ referred toVirginia Mathematics Teacher vol. 44, no. 1 14
above), the choice n = 4/3 yields, from expression ly wider) dispersion occurs for the secondary bow,(11), but the additional reflection reverses the sequence of colors, so the red color in this bow is on theD ic D arccos 9 20 3/2 138 . (13) inside edge of the arc. In principle more than two 16 27 internal reflections may take place inside each raindrop, so higher-order rainbows, i.e. tertiary (k = 3), quaternary, (k = 4) etc., are possible. Each addi-The supplement of this angle (≈ 42o) is the semi- tional reflection of course is accompanied by a loss of light intensity because of transmission out of theangle of the rainbow ‘cone’ formed with apex at drop at that point, so on these grounds alone, it would be expected that even the tertiary rainbow (kthe observer's eye, the axis being along the line = 3) would be difficult to observe or photograph without sophisticated equipment; however recentlyjoining the sun to the eye, extended to the antisolar several orders beyond the secondary have been identified and photographed (see the cited articlespoint (see Figure 3). by Edens, and Edens & Können). The reader’s attention is also drawn to the superb website onFigure 3. The ‘rainbow cone’ for the primary rainbow. For atmospheric optics, in particular the following link:the secondary bow (k = 2) the cone semi-angle is approxi- http://www.atoptics.co.uk/rainbows/ord34.htm.mately 51o, as may be calculated from equations (9) and(10). It is possible to derive the angular size of such a rainbow after any given number of reflec- So what happened to the colors of the rain- tions using equations (9) and (10) (Newton was thebow? They have of course been there all along, and first to do this). Newton’s contemporary, Edmundall we need to do is to utilize the fact that the re- Halley, noted that the third rainbow arc shouldfractive index n is slightly different for each wave- appear as a circle of angular radius nearly 40olength of light. Blue and violet light get refracted around the sun itself. The fact that the sky back-more than red light; the actual amount depends onthe index of refraction of the raindrop, and the Figure 4. The graph of the deviation angle D(i) for the pri-calculations thereof vary a little in the literature, mary bow from equation (3) as a function of the angle ofbecause the wavelengths chosen for red and violet incidence (both in radians). Note that the minimum devia-may differ slightly. Thus, for red light with a wave- tion of approximately 2.4 radians (≈ 138o) occurs wherelength of 656 nm (1 nm = 10–9 m), the cone semi- the critical angle of incidence ic ≈ 1.04 radians (≈ 59.4o).angle is about 42.3o, whereas for violet light of 405 These values may be calculated directly using equationsnm wavelength, the cone semi-angle is about 40.6o (3) and (6). The graph shows that above and below ic therean angular spread of about 1.7o for the primary are rays deviated by the same amount (via the horizontalbow. (This is more than three times the angular line test), indicating that at ic these two rays coalesce towidth of a full moon!) The corresponding values of produce the region of high intensity we call the rainbow.the refractive index differ very slightly: n ≈ 1.3318 15for the red light and n ≈ 1.3435 for the violet – lessthan a one percent increase! Similar (though slight-Virginia Mathematics Teacher vol. 44, no. 1
ground is so bright in this vicinity, coupled with Note that in the list of topics below eachthe intrinsic faintness of the bow itself, would meteorological phenomenon can be examined as amake such a bow almost, if not, impossible to see topic in mathematical physics because the subjector find without sophisticated optical equipment. of optics is very mathematical. At times, it requiredExercise for the student: Use the gener ic value very sophisticated mathematics. The author recom-for the refractive index of water, n = 4/3, in equa- mends another enrichment activity in which stu-tions (9) and (10) to show for the tertiary rainbow dents search for each of the topics (and others)(k = 3) that ic ≈ 70.6o and D(ic) ≈ 321o, so the below on the ‘Atmospheric Optics’ website men-‘bow’ is at about an angle of 39o from the incident tioned above: http://atoptics.co.uk/. There is a vastlight direction. In fact, this will appear behind the selection of topics (with many photographs) toobserver as a ring around the sun! choose from, including shadows, ice crystal halosExercise for the student: Calculate the angular around the sun or moon, ‘sundogs,’ reflections,width subtended at the eye by a ‘baby aspirin’ held mirages, coronas, glories, sun pillars as well as, ofat arm’s length. Then see if you can ‘cover’ the full course, rainbows. The advantage of this site (andmoon by extending your arm while holding the its ‘sister’ site, Optics Picture of the Day (OPOD:aspirin! http://atoptics.co.uk/opod.htm)) is that students atAn experiment: “road-bows.” all levels, elementary, middle and high school, will be able to find material of interest to them. These Have you ever noticed a rainbow-like re- sites are replete with straightforward physical ex-flection from a road sign when you walk or drive planations and illustrations of the phenomena, butby it during the day? Tiny, highly reflective there is little, if any, mathematical discussion sospheres are used in road signs, sometimes mixed in they can be appreciated in a scientifically accuratepaint, or sometimes sprayed on the sign. Occasion- way by students at any level of mathematical profi-ally, after a new sign has been erected, quantities ciency. The book A Mathematical Nature W alk,of such ‘microspheres’ can be found on the road together with the more advanced Mathematics innear the sign (see http://apod.nasa.gov/apod/ Nature cited in the bibliography, can provide aap040913.html for an excellent picture of such a starting point for both teachers and students inter-bow). I have had my attention drawn to such a find ested in pursuing some of the mathematical aspectsby an observant student! It is possible to get sam- of these phenomena. As a further example, a veryples of these tiny spheres directly from the manu- brief description of glories (with an associatedfacturers, and reproduce some of the reflective ‘student teaser’) is provided below.phenomena associated with them. In particular, forglass spheres with refractive index n ≈ 1.51 scat- Although ice-crystal halos are only brieflytered uniformly over a dark matte plane surface, a mentioned in the preceding paragraph, students atsmall bright penlight provides the opportunity to all levels can be encouraged to look for them.see a beautiful near-circular bow with an angular These can appear around the sun or full moon withradius of about 22o (almost half that of an atmos- surprising frequency (though it must be empha-pheric rainbow). In such an experiment this bow sized again that you should never look directly atappears to be suspended above the plane as a result the sun; block it off with your hand or a convenientof the stereoscopic effects because the observer’s chimney!). They are formed by sunlight passingeyes are so close (relatively) to the spheres com- through myriads randomly oriented, nearly regularpared with passing several yards from a road sign. hexagonal prismatic ice crystals, composing cirrusMore details of the mathematics can be found in clouds, the very highest type of cloud we generallythe article by Crawford (1998) and Chapter 20 of see (during the day at least). In distinction to rain-Adam (2012). bows, the most common halos are smaller in angu- lar radius (about 22o as opposed to 42o) and exhibitRelated topics in meteorological optics. a reddish tinge on the inside of the arc – a reversal of colors compared with the primary bow. This isVirginia Mathematics Teacher vol. 44, no. 1 16
because, unlike the mechanism producing the pri- look) from airplanes. This is the meteorologicalmary bow, there is no reflection occurring within optics phenomenon known as a glory. Cloud drop-the crystals to produce these particular halos, only lets essentially ‘backscatter’ sunlight back towardsrefraction. I live about a mile from Old Dominion the observer in a mechanism similar in part to thatUniversity and walk to my office; as a result I gen- for the rainbow. The glory, it is sometimeserally see such halos (and other types also) several claimed, is formed as a result of a ray of light tan-times a month; sadly, far more frequently than I gentially incident on a spherical raindrop beingwitness rainbows. On an otherwise clear night, a refracted into the drop, reflected from the backfull moon embedded in a thin cirrus cloud may surface and reemerging from the drop in an exactlyexhibit similar such halos, which can be quite antiparallel direction (i.e. 180o) into the eye of theprominent by virtue of the moon being so much observer, but this is actually incorrect (see theless bright than the sun. Indeed, I have frequently ‘student teaser’ below).been contacted by friends and students who witness Student teaser: Why is the r ay path allegedlythe latter but have never noticed a halo around the associated with the formation of a ‘glory’ as illus-sun! trated in Figure 5 (and in some meteorology text-Exercise for the student: Imagine a r egular books) incorrect? Use equation (3) to investigatehexagonal prism with a light ray entering side ‘1’, this.and exiting side ‘3’ (see the A tmospheric Opticswebsite for more details and its interactive ‘mouse’ Figure 5. An incorrect ray path explanation for the glory.tasks to discover the minimum angle of deviationfor both rainbows and halos). Using the same geo- Conclusion.metric, trigonometric and calculus concepts applied This article presents some of the basicto rainbows in the body of the article, show that theminimum angle of deviation for such rays is about mathematical concepts and techniques undergird-22o, the angular radius of the most commonly visi- ing a relatively common (and beautiful) phenome-ble halo. non in meteorological optics. The analysis present-Student teaser: When I lived in England I saw ed here does not contain new mathematics; it canmany more rainbows than I do living in Norfolk, be found from many sources because the subject ofVirginia. Why do you think this was? (No, it was meteorological optics has been around for a verynot because the annual rainfall where I lived was long time! What is emphasized, however, is themore than it is in Norfolk – in fact it’s rather less!). presentation of these ideas as a potential enrich-Think about latitude: I lived at about 52oN; now I ment topic for (i) ‘qualitative’ mathematical model-live at about 37oN, fifteen degrees further south. ing in elementary classrooms and (ii) more quanti-(You can imagine how excited I was to see the tative approaches in middle and high school class-constellation of Orion and Sirius (the ‘Dog star’) so rooms. It should also be noted that the many moremuch higher in the winter night sky than when I subtle features associated with these and otherlived in England!) optical effects in the atmosphere require far moreGlories. powerful and sophisticated mathematical tools to explain them. Nevertheless (though space does not Mountaineers and hill climbers have no- permit it), aspects of the above-mentioned phenom-ticed on occasion that when they stand with their ena of ice crystal halos and glories may also bebacks to the low-lying sun and look into a thick discussed at the level presented here. More detailsmist below them, they may see a set of coloredconcentric circular rings (or arcs thereof) surround- 17ing the shadow of their heads. Although an individ-ual may see the shadow of a companion, the ob-server will see the rings only around his or herhead. They may also be seen (if you know where toVirginia Mathematics Teacher vol. 44, no. 1
may be found in the references listed. It is hoped natural fifth-order rainbow, A pplied Opticsthat this article will also ‘whet’ the appetite of 54, B26–B34.interested instructors and students to pursue these Edens, H. E. and Können, G. P. (2015). Probableaspects in more detail. photographic detection of the natural seventh-order rainbow, A pplied Optics 54, A further suggestion may be made. The B93–B96.website Earth Science Picture of the Day (EPOD: Stewart, J. (1998) Calculus: Concepts and Contextsepod.usra.edu), which is a service of the Universi- (Single variable). Pacific Grove, CA:ties Space Research Association (USRA), publish- Brookes/Cole Publishing Company.es photographs from a variety of subject areas: Appendix: More mathematical patterns in na-geology, oceanography, space physics, meteorolog- ture.ical optics, agriculture, and many more. Anyone is What follows below is obviously only ainvited to submit their photograph of an interesting partial list of patterns that the attentive observeroptical or geological phenomenon, and is encour- might see on a “nature walk,” and could form theaged to write a short summary for the layman ex- basis of enrichment activities at all levels of studentplaining the picture and, where possible, the basic exposure to mathematics. The elementary, middlescience behind it. A recent submission by the au- or high school teacher could adapt the material forthor (August 15th, 2016), for example, uses simple his or her own students. Here are some possibleproportion to estimate the height of a tree canopy topics:using the ‘pinhole’ elliptical patches of light cast Basic two dimensional geometric shapeson the ground by gaps in the leaves of the tree that occur (approximately) in nature can be identi-(http://epod.usra.edu/blog/2016/08/estimating-tree- fied:height-using-natural-pinhole-cameras.html). For a Waves on the surfaces of ponds or puddlespropos that is the topic of this article, see http:// expand as circles;epod.usra.edu/blog/2017/07/streaky-rainbow-in- Ice crystal halos commonly visible around thezion-national-park.html. The site provides useful sun are generally circular;educational links for the daily pictures and is a Rainbows have the shape of circular arcs (asvaluable resource for teachers and students alike. noted already); Tree growth rings are almost circular.References But there are many other obviously non-circularAdam, J. A. (2006). Mathematics in Nature: Mod- and non-planar patterns: Hexagons: snowflakes generally possess hexag- eling Patterns in the Natural World. onal symmetry; Princeton, NJ: Princeton University Press. Pinecones, sunflowers and daisies (amongstAdam, J. A. (2008). Rainbows, Geometrical Opt- other flora) have spiral patterns associated ics, and a Generalization of a result of with the well-known Fibonacci sequence; Huygens, A pplied Optics, 47, H11 - H13. Ponds, puddles and lakes give scenes of ap-Adam, J. A. (2009). A Mathematical Nature W alk. proximate reflection symmetry (depending on Princeton, NJ: Princeton University Press. the position of the observer);Adam, J. A. (2012). X and the City: Modeling Cross-sections of various fruits also exhibit Aspects of Urban Life. Princeton, NJ: interesting symmetries; Princeton University Press. Spider webs have polygonal, radial and spiral-Austin, J.D. and Dunning, F. B., (1991). Mathe- like features; matics of the rainbow. In A pplications of Long bendy grass has an approximately para- Secondary School Mathematics (Readings bolic shape; from the Mathematics Teacher), 271 – 275. Starfish (suitably arranged) exhibit pentagonalCrawford, F.S., (1988). “Rainbow dust.” American symmetry; Journal of Physics 56, 1006 – 1009.Edens, H. E. (2015). Photographic observation of a 18Virginia Mathematics Teacher vol. 44, no. 1
The raindrops that scatter \"rainbow\" light into applies in particular to the size of land animals; the the eye of an observer essentially lie on cones relationship of surface area to volume, and its im- with vertices at the eye (as discussed in this plications for the relative strength of animals. By article); considering (and constructing) cubes of various sizes, much insight can be gained about basic bio- Cloud patterns, mud cracks and also cracks on mechanics in the animal kingdom, and much fun tree bark can exhibit polygonal patterns; (and learning!) may be had by thinking about such questions as: Clouds can also form wavelike \"billow\" struc- Why King Kong could not really exist, and tures with well-defined wavelengths, just as Why elephants are not just large mice. with ripples that form around rocks in a swiftly flowing stream; Furthermore, simple ideas such as scale enable us to compare, at an elementary level, me- In three dimensions, snail shells and many tabolism and other biological features (such as seashells and curled-up leaves are helical in strength) in connection with pygmy shrews, hum- shape and tree trunks are approximately cylin- mingbirds, beetles, flies and other bugs, ants and drical. African elephants to name a few groups! In view of these patterns, even at an ele- Note: This appendix is adapted fr om A MATHEMATI-mentary level, many pedagogic mathematical in- CAL NATURE WALK by John A. Adam. Copyright © 2009vestigations can be developed to describe such by Princeton University Press. Reprinted by permission.patterns - for example estimation, measurement, Note: Figur es 1, 2, 4 and 5 ar e r epr oduced fr om Ageometry, functions, algebra, trigonometry and MATHEMATICAL NATURE WALK by John A. Adam.calculus of a single variable. Basic examples might Copyright © 2009 by Princeton University Press. Reprintedinclude: by permission. Figure 3 is reproduced from X AND THE The use of similar triangles and simple propor- CITY: MODELING ASPECTS OF URBAN LIFE by John A. Adam. Copyright © 2012 by Princeton University Press. tion; Reprinted by permission. A table of tangents to estimate the height of John A. Adam trees; Professor Measuring inaccessible horizontal distances Old Dominion University [email protected] using congruent triangles.Simple proportion can again be used in estimation The author would like to thank theproblems, such as: reviewers for their detailed and Finding the number of blades of grass in a constructive criticism that resulted in a much improved version of the certain area, or the number of leaves on a tree. article.More geometric ideas appear when studying topicssuch as: 19 The relationship between the branching of some plants, such as sneezewort (Achillea ptar- mica), and the Fibonacci sequence can be in- vestigated; The related \"golden angle\" can be studied, and its occurrence on many plants (such as laurel) investigated; The angles subtended by the fist, and the out- stretched hand, at arm's length can be estimat- ed and used to identify the location of \"sundogs\" (parhelia) and ice crystal halos on days with cirrus clouds near the sun. Consequences of \"the problem of scale\" andgeometric similarity can also be investigated. ThisVirginia Mathematics Teacher vol. 44, no. 1
Organization Membership Information 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 Current Virginia Council of Teachers of Mathematics Membership: 1400 MembersVirginia Mathematics Teacher vol. 44, no. 1 20
MathGirls:The Invisibility of Black Girls in Mathematics Nicole M. Joseph Partnering with students to support them in tween Black and White students, I wonder whatbecoming educated, productive, and responsible Virginia plans to do for its Black girls in particular.citizens is at the core of the Virginia Department of The 2015-2016 Virginia State Standards of Learn-Education’s mission, and I wonder how Black girls ing (SOL) Assessment data shows the followingare experiencing this mission in their K-12 trajecto- pass rates for Black girls from third- throughries. The 2015 NAEP mathematics data for Virgin- twelfth-grade (see Table 1).ia shows a 22-point and 30-point difference be-tween Black and White fourth- and eighth-graders So just what is the state of Black girls’respectively (National Center for Education Statis- experiences in mathematics in Virginia? It is diffi-tics). Although Board of Education President Billy cult to answer this question in part because mostK. Cannaday Jr. notes that the state board’s top states rarely disaggregate their assessment data atpriority is to continue to work toward narrowing intersections of race and gender, but also becauseand ultimately closing the achievement gap be- there is a limited focus on contextualized studies that examine Black girls and their mathematicsTable 1. 2015-2016 Virginia State Standards of Learning (SOL) Assessment Pass Rate for Black Girls, 3rd – 12th3rd 4th 5th 6th 7th 8th 9th 10th 11th 12thBlack Girls 64.48% 73% 69.9% 74.15% 62.18% 71.60% 79.97% 72.32% 69.94% 67.03% Source: Virginia SOL Assessment Build-A-Table (http://bi.virginia.gov/BuildATab/rdPage.aspx) 21Virginia Mathematics Teacher vol. 44, no. 1
Table 2. United States’ Mathematical Sciences Doctorates: Females Compared to White Males 2005-2014 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014Total (M/W) 540 583 645 671 788 863 849 852 912 948Total Women 152 151 193 218 235 245 230 224 242 254Black 9 5 5 11 16 9 9 10 6 9Hispanic or Latino 6 11 4 5 12 8 9 11 6 7Native American 0 0 1 0 1 0 0 0 11White 101 102 132 161 154 168 155 163 170 179Asian NA NA NA NA NA NA 38 22 34 32Asian or Pacific Islander 26 20 29 24 27 39 NA NA NA NAHawaiian/ Pacific Islander NA NA NA NA NA NA 0 0 00Other 10 13 22 17 25 21 18 15 22 22Two or More Races NA NA NA NA NA NA 1 4 34Total Men 388 432 452 453 553 618 619 628 670 694Source: W omen, Minorities, and Persons with Disabilities in Science and Engineering: 2017 (table 7-7)education experiences. It is not just a Virginia tions than their White and Latina female peersproblem; it is a national problem. Black girls’ ex- (Riegle-Crumb, Moore, & Ramos-Wada, 2011),periences in mathematics remain invisible and yet, few make it to the doctoral level. Table 2largely untheorized and this invisibility produces shows the number of mathematics doctoratesobscurity to most mathematics teachers; conse- awarded to American citizens over the last 10quently, program and learning design efforts re- years. What we notice is that over the last 10 years,main non-existent. Interventions that are put in White males earned roughly 75% of the mathemat-place are usually single-axis and assume that either ics doctorate degrees in comparison to all women.all girls or all African-American students have the We also notice that in 2014, all women, with ra-same needs. Intersectional interventions can pro- cialized women yielding eight percent, and Blackvide promise for Black girls and other racialized women roughly one percent, earned 27% of theminorities. Such interventions would not only take math doctorates.into account Black girls’ particular learning needsfor persistence in mathematics (Joseph, Hailu, & These numbers can be explained by severalBoston, 2017), they would also account for issues factors; here, I discuss a few. We know from thirtyof power, oppression, and center social justice years of research by Jeannie Oakes (1985, 1990a,(Collins & Bilge, 2016). 1990b) that mathematics tracking substantially limits African-American students’ access to ad- Understanding Black girls’ experiences in vanced mathematics courses. Mathematics trackingmathematics during their K-12 trajectories can shed is the process of organizing students into mathe-light on their underrepresentation in higher educa- matics courses that are based on their ability, and ittion majors and careers that require mathematics has been an accepted practice in U.S. schools fordegrees. For example, studies have shown that nearly a century (Rubin, 2008).Black girls have higher mathematics career aspira- Teachers’ low expectations and overallVirginia Mathematics Teacher vol. 44, no. 1 22
assumptions about Black girls in society impede High School Experiences in Mathematics Shapethe opportunity for Black girls to learn in mathe- Higher Education Success.matics classrooms (Rist, 2000). Teachers hold lowexpectations of low-income Black girls in upper I recently conducted a pilot study that ex-elementary classrooms who are perceived as hav- plored mathematics identity among undergraduateing limited knowledge and bring social challenges Black women STEM majors at different institutionto the learning environment (Pringle et al., 2012). types. Seven Black women participated in a three-Black girls’ early confidence in and value of math- part study that included a one-on-one semi-ematics often fails to translate when it comes to structured interview, the creation of an annotatedinteractions with their mathematics teachers. Battey symbolic artifact of mathematics identity affirma-and Levya (2013) found positive and negative tion, and a focus group based on the movie Hiddeneffects on Black girls’ mathematics achievement in Figures. In this article, I share just a couple of theterms of relational interactions with their teachers. things that I learned from these women in theirThere is also a deep-seeded historical and societal conversations about Hidden Figures. We discussedmyth that Black girls and mathematics are incom- what they noticed about the Black women and theirpatible (Gholson, 2016; Hottinger, 2016; Joseph, roles as mathematicians, what our mathematics2016). In addition to the damaging stereotypes that education system does right and wrong for Blackcan lead to disidentification with the discipline of girls, and what is needed to promote robust mathe-mathematics, Black girls often lack access to high matics identities among Black girls. The twoquality, advanced mathematics and science courses themes below were salient in their discussion andin schools located in their communities (National have implications for mathematics teachers.Women’s Law Center, 2014). Limited Access to High Quality Mathematics Instruction. My own work examines how Black girlsdevelop mathematics identities since we know that The young women articulated a positionproductive and robust mathematics identities con- about the inequality of schools and the relatedtribute to longer-term persistence in mathematics mathematics instruction they received as a result of(Boaler & Greeno, 2000; Joseph, Hailu, & Boston, attending such schools. One participant stated:2017; McGee, 2015). Specifically, I aim to exam-ine the roles race, class, gender and other socially I talked to a [White] girl, she said theyconstructed identities as well as interlocking sys- didn’t have desks – they sat in a Socratictems of oppression play in shaping their mathemat- style. They sat around a table, and theyics identities. Understanding what factors contrib- were forced to think out – it wasn’t justute to robust mathematics identities for Black girls written tests, and that’s how you have tois important for the national discourse about un- think in order to succeed at these top insti-derrepresentation of racialized minorities in STEM tutions. It’s more than a paper test. Youbroadly. Next, I share some findings from a pilot have to be able to think outside the box, andstudy that explored these ideas with undergraduate if you haven’t been thinking like that andBlack women in STEM majors at both Predomi- they’ve been thinking like that since ninthnately White Institutions and Historically Black grade, you’re not gonna perform at theColleges and Universities. I close the essay by level.invoking Virginia’s unique history of Black excel- This quote suggests that Black girls experiencelence in STEM, as conceptualized in Hidden Fig- mathematics instruction that is more traditional andures. This is one way to center possibilities and rarely have access to high quality mathematicsbuild upon a legacy and tradition to transform teachers, specifically, the type of mathematicsBlack girls’ mathematics experiences and their teachers that prepare students to be critical thinkersoverall lives in the educational system of Virginia. (Oakes, 1990a). Oftentimes the schools’ best math- ematics teachers are given choices of which cours-Virginia Mathematics Teacher vol. 44, no. 1 es to teach, and many of them choose to teach advanced mathematics courses and these courses 23
can include a strikingly different pedagogy. These students struggle academically because of theircourses can be more student-centered and include perceptions of low teacher expectations and themore opportunities for critical thinking, rather than teacher-student relationship (Ferguson, 2003;drill and kill and worksheets, which often can be Noguera, 2003). Overall, low expectations canseen in many of the lower-level courses (Oakes et significantly decrease Black girls’ chances to suc-al, 2004). We know that the quality of a teacher has ceed academically. To counteract this, mathematicsbeen consistently identified as the most important teachers need to be made aware of how their per-school-based factor in student achievement ceptions and beliefs impact Black girls’ academic(McCaffrey, Lockwood, Koretz, & Hamilton, achievement. We need teachers who have already2003; Rowan, Correnti & Miller, 2002), and that or can adopt a disposition of high expectations andteacher effects on student learning have been found unequivocal faith in their Black girls to succeed into be cumulative and long-lasting (McCaffrey et learning rigorous mathematics. This can be chal-al., 2003; Mendro, Jordan, Gomez, Anderson, lenging because as mathematics teachers, we haveBembry, & Schools, 1998). Therefore, mathemat- cultural scripts (Stigler & Heibert, 1998) aboutics teachers can play a role in changing the type of what represents success in mathematics, and wheninstruction Black girls receive in their classrooms, Black girls do not align with that script, we makeas well as promoting Black girls’ enrollment into assumptions that something is wrong with them,advanced mathematics courses. rather than our own teaching and more importantly,Low Expectations from Others. our system. Overall, these themes help us under- stand the viewpoints of some Black girls’ experi- Not being expected to accomplish greatness ences in mathematics and that our mathematicswas another salient theme that I learned from the education system needs to change if we care aboutfocus groups. One woman stated emphatically: Black girls succeeding. Black Girls Becoming Visible in Mathematics: We expect White people to succeed and we The Affordances of Virginia’s History of Black expect Black people to fail inherently. STEM Excellence.A different young woman expressed her thoughtsabout low expectations when she responded: Virginia has a robust history of Black ex- I think it’s about expectations because they cellence in STEM as seen through the historical [White students]—when White kids whose analysis in Shetterly’s (2016) book, Hidden Fig- families have—like they aren’t the first in ures. We now know that over 20 Black women their family to go to college, wherever they worked as human computers or mathematicians at go, it’s expected of them to finish. If they NASA and contributed to the United States’ suc- come into engineering, it’s not like, oh, you cess in the space program. Shetterly recalled: “that [Black girl] may change your major, and so many of them were African-American, many of this and that. It’s expected that they’re them my grandmothers’ age, struck me as simply a [White students] gonna finish. Why is it part of the natural order of things. Growing up in that when Black kids go to college and they Hampton, the face of science was brown like may be the first in their family, it’s like, are mine” (p.xiii). She continued, “I knew so many you sure you gonna be able to do that African-Americans working in science, math, and [engineering]? engineering that I thought that’s just what BlackThese findings suggest that Black girls perceive folks did” (p. xiii). This history gives mathematicsthat their mathematics teachers and other school teachers a way to center and engage Black girls inpersonnel do not believe they will succeed in mathematics in a meaningful way—helping BlackSTEM fields. Other studies have found that low- girls understand the greatness from which theyexpectation teachers blame Black students, their come and can draw upon.families, and their communities for achieving atlower rates than Whites (Lynn et al., 2010; West- Virginia is an epicenter of Historically BlackOlatunji et al., 2010). Some middle-school Black Colleges and Universities focused on STEM andVirginia Mathematics Teacher vol. 44, no. 1 24
can serve as spaces to transform the mathematics personal, local, national, or global communityteaching and learning experiences of Black girls. contexts to pose problems that are meaningfulBelow, I list a few recommendations about what to them. They should be able to produce prob-mathematics teachers can do to disrupt Black girls’ lems that are open-ended, relevant, and co-invisibility in the discipline: created with others. This type of learning experience should be employed from day one a. Encourage all Black girls to take more mathe- of school and developed all throughout the matics, not less, pushing back against the year in order to see sustainability. tracking system. This is important because it d. Partner with local HBCU STEM departments can open up more opportunities for the pur- to create programming that provide opportu- suit of STEM degrees in college (Tyson et al., nities for research and mentoring. This is 2007). Ways mathematics teachers can do this important because it can increase interest in is by working with school counselors to find STEM fields and college going rates “bubble” and high-achieving Black girls and (Schneider et al., 2013), and graduate school strongly encourage them to take additional aspirations (Odera et al., 2015). This can math (if they are not already doing so). It is create multiple pathways through the pipeline. more than just telling them to take more math, These recommendations are not silver bullet an- but it is about pulling in their families to dis- swers, as this phenomenon is complex. Our nation cuss together the benefits of such actions as must work to disrupt deficit narratives about Black well as the future consequences for not doing girls and the associated myths about mathematics. so. Virginia mathematics teachers have a unique op- portunity to be the “first” to lead this effort in a b. Invite Black women in STEM careers from systemic way. local Virginia universities/industries to dia- logue with Black girls in small groups. Many References Black girls, such as those in my study, may Battey, D. & Leyva, L. (2013). Rethinking mathe- not even know about or understand different STEM careers (i.e. engineers, actuaries, sci- matics instruction: An analysis of relational entists, etc.). Finding other math teachers in interactions and mathematics achievement your local school district and working togeth- in elementary. In Martinez, M., & Super- er to locate these Black women professionals fine, A. (Eds.), Proceedings of the 35th is one idea for getting at this recommenda- annual meeting of the North American tion. Mathematics and science teachers also Chapter of the International Group for the might survey their Blacks girls to see if they Psychology of Mathematics Education (pp. have family members and/or friends who 980-987). Chicago: University of Illinois at work as STEM professionals to come share Chicago. their stories. Boaler, J., & Greeno, J. G. (2000). Identity, agen- cy, and knowing in mathematics worlds. In c. Teach Black girls mathematics through prob- J. Boaler (Ed.), Multiple perspectives on lem posing and discovery, rather than tradi- mathematics teaching and learning, 171- tional procedures, allowing them to bring 200. Westport, CT: ABLEX Publishing. their full identities to the classroom. Complet- Collins, P. H., & Bilge, S. (2016). Intersectionality. ing worksheets usually does not stimulate Malden, MA: Polity Press. students’ minds for discovery and problem Dunleavy, T., Joseph, N. M., Zavala, M. (2016, posing. My work with high school Black girls November). Black girls in high school suggests that they value mathematics teachers mathematics: Crossing the border of deficit who promote academic and social integration, discourses. In M. B. Wood, E. E. Turner, while learning mathematics (Dunleavy et al.). M. Civil, & J. A. Eli. (Eds.). Proceedings of What this might look like in a classroom is the teacher promoting Black girls to utilize 25Virginia Mathematics Teacher vol. 44, no. 1
the 38th annual meeting of the North Amer matical identities: A framework for explor- ican Chapter of the International Group for ing racialized experiences and high the Psychology of Mathematics Educa achievement among black college stud- tion, (p. 1487-1494). Tucson, AZ: The ents. Journal for Research in Mathematics University of Arizona. Online. 2016-11-3, Education, 46(5), 599-625. http://www.pmena.org/pmenaproceedings/ Mendro, R., Jordan, H., Gomez, E., Anderson, M., PMENA%2038%202016% Bembry, K., & Schools, D. P. (1998, April). 20Proceedings.pdf An application of multiple linear regressionFerguson, R. F. (2003). Teachers’ perceptions and in determining longitudinal teacher effec- expectations and the Black-White test score tiveness. In Annual Meeting of the Ameri- gap. Urban Education, 38, 460-507. can Educational Research Association, SanGholson, M. L. (2016). Clean corners and Algebra: Diego, CA. A critical examination of the constructed National Women’s Law Center. (2014). Unlocking invisibility of Black girls and women in opportunity for African American girls: A mathematics. The Journal of Negro Educa- call to action for educational equity. New tion, 85(3), 290-301. York: NWLC.Hottinger, S. N. (2016). Inventing the mathemati- National Center for Education Statistics (no date). cian: Gender, race, and our cultural under- Retrieved from https://nces.ed.gov/. standing of mathematics. New York, NY: Noguera, P. A. (2003). The trouble with Black SUNY Press. boys: The role and influence of environ-Joseph, N. M. (2016). What Plato took for granted: mental and cultural factors on the academic Examining the biographies of the first five performance of African American males. African American female mathematicians Urban Education, 38, 431-459. and what that says about resistance to the Oakes, J. (1985). Keeping track. New Haven, CT: western epistemological cannon. In B. Yale University Press. Polnick, B. Irby, & J. Ballenger (Eds.), Oakes, J. (1990a). Multiplying inequalities: The Women of Color in STEM: Navigating the effects of race, social class, and tracking on Workforce (pp. 3-38). Charlotte, NC: Infor- opportunities to learn mathematics and sci- mation Age Publishing Inc. ence. Santa Monica, CA: Rand.Joseph, N.M., Hailu, M. & Boston, D. L. (2017). Oakes, J. (1990b). Opportunities, achievement, and Black women’s and girls’ persistence in the choice: Women and minority students in P-20 mathematics pipeline: Two decades of science and mathematics. Review of Re- children, youth, and adult education re- search in Education, 16, 153–222. search. Review of Research, 41, 203-227. Oakes, J., Joseph, R., & Muir, K. (2004). AccessLynn, M., Bacon, J.N., Totten, T. L., Bridges, T. L. and achievement in mathematics and sci- & Jennings, M.E. (2010). Examining teach- ence: Inequalities that endure and change. ers' beliefs about African American male In J. A. Banks & C. A. M. Banks (Eds.), students in a low-performing high school Handbook of Research on Multicultural in an African American School Dis- Education, 2nd Edition, (pp. 69-90). San trict. Teachers College Record, 112(1), 289 Francisco, CA: Jossey-Bass. -330. Odera, E., Lamm, A., Odera, L., Duryea, M. & DaMcCaffrey, D. F., Lockwood, J. R., Koretz, D. M., vis, J. (2015). Understanding how research & Hamilton, L. S. (2003). Evaluating value experiences foster undergraduate research -added models for teacher accountability. skill development and influence STEM ca- Monograph. RAND Corporation. PO Box reer choice. NACTA Journal, 59(3), 180- 2138, Santa Monica, CA 90407-2138. 188.McGee, E. O. (2015). Robust and fragile mathe- Pringle, R. M., Brkich, K. M., Adams, T. L., West-Virginia Mathematics Teacher vol. 44, no. 1 26
Olatunji, C., & Archer-Banks, D. A. (2012). ing is a cultural activity. A merican Educa- Factors influencing elementary teachers’ tor, 1-10. positioning of African American girls as Tyson, W., Lee, R., Borman, K. M., & Hanson, M. science and mathematics learners. School A. (2007). Science, technology, engineer- Science and Mathematics, 112, 217-229. ing, and mathematics (STEM) pathways:Riegle-Crumb, C., Moore, C., & Ramos-Wada, A. High school science and math coursework (2011). Who wants to have a career in sci- and postsecondary degree attain- ence or math? Exploring adolescents’ future ment. Journal of Education for Students aspirations by gender and race/ethnicity. Placed at Risk, 12(3), 243-270. Science Education, 95, 458–476. West-Olatunji, C., Shure, L., Pringle, R., Adams,Rist, R. C. (2000). HER classic: Student social T., Lewis, D. & Cholewa, B. (2010). Ex- class and teacher expectations: The self- ploring how school counselors position low fulfilling prophecy in ghetto education. income African American girls as mathe- Harvard Educational Review, 70, 257–301. matics and science learners. (Report). Pro-Rowan, B., Correnti, R., & Miller, R. J. (2002). fessional School Counseling, 13(3), 184- What large-scale, survey research tells us 195. about teacher effects on student achieve- ment: Insights from the \"Prospects\" study Nicole M. Joseph of elementary schools. CPRE Research Re- Assistant Professor of Mathematics port Series. EducationRubin, B. (2008). Detracking in context: How local Vanderbilt University constructions of ability complicate equity- Peabody College of Education and geared reform. Teachers College Record, Human Development 110(3), 646-699. [email protected], B., Broda, M., Judy, J., & Burkander, K. (2013). Pathways to college and STEM careers: Enhancing the high school experi- ence. New Directions for Youth Develop- ment, 2013(140), 9-29.Stigler, J.W. & Hiebert, J. (1998, Winter). Teach- Math JokesWhat do you get if you divide the circumference of a jack-o-lantern by its diameter? Pumpkin Pi.A \"combination lock\" should really be called a \"permutation lock.\" The order you put the numbersin matters.Why did the mutually exclusive events break up? They had nothing in common.People who take a long time computing the ratio of rise to run are slope pokes.Why did I divide sin by tan? Just cos.Virginia Mathematics Teacher vol. 44, no. 1 27
Grant OpportunitiesFlanagan Innovation in Mathematics Education Grant: The Virginia Council of November 4, 2017Teachers of Mathematics is offering Virginia mathematics educators a one-year December 2, 2017grant worth up to $500. This grant is designed to support educators who wish tocreate a meaningful, innovative project that enhances some aspect of the K-12 math-ematics curriculum.Karen Dee Michalowicz First Timers Grant: This grant is name for Karen DeeMichalowicz who actively promoted and supported conference development andattendance. The purpose of this grant is to provide funding support for: VCTM mem-bers who have NOT previously attended but wish to attend a regional or annualNCTM meeting; or any Virginia teacher (including non-members) who wish to at-tend a VCTM Annual Conference or VCTM Academy for the first time. Threegrants are available. One award is given for the regional or annual NCTM confer-ence of $800 and two awards of $400 each for the state conference or academy.For more information about VCTM grants, please visit: vctm.org/Grants NCTM deadlines startFor information about grants through NCTM, please visit: nctm.org/Grants/ November 3, 2017Busting Blockbusters!Many Hollywood movies have scenes that In the Star Trek episode “Court Martial,” theseem mathematically inaccurate, if not, crew is searching for a supposedly deadimpossible. Are these scenes truly impossi- crew member. Captain Kirk beams the crewble, or are they more plausible than they down to the surface and uses a “boost” of 14seem? The goal of the contest is to provide power to increase the auditory sensors tothe best mathematical explanations for the hear the heartbeats of everyone on board.following scene. The solutions that best How much did the auditory sensors actuallyexplain a scene’s possibility or impossibility increase? Would it be enough to hear aand the different elements that help form this heartbeat?will receive an award and the winner will befeatured in the Spring issue. Answers may https://www.youtube.com/watch?v=qyNxjtVKUT4be submitted by November 31, 2017 [email protected] with the subject line: This issue of Busting Block Busters isBusting Blockbusters entry. contributed by Dr. Neil Sigmon, RU.Virginia Mathematics Teacher vol. 44, no. 1 28
Virginia Mathematics Teacher vol. 44, no. 1 29
HEXAChallenge Problems created by: Dr. Oscar TagiyevOctober Challenge:In the triangle ABC, angle C is obtuse. The side AB equals x cm. A circlegoes through points A, B, and the point of intersection of the heights(orthocenter) of the triangle ABC. Find the radius of such circle.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)?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.Virginia Mathematics Teacher vol. 44, no. 1 30
Please be sure to state your Contest Alert! assumptions as you solve Virginia Mathematics Teacher is each problem. conducting a contest for educators Answers to the Fall 2017 and students who can solve the great-Hexa Challenge Problems est number of problems correctly by 02/28/2018. The winner will receive a will be featured in the prize and will be featured in the next Spring 2018 Issue of issue of the VMT. Send your solu- Virginia Mathematics tions to [email protected] Teacher. with the email subject line: Hexa ChallengeJanuary Challenge:Find all the integer pairs (X; Y) that satisfy the equation X2 ‒ 5XY + 6Y² = 11.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) geteaten. However, you allow the female ducklings to mature and have 10 eggs each. Once they lay ten eggs, youeat the duck. The same principle continues: male ducklings become eaten as early as possible, but you allowfemale ducklings to mature and have 10 eggs, only then you eat the female ducks. Unfortunately, one day onlydrakes (male ducklings) hatch from all eggs. You eat them all. You have counted that all together you haveeaten exactly 1000 drakes. How many ducks (female) have you eaten?March Challenge:There is a coordinate system with the origin O, where OA = 5 units,OB = 6 units, and OC = 7 units. Estimate the area of the triangle ABC.Virginia Mathematics Teacher vol. 44, no. 1 31
Teaching Dilemmas:Understanding the Concept of \"Slope\" Through Graphs and Storytelling Denise Wilkinson NCTM’s Principles to A ctions: Ensuring algebra class on the day that I first introduce theMathematical Success for All stresses the im- topic of slope. The activity presents a relatableportance of teaching mathematical topics to stu- example of a student, named Joe, who is travellingdents in relatable ways that help clarify concepts from his dorm room to his math class. Students areand promote mathematical reasoning through stu- instructed to find the rate of the change of the dis-dent engagement and collaboration (NTCM, 2014.) tance Joe travelled (feet) over the change in timeMathematics teachers additionally recognize the lapsed (minutes) during each leg of Joe’s journeyvalue of incorporating real world problems or prob- from his dorm room to his math class. Joe’s jour-lems that are experientially real and familiar to ney is presented as a graph of various connectedstudents into our algebra classes to enhance the line segments in the first quadrant of a coordinatelearning process (Gravemeijer, 1999). I have strug- system. The graph is shown below.gled with recognizing whether or not students trulyunderstand mathematical concepts through the I organize the students into groups of threeintroduction to a topic, examples and problems. By and give each group a copy of the graph. Studentsconnecting the concepts to relatable and applicable are initially asked to discuss their interpretation ofexamples as supported by these views, I have found the graph and the meaning of the line segments onthat students tend to gain a stronger and deeper the graph. This introduction provides me with aunderstanding and have a better chance of retaining base line of comprehension. Students tend to un-that understanding. derstand that the graph is a relationship between distance and time and that, in some context, Joe is A specific algebra topic that my students travelling from his dorm room to his math class.have found conceptually challenging is slope as a However, when interpreting the meaning of the linerate of change. When teaching this topic usingmore traditional teaching practices, I have foundthat students are often able to memorize the formu-la to find the slope of a line by calculating thechange in “Y” divided by the change in “X.” Addi-tionally, they seem to understand that the formularepresents “Rise over Run.” I have also observedthat students appear to recognize that slope tells usthe steepness of a line. However, many studentsseem to have difficulty wrapping their brainsaround the concept of the slope as the rate ofchange of two measurements in an application-based problem. A while back, I became inspired bya series of examples in an algebra workbook thatincluded graphs that told stories (The Consortiumfor Foundation Mathematics, 2012). In hopes ofachieving success with a new teaching approach, Ireflected on the “story” concept and implemented a“Stories and Slopes” group work activity in myVirginia Mathematics Teacher vol. 44, no. 1 32
segments, it is not unusual for students to incorrect- that the fictitious college student, Joe, encountersly visualize each line segment as a type of terrain on his journey to his math class. They are instruct-on which Joe travels. For example, they might ed to tell Joe’s story by interpreting the slopes ofinterpret a line segment with an increasing slope as the line segments on the graph. To give students aJoe climbing a hill and a negative slope might be starting point, the first sentence with a fill-in spaceinterpreted as Joe descending a hill. Additionally, a for Joe’s rate of change is presented in the instruc-line segment with a zero slope may appear to repre- tions. Together, they work in groups to tell Joe’ssent level ground. story using the slope of each of the six line seg- ments on the graph as a basis for each related sen- Once the observation process has been tence. To include all members of the group, eachcompleted, I present information on the definition student in the group must create (and share) a de-of a slope, the formula to find the slope of a line, scription of at least one of the six legs of Joe’sand an explanation of how to find the slope of a journeys. Students collaborate with their groupline segment using the “rise over run” process. I members to write their story of Joe’s journey andthen ask students to work in groups to complete a share it with the rest of the class. Through thistable of questions related to finding the slope of activity, I have observed that students tend to gaineach line segment. They are asked to: determine a stronger and more concrete understanding of thewhether each line segment represents an increas- concept of slope and rate of change. Positive anding, decreasing or zero slope; name the direction negative slopes seem to be more easily identifiedJoe is walking (away from his dorm, toward his by students as they make the connection of Joedorm or standing still); and find the slope of each moving towards his math class with a positiveline segment on the “Stories and Slopes” graph slope and Joe travelling back to his room with ausing the slope formula that was introduced and the negative slope. Students are also in a position toprocess to find the “rise over run.” Students are better comprehend a “zero” slope and recognizereminded to label their answer as a rate of change that the horizontal lines in the graph depict Joe(feet/minutes). The goal of the activity is to help standing still since he is not traveling a distancestudents gain a stronger understanding of the rate while time is still continuing to pass.of change in distance (Y2-Y1) over the change intime (X2-X1). The questions and the solutions are Perhaps, the most unexpected benefit that Ishown below. discovered through the inclusion of this activity was the level of student engagement in the story- To solidify the concept, students are then telling component. Telling stories allows us toasked to create a story that describes the adventures Stories and Slopes1. Answer the following questions. Place your results in the table: a. Determine whether each segment is increasing, decreasing or constant b. Determine in which direction Joe is walking for each segment. c. Find the slope of each segment. Be sure to include units in your answers.Segment Inc/dec/constant Direction Joe is walking Slope (label ft/min) 1 Increasing Away from Joe’s dorm room (40 ft)/(2 min) = 20 ft/min 2 Constant 3 Decreasing Standing still (0 ft)/(2 min) = 0 ft/min 4 Constant Walking back to Joe’s dorm (-40 ft)/(2 min) = -20 ft/min 5 Increasing Increasing room (0 ft)/(2 min) = 0 ft/min 6 Standing still (30 ft)/(3 min) = 10 ft/min Away from Joe’s dorm room (100 ft)/(2 min) = 50 ft/min Away from Joe’s dorm roomVirginia Mathematics Teacher vol. 44, no. 1 33
entertain, motivate our audience, communicate fastest?information, foster creativity, connect with new How far was Joe’s math class from his room?friends and helps us understand the world. With How long was Joe standing still and not mov-these beneficial aspects in mind, the engagement ofstorytelling may enhance student learning (Alterio, ing?2003). Although the original intent of asking stu- The activity may also be expanded to include stu-dents to interpret the graph of Joe’s journey was to dent exploration and investigation in an out-of-reinforce the concepts of slope and rate of change, I class lab, in which students first create and writesoon realized that students were in fact participat- their own travel story that includes a starting loca-ing in the process of storytelling. As a result, I have tion and destination. Students would next create anoted that students are engaged as they work cohe- labeled graph of line segments with indicatedsively and creatively together in groups to create slopes that models their story.and tell their version of Joe’s story. The storytell-ing question in the activity with an example of a My experience with the inclusion of thisgroup’s response is shown below. activity into my algebra class has been overwhelm- ingly positive. Quiz results on students’ under- Stories and Slopes standing supported the value of the activity. After the activity, students were given an application- 2. In your groups, create and write your story of Joe based quiz problem on the slope concept to solve walking to class as reflected by the graph in the text individually. Ninety-one percent of the students box below. You should include six sentences – each who completed the quiz mastered the problem and describing one of the six intervals. Your first sentence answered it correctly. Student feedback from an should read “Joe left his dorm room and travelled to anonymous survey on their “Stories and Slopes” his math class at a rate of ______ft/min.” experiences also supported my observations. All students who completed the survey indicated that Joe left his dorm room and travelled to his math class at a their understanding of the material had improved or rate of 20 ft/min, but ran into Sally who needed his notes was reinforced through their participation in the for Biology class. Joe stopped to talk with Sally for 3 activity. Students found the hands-on approach to minutes and then realized that he left his graphing calcula- be engaging and commented that it helped them tor in his dorm room. Because he was still on time for remember the information. They reported that they class, Joe returned to his room at the same rate of 20 feet/ could better recognize how the concept of “slope” min. It took him 2 minutes to find his graphing calculator can be applicable in the real world and applied to where he ran into Jeff, his roommate. Together Joe and everyday life. Responses to the storytelling compo- Jeff walked to the math building at a leisurely rate of 10 nent were also positive. Students shared that this feet/min. After 3 minutes, Joe looked at his Fit Bit and aspect made it possible for them to visualize the noticed that he was late for class so he increased his pace “slope” concept in a different and helpful way. In to 50 ft/min and hurried to his math class. fact, one student commented that storytelling has been an important part of his/her culture. I have also expanded the “Stories andSlopes” activity to include additional concepts on Finally, student comments indicated theirslopes and lines, such as the Slope-Intercept formu- appreciation for the creative aspect and group inter-la and the Point-Slope formula and have asked action of telling a group’s story, as well as thestudents to find the equation of each line segment. value of the writing component. Students have alsoThe table of questions and key are listed on the responded on course evaluations that through thenext page. participation of activities, including “Stories and Slopes,” they are more comfortable explaining I have also continued the “Stories and answers in class and are more inclined to ask groupSlopes” activity by including additional questions members to work on homework assignments andthat require students to make connections between prepare for tests outside of class.the graph and the context of Joe’s journey. Exam-ples of these questions include: Each semester, I look forward to the presen- During what period of time did Joe travel the 34Virginia Mathematics Teacher vol. 44, no. 1
Stories and Slopes3. First, record the slope in the appropriate column. Second, find the equation of the first segment using only the y = mx + b form. (Notice the y-intercept is given for segment “1” on the graph above.) Third, record the ordered pairs of the first and last point of each segment in the table below. Fourth, using the point-slope formula, y = y1 + m(x – x1), find the equations for each of the remaining line segments. Finally, place each equation in y = mx + b form in the table below. Number and show your work below the table, clearly. Begin by writing the formula, y = y1 + m(x – x1).Segment Slope (label ft/min) Ordered pair of first Ordered pair of last point of Equation of line: 1 point of segment segment y = mx+b 2 20 ft/min (0, 0) (2, 40) Y = 20x + 0 3 0 ft/min (2, 40) (5, 40) Y = 40 4 -20 ft/min (5, 40) (7, 0) 5 0 ft/min (7, 0) (9, 0) Y = -20x + 140 6 10 ft/min (9, 0) (12, 30) Y=0 50 ft/min (12, 30) (14, 130) Y = 10x - 90 Y = 50x - 570tation of the topic of slopes and this activity and to National Council of Teachers of Mathematics.witnessing the energy and engagement of students, (2014). Principles to actions: ensuringmany of whom have been quiet most of the semes- mathematical success for all. Reston, VA:ter. Through this interactive learning component of NCTM, National Council of teachers ofcollaboration and storytelling while participating in Mathematicsan application-based activity, students are in aposition to gain a stronger and deeper understand- The Consortium for Foundation Mathematics.ing of an important and relevant mathematics topic. (2012). Mathematics in action. Boston, MA: Pearson Education, Inc.ReferencesAlterio, M., & McDrury, J. (2003). Learning . through storytelling in higher education: Denise Wilkinson Using reflection and experience to improve Professor of Mathematics learning. Routledge. Associate Dean of InnovativeGravemeijer, K., & Doorman, M. (1999). Context Teaching and Engaged Learn- problems in realistic mathematics educa- ing tion: A calculus course as an example. Virginia Wesleyan University Educational studies in mathematics, 39 [email protected] (1-3), 111-129.Virginia Mathematics Teacher vol. 44, no. 1 35
Solutions to Spring 2017 HEXA Challenge ProblemsApril Challenge:Solve xlog(5x) = 40 algebraically.SOLUTION:(log(x)) stands for the logarithm of x. Take the logarithm of both sides: log(xlog(5·x)) = log(40)The left-hand side is equal to: log(5·x) · log(x) = log((10/2) ∙ x) ∙ log(x) = (1 - log(2) + log(x)) ∙ log(x)Respectively, the right hand side is: log(10 ∙ 4) = log(10 ∙ 22) = log(10) + log(22) = 1 + 2 ∙ log(2)Therefore: (1 - log(2) + log(x)) ∙ log(x) = 1 + 2 ∙ log(2)Let’s say z = log(x). Then: (1 - log(2) + z) ∙ z = 1 + 2 ∙ log(2)This reduces to a quadratic equation z2 + (1 - log(2)) ∙ z = 1 + 2 ∙ log(2)whose roots are z1 = , z2 = where z1 = log(x) and z2 = log(x).Hence the roots of the primarily suggested equation are: x1 ≈ 0.056; x2 ≈ 3.588May Challenge:Construct a right triangle ABC where AC is the hypotenuse. Create a point D, such that BD is the height, per-pendicular to AC. In ∆ABD and ∆BCD, there are inscribed circles with radii r1 and r2, respectively. Find theradius of the circle inscribed in ∆ABC.SOLUTION:There are three similar triangles: ∆ADB ~ ∆BDC ~ ∆ABC. Therefore,Virginia Mathematics Teacher vol. 44, no. 1 36
Solutions to Spring 2017 HEXA Challenge Problems Figure 1. The bigger radius, r, inscribed into the triangle ABC.Next, we add the last two proportions to obtain:June Challenge:If you were to write all of the whole numbers from 1 to 1,000,000 and add them together, to what would theysum?SOLUTION:1; 2; … ; 999,999; 1,000,0001 + 1,000,000 = 1,000,0012 + 999,999 = 1,000,001…500,000 + 500,001 = 1,000,001You add 500,000 cases of 1,000,0001500,000 x 1,000,001 = 500,000,500,000July Challenge:A man owes $10,000. At the moment, he has nothing, and has just started a new job. His schedule has himwork only every other month. During the month he is off, it takes him two weeks to cash the check for hisVirginia Mathematics Teacher vol. 44, no. 1 37
Solutions to Spring 2017 HEXA Challenge Problemsprevious month's work, and from that amount, at the end of the month, he has $1,000 left. When should thecollection agency next meet with him to settle his debt?SOLUTION:The problem is purposely ambiguous. Several simplifying assumptions have to be made in order to solve it.The information on the monthly pay is not provided. According to the U.S. Census Bureau (https://www.census.gov/topics/income-poverty/income.html), income for a single person is about $3,000. We arealso making an assumption that he does not spend during the month he is working but he gets paid.Therefore:Months Total Funds The debt collector should be able to collect 1 $3,000 money at the end of the fifth month. 2 $3,000 + $1,0003 $4,000 + $3,0004 $7,000 + $1,0005 $8,000 + $3,000August Challenge:The shadows of a three-dimensional object cast on three planes perpendicular to each other create a circle, asquare, and an equilateral triangle. What is the shape? SOLUTION: Imagine a cylinder, whose base diameter and height are equal to each other. AB and CD are mutually perpendicular base diameters lying respectively in the upper and lower bases of the cylinder. The cylin- der has been cut so that the 'shoulders' could be taken off (Figure 1). Let's see what remains:Figure 1. The cylinder with the ‘shoulder’ cut lines.Virginia Mathematics Teacher vol. 44, no. 1 38
Solutions to Spring 2017 HEXA Challenge Problems Figure 2. Figure 3. Figure 4.Circular Shadow Square Shadow Triangular ShadowSeptember Challenge:It is currently midnight. How long is it until the minute hand and the hour hand are perpendicular to each oth-er?SOLUTION:The speed of rotation of the hour hand isThe speed of rotation of the minute hand isSince both hands start moving at 12:00, the time interval ‘t’ that measures how long it will take for bothhands to be perpendicular to each other can be calculated by the following equation: t ≈ 16.36 minutesVirginia Mathematics Teacher vol. 44, no. 1 39
Upcoming Math CompetitionsName of Organization Website Dates of CompetitionsMath Kangaroo in the USA http://www.mathkangaroo.org TBA Registration starting October 22, 2017Noetic Learning Math Contest http://www.noetic-learing.com/mathcontest/ November 9-22, 2017 Register by October 27, 2017Huntington University Middle School https://www.huntington.edu/math/mathematics-competition/ April 11, 2018 Mathematics Competition middle-school-competition Register by October 31, 2017American Math Competition 8 https://www.maa.org/math-competitions/amc-8 November 14, 2017Who Wants to Be a Mathematician http://www.ams.org/programs/students/wwtbam/wwtbam November 15, 2017Rocket City Math http://www.rocketcitymath.org/ November 17, 2017Math Con http://www.mathcon.org/ May 5, 2018 Register by November 22, 2017Continental Math League https://www.cmleague.com January 4, 2018 Register by December 10, 2017 Math Counts http://www.mathcounts.org/programs/competition-series/ December 15, 2017 The Math League competition-faq Month depends on grade levelPurple Comet Math Meet http://www.mathleague.com April 10-19, 2018 http://purplecomet.org/home/home Rochester Institute of Technology's http://www.rit.edu/ntid/mathcompetition/ April 13-15, 2018Math Competitions for Students Who Are Deaf or Hard-Of-HearingAmerican Regions Math League http://mathleague.org/arml.php Register through your Regional TeamMandelbrot http://www.mandelbrot.org/ TBAUSA Math Talent Search http://www.usamts.org/TipsFAQ/U_Tips.php Online, month-long contestSUM Dog Online http://www.sumdog.com/enter_contest/ Online competitionVirginia Mathematics Teacher vol. 44, no. 1 40
Technology ReviewSection 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 51.Graphing with DesmosAbby Smith Desmos is a free online graphing website Figure 1. An example of how to use Desmos to help studentsaccessible to students and teachers on their com- explore quadratic functions and the transformations that oc-puters, tablets, and phones. The website is designed cur to the graph given different values of a, b, and c.to allow students to interact with graphical repre-sentations of equations and tables. Not only does pears. The website lends itself well to inquiry-the website allow students to examine graphs, but based learning, meaningful mathematical conversa-there are also pre-designed class activities that tions, and a way for struggling students to connectteachers can freely use in their classroom. to the content; however, there are downsides to using the software as well. When www.desmos.com/calculator is open-ed, students are led to a page where they can type The Desmos software is user-friendly forin two-variable equations and a graph appears of students and teachers. Many of the students I havethe equation. Multiple equations can be graphed at worked with download the Desmos app to theirthe same time. Students can even remove the visu- phones after I show them how to use the site. As aals of a specific equation without deleting it by teacher, I understand that this is a blessing and aclicking the circle to the left of the equation. In curse. On the bright side, students can use the gra-addition to equations, students can type in func- phing tools to make their own conclusions on howtions, inequalities, and tables of values. The main specific equations are graphed and why the graphsdrawback is that students are restricted to two vari- take on the specific shapes. Sadly, not all studentsables. In order to help students examine howgraphs change based on the variables included inthe equation, sliders (which change the value of aspecific variable) are also available for students tomanipulate their equations. While it is not neces-sary to set up an account with Desmos, if studentslog in, they can save their work to return to it at alater date. As a middle and high school math teacher, Ialways have enjoyed the lessons where we breakout the tablets or visit the computer lab to spend aday exploring specific graphs of functions, discuss-ing the constraints on the graph, or wondering howtransformations change the way the graph ap-Virginia Mathematics Teacher vol. 44, no. 1 41
are as curious about how the equations are graphed During my time teaching at-risk 8th gradeand simply desire to finish their work with a mini- students, we had additional time after a day ofmal amount of effort. Just as with Photomath, stu- testing and I had the students explore the Desmosdents can use the website to answer their home- page. As the kids used the website, they beganwork problems and avoid the struggle of graphing exploring the graphs of trigonometric functions andfunctions on their own. posed questions about asymptotes. Some students realized how to graph a circle (which soon became The app can be downloaded to your phone the talk of the class) and others noticed the differ-or other device, so it can take the place of the ex- ences between an equation's graph and an inequali-pensive TI-84 which can cost students $100. Some ty’s graph. Students who are traditionally disen-standardized testing websites are beginning to gaged with the material were asking wonderfulincorporate the Desmos software into their test to questions and were sad to pack up at the end of thesave students the burden of purchasing an expen- day. If you have not had the opportunity to usesive, out-of-date, piece of technology. During the Desmos in your classroom, I strongly encouragetest, students will be restricted to the use of the you to sign up for a computer lab day and see whatgraphing software and the testing site (King, you and your students have been missing.2017). In the classroom lessons, however, studentscan still become side-tracked with distractions from Referencestheir phones or other websites. In my experience, King, Ian. “Startup Targets The TI Calculatorsthe students find the software enjoyable to usewhich increases their likelihood of staying on task Your Kid Lugs to Class.” Bloomberg.com,and avoiding the myriad of other technological Bloomberg, 12 May 2017, www.bloombergdistractions. .com/news/articles/2017-05-12/startup- targets-the-ti-calculators-your-kid-lugs-to- Along with the graphing software, several class.lessons have already been created by the Desmosteam to further students’ mathematical understand- Abby Smithing. To access all of the features of the pre- Mathematics Teacherdesigned lessons, teachers and students must create Pomona High Schoolan account with Desmos. Students must sign in [email protected] a Google account or provide their name and e-mail address. There are several activities that are 42appropriate for early middle school students all theway into trigonometry. Students can see other stu-dents’ answers to free response questions once theyhave submitted their own and submit work to theirteacher. As the teacher, you can see all of the stu-dent responses and restrict access to specific pagesif you would like to pace the group as a whole.Some of the questions during the lesson will al-ready be marked as correct or incorrect; however,to assign an accurate grade, the instructor wouldneed to review each student’s work individuallywhich could be quite time-consuming. The lessonshave come pre-designed and many cannot bechanged to be more appropriate for your students.Many of the lessons and activities have been de-signed to work on a tablet or computer, but not on aphone.Virginia Mathematics Teacher vol. 44, no. 1
Unsolved Mathematical Mysteries: Happy End Problem Kevin HartnettIn this section, we accept articles about fascinating mathematical problems that have not yet been solved. Weask the authors to describe the problem so that a middle school student could understand it. The piece shouldalso reference the progress that has been made by the mathematics community in solving the problem. The nickname of this problem \"Happy End\" able to prove that it took three points to guaranteecomes from not mathematical reasons, but the fact construction of a convex triangle, five points tothat Esther Klien and George Szekeres worked construct a convex quadrilateral, and nine points tocollaboratively on the problem in the mid-1930's guarantee a construction of a convex pentagon.which resulted in their marriage. Their friend Paul They also proposed a formula for counting theErdos also contributed to working on this problem. number of necessary points: 2(n-2) + 1, where n is the number of sides of the convex polygon. How- In 1933, a 23-year-old Esther, who lived in ever, this was a conjecture and not a proof.Budapest, Hungary, brought home a puzzle that shepresented to her friends, Paul and George. Here's The problem seems simple enough to un-the puzzle: derstand; many mathematicians worked on it for several decades. \"Yet as the decades passed, math- Given five points, and assuming no three fall exactly ematicians made virtually no progress in proving on a line, prove that it is always possible to form a the conjecture. (The only other shape whose result convex quadrilateral — a four-sided shape that's is known is a hexagon, which requires at least 17 never indented, meaning that as you travel around it, points, as proved by Szekeres and Lindsay Peters you make either all left turns or all right turns in 2006.) Now, in work recently published in (Figure 1). the Journal of the American Mathematical Socie- ty, Andrew Suk of the University of Illinois, Chica- Esther derived a proof prior to presenting go, provides nearly decisive evidence that the intui-the puzzle to her friends and they were able to tion that guided Erdos and Szekeres more than 80show that Esther's proof was true. As mathemati- years ago was correct\" (Hartnett, 2017).cians always do, they decided to check if the proofis generalizable, meaning if this works for five There is still work to be done in solving thispoints that guarantee forming a quadrilateral by problem. We encourage you to introduce this con-connecting four of them, how many points are jecture to your students, and maybe one of themneeded to guarantee a formation of a convex poly- will provide the solution someday and will get togon with any number of sides (e.g. pentagon, hexa- publish their story in the V irginia Mathematicsgon, etc.)? Teacher. Paul and George did solve this problem for This is a summary of the article titled \"Atriangles, quadrilaterals, and pentagons. They were Puzzle of Clever Connections Nears a Happy End\" by Kevin Hartnett and is printed with permission Figure 1. The shape on the left is a convex polygon. The from Quanta Magazine, May 2017 issue. The link shape on the right is a concave (non-convex) polygon. to the original article can be found at https://www.quantamagazine.org/a-puzzle-of- clever-connections-nears-a-happy-end-20170530/.Virginia Mathematics Teacher vol. 44, no. 1 43
The Shape of Ordered Pairs Harold Mick and Benjamin Bazak The purpose of this article is to present an The mathematical playground for this arti-alternative approach that applies transformations cle is the coordinate plane. The layout of the coor-(motions) to problems of the type given an equa- dinate plane with its two intersecting perpendiculartion find its graph and given a graph find its equa- axes is such that each point has an address givingtion that are dealt with in the secondary school its location in the plane. We write these addressesmathematics curricula. By an alternative approach in the form of ordered pairs. For instance, the or-we mean a decidedly different way to deal with dered pair (u, v) consists of two coordinates u andthese problems than is found in current textbooks v. The first coordinate u is a real number repre-and reflected in Virginia’s Standards of Learning. senting the directed distance from the Y -axis. TheMost of this article is spent laying the foundation second coordinate v is a real number representingof our approach. In the beginning, we focus on the directed distance from the X-axis. The intersec-problems of the type given a graph find its equation tion of these two directed distances locates a pointand then continue to problems of the type given an with ordered pair (u, v).equation find its graph but stop short of actually Shape of Ordered Pairs.sketching the graph. Instead we challenge you, thereader, to complete the “last step” of our approach. A parabola has a geometric shape; its ver- tex may be anywhere in the coordinate plane, it Our audience includes secondary mathe- may open up or down, it may be narrow or wide.matics teachers, mathematics educators, mathemat- With all this variance of geometric shapes the cor-ics supervisors, secondary mathematics curriculum responding ordered pairs vary too. Suppose you arespecialists and curious readers. However we direct “standing” on a parabola. Now look down andour attention primarily to high school mathematics follow ordered pairs of points on the parabola.teachers. Our intent is to help teachers develop a What do you see? Clearly the individual first anddeeper level of understanding of the mathematics second coordinates of each ordered pair change asassociated with applying motions as a way to ap- points move along the path of the curve. You canproach equations and graphs. In short, we address become dizzy watching these changes. But as youteachers as learners. look at these changing ordered pairs you may no- tice something that remains the same. What is it We invite you (the reader) to put on your you may ask? As we closely examine these ordered“math magic” hat and stretch your imagination as pairs we see a relationship between first and secondyou accompany us on our exploration. We intro- coordinates that remains the same as the orderedduce our approach by exploring “shapes” of or- pairs move along the curvature of the parabola. Wedered pairs. We are going to put a geometric spin call this relationship the shape of ordered pairs.on what is usually considered an algebraic concept.We restrict our discussion to parabolas. Our inspi- For instance, points with ordered pair (u, v),ration will come from a parabola’s location, orien- where the second coordinate is the square of thetation (up or down) and geometric shape (narrow or first coordinate, form a parabola with its vertex atwide). However, we will concentrate on what a the origin. From this geometric shape we extract anparabola “looks like” algebraically. What can we algebraic shape of ordered pair (u, v) in the formsay about the “shape” of ordered pairs? How are of statement v = u2. This algebraic shape or rela-complicated “shapes” of ordered pairs understood tionship remains the same as points with orderedby comparing them to simpler “parent shapes”? pair (u, v) move along this special parabola. This shape, v = u2, shows how first coordinates u andVirginia Mathematics Teacher vol. 44, no. 1 44
second coordinates v are related for exactly those tions and ordered pairs: “X” stands for “take thepoints on this parabola: “to get second coordinate v first coordinate” and u is the coordinate taken.take first coordinate u and square.” We describe Please note that almost our entire article takes placethis shape by writing equation Y = X2, where X in the middle row representation of ordered pairsrepresents first coordinates and Y represents sec- (Figure 1). That is most unusual and will take someond coordinates. We call this particular parabola adjustment on your part (our readers).the parent parabola because of its basic structure. In secondary mathematics textbooks, in In the paragraph above we made a subtle general, the equation of the parent parabola is writ-distinction between the equation, Y = X2, and the ten with small letters as y = x2. Substitution oralgebraic shape v = u2. To heighten the role repre- replacement rules are applied to equations at thesentations play in our article, we illustrate three same representation level and almost always origi-representations in Figure 1: equations in the top nate with the parent. These authors do not pick arow, order pairs in the middle row, and graphs in generic point on the parent parabola with say or-the bottom row. dered pair (x, y) and move it about the coordinate plane. Consequently they have no need to distin- Figure 1. guish between equation variables and ordered pair variables. We digress for a moment to discuss differ-ences between our approach and the approach of We use small letters “u” and “v” instead ofsecondary mathematics textbooks in general. To small letters “x” and “y” to avoid confusion withemphasize the distinction between equation repre- our use of capital letters “X” and “Y”, as well assentation and ordered pair representation we use textbooks’ use of small letters “x” and “y” for writ-capital letters for equations and small letters for ing equations.ordered pairs. For us “X” stands for “take the first Motions.coordinate” and “Y” stands for “take the secondcoordinate.” To say that Y = X2 describes the equa- Before we illustrate our approach with ation of the parent parabola is to say that a point is problem, we go over the actions and labels associ-on the parabola if and only if the square of its first ated with the motions we apply in this article. Wecoordinate is equal to its second coordinate. So if apply three types of motions with each havingwe take a point with ordered pair (u, v) that lies on horizontal and vertical components. We call thesethe parent parabola then the square of first coordi- three types: shifts, flips and scales. (We define andnate u is equal to the second coordinate v. In sym- label these motions specific to the coordinate planebols we write v = u2. Here is another way to look at to avoid confusion with transformations such asthe distinction between representations of equa- translations, reflections and dilations that are de- fined in the Euclidean plane.)Virginia Mathematics Teacher vol. 44, no. 1 Types of Motions Xshift b Horizontal shift (u, v) → (u + b, v) Yshift d Vertical shift (u, v) → (u, v + d) Xflip Horizontal flip (u, v) → (−u, v) Yflip Vertical flip (u, v) → (u, −v) Xscale a Horizontal scale (u, v) → (au, v) Yscale c Vertical scale (u, v) → (u, cv) where a, b, c, d are real numbers, a > 0 and c > 0 Notice that each of these six basic motions is associated with an arithmetic operation, either addition or multiplication: shifts are associated with addition by a real number, flips are associated with multiplication by –1 and scales are associated 45
with multiplication by a positive real number. We those points making up the parabola’s geometricuse a dual language system of both geometric and picture; that is, what is the shape of ordered pairalgebraic designations to communicate motions in (u, v) for points on the solid parabola? If the solidthe coordinate plane. parabola had only been the parent parabola we would be done for then each second coordinate We form a sequence of motions by follow- would be the square of its first coordinate. We areing one motion with another. For instance a hori- not so lucky. But all is not lost.zontal shift 2 units followed with a horizontal scaleby factor 3 moves (u, v) to (u + 2, v) and (u + 2, v) We will move the solid parabola to theto (3(u + 2), v). These changes of ordered pairs parent parabola where we know the shape of or-show the order of the arithmetic operations addi- dered pairs (see dotted parent parabola in Figure 3).tion and multiplication which build up algebraic What program can we write to move the solid pa-expressions of coordinates; this is algebraic lan- rabola to the dotted parent parabola? In comparingguage. We write programs for sequences of mo- the two parabolas there appear to be several se-tions (compositions) by listing their individual quences of motions (actually there are infinitelylabels in the order they are applied; in other words, many sequences but only a few practical ones). Weprograms are ordered lists of motion labels. In this discuss two such sequences and their programs.instance we write program X shift2|Xscale3; this is Program (1): Suppose we move the vertex of thegeometric language. Programs may be of any solid parabola to the origin. To do this we shift thelength. We display programs horizontally, vertical- solid parabola 2 units horizontally and −1 unitly or in whatever arrangement is convenient; it’s vertically (see dashed parabola in Figure 4). Thenthe ordered sequencing that is important. We dis- we flip the dashed parabola vertically (see dashed/cuss programs in more detail under the section dotted parabola in Figure 4) and finally we scaleentitled Programs. that image vertically by a factor of 4 (see dottedFrom Motions to Equations. parent parabola in Figure 4). To illustrate our approach we consider the We write the corresponding programfollowing problem: Xshift2|Yshift−1|Yflip|Yscale4 to move (u, v) toProblem 1: Find an equation for the solid pa- (u + 2, −4(v − 1)) which lies on the dotted parentrabola shown in Figure 2. parabola. The point of recording this sequence of ordered pairs To find an equation for the solid parabolameans we seek the shape of its ordered pairs. To (u, v) → (u + 2, v − 1) → (u + 2, −4(v − 1))this end take points with ordered pair (u, v) on the is that we know the shape of ordered pairs on thesolid parabola. How are u and v related for exactly parent parabola; namely, second coordinates are the square of first coordinates. Since FFigiugreu22r. e FiFguireg3u. re 3Virginia Mathematics Teacher vol. 44, no. 1 46
FFigiugreu44.re FFiguigreu55.re(u + 2, −4(v − 1)) lies on the parent parabola the Program (2): Again we compare the given solidsecond coordinate −4(v − 1) is the square of first parabola with the dotted parent parabola in Fig-coordinate u + 2. More simply put, ure 3. For this second sequence of motions we begin by shifting the solid parabola vertically −1 4(v − 1) = (u + 2)2. unit followed with a vertical flip (see dashed parab- To emphasize that we are describing the ola in Figure 5). Rather than scaling the secondshape of ordered pair (u + 2, −4(v − 1)) on the coordinates by a factor of 4, we scale the first coor-parent parabola in the statement −4(v − 1)= (u + 2)2 dinates by a factor of (1/2) (see dashed/dotted pa-we enclose first coordinate u + 2 and second coor- rabola in Figure 5). Finally we apply a horizontaldinate −4(v − 1) in the statement, shift 1 unit (see dotted parabola in Figure 5). This completes the move from the solid parabola to the −4(v − 1) = ( u + 2 )2 . parent parabola. Why is this statement valuable? Recall that We write the corresponding program,we are looking for the shape of ordered pair (u, v) Yshift−1|Yflip|Xscale(1/2)|Xshift1 to move (u, v) toon the solid parabola. If we interpret the statement, ((1/2)u + 1, −(v − 1)). Since ((1/2)u + 1, −(v − 1))−4(v − 1) = (u + 2)2, from the perspective of lies on the parent parabola, the second coordinate“standing” on the solid parabola, we see a relation- −(v − 1) is the square of first coordinate (1/2)u + 1 .ship between coordinates u and v of ordered pair More simply put, −(v − 1) = ((1/2)u + 1)2. We(u, v) on the solid parabola. To emphasize this change perspectives from the parent parabola backrelationship we enclose u and v in statement to the mystery parabola by changing shapes from −4( v − 1) = ( u + 2)2. −(v − 1) = ((1/2)u + 1 )2 toWe describe this shape of (u, v) on the solid parab- −( v − 1) = ((1/2) u + 1)2.ola in words: take second coordinate v and add −1, The statement −( v − 1) = ((1/2) u + 1)2, with u andthen multiply this quantity by −4 to get the square v enclosed for emphasis, describes the shape ofof the quantity take first coordinate u and add 2. ordered pair (u, v) for points on the solid parabola:Using X to stand for “take the first coordinate” and take second coordinate v and add −1, multiply thisY to stand for “take the second coordinate” we get quantity by −1 to get the square of the quantity,equation take first coordinate u multi- ply by (1/2) then add 1. The −4(Y − 1) = (X + 2)2 equation for the solid parabolafor the solid parabola. is −(Y − 1) = ((1/2)X + 1)2. We show that the two equa- Upon reflection, we took points with or- tions derived from the twodered pair (u, v) on the solid parabola. Then we programs are equivalent:moved those points to the parent parabola wherewe knew the shape of ordered pairs, wrote a state- 47ment, and then re-interpreted the statement interms of the chosen ordered pair (u, v) on the solidparabola.Virginia Mathematics Teacher vol. 44, no. 1
Programs. FiFguirge u66.re Programs play a central role in our ap- (we could multiply by −1 first and multiply by 4proach. Underlying these programs are the order of second), add 4. The corresponding motions are:operations associated with the order of individual scale vertically by factor 4, flip vertically, shiftmotions. It is the application of operations that vertically 4 units. We move the dashed parabola tocreate the algebraic expressions for coordinates. the parent parabola by applying program Yscale4|Yflip|Yshift4. Suppose we apply a horizontal shift 2 unitsto the solid parabola in Figure 2 so that the image We observe that this sixth program is yetparabola has its vertex at (0, 1) on the Y -axis (see another program distinct from the others but it stilldashed parabola in Figure 6). Let this be our new moves the dashed parabola to the parent parabola.starting location; in other words, our task is to write In short, each of these four algebraic expressionsan equation for the dashed parabola shown in Fig- expresses a different program that we can uncoverure 6. To this end we write three different programs by interpreting the corresponding expressions’to move this dashed parabola to the dotted parent order of operations.parabola. (Note that we are dealing only with sec- Parent Form.ond coordinates to make the moves.) We numberthese programs (3), (4) and (5). In general, suppose we are given a parabolaProgram (3): we apply Yshift−1|Yflip|Yscale4. The and suppose we take (u, v) on this mystery parabo-order of arithmetic operations is: add −1, multiply la and move its points to the parent parabola byby −1, and multiply by 4. The corresponding alge- applying successive basic motions. We may applybraic expression is 4(−1)(v + (−1)) = −4(v − 1). any number of motions, we may shift, flip andProgram (4): we apply Yflip|Yshift1|Yscale4. The scale all over the coordinate plane but eventuallyorder of arithmetic operations is: multiply by −1, we arrive at the parent parabola. The associatedadd 1, and multiply by 4. The corresponding alge- operations of addition and multiplication simplifybraic expression is 4((−1)v + 1) = 4(−v + 1). to algebraic expressions of the form au + b for firstProgram (5): we apply Yscale4|Yshift−4|Yflip. The coordinates and cv + d for second coordinates,order of arithmetic operations is: multiply by 4, add where a ≠ 0, b, c ≠ 0, d are real numbers (flips−4, and multiply by −1. The corresponding alge- occur when a or c are negative). These individualbraic expression is (−1)(4v + (−4)) = −(4v − 4). motions make up programs that move (u, v) to (au + b, cv + d) that lie on the parent. The resulting In our discussion of programs (3), (4) and shape of ordered pairs (au + b, cv + d) on the par-(5) we went from programs to expressions. Now ent parabola is cv + d = (au + b)2. At this point inwe go from expressions to programs. Consider our approach we change perspectives from theexpression (3): −4(v − 1). We begin with second parent parabola to the mystery parabola and con-coordinate v. The first operation applied to v is add sider the shape of (u, v) on the mystery parabola.−1. The result is v − 1. Next we multiply by 4. The The equation isresult is 4(v − 1). Finally we multiply by −1. Theresult is −4(v − 1). The order of operations is: add cY + d = (aX + b)2.−1, multiply by 4, multiply by −1. The correspond-ing order of motions is shift vertically −1, scale 48vertically by factor 4, flip vertically. The corre-sponding program is Y shift−1|Yscale4|Yflip. Weread expressions 4(−v + 1) and −(4v − 4) similarlyfor programs (4) and (5). These three expressions are mathematicallyequivalent to expression (6) −4v + 4. We interpretthe order of operations for expression (6): takesecond coordinate v, multiply by 4, multiply by −1Virginia Mathematics Teacher vol. 44, no. 1
Sometimes we lump together Y expressions and the mystery parabola to first coordinate u + 2 onjust call them Y stuff and we call X expressions parent parabola by adding 2 which corresponds toXstuff. With this language we write equations in applying X shift2. We move second coordinate v onparent form as Ystuff = (Xstuff)2. the mystery parabola to second coordinateFrom Equations to Motions. −4(v − 1) on the parent parabola by adding −1, multiplying by −1 and multiplying by 4 which We have shown that equations for parabolas corresponds to applying Y shift−1|Yflip|Yscale4.can be built up and described in parent form Joining these programs together we move (u, v) tocY + d = (aX + b)2. Now we pose the converse (u + 2, −4(v − 1)) by applyingsituation: can we begin with an equation with itsmystery parabola, uncover a sequence of motions Xshift2|Yshift−1|Yflip|Yscale4.embedded in the equation, and from this sequence By uncovering moving directions in anwrite a program that moves the mystery parabola to equation we learn to see in a new way. All thisthe parent parabola? information — the two parabolas, two shapes ofResponse. We take another look at the fir st ordered pairs and directions for moving from oneprogram’s equation −4(Y − 1) = (X + 2)2 that has a parabola to the other — is embedded in the equa-parabola for its graph. We know the parabola (see tion. We just have to see it.Figure 2) but we pretend this is a new example Problem 2: Let Y = 4(X − 3)2 − 5 be an equationwhere we are given an equation and seek to uncov- for the graph of a mystery parabola. Write aer the motions that lead to its formation. We call program that moves this mystery parabola tothis “unknown” parabola the mystery parabola. the parent parabola.Take points (u, v) on −4(Y − 1) = (X + 2)2 so that First we sweep all the Y stuff to the left side−4(v − 1) = (u + 2)2. This statement shows the of the equation:shape of ordered pair (u, v) on the mystery parabo-la. We emphasize this fact by enclosing u and v in (1/4)(Y + 5) = (X − 3)2.−4( v − 1) = ( u + 2)2. This puts the original equation in parent form: The next step involves a change of per- Ystuff=(Xstuff)2.spectives only this time the change goes the other Next take (u, v) on (1/4)(Y + 5)= (X − 3)2 so thatway. After looking at statement −4(v − 1)= (u + 2)2from the perspective of “standing” on the mystery (1/4)(v + 5) = (u − 3)2.parabola, we “hop” to the parent parabola and see Using our “eye-training” and “hopping” skills wethe shape of ordered pairs (u + 2, −4(v − 1)) where see two shapes of ordered pairs:−4(v − 1) is the square of u + 2. We emphasize thischange of perspectives from the mystery parabola (u, v)to the parent parabola by changing shapes from the on the mystery parabola andstatement −4( v − 1) = ( u + 2)2 to−4(v − 1) = ( u + 2 )2. (u − 3, (1/4)(v + 5)) on the parent parabola. To move (u, v) to In other words we see two shapes of or- (u − 3, (1/4)(v + 5)) we move u to u − 3 by addingdered pairs simultaneously in statement −3 which corresponds to X shift−3, and we move v−4(v − 1) = (u + 2)2: one shape for the mystery to (1/4)(v + 5) by adding 5 and multiplying by (1/4)parabola and the other shape for the parent parabo- which corresponds to Y shift5|Yscale(1/4). Joiningla (changes we see as we “hop” back and forth these two programs together we move the mysterybetween parabolas). Knowing these two shapes, we parabola to the parent parabola underuncover moving directions to move from the mys-tery parabola to the parent parabola by interpreting Xshift−3|Yshift5|Yscale(1/4).the order of arithmetic operations that move (u, v) Wrapping Up.to (u + 2, −4(v − 1)). We move first coordinate u on We call the relationship between coordi-Virginia Mathematics Teacher vol. 44, no. 1 nates of ordered pairs on a parabola the shape of ordered pairs; it’s the algebraic shape of parabolas. Given a parabola we have shown how to build up its equation by moving the given parabola to the parent parabola where we know the shape of or- 49
dered pairs. Conversely we have shown how to Figure 7.look at an equation of a parabola, re-write it inparent form if necessary, and then see the shapes of how the richness of our approach fits in nicely withordered pairs for both the given parabola and the integrating big ideas in mathematics like inverseparent parabola embedded in the equation. (We motions (transformations and functions) and sym-“connect” these two shapes with two small hori- metry.zontal arrows in the middle of Figure 7.) These twoshapes of ordered pairs enable us to uncover direc- Harold Mick Benjamin Bazaktions for moving the given parabola to the parent Retired Faculty Mathematics Teacherparabola by following the order of operations in the Virginia Tech Patrick Henry High Schoolordered pairs’ coordinate expressions (see the sin-gle horizontal bold arrow in the middle of Fig- [email protected] [email protected] 7).Last Step. We leave you, the reader, with a challenge.We have shown that given an equation of a parabo-la in parent form we can uncover moving directionsin the equation to write a program that moves themystery parabola to the parent parabola. Referringto Problem 2, what might be a last step to sketchthe given equation’s mystery parabola? More spe-cifically, your task is to sketch the graph of Y = 4(X − 3)2 − 5knowing that program X shift−3|Yshift5|Yscale(1/4)moves the graph of the mystery parabola to theparent parabola. Here is a hint: Develop an inverseprogram that will enable you to complete the task.This development will not only add the missingbold arrow in Figure 7, but will help you discoverVirginia Mathematics Teacher vol. 44, no. 1 50
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