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Virginia Council of Teachers of Mathematics |www.vctm.org

V IRGINIA M ATHEMATICS T EACHER Vol. 44, No. 1 Fall 2017

Commun i t y of Heroes !

Virginia Mathematics Teacher vol. 44, no. 1

Editorial Staff

Dr. Agida Manizade Editor-in-Chief Radford University vmt@radford.edu

Dr. Jean Mistele Associate Editor Radford University jmistele@radford.edu

Ms. Alexandra Largen Assistant Editor Radford University

Ms. Cameron Leo Assistant Editor Radford University

Printed by Wordsprint Blacksburg, 2200 Kraft Drive, Suite 2050

Virginia 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 Tech

Membership Chair: Ruth Harbin-Miles

Virginia Lewis, Longwood University

Treasurer: Virginia Lewis

John McGee, Radford University

Webmaster: Ian Shenk

Laura Moss, Radford University

NCTM Representative: Lisa Hall

Andy Norton, Virginia Tech

Elementary Representatives: Meghann Cope; Vicki Bohidar

Matthew Reames, University of Virginia

Middle School Representatives: Melanie Pruett; Skip Tyler

Padhu Seshaiyer, George Mason University

Secondary Representatives: Pat Gabriel; Lynn Reed

Ryan Smith, Radford University

Math Specialist Representative: Spencer Jamieson

Maria Timmerman, Longwood University

2 Year College: Joe Joyner

Katy Ulrich, Virginia Tech

4 Year College: Ann Wallace; Courtney Baker

Jay Wilkins, Virginia Tech

VMT 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. The high-quality journal is only possible because of your dedication and hard work.

This issue had a 20% acceptance rate

Virginia Mathematics Teacher vol. 44, no. 1

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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 Math Girls ............................................................... 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

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Featured Awards

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 County

Virginia Mathematics Teacher vol. 44, no. 1

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Superheroes at Work: Note from the Editor Dr. Agida Manizade

In this issue, we are celebrating the contri- butions by the community of superheroes, individ- uals with extraordinary skills and abilities: our mathematics teachers. Every day, this group of exceptional professionals engage in incredible activities, such as  Helping young mathematicians to develop deep conceptual understandings of the subject matter through hands-on experiences, as described in Dana Johnson’s piece;  Motivating students to do mathematics, regard- less of existing challenges, as explained by Nicole Joseph’s research;  Finding and analyzing great examples of math- ematics in nature and the world around us, as presented by John Adam in his article about rainbows;  Designing high cognitive demand tasks, lessons and activities that promote mathematics dis- course in the classroom, similar to ideas by Denise Wilkinson; and  Inspiring children to see the beauty of mathe- matics as well as developing their problem- solving and critical thinking skills.

engage on a daily basis while dealing with pres- sures of standardized testing, often the lack of resources, and many other factors, not under their control, that can affect the teaching and learning process. These superheroes also participate in ongo- ing professional development in order to continue improving their own skills and abilities. We en- courage our readers to consider professional devel- opment (PD) opportunities discussed in this issue, as well as grant opportunities that can fund addi- tional PD. Our mathematics teachers are phenomenal individuals that continue their work and commit- ment to excellence in teaching mathematics to our children. We thank them for their invaluable ser- vice in shaping future generations.

Agida Manizade Editor in Chief, Virginia Mathematics Teacher vmt@radford.edu

These are just a few examples of extraordi- nary activities in which our mathematics teachers

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Message from the President Jamey Lovin

Yesterday my neighbor told me that her son had just finished his “What I Did This Summer” essay for the first day of school. She mentioned he referred to having done lots of math practice because THE MATH LADY lived next door! I was pleased, not

iate issue. Our issue? Providing services that attract and retain membership. How? Seeking a platform that reflects the diverse needs of Virginia educa- tors. We weren’t surprised to see many groups sharing the same goal for their organization. In fact, Matt Larsen, in his July 19, 2017 NCTM Pres- ident’s message, shared the following from the very first NCTM president: “Unless we give value received for the membership fee no one will be- come a member a second time” - that was 1921 - almost 100 years ago. In August, the Executive Board met to plan initiatives of “value” for educators in Virginia. We are very excited with the activities we have planned for this year and beyond. First on our planning list is the 2018 conference at Radford University. The theme will be Moving Mountains with Mathemat- ics . We will be inviting proposals that address four conference strands: teaching with the new standards, access and equity, enhancing teaching and learning, and enhancing instruction with tech- nology. We are also in the process of revamping our award and grant applications to make them more accessible. We promise to keep you updated about this and many more upcoming events spon- sored by VCTM and its affiliates. Check our web- site, www.vctm.org, and watch for email blasts and updates!

only because he had been working to keep his math skills sharp, but also because he had provided me with the perfect topic for this message ... VCTM: What We Did This Summer! In July, NCTM hosted a Leadership Con- ference in Baltimore, Maryland. The theme was Intent to Impact: Addressing Access, Equity, and Advocacy in Your Affiliate . Affiliate members from all over the United States discussed ways we can take action to provide a high-quality mathemat- ics education for each and every learner. In a presentation from Marilyn Strutchens, we were challenged not to tell a different version of the equity story, but to create one that was truly our own. The afternoon was filled with dynamic con- versations that shaped our discussions for the re- mainder of the conference.

Here’s to an awesome year!

Jamey Lovin, VCTM President Jamey.Lovin@vbschools.com

Figure 1. See https://www.storybasedstrategy.org/ the4box.html for information about this graphic and more access, equity, and advocacy conversations.

Cathy Blair, Chelsea Prue, Melissa Demlein (GRCTM), Lisa LoConte-Allen (TCTM), and I formulated an action plan to address our state Affil-

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Note from the Virginia Department of Education: Teaching Mathematics During the 2017-2018 Crosswalk Year Tina Mazzacane

During the 2017-2018 school year, Virginia mathematics teachers will include the 2009 and 2016 Mathematics Standards of Learning in both the written and taught curricula. This is known as a “crosswalk” year and requires curricula to blend two sets of standards, leading to full implementa- tion of the 2016 Mathematics Standards of Learn- ing in 2018-2019. The chart below summarizes the instructional and assessment implications for im- plementation of the 2016 Mathematics Standards of Learning: 2017-2018 School Year – Crosswalk Year  2009 Mathematics Standards of Learning and 2016 Mathematics Standards of Learning are included in the written and taught curricula.  Fall 2017 Standards of Learning Assessments measure the 2009 Mathematics Standards of Learning, but will not include field test items measuring the 2016 Mathe- matics Standards of Learning.  Spring 2018 Standards of Learning assessments meas- ure the 2009 Mathematics Standards of Learning and will include field test items measuring the 2016 Mathe- matics Standards of Learning . 2018-2019 School Year – Full-Implementation Year  Written and taught curricula reflect the 2016 Mathemat- ics Standards of Learning .  Fall 2018 End-of-Course (Algebra I, Geometry, and Algebra 2) and Spring 2019 (Grades 3-8 and EOC) Standards of Learning assessments measure the 2016 Mathematics Standards of Learning . The Virginia Department of Education provides several resources to assist teachers as they navigate the crosswalk year. The following re- sources are currently available on the 2016 Mathe- matics Standards of Learning (SOL) and Testing webpage:  2016 Mathematics Standards of Learning  2016 Mathematics Standards Curriculum Frameworks

 2009 to 2016 Crosswalk (summary of revi- sions) documents  2016 Mathematics SOL Video Playlist (Overview, Vertical Progression & Support, Implementation and Resources)  Narrated Crosswalk Presentations Teachers may also find resources from the 2017 Mathematics SOL Institutes useful in plan- ning instruction during the Crosswalk Year. The Virginia Department of Education is currently working to revise the following resources and anticipates posting updated electronic versions on the 2016 Mathematics Standards of Learning (SOL) and Testing webpage in 2018 as the work is finalized:  Updated and New Lesson Plans (Enhanced Scope and Sequence Lessons)  Updated Vocabulary Word Wall Cards  Virginia Board of Education Textbook Approv- al List  Instructional Video Resources for Teachers Educators with questions about the implementation of the 2016 Mathematics Standards of Learning should email mathematics@doe.virginia.gov.

Tina Mazzacane Mathematics Coordinator Virginia Department of Education

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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 teaching activity that is useful and powerful for instruction, motivation of students, and enrichment in math classrooms for grades 3 – 8. It is also a great game to teach to families. The game of

can be found through internet search terms “NCTM and Krypto.” Mental arithmetic is the preferred method of solving the hands. Some students may be al- lowed to work with paper and pencil. For young

students or those who need a more concrete approach, you may have them write the six numbers on a long strip of scrap paper. Then they tear the numbers apart, thus creating their own set of mini-cards for the hand. Some students are more suc- cessful when they can physically rearrange the numbers. When you work with young students, you may want to use only the numbers 1-10. For primary grade stu-

Krypto is a card game that consists of 56 cards. In the deck there are three each of the numbers 1 – 6, four each of 7 – 10, two each of 11 – 17, and one each of 18 – 25. Play begins by dealing a set of six numbers. The first five are combined in any order along with any of the operations +, –, x, or ÷ to obtain a result equal to the sixth number (called the objective or target num- ber).

For example, suppose the numbers dealt are 20, 15, 17, 3, 9, and 4. Here is a solution: 20 ÷ [(17 -15) + (9 ÷ 3)] = 4. This game is similar to the game called “24,” which uses four numbers that are printed on a card to get the objective number 24. In Krypto, the purpose of the cards is to generate numbers for use in the game. If you do not have cards in your classroom and you are playing as a whole-class activity, you may simply ask six students to choose a number between 1 and 25. Write the numbers on the board and have everyone work to find a solution. When a student finds a solution, s/he calls “Krypto” and explains it to the class. Students may also play in small groups with a deck of cards. The game is also an excellent soli- taire game. The National Council of Teachers of Mathematics has an online version of the game that

dents, you may allow them to solve the hand using fewer than all five numbers. For example, suppose the numbers dealt are 4, 3, 6, 8, 1 with an objective number of 7. Students may find solutions such as: 2-number solutions: 4 + 3 = 7 6 + 1 = 7 8 - 1 = 7 3-number solution: 4 + 6 – 3 = 7 4-number solution: 6 ÷ 3 x 4 – 1 = 7 5- number solution: 6 ÷ (8 ÷ 4) + 3 + 1 = 7 This strategy allows students to differentiate the activity for their own level of comfort. Over the last few decades I have played this game with students in grades 3 to 12. No one has ever cared much about scoring. The satisfaction seems to be in finding a solution or seeing someone else find one. Sometimes several students share

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different solutions and no one seems to care who gets points. But if you want to score, small groups can give one point to the first person to solve the hand correctly. I use a fun scoring method for whole-class teams – I divide the class in half and write the numbers for each hand on the board. The first person with a correct solution earns many points for the team’s score – the sum of the six numbers! If they call “Krypto” but cannot produce a correct solution, they have the sum of the six numbers SUBTRACTED from their team’s score. This minimizes impulsive, false claims. There are many possible benefits to playing this game in your classroom. May (1995) enthusi- astically describes and recommends the game of Krypto in an article on motivating activities for the math classroom. Way (2011) describes additional benefits of games to support mathematical cogni- tive objectives, including application of math skills in a context that is meaningful to students, building of positive attitudes towards math, increased skill levels, opportunities for students to participate at various levels of thinking, and opportunities to connect with families as students share the games at home. Lach and Sakshaug (2005) discuss their action research on games in a sixth grade class- room. Two of them, Muggins and 24, are similar to Krypto. After 12 weeks of playing math games the authors found students scored better in an assess- ment of algebraic reasoning. The game of Krypto does not present facts in the way flash cards do, but incorporates problem solving and pattern searching into fact practice. Beyond the obvious practice in mental arithmetic and developing number sense, it can be an environ- ment for applying properties of real numbers and the rules for order of operations. This game pro- motes the kind of number juggling used in factor- ing quadratic trinomials. Here are some examples:  Factoring quadratic trinomials . When we factor x 2 – 8x + 12 we are looking for numbers whose product is 12 and whose sum is -8. Once when I was teaching factoring to an 8 th grade algebra class, a student blurted out, “It’s easy. It’s just like Krypto!”

 Order of operations. Ask students to wr ite their solutions in correct notation, using rules for order of operations or “algebraic logic.” Once a student writes a solution, others can check. For example, if one student incorrectly writes 2 + 3 x 6 – (7 + 3) = 20 then others should note that parentheses are required around 2 + 3.  Commutative and Associative Properties . In comparing solutions, students will see that variations in grouping and order may produce the same result. For example, one student may write (3 + 2) x 1 + 4 + 5 = 14. Another may claim to have a different solution: 1 x (2 + 3) + 4 + 5 = 14. Students should recognize that two instances of commutative property are used – one for addition and one for multiplication. This can lead to a good discussion. Some stu- dents will say it is really the same solution and others will say they are different solutions.  Multiplicative Property of Zero. If you can make a zero from two of the numbers, you can eliminate other numbers that you don’t need.

Example: 6 4 3 6 22 Objective number is 7. Solution: (3 + 4) + (6 – 6) x 22 = 7

 Identity elements. If you can get the objec- tive number with one or two numbers, then try to get a zero or 1 with the remaining numbers. Multiply by one or add zero.

Example: 7 9 2 15 20 Objective number is 8.

Notice that 15 – 7 = 8. Can you get a 1 from the other three numbers? Yes, 20 ÷ 2 - 9. So the identity element for multiplication helps with a solution of (15 – 7)(20 ÷ 2 – 9) = 8 or 8 x 1 = 8.

Example: 2 4 5 3 7 Objective number is 5.

Notice that you are given a 5 in the middle of the set of five numbers. Can you get a

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zero from the other four numbers? One way is 2 [(4 + 3) – 7] = 0. The solution be- comes 5 + 2 [(4 + 3) – 7] = 5 or 5 + 0 = 5. Is it always possible to find a solution using all five numbers to get the objective? It is rare, but sometimes there is no solution. Here are the only two examples I have encountered in the 45 years I have been playing the game. 3, 23, 13, 16, 1 Objective = 20 9, 9, 7, 16, 4 Objective = 25 If you search the internet for a “Krypto solver” you will find some sites that will check your hand for you. If it is solvable, they do it for you. If not, they will tell you it is unsolvable. Whenever we hit a tough hand in a classroom that no one seems to be able to solve, I write it on the corner of the board. By the next day, it always seems to be solved by someone! As a challenge for middle school students who seem adept at this game, I give five numbers but no objective. They write the five numbers at the top of a piece of lined notebook paper. They write the numbers 1-25 down the margin on the left. This gives them 25 possible objectives. An example is given in the next column. This activity can be done by individuals or groups. It is an excellent setting to motivate the use of order of operations notation. The game of Krypto can also be used as a classroom management activity. I use it to fill the last minute or two before dismissal. I tell students to cross their arms when they are ready to leave. I quickly write the numbers on the board and ask them to solve the hand using only mental arithme- tic. They stay focused and quiet while thinking. If it is solved before the bell, I generate a new hand. This strategy helps make every minute count in math class! References Lach, 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.

Way, J. (2011). Learning mathematics through games. Series 1: Why games? Retrieved from http://nrich.maths.org/2489.

Use these five numbers to get each of the objectives listed on the left. Use correct Order of Operations symbols! Two of them are done for you. 3 6 8 25 22

1 =

2 =

(25 – 22) ÷ 3 x (8 – 6)

3 =

4 =

22 ÷ (25 – 14) + 8 – 6

5 =

6 =

7 =

8 =

9 =

10 =

. . .

23 =

24 =

25 =

Dana T. Johnson Retired Faculty College of William and Mary dtjohn@wm.edu

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An Example of Nature’s Mathematics: The Rainbow John A. Adam

with the horizontal, just as the direction to the top of a tree makes an angle with the direction of its shadow on the ground (in fact that angle is exactly the solar altitude!). By making a large paper cone to mimic the ‘rainbow cone’ and varying the angle at which students hold it, (Figure 3), they will see that if the sun is very low (i.e. close to the horizon), then the rainbow arc is almost a complete semicir- cle, whereas if the sun is too high (altitude greater than 42 o ), then the top of the rainbow is below the horizon and therefore not visible (unless the ob- server is on a hill or in flight; see http:// www.slate.com/content/dam/slate/blogs/ bad_astronomy/2014/09/01/ circular_rainbow.jpg.CROP.original-original.jpg). If the student (or anyone!) is fortunate enough to see a nearly semicircular rainbow, then the angle between the two ‘ends’ of the rainbow and the observer – its ‘angular diameter’ – is twice 42 o , which is not far from a right angle! What about middle-school students? In the summer of 2015 I was privileged to teach a dozen specially selected 6 th – 8 th grade students in the Virginia STEAM Academy at Old Dominion Uni- versity. The acronym refers to Science, Technolo-

Introduction.

It is the author’s contention that ‘nature’ is a wonderful resource and vehicle for teaching stu- dents at all levels about mathematics, be it qualita- tively at elementary schools (shapes, circular arcs, polygonal patterns) or more quantitatively at mid- dle and high schools (geometrical concepts, alge- bra, trigonometry and calculus of a single variable). This was the motivation for writing A Mathemati- cal Nature Walk (as well as the somewhat more advanced Mathematics in Nature ). Within the realm of nature the subject of meteorological optics is a particularly fascinating one; it includes the study of the rainbow as well as others such as ice crystal halos and glories. Obviously there is some physics involved in the explanation of these phe- nomena, but fortunately it is not necessary to go into a lot of physical detail in order to appreciate the value of geometry, trigonometry and high- school calculus concepts used in modeling the beautiful rainbow arcs in the sky. For students in elementary school there is a variety of angle-based concepts that can be ad- dressed when discussing rainbows. Thus, ‘solar altitude’ is the angle the direction to the sun makes

Virginia Mathematics Teacher vol. 44, no. 1

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in terms of the angles of incidence ( i ) and reflec- tion ( r ) respectively (see Figure 1 where the inci- dent ray is refracted and reflected inside the spheri- cal drop; Figure 2 illustrates the ray path for the secondary bow). But what is this angle ? Essentially it is the direction through which an incoming ray from the sun is ‘bent’ by its interaction with the drop to reach the observer’s eye (the reader is re- ferred to the caption for Figure 1 for more details). The angle of refraction inside the drop is a function of the angle of incidence of the incoming ray. This relationship is being expressed in terms of Snell's famous law of refraction, namely sin i = n sin r , where n is the relative index of refraction (of water, in this case). This relative index is defined as the ratio of the speed of light in medium I (air) to the speed of light in medium II (water); note that n > 1; in fact n ≈ 4/3 for the rainbow, but it does depend slightly on wavelength (this is the phenom- enon of dispersion, and without it we would only have bright ‘whitebows’!). The article by Austin & Dunning (1991) provides a helpful summary of the ‘calculus of rainbows,’ as does the even briefer ‘Applied Project’ in Chapter 4 of Stewart (1998). In view of Snell’s law the high school stu- dent should attempt to write the angle of refraction r in terms of the angle of incidence i using the inverse sine function, thus:

gy, Engineering and Applied Mathematics. The topics covered included rainbows, ice crystal halos, water waves, glitter paths and sunbeams; addition- ally, the topics ‘Guesstimation’ (i.e. back-of-the- envelope problems that require estimation) and ‘dimensional analysis’ (i.e. what happens as things get bigger?) were incorporated into the week-long class. Given that the mathematical background of these students included algebra, geometry and trigonometry, much of the material discussed in this article was covered, and the results from the calculus-based topics were presented qualitatively (and very successfully) by engaging the students on their understanding of maxima and minima, and applying those ideas in this context. Doing the mathematics. The primary rainbow is caused by light from the sun entering the observer's eye after it has undergone one reflection and two refractions in myriads of raindrops. An additional internal reflec- tion produces a frequently-observed secondary bow, and so forth (but tertiary and higher bows are rarely, if ever, seen with the naked eye for reasons discussed below). By adding all the contributions to angular deviations of the ray from its original direction, the middle- or high-school student can verify that for a primary bow the ray undergoes a total deviation of D ( i ) radians, where

   2

 



 

D i

2         i r r 

2 4 , (1) i r i

i

sin



  

r

arcsin  

. (2)

n



Hence equation (1) may be rewritten as

i

sin

  

  

 

D i

2 4arcsin i

   

. (3)

n

Figure 1. The path of a ray inside a spherical raindrop which, along with myriads of other such drops, contributes to the formation of a primary rainbow ( k = 1). The devia- tion angle D ( i ) referred to in the text (see equation (1)) is the obtuse angle between the extension of the horizontal ray from the sun and the extension of the ray entering the observer’s eye. Its value is approximately 138 o . Its supple- ment, 42 o , is the semi-angle of the ‘rainbow cone’ in Figure 3.

Figure 2. The corresponding ray path for the secondary rainbow, arising because of a second reflection within the raindrops.

Virginia Mathematics Teacher vol. 44, no. 1

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By examining the graph of D ( i ) in Figure 4 [0, π/2] (which is the only interval of interest for physical reasons), the condi- tion for an extremum (minimum in this case) im- plies there exists a critical angle of incidence i c such that D '( i c ) = 0. To prove this the student may either use implicit differentiation of equation (1) (with subsequent use of Snell’s law) to obtain   0, / 2 i   it is seen that for i

the expression

   2

 





  

D i

2         2 2 1 i r k r k i  

k r i

i

sin

  

  





k i 

2 2 1 arcsin k

=

  

. (9)

n

Note that the result in equation (9) is modulo 2π. Although realistically k ≤ 2 (see below for details), with k internal reflections the corresponding result for the critical angle of incidence that gives rise to the minimum deviation is

cos i di n r  cos dr

, (4)

1/2

or directly differentiate the expression (3) and equate it to zero to find the critical angle i c from the resulting expression below, i.e.

2

 

 

n

1

(10)

i



arccos

.

   

c

k k

( 2)

This result is established using exactly the same method to arrive at equation (6). For the primary bow ( k = 1) this reduces (as it should) to equation (6) above. Additionally, equation (9) reduces to equation (3) for k = 1 since k ( k + 2) = 3. It is an interesting trigonometric exercise to eliminate all dependence on the angle of incidence (as Kepler did in 1652) to prove from equation (8) that

2

cos 1 cos i

1



, (5)

2

2

n

i

4

 

from which it can be found that

1/2

2

  



n

1

i i

 

arccos

. (6)

 

c

3

Thus, with a generic value for n of 4/3, i c radians, or about 59.4 o for the primary bow. As noted above this extremum is a mini- ) > 0, as can be shown by differenti- ating the expression (1) a second time. In fact, by noting from equation (1) that D "( i ) = –4 r "( i ) and utilizing equation (4) it follows that (after some algebraic manipulation) mum , i.e. D "( i c ≈ 1.04

3/2

   

   

2

2 1 4  

 

n

  c

D i



2arccos

.

(11)

 

n

3



To achieve this, rewrite equation (8) as

1/2

1/2

2

2

  

  

  

  

D i

( )





n

n



1

4



c

A  



arccos



2arcsin

2

2

n

2

3

3





2

n



1 sin

2

d r

i

(7)

 



0,

≡ A – 2 B , (12).

2

3

3

di

n

r

cos

Then, by expanding the equation

So, D "( i c

) > 0 as indicated. Note that in this in-

stance it was not necessary to specify i c so the re- sult is a global one, i.e. the concavity of the graph of D ( i ) does not change in [0, π / 2], the interval of physical interest. Exercise for the student: Using equations (3) and (6) show that the minimum angle of deviation (the ‘rainbow angle’) is Each internal reflection adds an amount of π - 2 r radians to the total deviation of the incident ray. Thus, for k internal reflections within a raindrop, a term k (π - 2 r ) is added to the angle through which an incident ray is deviated, (see Figure 2, for the secondary bow, k = 2), yielding 1/2 1/2 2 2 2 1 4 ( ) 2arccos 4arcsin . 3 3 c n n D i n                   (8)

D i

( )

   

c

sin( 2 ), A B

sin

 

 

 

2

it is possible to write cos[ D ( i c )] in terms of sin A , cos A , sin B and cos B , each of which can be found easily from the definitions of A and B in equation (12). The result is a rather nasty expression which can be reduced algebraically to equation (11). Voilà! This has been generalized to higher-order bows (see Adam 2008), but it would take us too far afield to describe here; essentially the same ideas are involved. Some numerical values. Thus far, we have been describing a gener- ic, colorless type of rainbow. For a ‘generic’ mono- chromatic rainbow (the ‘whitebow’ referred to

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above), the choice n = 4/3 yields, from expression (11),

ly wider) dispersion occurs for the secondary bow, but the additional reflection reverses the sequence of colors, so the red color in this bow is on the inside edge of the arc. In principle more than two 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- tional reflection of course is accompanied by a loss of light intensity because of transmission out of the drop at that point, so on these grounds alone, it would be expected that even the tertiary rainbow ( k = 3) would be difficult to observe or photograph without sophisticated equipment; however recently several orders beyond the secondary have been identified and photographed (see the cited articles by Edens, and Edens & Können). The reader’s attention is also drawn to the superb website on atmospheric optics, in particular the following link: http://www.atoptics.co.uk/rainbows/ord34.htm. It is possible to derive the angular size of such a rainbow after any given number of reflec- tions using equations (9) and (10) (Newton was the first to do this). Newton’s contemporary, Edmund Halley, noted that the third rainbow arc should appear as a circle of angular radius nearly 40 o around the sun itself. The fact that the sky back-

3/2

   

   

   

   

9 20

     

  c

D i

D



arccos



138 .

(13)

16 27

The supplement of this angle ( ≈ 42 o ) is the semi- angle of the rainbow ‘cone’ formed with apex at the observer's eye, the axis being along the line joining the sun to the eye, extended to the antisolar point (see Figure 3).

So what happened to the colors of the rain- bow? They have of course been there all along, and all we need to do is to utilize the fact that the re- fractive index n is slightly different for each wave- length of light. Blue and violet light get refracted more than red light; the actual amount depends on the index of refraction of the raindrop, and the calculations thereof vary a little in the literature, because the wavelengths chosen for red and violet may differ slightly. Thus, for red light with a wave- length of 656 nm (1 nm = 10 –9 m), the cone semi- angle is about 42.3 o , whereas for violet light of 405 nm wavelength, the cone semi-angle is about 40.6 o an angular spread of about 1.7 o for the primary bow. (This is more than three times the angular width of a full moon!) The corresponding values of the refractive index differ very slightly: n ≈ 1.3318 for the red light and n ≈ 1.3435 for the violet – less than a one percent increase! Similar (though slight- Figure 3. The ‘rainbow cone’ for the primary rainbow. For the secondary bow ( k = 2) the cone semi-angle is approxi- mately 51 o , as may be calculated from equations (9) and (10).

Figure 4. The graph of the deviation angle D ( i ) for the pri- mary bow from equation (3) as a function of the angle of incidence (both in radians). Note that the minimum devia- tion of approximately 2.4 radians (≈ 138 o ) occurs where the critical angle of incidence i c ≈ 1.04 radians (≈ 59.4 o ). These values may be calculated directly using equations (3) and (6). The graph shows that above and below i c there are rays deviated by the same amount (via the horizontal line test), indicating that at i c these two rays coalesce to produce the region of high intensity we call the rainbow.

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ground is so bright in this vicinity, coupled with the intrinsic faintness of the bow itself, would make such a bow almost, if not, impossible to see or find without sophisticated optical equipment. Exercise for the student: Use the gener ic value for the refractive index of water, n = 4/3, in equa- tions (9) and (10) to show for the tertiary rainbow ( k = 3) that i c ≈ 70.6 o and D ( i c ) ≈ 321 o , so the ‘bow’ is at about an angle of 39 o from the incident light direction. In fact, this will appear behind the observer as a ring around the sun! Exercise for the student: Calculate the angular width subtended at the eye by a ‘baby aspirin’ held at arm’s length. Then see if you can ‘cover’ the full moon by extending your arm while holding the aspirin! An experiment: “road-bows.” Have you ever noticed a rainbow-like re- flection from a road sign when you walk or drive by it during the day? Tiny, highly reflective spheres are used in road signs, sometimes mixed in paint, or sometimes sprayed on the sign. Occasion- ally, after a new sign has been erected, quantities of such ‘microspheres’ can be found on the road near the sign (see http://apod.nasa.gov/apod/ ap040913.html for an excellent picture of such a bow). I have had my attention drawn to such a find by an observant student! It is possible to get sam- ples of these tiny spheres directly from the manu- facturers, and reproduce some of the reflective phenomena associated with them. In particular, for glass spheres with refractive index n ≈ 1.51 scat- tered uniformly over a dark matte plane surface, a small bright penlight provides the opportunity to see a beautiful near-circular bow with an angular radius of about 22 o (almost half that of an atmos- pheric rainbow). In such an experiment this bow appears to be suspended above the plane as a result of the stereoscopic effects because the observer’s eyes are so close (relatively) to the spheres com- pared with passing several yards from a road sign. More details of the mathematics can be found in the article by Crawford (1998) and Chapter 20 of Adam (2012).

Note that in the list of topics below each meteorological phenomenon can be examined as a topic in mathematical physics because the subject of optics is very mathematical. At times, it required very sophisticated mathematics. The author recom- mends another enrichment activity in which stu- dents search for each of the topics (and others) below on the ‘Atmospheric Optics’ website men- tioned above: http://atoptics.co.uk/. There is a vast selection of topics (with many photographs) to choose from, including shadows, ice crystal halos around the sun or moon, ‘sundogs,’ reflections, mirages, coronas, glories, sun pillars as well as, of course, rainbows. The advantage of this site (and its ‘sister’ site, Optics Picture of the Day (OPOD: http://atoptics.co.uk/opod.htm)) is that students at all levels, elementary, middle and high school, will be able to find material of interest to them. These sites are replete with straightforward physical ex- planations and illustrations of the phenomena, but there is little, if any, mathematical discussion so they can be appreciated in a scientifically accurate way by students at any level of mathematical profi- ciency. The book A Mathematical Nature Walk , together with the more advanced Mathematics in Nature cited in the bibliography, can provide a starting point for both teachers and students inter- ested in pursuing some of the mathematical aspects of these phenomena. As a further example, a very brief description of glories (with an associated ‘student teaser’) is provided below. Although ice-crystal halos are only briefly mentioned in the preceding paragraph, students at all levels can be encouraged to look for them. These can appear around the sun or full moon with surprising frequency (though it must be empha- sized again that you should never look directly at the sun; block it off with your hand or a convenient chimney!). They are formed by sunlight passing through myriads randomly oriented, nearly regular hexagonal prismatic ice crystals, composing cirrus clouds, the very highest type of cloud we generally see (during the day at least). In distinction to rain- bows, the most common halos are smaller in angu- lar radius (about 22 o as opposed to 42 o ) and exhibit a reddish tinge on the inside of the arc – a reversal of colors compared with the primary bow. This is

Related topics in meteorological optics.

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because, unlike the mechanism producing the pri- mary bow, there is no reflection occurring within the crystals to produce these particular halos, only refraction. I live about a mile from Old Dominion University and walk to my office; as a result I gen- erally see such halos (and other types also) several times a month; sadly, far more frequently than I witness rainbows. On an otherwise clear night, a full moon embedded in a thin cirrus cloud may exhibit similar such halos, which can be quite prominent by virtue of the moon being so much less bright than the sun. Indeed, I have frequently been contacted by friends and students who witness the latter but have never noticed a halo around the sun! Exercise for the student: Imagine a r egular hexagonal prism with a light ray entering side ‘1’, and exiting side ‘3’ (see the Atmospheric Optics website for more details and its interactive ‘mouse’ tasks to discover the minimum angle of deviation for both rainbows and halos). Using the same geo- metric, trigonometric and calculus concepts applied to rainbows in the body of the article, show that the minimum angle of deviation for such rays is about 22 o , the angular radius of the most commonly visi- ble halo. Student teaser: When I lived in England I saw many more rainbows than I do living in Norfolk, Virginia. Why do you think this was? (No, it was not because the annual rainfall where I lived was more than it is in Norfolk – in fact it’s rather less!). Think about latitude : I lived at about 52 o N; now I live at about 37 o N, fifteen degrees further south. (You can imagine how excited I was to see the constellation of Orion and Sirius (the ‘Dog star’) so much higher in the winter night sky than when I lived in England!) Glories. Mountaineers and hill climbers have no- ticed on occasion that when they stand with their backs to the low-lying sun and look into a thick mist below them, they may see a set of colored concentric circular rings (or arcs thereof) surround- ing 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 her head. They may also be seen (if you know where to

look) from airplanes. This is the meteorological optics phenomenon known as a glory. Cloud drop- lets essentially ‘backscatter’ sunlight back towards the observer in a mechanism similar in part to that for the rainbow. The glory, it is sometimes claimed, is formed as a result of a ray of light tan- gentially incident on a spherical raindrop being refracted into the drop, reflected from the back surface and reemerging from the drop in an exactly antiparallel direction (i.e. 180 o ) into the eye of the observer, but this is actually incorrect (see the ‘student teaser’ below). Student teaser: Why is the r ay path allegedly associated with the formation of a ‘glory’ as illus- trated in Figure 5 (and in some meteorology text- books) incorrect ? Use equation (3) to investigate this.

Figure 5. An incorrect ray path explanation for the glory.

Conclusion.

This article presents some of the basic mathematical concepts and techniques undergird- ing a relatively common (and beautiful) phenome- non in meteorological optics. The analysis present- ed here does not contain new mathematics; it can be found from many sources because the subject of meteorological optics has been around for a very long time! What is emphasized, however, is the presentation of these ideas as a potential enrich- ment topic for (i) ‘qualitative’ mathematical model- ing in elementary classrooms and (ii) more quanti- tative approaches in middle and high school class- rooms. It should also be noted that the many more subtle features associated with these and other optical effects in the atmosphere require far more powerful and sophisticated mathematical tools to explain them. Nevertheless (though space does not permit it), aspects of the above-mentioned phenom- ena of ice crystal halos and glories may also be discussed at the level presented here. More details

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natural fifth-order rainbow, Applied Optics 54, B26–B34. Edens, H. E. and Können, G. P. (2015). Probable photographic detection of the natural seventh-order rainbow, Applied Optics 54, B93–B96. Stewart, J. (1998) Calculus: Concepts and Contexts (Single variable). Pacific Grove, CA: Brookes/Cole Publishing Company. Appendix: More mathematical patterns in na- ture. What follows below is obviously only a partial list of patterns that the attentive observer might see on a “nature walk,” and could form the basis of enrichment activities at all levels of student exposure to mathematics. The elementary, middle or high school teacher could adapt the material for his or her own students. Here are some possible topics: Basic two dimensional geometric shapes that occur (approximately) in nature can be identi- fied:  Waves on the surfaces of ponds or puddles expand as circles;  Ice crystal halos commonly visible around the sun are generally circular;  Rainbows have the shape of circular arcs ( as noted already );  Tree growth rings are almost circular. But there are many other obviously non-circular and non-planar patterns:  Hexagons: snowflakes generally possess hexag- onal symmetry;  Pinecones, sunflowers and daisies (amongst other flora) have spiral patterns associated with the well-known Fibonacci sequence;  Ponds, puddles and lakes give scenes of ap- proximate reflection symmetry (depending on the position of the observer);  Cross-sections of various fruits also exhibit interesting symmetries;  Spider webs have polygonal, radial and spiral- like features;  Long bendy grass has an approximately para- bolic shape;  Starfish (suitably arranged) exhibit pentagonal symmetry;

may be found in the references listed. It is hoped that this article will also ‘whet’ the appetite of interested instructors and students to pursue these aspects in more detail. A further suggestion may be made. The website Earth Science Picture of the Day (EPOD: epod.usra.edu), which is a service of the Universi- ties Space Research Association (USRA), publish- es photographs from a variety of subject areas: geology, oceanography, space physics, meteorolog- ical optics, agriculture, and many more. Anyone is invited to submit their photograph of an interesting optical or geological phenomenon, and is encour- aged to write a short summary for the layman ex- plaining the picture and, where possible, the basic science behind it. A recent submission by the au- thor (August 15 th , 2016), for example, uses simple proportion to estimate the height of a tree canopy using the ‘pinhole’ elliptical patches of light cast on the ground by gaps in the leaves of the tree (http://epod.usra.edu/blog/2016/08/estimating-tree- height-using-natural-pinhole-cameras.html). For a propos that is the topic of this article, see http:// epod.usra.edu/blog/2017/07/streaky-rainbow-in- zion-national-park.html. The site provides useful educational links for the daily pictures and is a valuable resource for teachers and students alike. References Adam, J. A. (2006). Mathematics in Nature: Mod- eling Patterns in the Natural World . Princeton, NJ: Princeton University Press. Adam, J. A. (2008). Rainbows, Geometrical Opt- ics, and a Generalization of a result of Huygens, Applied Optics , 47, H11 - H13. Adam, J. A. (2009). A Mathematical Nature Walk . Princeton, NJ: Princeton University Press. Adam, J. A. (2012). X and the City: Modeling Aspects of Urban Life . Princeton, NJ: Princeton University Press. Austin, J.D. and Dunning, F. B., (1991). Mathe- matics of the rainbow. In Applications of Secondary School Mathematics (Readings from the Mathematics Teacher) , 271 – 275. Crawford, F.S., (1988). “ Rainbow dust .” A merican Journal of Physics 56, 1006 – 1009. Edens, H. E. (2015). Photographic observation of a

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 The raindrops that scatter "rainbow" light into the eye of an observer essentially lie on cones with vertices at the eye ( as discussed in this article ) ;  Cloud patterns, mud cracks and also cracks on tree bark can exhibit polygonal patterns;  Clouds can also form wavelike "billow" struc- tures with well-defined wavelengths, just as with ripples that form around rocks in a swiftly flowing stream;  In three dimensions, snail shells and many seashells and curled-up leaves are helical in shape and tree trunks are approximately cylin- drical. In view of these patterns, even at an ele- mentary level, many pedagogic mathematical in- vestigations can be developed to describe such patterns - for example estimation, measurement, geometry, functions, algebra, trigonometry and calculus of a single variable. Basic examples might include:  The use of similar triangles and simple propor- tion;  A table of tangents to estimate the height of trees;  Measuring inaccessible horizontal distances using congruent triangles. Simple proportion can again be used in estimation problems, such as:  Finding the number of blades of grass in a certain area, or the number of leaves on a tree. More geometric ideas appear when studying topics such as:  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" and geometric similarity can also be investigated. This

applies in particular to the size of land animals; the relationship of surface area to volume, and its im- plications for the relative strength of animals. By considering (and constructing) cubes of various sizes, much insight can be gained about basic bio- mechanics in the animal kingdom, and much fun (and learning!) may be had by thinking about such questions as:  Why King Kong could not really exist, and  Why elephants are not just large mice. Furthermore, simple ideas such as scale enable us to compare, at an elementary level, me- tabolism and other biological features (such as strength) in connection with pygmy shrews, hum- mingbirds, beetles, flies and other bugs, ants and African elephants to name a few groups! Note: This appendix is adapted fr om A MATHEMATI- CAL NATURE WALK by John A. Adam. Copyright © 2009 by Princeton University Press. Reprinted by permission. Note: Figur es 1, 2, 4 and 5 ar e r epr oduced fr om A MATHEMATICAL NATURE WALK by John A. Adam. Copyright © 2009 by Princeton University Press. Reprinted by permission. Figure 3 is reproduced from X AND THE CITY: MODELING ASPECTS OF URBAN LIFE by John A. Adam. Copyright © 2012 by Princeton University Press. Reprinted by permission.

John A. Adam Professor Old Dominion University jadam@odu.edu

The author would like to thank the reviewers for their detailed and constructive criticism that resulted in a much improved version of the article.

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