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

VIRGINIA MATHEMATICS

TEACHER

Vol. 47 No. 2

Mathematics in Nature

Virginia Mathematics Teacher vol. 47, no. 2

Editorial Staff

Dr. Agida Manizade Editor - in - Chief Radford University

Dr. Jean Mistele Associate Editor Radford University

Ms. Grace Chaffin Assistant Editor Radford University

Ms. Mia Bialobreski Assistant Editor Radford University

Ms. Grace Hurst Assistant Editor Radford University

Ms. Adrienne Young Assistant Editor Radford University

Printed by Wordsprint Blacksburg, 2200 Kraft Drive, Suite 2050 Blacksburg, Virginia 24060

Virginia Council of Teachers of Mathematics

Many Thanks to our Reviewers for 47(2)

Jean Mistele, Radford University Darryl Corey, Radford University Anthony Dove, Radford University

President: Lynn Reed

President - Elect: Agida Manizade; Past Presidents: Pam Bailey,

Ann Wallace, James Madison University Sarah Ferguson, Old Dominion University Melva Grant, Old Dominion University Stephen Brazelle, Riverview & Livingston Elementary Jessica Hofer, Spotsylvania County Schools

Secretary: Kim Bender; Historian: Beth Williams

Treasurer: Virginia Lewis

Complete VMT Editorial Staff

Webmaster: Ian Shenk

Agida Manizade, Editor - in - Chief, Radford University

NCTM Representative: Skip Tyler

Jean Mistele, Associate Editor, Radford University

Darryl Corey, Section Editor, Radford University

Elementary Reprs: Tracy Profitt, Tammy Sanford, Scarlett Kibler

Alexander Moore, Section Editor, Virginia Tech

John Adam, Section Editor, Old Dominion University

Middle School Representatives: Lisa LoConte - Allen, Kathleen Stoebe

Eric Choate, Section Editor, Radford University

High School Reps: Regan Davis, Timothy Barnes, Fallon Graham

Grace Chaffin, Assistant Editor, Radford University

Grace Hurst, Assistant Editor, Radford University

Math Specialist Representative: Allison DePiro

Adrienne Young, Assistant Editor, Radford University

Mia Bialobreski, Assistant Editor, Radford University

Two - Year College: Doniray Brusaferro, Nikki Harris

Four - Year College: Theresa Wills, Toni Sorrell, Darryl Corey

VMT Editor in Chief: Dr. Agida Manizade

VMT Associate Editor: Dr. Jean Mistele

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Table of Contents:

Note from the Editors ............................................ 5

Message from the President .................................. 6

Introducing VCTM Scholarships Namesakes ........ 7

Culturally Responsive Teaching of Mathematics 11

All About The Number Line ................................. 17

Math Jokes ........................................................... 22

Mathematics in Nature ........................................ 23

Notes from the Field …. ........................................ 33

NASA Resources ................................................. 37

Busting Blockbusters ........................................... 38

Technology Review .............................................. 39

Tips From Practitioners ...................................... 43

Math Jokes ........................................................... 47

Meet Your Representatives .................................. 48

Affiliate Information ............................................ 49

Good Reads ......................................................... 50

Call For Manuscripts .......................................... 54

HEXA Challenge ................................................. 55

Grant and Scholarship Opportunities ................. 57

Note From the VDOE ......................................... 58

Key to the 47(1) Puzzlemaker .............................. 62

Solutions to 47(1) HEXA Challenge .................... 63

The Puzzlemaker .................................................. 67

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Mathematics in Nature: Note From the Editors

In this volume of the journal, we are excited to in troduce two new sections: Mathematics in Nature, edited by the section editor Dr. John Adam from Old Dominion University, and Culturally Respon sive Teaching (CRT) of Mathematics, edited by Dr. Darryl Corey from Radford University. Mathematics in Nature is the section in which we feature articles that examine nature through the lens of mathematics. These articles provide a con text in which mathematics can be taught in K - 14 classrooms. In the current issue, the article featured in the section is titled, Modeling Climate Change and is appropriate for high school students as well as for community college level mathematics. This article focuses on modeling climate change and includes topics from elementary algebra, quadratic equations, geometric series, introductory calculus, and introductory differentials. The ideas presented here can be incorporated not only in the mathemat ics classroom, but also into physics and chemistry lessons, which makes this section appropriate as a resource for any STEM classroom. Culturally Responsive Teaching of Mathematics focuses on research - based approaches on teaching mathematics, such as CRT. The articles presented here connect students ’ cultures, languages, and life experiences with what they learn in school mathe matics with the goal to develop higher - level aca demic skills through equitable practices in the classroom. In this volume, we are featuring the ar ticle titled Implementing a Rethinking Schools Les son Plan in Middle School Mathematics by Alex Moore. The article is on a lesson plan that allows students to explore the reality of living and paying

for expenses if one works at minimum wage job.

In addition, this volume presents a wide range of articles that are appropriate for elementary through college - level mathematics, including articles: All About the Number Line by Stiffler, Hofer, and Bra zelle, Graspable Math as a Visual - Dynamic Ex pressions and Equations Manipulator by Moore, Engaging Students in the Digital Math Classroom by Badgett and Amis, Good Reads by Gretchen Lee, Ashley Dreesch, and Ashley Carneal focused The Five Practices for Orchestrating Mathematics Discussion , as well as a multitude of resources from NASA and VDOE. We open this issue by introducing five prominent Virginia educators who offered scholarships and grants for students and teachers across the Com monwealth. Their stories about educators are shared in the, Introducing VCTM Scholarships Namesakes, section. We hope you enjoy this issue and continue engaging with the Virginia Council of Teachers of Mathematics.

Dr. Agida Manizade & Dr. Jean Mistele

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Introducing VCTM Scholarships ’ Namesakes This section is dedicated to recognizing namesakes who provide funding for students and teachers displaying financial need. We greatly appreciate all of our donors' dedication and support. For more information on grants and scholarships offered through VCTM and NCTM, please see the section "Grant and Scholarship Op portunities".

on the front cover with a full - page describing his professional and personal life. Each year, VCTM awards up to three scholarships of $2,000 each for college students who are pursuing a career as a teacher of mathematics to honor and remember the work of our colleague and friend.

Ed Anderson

Ed Anderson began his career at the Northern Vir ginia Community College, Manassas Campus in 1989 as an adjunct faculty member. He became an associate professor in the math department in 1993 and served there until his retirement in March 2011. His love for mathematics and for teaching was demonstrated by his many years of dedicated service. Ed served on the Board of the Virginia Council of Teachers of Mathematics for many years. He chaired the Mathematics Scholarship committee and served as the Parliamentarian. He knew Rob erts Rules of Order, and kept all Board meetings in compliance with them. Anderson also served as co advisor for Phi Theta Kappa, Honor Society for many years. Ed loved people. For many years he played the role, very well, of Santa Claus at the annual Senior Citizen ’ s Holiday Luncheon at the Manassas Cam pus. He will be remembered as a kind, gentle and loving person, and a wonderful math instructor. After heart surgery in 2010, Anderson was unable to continue teaching. However, he did continue to mentor his colleagues and offer tutoring assistance to his students until his death in August of 2011. In his memory, VCTM honored Mathematics Pro fessor Anderson by dedicating its spring 2012 jour nal issue to him (31(2)). Anderson ’ s photo appears

Stuart Flanagan

Dr. S. Stuart Flanagan, a life - long educator, always recognized the need for education. He says, "Giving to education is a way to have a chance at improving mankind. We need leaders in this coun try who can speak and think well. We need stu dents from all backgrounds who are able to be those leaders and make contributions to our socie ty. ” Dr. Flanagan has modeled this for all of us. Dr. Flanagan grew up in Fluvanna County. He spoke often about his parents. They both placed high emphasis on education for all of their chil dren, and expected them to give back to their com munity. Dr Flanagan took these teachings to heart throughout his life.

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He attended Washington and Lee University and attended the University of Virginia to earn a M.Ed., and an Ed.D. He began his teaching career at Varina High School in Henrico County. Afterwards, he taught at St. Christopher ’ s School in Richmond, Virginia, and served as chair of the mathematics department for ten years. He also taught mathematics at the University of Virginia and Virginia Common wealth University. After earning his doctorate, he became a professor at the College of William and Mary, where he taught for 30 years. During this time, he taught mathematics and mathematics edu cation courses including research, testing, and cur riculum development. On retirement, he earned the Professor Emeritus status. During his tenure at the College of William and Mary, he developed test items for several projects at the local and state level, directed several NSF grants, consulted with school divisions statewide, authored numerous articles, and was a senior con sultant to CBS Publications for the mathematics series Mathematics Unlimited. Additionally, he was a grader for the ETS Advanced Placement Program. During the late '80s and early '90s, he was known throughout Virginia for his highly suc cessful Literacy Passport Test materials, used ex tensively throughout the Commonwealth of Virgin ia. These materials, provided data for each child on every standard, enabled a number of districts to dramatically increase their student - passing rate. The conceptual model for these materials is the basis for the Tests for Higher Standards. Dr. Flanagan was the FIRST President of the Vir ginia Council of Teachers of Mathematics. In 1991 he received the 1991 Outstanding College Mathe matics Teacher Award from VCTM. He said, “ Getting an award is always an honor, but getting one from your peers is even more special, ” LaRay Mason, then chairwoman of the Council ’ s Teacher of the Year Committee spoke when Dr. Flanagan received this award. She said, “ Dr. Flanagan's stu dents consistently evaluate him as an outstanding professor in all areas. He remains very knowledge able of subject matter, open to diverse opinion, available for consultation and very effective in

overall teaching. Their comments make it evident Dr. Flanagan is a master teacher of teachers. ”

In March of 2012, Dr. Flanagan generously started providing funds for one teacher, or for a team of teachers to design and fund a project that meets the needs of the students in their classrooms. The Flanagan Grant is managed through VCTM and is awarded yearly, providing resources that are not usually available to classroom teachers.

This grant supports K - 12 educators by offering them a one - year grant worth up to $5,000.

Ena Gross

When reflecting on our formative development as a mathematics teacher, most can point to an individ ual or two who deeply influenced them. Many mathematics teachers in Virginia would point to Dr. Ena Gross. After receiving her Ph.D. from Georgia State Uni versity, Dr. Ena Gross served the VCU School of Education for 34 years as a nationally recognized professor of middle and secondary school math education. At VCU, Dr. Gross was the recipient of numerous School of Education awards for her work (two awards for teaching and one for service) and also received the University ’ s Distinguished Teaching Award, becoming one of a handful of School of Education faculty so honored with the university level of recognition. Dr. Gross ’ passion for excellence in math education was infectious in her classroom. She was dedicated to helping teach ers understand the mathematics they teach, how their students learn mathematics and how to facili tate that learning. She mentored and influenced hundreds of phenomenal Virginia middle and sec ondary math educators.

Dr. Gross ’ s advocacy and passion was evidenced

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by her leadership in the Greater Richmond Council of Teachers of Mathematics, the Virginia Council of Teachers of Mathematics, the Virginia Council for Mathematics Supervision, the Virginia Mathe matics and Science Coalition, and the National Council of Teachers of Mathematics. She served on the Greater Richmond Council of Teachers of Mathematics board as president and was always on the conference program as either a keynote speaker or a workshop leader. As a member of VCTM, she was on the scholarship committee, on the confer ence program committee and had won the VCTM Outstanding Faculty award. Dr. Gross served as co - chair of the first Richmond NCTM Regional Conference and co - chaired an administrators ’ conference. She served as the meet ings chair for the national NCTM conference and presented nationally and internationally. Dr. Gross believed that understanding mathematics led to un derstanding the world. She was an inspiration to all because she was so knowledgeable, so creative and so willing to share. She was also responsible for numerous state and national grant programs such as a 5 - year NSF statistic grant specifically for Virgin ia ’ s middle school teachers and with Virginia ’ s NSF - Mathematics Specialist Program. In her per sonal and professional life, Dr. Gross modeled courage, grace, strength, and innovation. Her pas sion for excellence in mathematics education and developing teachers and coaches that valued con nected, personalized mathematics instruction was only eclipsed by her kindness and compassion. Dr. Gross ’ scholarship awards two students up to $2,000 per year for reimbursement upon complet ing a mathematics or mathematics education course.

William C. Lowry

Dr. Lowry received his B.S in Education with hon ors from Ohio University, majoring in mathemat ics. He received his M.Ed. and Ph.D. degrees from Ohio State. Prior to teaching, he served in the armed forces of the United States as a glider pilot. He taught at a couple of high schools before turn ing to higher education. He was an instructor at St. Helena, a branch of the College of William and Mary, Kent State University, and Ohio State Uni versity. In 1955, Dr. Lowry came to the University of Virginia as an Assistant Professor of Education. He was promoted to Associate Professor of Educa tion in 1966. Upon his retirement, Dr. Lowry was appointed Professor Emeritus. It is noted that Dr. Lowry was not only respected by colleagues in the Curry School of Education, but he was also highly respected by faculty mem bers in the Mathematics Department in the School of Arts and Sciences at the University of Virginia. Dr. Lowry was a teacher, whose life work was de voted to promoting the learning of mathematics. He published more than 20 articles, and wrote many book reviews. According to Dr. James D. Gates, Executive Director of NCTM, “ Bill Lowry has made significant contributions to the efforts of the National Council of Teachers of Mathematics. His work in the area of publication, particularly yearbooks, has been widely acclaimed. Many teachers across our nation have benefitted profes sionally from his publications. ” Dr. Lowry ’ s professional activities was director for thirteen annual University of Virginia School of Education Conferences for Teachers of Mathemat ics. He spoke at over 100 national, state, and local meetings for mathematics educators. Dr. Lowry also mentored the next generation of speakers and presenters. An analysis of the NCTM Atlanta pro gram in 1970 shows that half of the presenters were former students, or students of former students, of Dr. Lowry. Dr. Lowry served as the associate director and di rector for the National Science Foundation aca demic year and summer institutes held at the Uni-

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versity of Virginia. In addition to directing these institutes for over a decade, he traveled across the Commonwealth to teach evening classes linked to these institutes. In July of 1982, Dr. Lowry was appointed as the first Executive Secretary of the Virginia Council of Teachers of Mathematics. His life ’ s work is the focus of the 1999 Winter edition of the Virginia Mathematics Teacher Journal.

Mathematics Teaching in 1990, the American As sociation of University Women's state teacher of the year award and the Virginia Council of Teach ers of Mathematics teacher of the year award. Ka ren Dee was a co - director of a National Science Foundation project. She helped to create the CD published in 2005 called Historical Models for the Teaching and Learning of Mathematics. Michalowicz was a member of the board of Math counts, a national enrichment, coaching and com petition organization. She was a member of the Na tional Research Council's U.S. National Commis sion on Mathematics Instruction and helped design a series of posters for the Benjamin Banneker As sociation and she contributed articles for its month ly newsletter. She was a member of the Virginia Council of Teachers of Mathematics for many years, and served the Board in the role of chairwoman for the First Timers grant. In the community she volunteered at So Others Might Eat, drove the elderly to church services, and played organ at St. Anthony's Catholic Church, where she was a parishioner for more than 40 years. Ms. Michalowicz's interests included math history, female mathematicians and the use of geometry in African artifacts, the topic of a book she reviewed for the Mathematical Association of America. She published numerous articles, and her collection of 500 old textbooks were published between 1529 and 1899. Upon her death in 2006, her collection was gifted to the University Library at the Ameri can University in Washington. An endowment fund was established at the Langley School in 2006 to honor her. This fund provides a grant in her memory, which is awarded each year to a faculty member who plans to undertake a new challenge to further his or her professional or per sonal growth. This grant provides one $1,000 prize and two $400 prizes to VCTM members who have not previously attended, but wish to attend, a re gional or annual NCTM meeting.

One educator from each of the five categories (elementary, middle, sec ondary, college/university, and math specialist/coach) may be granted the Wil liam C. Lowry award in his memory.

Karen Dee Michalowicz

Karen Dee Michalowicz taught middle school mathematics for nearly 40 years. She was the Chair of the Upper School Mathematics Department at the Langley School in McLean, Virginia, and she was dedicated to her students. She taught fifth graders advanced concepts such as the golden ratio and the Fibonacci sequence by pointing out how shells develop and how plants grow leaves. Her demonstration of the concept in nature triggered fascination with math among many students. "She was very good at showing how math is inter esting, how it matters and how it pops up . . . in stead of having the class do long division and mul tiplication," said Nathan Curtis, one of her former students, who competed in the international Math Olympiad. He told The Washington Post in 1997 that "Ms. Mikey" taught him how the rules of mathematics govern accurate drawings of plants, clouds, trees and mountains. Karen Dee was a mathematics adjunct professor at George Mason University. She edited a multimedia program on math history and presented talks at more than 75 workshops. She received the National Presidential Award for Excellence in Science and

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Culturally Responsive Teaching of Mathematics Section Editor: Darryl Corey

In light of recent events, the Virginia Board of Education has added a standard on culturally responsive teaching and equitable practices in the classroom. Culturally responsive teaching is a research - based approach to teaching. It connects students' cultures, languages, and life experiences with what they learn in school. These connections help students access rigorous curriculum and develop higher - level academic skills. Submissions for this section should be sent to dcorey3@radford.edu Implementing a Rethinking Schools Lesson Plan in Middle School Mathematics

Alexander S. Moore

Culturally Responsive (CR) pedagogy has gained increasing attention in recent years due to its well developed philosophy (Ladson - Billings, 1995, 2014) and well - documented benefits (Averill et al., 2009; Liu, 2020). Despite this, CR can seem daunt ing to learn about, much less adopt as one ’ s curric ulum and instruction philosophy. CR lesson plans may seem contextually foreign to a teacher, may seem difficult to justify to colleagues and adminis trators, or just seem to be difficult to understand and implement effectively. However, these con cerns should not preclude one from becoming a CR teacher because the result is extremely empow ering, if approached with enough support. One of the primary hurdles to overcome is learning how to write a CR lesson plan. Novice teachers in the CR realm can find a lesson plan from a published re source such as Rethinking Schools , and then adapt ing that lesson to fit one ’ s particular needs in terms of class structure, content, and so forth. In this arti cle, I discuss a CR lesson plan from the book Re thinking Mathematics and how I adapted it to fit my particular class needs as an Algebra I teacher. Rethinking Mathematics: Teaching Social Justice by the Numbers (“ RM ”; Gutstein & Peterson, 2013) is a teacher resource book containing lesson ideas that have been turned into articles by teach ers who either developed the ideas and/or imple

mented them. One of the articles (Gutstein, 2013) focuses on the living reality of minimum wage, an issue that is especially important to discuss with students in the 2021 aftermath of COVID - 19 and the “ work shortage ” that minimum wage business es are currently facing. While the media posits that the problem is with the workers (i.e., that they can make more money on unemployment so that ’ s why no one wants to work), this lesson plan reveals that the actual issue lies with the minimum wage itself. In the lesson, students explore the reality of living and paying for expenses if one has a minimum wage job, the exact jobs that are currently experi encing a worker shortage. Through the lesson, stu dents learn that it is not actually an issue with find ing workers (cf. those who would rather stay home and receive unemployment), but rather it is an is sue of finding people who are willing to work and simultaneously be locked into a life of poverty (viz. work for minimum wage). The fact that this paradox is one of the symptoms of late - stage capi talism is beyond the scope of the present article, but this lesson can certainly be extended to have such a discussion afterwards, perhaps as an inter disciplinary lesson with a social studies or history teacher. I taught this lesson in a 7th grade Algebra I class to coincide with a unit on functions and data analysis. The related SOLs are listed in the lesson plan. To

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plan the lesson 1 , I took the RM chapter and began to reverse engineer the lesson plan to produce what was being described in the chapter. This process included an iterative and reflexive process of think ing, “ What would I have to do as a teacher to set up my classroom and my instruction so that [this out come printed in the book chapter] was being pro duced by my students? ” I also adjusted the particu lars of the lesson, for example, adjusting the mini mum wage from $8.25 in the RM chapter down to $7.25 in my lesson plan, to contextualize it for the state of Virginia in the year 2016, when I originally wrote the lesson plan. I spread the lesson over four 55 - minute class peri ods. The day before the lesson began, I reserved the Computer Lab so that we would have the first two days for research and investigation; I also de cided on partner groups and intentionally paired up strong students with ones who need extra support. When students came into class the first day, I in structed them that we were beginning an inquiry project and that I was putting them into groups. I announced the partner groups and instructed stu dents to sit with their partner. I then began handing out copies of the Student Handout (see Lesson Plan file) and introduced the lesson with the introducto ry questions provided above. We had a brief Q&A about jobs, making money, and what it meant to have a certain “ quality of life. ” Students offered varied information about their own lives and the choices their families made, but all agreed that it ’ s very important to have enough money in life to support one ’ s livelihood and family. Next, I asked some leading questions about Wal - Mart and Chick Fil - A: “ Who here likes Chick - Fil - A? Who shops at Wal - Mart? Who shops at Target? All those work ers who serve you your food or ring up your items at the store make minimum wage. This project is about them. As you ’ re going through the calcula tions, think about them. This is what their lives are like. ” Implementation of the Lesson Day 1

I explained the expectations of the lesson under the outline of the three tasks: (1) income, (2) bills, and (3) how their theoretical minimum wage income would compare to the average earner in our city. I explained that the students would imagine they were a fast - food worker, retail cashier, or some other minimum wage worker, and they needed to envision what life would be like if they were that person. Would they be able to live the life they wanted? Would they have enough money to get by? To create a context for these thoughts, I asked the class, “ Imagine you worked a minimum wage job and made $7.25 per hour. How much would you make in one hour? ” The class answered $7.25 confidently. Then I asked, “ Then how much in two hours? ” After a second of thinking, most said $14.50. “ Ok, how about three hours? ” A bit more hesitation yielded some $21.75 answers. Then they were ready to take the reins: “ Ok, now you ’ re on your own. Make sure you read everything on the Handout and follow all the directions. It ’ s an in quiry project, so expect to face some concepts you aren ’ t totally sure about. ” As an inquiry - based teacher, it is important for me to be sure the students remember that this lesson activity is inquiry, meaning that each group ’ s meth od or process may be different. It doesn ’ t matter what order they address the tasks, just that they get them all done and incorporated into the final presentation. I told the class that their homework for that night (the night of the First Day) was to talk to parents or caregivers and discuss, in detail if possible, what their bills and expenses were. I made sure to underscore that income was a person al matter and students should never ask their par ents about their income; rather, they were to only ask the parents about bills and expenses. I want the students to learn what it costs for their parents to provide the life they live. The class then walked over to the Computer Lab and began working. It wasn ’ t long before I heard discussions commence and questions being asked. “ Ok, so how much do we make each day? Ok what about the week? The month? ” and “ Is that all? That doesn ’ t seem like a lot. I wonder how much our parents make? What does it cost for us to live? ” Students began opening the Wikipedia site to look

1 The Lesson Plan file is available online at http:// amoore.net/files/couldyousurvive.pdf

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at the per capita data in Task 3. I was occasionally approached with a procedural question or detail that needed explaining. Overall, the students un derstood the expectations and proceeded with un easiness and blind faith that inquiry brings forth.

apartments for as little as $390 and were pleased that it was so affordable. As the discussion turned to their own expenses in the theoretical minimum wage life, students began to realize the necessity of paring down their current living expenses due to the reality of their minimum - wage jobs. One stu dent even announced to the class (with great dis tress) that she was “ Already in debt $430 and it wasn ’ t even the end of the month yet! ” This same student then asked me if it was allowed for her to live in a homeless shelter instead. I laughed and told her it wouldn ’ t be in the spirit of the project, but rather asked her to reflect on what she was learning. She said, “ It ’ s so expensive to live, I don ’ t see how someone making minimum wage could do it! ” My follow - up comment was, “ And yet we love getting fast food and expect to have low prices when we go to Target to shop. ” Other students turned to Excel to create graphs of the da ta points they were generating in their calculations. I informed the class they could hand - draw or use Excel for any graphs they were making and assist ed some groups with programming in Excel if they chose to use it. I closed the class session by explaining homework expectations for the night: everyone was to work more on their income and bills calculations de pending on what they still needed to accomplish. I explained that the next day they would need to be ready to create their presentations, so they needed to come to class ready to do this. I also suggested that they think about what they wanted to include in their presentation: PowerPoint, a hand - drawn poster on a giant sticky note, Excel graphs, hand drawn graphs, a short skit or demonstration. Stu dents were to think about these options overnight and discuss with their partner so that class on the third day could begin with an action plan in place. The third day was reserved for tying up loose ends and creating presentations. The presentations would be held on the fourth day, when students would also be self - assessing using the Assessment Rubric (see Lesson Plan file). Students came to class ready to get started, since they had worked hard to gain momentum on Day 2. Most of the stu- Day 3

Day 2

Students on the second day were already in the groove of the project. Their homework on the first day was to discuss their family ’ s bills and expenses with parents; they came into class on the second day talking about this information. Things started much quicker and we went directly to the Comput er Lab to begin working. Students were asking how their family ’ s expense data translated to a sin gle person and I helped explain the differences in cost for a standard single person in our city. The students started recognizing the relationship of their income and expense data to graphs, and some asked questions about how they would graph the information. The most common misconception was the use of bar graphs. I explained to these students that a bar graph wasn ’ t appropriate because it showed that each day you made the amount of money of that bar, so every day would be the same size bar. This didn ’ t, however, show them the in creasing money in their bank account as the work ing days of the month went by. I explained that it would be much more useful to visualize the entire amount of money in their bank account by looking at the cumulative increase over time; they started to understand the relationship to a line graph and plotting points that corresponded to money versus time. Most students then made the inverse realiza tion that if their income had a positive slope, then their expenses would have a negative slope. Look ing back, I had not anticipated their need for this handholding to make the leap between the money calculations to the line graph. All the students in the class had seen financial graphs before, because I taught them last year and did a stock market pro ject. However, an inquiry - based teacher must al ways be flexible and adapt to unexpected behavior from the students. I was most impressed with one group that logged onto ApartmentFinder.com to look up actual rent costs for different areas of the city. They found

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dents were self - directed at this point, but the pur pose and direction of the inquiry periodically be came unclear for some students. With light guid ance and clarification, they were all able to get back on track. The most indelible comment from the class period was a student who, seemingly con fused about the project ’ s purpose, came to me and said, “ I don ’ t understand [this project]. We are sup posed to have a minimum wage job and then take on all these bills like a real person would, but I don ’ t see how it ’ s possible. How do these [low wage] people make it work? ” My response was candid: “ Well, a lot of times they don ’ t make it. Or they must work multiple jobs because their boss won ’ t pay for overtime. ” She responded, “ So what are we supposed to learn here? ” I said, “ That is for you to decide. What have you learned so far? ” She replied, “ That minimum wage isn ’ t high enough for people to live! ” I closed the conversation with the question, “ So could you survive on $7.25?” She exclaimed with great authority, “ NO! ” Throughout the class period, others came forward with realizations about the undertones of the pro ject. One student said, “ This whole project is using real numbers for what you make at a [low wage] job and real prices for things you need to live. Is it legal for people to make that little? Shouldn ’ t the employers pay their workers more? I don ’ t think it ’ s fair. ” Another student remarked that, “ I don ’ t think the employers know how much peoples ’ bills are, or they would be paying their workers more. ” Reflections like this peppered the class period as the groups were marching forward with creating their presentations. Most groups chose to use the giant sticky notes and printed out graphs from Ex cel which they taped to the sticky note. Groups decorated their posters with narratives of questions and answers. As I circulated to discuss progress with each group, I was mostly impressed with my students ’ ability to discuss mathematics and at tempt to synthesize information. My students are comfortable with the unknown ethos of “ inquiring within ” when faced with an unfamiliar idea. This is the result of working hard to develop a culture of inquiry; I had made it a regular and integrated part of the class structure and the students become com fortable with the unknown by grappling with it reg ularly.

As the class period ended, students were busy drawing with rulers and yard sticks and working to make exemplary displays of their labor the past two days. I announced that homework for the night was only to keep working if they still needed to finish their presentations. Class on Day 4 was re served for presenting and self - assessing using the Rubric. I reminded the students that they would be grading themselves using the same criteria as I would, and that I would compare their self assessment with my assessment when determining the final grade for the project. I should also remind the readers of this article: in quiry - based learning is up to the student to direct. In many cases, the students do not end up where you intended. “ Could You Survive ” is a good ex ample of this. I intended students to arrive at a real ization about the slope of their income and expens es, and to write linear equations of the two func tions using slope. For example, if a minimum wage worker makes $58 per day, I expected students to realize they could write y=58x as a basic example of the linear equations we had been learning about in class in the weeks prior. However, only one stu dent made this conclusion. There were plenty of other good connections being made between mate rial, but I expected this one, but it didn ’ t end up happening. Students put the finishing touches on their presen tations on the fourth day. There were some disa greements over mathematical process between two of the groups, but one finally conceded that they had made some mistakes in their calculations. Two groups ’ examples are shown to represent some of the work the students produced. Overall, I was satisfied with the outcome of this lesson, but I had expected the students to use more mathematics in their results. One group (see Figure 2) was the only group to write an equation of the earnings, and brilliantly incorporated regular and overtime pay in the same equation. Another group was the only group that made the pay rate connec- Day 4 Reflection

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that this lack of conceptual connection to points also extended to the students not understanding the connection to slope. I had to help all the groups understand that the data points they were defining corresponded to a slope, just like we had been learning in the weeks prior, which could be found by the change in each variable, respectively. How ever, if the students couldn ’ t make the connection to the points, then I wouldn ’ t expect them to make the connection to slope or other graphic or relation al properties of lines. Additionally, I saw a lot of groups confusing the use of a scatter plot with a histogram or bar graph. While most groups contin ued to use bar graphs in their presentation, I ex plained to each of them the difference in the appro priate use of bar graphs and line graphs. The expla nation seemed to resonate with them. Teaching by inquiry always comes with a gam bling element. It is like making the students jump across a ravine, and there is a chance that they jump and miss. Sometimes recovering from a failed “ jump ” is painstaking, but I believe it is worth the risk to be an inquiry teacher. Throughout my practice of teaching by inquiry, I have seen so much greater benefit than disaster come from it. However, one must be ready to recognize when a student is falling short of that jump when they are out on the waters of uncertainty on their own. It may be a misconception or just a misguided think ing process, but the teacher needs to be poised to gently help the students back on track. This process can be exhaustive when monitoring an entire class. My largest class size for an inquiry lesson is 17 people. I am proud of the practical conclusions the students made as they went through this project. All groups used deductive reasoning to synthesize the information they gathered from parents and from the figures they calculated together, to pro duce authentic meaning from the project.

tion to slope, which surprised me, but indicated that I need to do more conceptual development of slope with the entire class. Specifically, the stu dents quickly recognized the method for calculat ing earnings per day, week, and month, but only one made the connection between this rate and gra phing, i.e., I had to help all but one group make the connection between time - and - earnings to data points. An example would be that students under stood one hour yielded $7.25, two hours yielded $14.50, and so on, but only one group (see Figure 2) made the connection between these figures and the points (1, 7.25), (2, 14.50), etc. and how that related to an equation. I had expected students to make this conceptual jump because of the in - depth stock market project I had done with them last year. They had plenty of exposure to graphs and earnings in that project. It stands to reason, then, Figure 1: Example group work showing how some students provided lots of information and was the only group to graph the expenses.

Conclusion

In conclusion, I was pleased with the way this pro ject turned out, but as with all inquiry teaching, the teacher must be prepared to have their expectations altered depending on the level of student success. This flexibility is critical if the inquiry teacher is going to successfully use inquiry in their teaching

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search in Mathematics Education , 40 (2), 157 – 186.

Gutstein, E. (2013). I can ’ t survive on $8.25: Using math to investigate minimum wage, CEO pay, and more. In E. Gustein & B. Peterson (Eds.), Rethinking mathematics: Teaching social justice by the numbers (2nd Ed., pp. 75 – 77). Rethinking Schools. Gutstein, E., & Peterson, B. (Eds.) (2013). Re thinking mathematics: Teaching social jus tice by the numbers (2nd Ed.). Rethinking Schools. Ladson - Billings, G. (1995). Toward a theory of culturally relevant pedagogy. American Ed ucational Research Journal , 32 (3), 465 – 491. Ladson - Billings, G. (2014). Culturally relevant pedagogy 2.0: A.k.a. the remix. Harvard Educational Review , 84 (1), 74 – 84. Liu, R. (2020). Cultivating cosmopolitans: Cultur ally relevant pedagogy in an age of instru mentalism. Anthropology & Education Quarterly , 51 (1), 90 – 111. https:// dx.doi.org/10.1111/aeq.12321

practice. Any teacher wishing to embark on such a lesson must remember two things: (1) the purpose of CR teaching is to introduce students to critical ideas that they may not have encountered before, and to wrestle with these ideas from their social standpoint and from their mathematical content, and (2) inquiry - based learning is a messy and a non - linear process. What students glean from the experience is primarily a product of how the teach er has structured and operates the inquiry - oriented environment of their classroom, and what norms and expectations the students have been socialized to agree to. These are non - trivial aspects of CR in quiry teaching, but as I have shown in the present article, they are not reasons to avoid committing oneself to CR teaching. Indeed, the present situa tion in our country necessitates CR teaching in a way that historically has not been seen in recent decades. Now is the time to become a CR teacher. Figure 2: Example group work that shows how some students made connection to slope and writ ing equations.

Alexander S. Moore School of Education

Virginia Tech asm1@vt.edu

References

Averill, R., Anderson, D., Easton, H., Te Maro, P., Smith, D., & Hynds, A. (2009). Culturally responsive teaching of mathematics: Three models from linked studies. Journal for Re

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All About the Number Line

Derek Stiffler, Jessica Hofer, & Stephen Brazelle

A high percentage of U.S. students lack conceptual understanding of fractions, even after studying fractions for several years; this, in turn, limits stu dents ’ ability to solve problems with fractions and to learn and apply computational procedures in volving fractions. (Siegler et al., 2010). Millions of students across the United States strug gle with numbers and number sense, which directly leads to a struggle with concepts related to rational numbers and negative numbers. When students struggle with whole numbers and operations, that struggle then carries over to rational numbers and negative numbers. One of the most useful representations to show a number is on the number line. The number line is important as it can show the connection between decimals, fractions, and other types of numbers and to develop a sense of relative size among the num bers. It is an especially important representation because, unlike many other representations and models used in teaching, the number line plays an important role in mathematics to the most ad vanced levels. It is used for measuring scales and Cartesian axes, as well as embodying the abstract set of real numbers. (Widjaja, Stacey, & Steinle, 2011) The number line promotes conceptual understand ing of rational numbers. From an early age, stu dents are directed to using a number line to count whole numbers, compare whole numbers, and

model early elementary operations such as addition and subtraction. At some point during their later elementary math journey, some teachers tend to shift their focus to procedural fluency, rather than maintaining a focus on number sense and the num ber line. Number lines form strong connections to students ’ learning and understanding when com paring rational numbers starting in primary grades, through upper elementary, into their middle school years and beyond. Why are number lines so helpful for rational num bers? Number lines can show “... the continuity as pect of rational numbers, ” (Diezmann & Lowrie, 2006). From elementary, focus on whole numbers to upper elementary and middle school focus on fraction, decimal and percentages, to the higher level courses in upper middle school that focus on scientific notation and other aspects of the rational number system: students ’ knowledge of the num ber line can only help build their vocabulary and knowledge of these number sets. The number line helps to connect to other models used with the ra tional number system such as the array and set model to show there are multiple ways to show these numbers that have multiple forms. The un derstanding of the number line can be crucial for students in the younger grades and will continue to add on to their math knowledge when more subsets of the rational number system are added to their mathematical vocabulary.

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Many students have little or no understanding of the magnitude of a rational number and thus every operation has no meaning. Rational numbers are part of many higher level mathematics concepts. This weak knowledge of rational number arithme tic impacts advance learning of mathematics and science. Students ’ understanding is weak which sets them up for more struggles in the workforce beyond high school or college. In both the U.S. and the U.K., 5th graders ’ knowledge of fractions predicts 10th graders ’ over all math achievement. Their knowledge of algebra, when controlling for knowledge of whole number arithmetic, IQ, working memory, parental educa tion and income, and a host of other relevant varia bles. (Siegler et al., 2010) As children develop their sense of number, one of the early tools that they use is a number line. Chil dren learn to count on, compare and even add and subtract numbers on a number line. Somewhere along the way, as procedures become more com plex, many teachers can easily fall into the trap of moving away from visual representation of num bers to focus more on procedural fluency through the use of algorithms. While the algorithms are beneficial for fluency, they do not promote con ceptual understanding. If visual representations and manipulatives are only used in primary grades for whole number identification and calculation, the transition to problem solving with fractions can be a struggle for many students. The procedures that students use and the meaning of the numbers completely change and if there is no conceptual understanding of what fractions and mixed num bers represent. These algorithms become robotic repetition for students and can quickly cause them to become confused and disinterested in mathemat ics. The number line is a great introduction to frac tions, as it is a connection between whole numbers. As one reads a number line, the space between whole numbers is connected with a line, represent ing the numeric significance between each whole number. Even as early as kindergarten, students are Number Lines in Upper Elementary School

learning to fair share objects into two equal groups. The number line can easily help children make connections between a physical object, such as a brownie or cookie, and an equal distance between two whole numbers on a number line. This visuali zation connects to a concrete representation and helps show the importance of halves by partition ing the number line into two equal parts between the whole numbers. One of the foundational under standings of fractions is the value of a unit frac tion. As students continue to learn about fractions in elementary school, this idea of using number lines to fair share or partition the line into equal seg ments also helps the students when comparing fractions and mixed numbers. As students become more comfortable with common fractions such as halves and fourths, as they learn to use those as benchmarks when comparing rational numbers. As students start to learn how to represent whole num bers on a number line, they start to see patterns that they can use to compare numbers without having to use a pictorial representation. Such patterns for whole numbers could include looking for the num ber that goes out to the greatest place or finding the larger digit of two 2 - digit numbers in the tens place. The value of denominators of fractions is different from whole numbers. As the denominator gets larger the smaller the piece and it has less val ue. This change can be very difficult for students to grasp, and using these common benchmark frac tions on a number line can help students compare fractions and mixed numbers with greater accura cy. The more familiar they become in partitioning a number line to represent a fraction or mixed number, the more it can help them estimate wheth er a fraction is closer to zero, one half or to one whole. In addition, students in the upper elementary grades begin to study decimals. While decimals represent parts of a whole like fractions, their sym bolic representation is very different. While deci mals bring out a variety of new challenges for stu dents, number lines are still a representation that can help students make sense of many ‘ rules ’ that they had previously learned about whole numbers that may no longer apply to rational numbers. One

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