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

V IRGINIA

M ATHEMATICS T EACHER

Vol. 46 No. 2

Teaching Mathematics Relevant to Our Students!

Virginia Mathematics Teacher vol. 46, no. 2

Editorial Staff

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

Dr. Jean Mistele Associate Editor Radford University

Mr. Alexander Burnley Assistant Editor Radford University

Ms. Grace Chaffin 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 46(2)

President: Lynn Reed

John Adam, Old Dominion University

Pam Bailey, Mary Baldwin University

Past President: Pam Bailey

Eric Choate, Radford University

Harold Mick, Virginia Tech

Secretary: Kim Bender; Historian: Beth Williams

Alexander Moore, Virginia Tech

Shannan Chappell Moots, Old Dominion University

Treasurer: Virginia Lewis

Dale Parris, Radford University

Webmaster: Ian Shenk

Ann Wallace, James Madison University

Darryl Corey, Radford University

NCTM Representative: Skip Tyler

Tiffany LaCroix, Virginia Tech

Spencer Jamieson, Fairfax County Public Schools

Elementary Representatives: Tammy Sanford

Laura Moss, Radford University

Middle School Representatives: Lisa Allen

Dana Johnson, College of William and Mary (R)

Rich Busi, James Madison University

Secondary Representatives: Timothy Barnes

Kateri Thunder, Charlottesville County Public Schools

Michael Coco, University of Lynchburg

Math Specialist Representative: Allison DePiro

Special thanks to all reviewers! We truly appreciate your time and service. The

Two - Year College: Doniray Brusaferro

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

Complete VMT Editorial Staff

Four - Year College: Darryl Corey

Agida Manizade, Radford University Jean Mistele, Radford University Alexander Burnley, Radford University Grace Chaffin, Radford University Alexander Moore, Virginia Tech Eric Choate, Radford University David Shoenthal, Longwood University Betti Krier, Virginia Tech

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 .................................. 7

PBL, Slopes, and Lines in Stained Glass .............. 8

Note from the VDOE ........................................... 14

Organization Membership Information …. .......... 15

How to Keep Students from Falling Off the Slope .................................................................... 16 Call for Manuscripts ............................................ 22 Math GIRLS .............................................................. 23

VCTM Representatives ........................................ 30

HEXA ................................................................... 32

NASA Resources For Elementary School Teachers and Students ......................................................... 33

Upcoming Math Competitions ............................. 35

Teaching Dilemmas ............................................. 36

Technology Review .............................................. 41

Good Reads ......................................................... 47

What ’ s Your Sphericity Index? ............................ 48

Grant and Scholarship Opportunities ................. 54

Vertical Number Lines are Important, Too! ........ 55

Is This Game Fair? Deciding with Simulation Data and Organized Lists .................................... 58

Affiliate Information ............................................ 62

Notes from the Field ............................................ 63

Key to the 46(1) Puzzlemaker .............................. 65

Math Jokes ........................................................... 65

Solutions to 46 (1) HEXA Challenge ................... 66

The Puzzlemaker .................................................. 71

Conferences of Interest ........................................ 72

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Teaching Mathematics Relevant to Our Students:

Note from the Editors Dr. Agida Manizade and Dr. Jean Mistele

During our vulnerable times, the focuses of this is sue is on teaching mathematics most relevant to our students. Our goal as educators is to continuously deliver high - quality instruction to our students. The challenge in mathematics education is that we as a community define “ good ” teaching and quality instruction in mathematics classrooms differently. In a high - stakes testing environment, mathematics teaching is considered “ good ” if particular learning outcomes, or test scores, are presented by students. On another hand, quality of mathematics instruc tion could be tied to the type of activities students engage with in their classrooms. These learning ex

periences—if relevant to students, appropriate for their backgrounds, and designed to engage them in rigorous thinking about challenging mathematics content—can constitute the definition of “ good ” mathematics teaching. Two good examples of this definition of a quality mathematics classroom are presented first in the article by Ferguson et al., ti tled “ PBL, Slopes, and Lines in Stained Glass, ” and next by Dawn Hakkenberg and Diana Fisher ti tled “ How to Keep Students from Falling Off the Slope. ” In these pieces, two teams of authors dis cuss hands - on, real - life activities in which students engaged during a lesson on slope. In another article by John Adam titled “ What ’ s Your Sphericity In-

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dex?, ” the author presents zoological context and more in activities designed from an integrated envi ronmentalist approach for teaching mathematics with the goal to motivate and engage students. Groth et al., in the article titled, “ Is This Game Fair? Deciding with Simulation Data and Orga nized Lists, ” explains how playing, simulating, and analyzing games of chance helped to develop un derstanding of content for simple probability, com pound probability, sample space, theoretical proba bility, and experimental probability. Alternatively, quality mathematics instruction could be connected to the best practices in which a mathematics teacher engages while being with her/ his students. In her article, Lorraine Howard dis cusses best practices identified by researchers for teaching mathematics to girls. We encourage you to consider the ideas laid out in the article present ed in the Math GIRLS section. Another example of this perspective on “ good ” mathematics teaching is shared by Zhenqiang Li, who discusses various ef fective ways a mathematics teacher can engage stu dents to improve their understanding of factoring trinomials. Finally, Katelyn Devine provides an ex ample with a focus on using multiple representa tions as one of the best practices in teaching mathe matics in her article titled “ Vertical Number Lines are Important, Too! ” Yet another way to consider the quality of mathe matics teaching is to focus on mathematics teacher activities such as lesson planning, assessment of

student work, and many other professional/teaching activities mathematics educators engage in while working outside of the classroom. In their articles, Alexander Moore and Kristin McKitrick - Rojas dis cuss the best way to plan and to blend the applica tions of DESMOS in mathematics instruction. You can find their contribution under the Technology Review section of the journal. Our goal in this journal is to provide a space for teachers to improve their professional knowledge, competencies, and skills by exploring topics and resources collected and discussed within the jour nal. We encourage you to explore resources for ele mentary teachers presented by NASA. Consider reading Rethinking Mathematics, this issue ’ s book reviewed under the Good Reads section. The Vir ginia Council of Teachers of Mathematics repre sentatives have also shared resources with our readers—you can find this information under Notes from the Field. We also have some competitions and puzzles that you can do on your own or with your students, and we will feature the winners in the 47(1) Special Is sue: Teaching During a Pandemic . Last, but not least, we would like to welcome our new VCTM president, Lynn Foshee Reed, and all of the newly elected representatives for the 2020/2021 year. We appreciate their service and commitment to mathematics educators of Virginia!

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Message from the President Lynn Foshee Reed

The word “ unprecedented ” certainly describes the state of mathematics education (and, of course, ed ucation in general) across our schools, our state, our nation, and our world. Like you, I had to fig ure out how to deliver virtual instruction to my stu dents in March. Was it perfect? Of course not, but it wasn ’ t bad, either, thanks to many suggestions and nuggets of wisdom from colleagues at my school, from within VCTM, and from larger com munities of AP and NBCT teachers. In turn, I hope I was also able to support colleagues by shar ing what was working for me, as well as what I quickly abandoned. This sense of “ we are in it to gether ” is so important, especially as we maintain social distancing (and mask - wearing, hand washing, etc.) to combat the spread of Covid - 19. The new school year will continue to challenge us all. New teachers, seasoned veterans, teachers with school - aged children of their own, teachers with spouses or parents in high - risk categories— the list goes on and on—face the coming year with excitement and trepidation. Some of you will con tinue to teach online while others journey back into the classroom, and some are doing a hybrid of both. To help our members meet these challenges, VCTM will develop virtual professional learning and support groups as well as create a virtual con ference for Spring 2021. We hope that such VCTM - sponsored opportunities to learn best prac tices for virtual instruction, to share methods of safe and secure assessment, to take the data from the pandemic to craft important math modeling and data analysis lessons, or to simply provide a sounding - board—just to name a few ideas—will be helpful to each of you as you navigate the 2020 21 school year! In addition to the pandemic ’ s upheaval of our lives, this summer has brought into focus the con tinuing struggle for racial and social justice. I am

proud that NCTM ’ s President Trena Wilkerson and Past President Robert Berry quickly and clear ly reminded us, “ As a mathematics education com munity, we must not tolerate acts of racism, hate, bias, or violence ” (statement on George Floyd, Breonna Taylor, and Ahmaud Arbery). Further more, they reiterated the 2017 call by Robert Berry and Matt Larson that:

We support the use of mathematics as an analytic tool to challenge power, privilege, and oppression.

We encourage all educators to challenge systems of oppression that privilege some while disad vantaging others. We encourage all educators to create socially and emotionally safe spaces for themselves, their stu dents, and colleagues. At the VCTM Board ’ s summer planning virtual re treat, Darryl Corey, Timothy Barnes, and Tracy Proffitt began crafting not only a statement of VCTM beliefs, but, more importantly, commit ments to this call. If you would like to be part of this work as a member of the new “ Justice and Eq uity ” committee, please send me an email at the following address (lynn.foshee.reed@gmail.com). Furthermore, if you are interested in working with the March 2021 virtual conference committee, reach out to Theresa Wills (twills@gmu.edu). If you have ideas for ongoing professional learning, then contact Shelly Pine, shelly.pine1@gmail.com. Finally, I was recently reminded of the Serenity Prayer by the American theologian, Reinhold Nie bur: Grant me the serenity to accept the things I cannot change, courage to change the things I can, and wisdom to know the difference.

Wishing you all the best! Stay safe!

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PBL, Slopes, & Lines in Stained - Glass Sarah C. Ferguson, Brittany Houdashelt , Deja Richardson, & Kelly Johnson

op their own answers. To demonstrate their under standing, students create a publicly presented prod uct to share with people within or beyond the class room. (Larmer et al., 2017) PBL environments are typically “ student - driven. ” During this lesson, the students had some prior knowledge of slopes but had not mastered calculat ing, identifying, or evaluating slopes. However, as math teachers know, understanding the concept of slopes in linear equations and graphically is crucial in the development of math knowledge and skills. Conducting a PBL lesson on foundational material, such as slopes, gives students the opportunity to enrich their understanding through research, explo ration and practical application. 5E Lesson Plan The 5E Instructional Model, or the 5Es, consists of the following phases: engagement, exploration, ex planation, elaboration, and evaluation. Each phase has a specific function and contributes to the teach er ’ s coherent instruction and to the learners ’ formu lation of a better understanding of scientific and technological knowledge, attitudes, and skills. (Bybee et al., 2006) Framing the PBL within a 5E lesson plan over the course of three days aligned rather well. The 5E In structional Model offered an organizational method to the PBL. The engagement and exploration phas

Introduction Understanding slopes in linear equations can be a challenging concept for students to grasp. Students ability to memorize and recall the mathematical definition of a slope is useful; however, being able to evoke deeper meaning by critical evaluation of an actual or simulated problem involving slopes can help students evolve their critical - thinking and problem - solving skills. Students aptitude to under stand linear equations and slopes can enable them to evaluate, assess and predict practical situations in the real world. This article describes a student - centered, inquiry driven lesson on slopes. Given a class of twenty 7 th grade Algebra I students, three 90 - minute classes, and a desire to bring some depth and fervor to slopes, a project - based learning (PBL) lesson was developed and paired with a 5E Lesson Plan. Project - Based Learning Project - based learning offers an approach to teach ing lessons that exposes students to “ why ” they are learning a concept while establishing to the stu dents that they “ need to know ” the concept. In a PBL lesson, students are gently guided through the content using a meaningful question to explore, an engaging real - world problem to solve, or through a challenge to design or create something. Typically, students first inquire into the topic and then devel

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es occurred on day one, the explanation and elabo ration phases on day two and the evaluation phase on day three. Lesson plans and further details on each phase of the 5E ’ s are available from the pro ject website: https://drive.google.com/open? id=1_FsIdDJYzKJoaFVYIj7QkH3qICK2VrFz Planning Planning a PBL lesson requires far more prepara tions than a standard lesson. It seems like “ student driven ” would mean that students are figuring out content by self - discovery with minimal teacher in teraction. However, PBL lessons require a massive amount of teacher interaction and planning. The planning is detailed and the teacher - student interac tions occur in the classroom by paying attention to students as they work and only offering guidance as needed.

were given approximately five minutes to work through the four problems. As the five minutes came to an end, the students were asked if anyone wanted to volunteer to show their work on the board for any of the problems. Students eagerly raised their hands and four students were chosen to demonstrate their solutions for the four problems. Showing student work was beneficial because it showed their classmates different ways to obtain a solution. After the board work, each problem was revisited through a brief class discussion lead by the students. Next, the students broke into three groups and giv en verbal instructions about the rotation flow be tween four work stations spread throughout the classroom. Students had fifteen minutes to work at each of the four stations. Once the groups had ro tated through the first three stations, everyone came back together as a whole class for the fourth sta tion. The first three stations served as the engage phase for day one in the 5E lesson plan model. At station one, the students researched stained - glass windows, on the classroom laptops, and wrote down some characteristics they saw on the “ Instructions for the Research Station ” worksheet. It was awesome to see students research stained glass windows and identifying their favorites (i.e., basketball, flowers, food, video games, etc.). Sta tion two had the “ Find the Mistakes ” worksheets, where the students had to find and explain mistakes for the incorrectly solved linear equations. Station three had the “ Graphing Station ” worksheets that had the students practice their graphing skills by finding the slope from the graph. Students worked with their group members to complete each station as the teacher circulated through the classroom, en suring everyone was on task, using probing ques tions for formative assessment, and answering questions as needed. When completed, the class came back together for the fourth E phase, explana tion, about the first three stations. Misconceptions emerged such using run/rise instead of rise/run, confusing the x and y - axes, and interpreting slope as a point. During the discussion, the teacher spe cifically talked about and demonstrated rise and run, axes, slopes and plotting points to ensure stu dents were accurately reviewing their prior content knowledge. Next, the class moved to station four where the students had to make connections be tween the concept of slope and real world situa tions. The class was taken outside, and the students were challenged to look for each type of slope in an everyday setting and write their findings on the “ Instructions for Slopes in the Real World ” work-

For this lesson, the student learning performance objectives were:

• Students will identify, classify and define all terms of a linear equation • Students will differentiate between different slope - intercept forms: standard form and point - slope form • Students will construct a graph for a linear equation

• Students will identify and define the slope

• Students will differentiate between positive, negative, zero and undefined slopes

Students will write the equation of a line when giv en a graph, two points on a line, or when given the slope and one point on the line. Implementation The PBL joined with the 5E model lesson led to an amazing three - day experience. The students were totally engaged and each day their understanding and connections between linear equations and slopes blossomed. One day one, the focus was on engaging students in the PBL lesson and ensuring that a strong under standing of foundational content was established. The “ Linear Equation Warm - Up ” worksheet was used as an individual warm - up, aligned with the 5E engagement activity model, that acted as a pre assessment to gauge student knowledge. Students

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sheet. The students were encouraged to use their cell phone to take pictures of each of the four slopes to document their findings. The stations of fered the students a chance to collaborate with peers and research real - world applications that use linear equations and slopes within the first day of the 5E lesson plan model. These connections to the content align with essential components found in the PBL framework, which give students explicit and concrete examples of linear equations and slopes. In addition, Larmer et al. (2017) explains that activities such as games, puzzles and physical activities can prepare the students to work effec tively in teams. These activities gave the students opportunities to build 21st century skills such as collaboration, communication, critical thinking, and enhance their technology skills, which will serve them well in the workplace and life (Larmer et al., 2010). The closing activity for day one, and the final E phase, evaluation, students performed the slope dance, in which students use their arms to mimic each type of slope as called out by the teacher. As the students danced out the door, it was exciting to hear their comments about the day ’ s activities as they walked out of the room towards their next class. While this was intended as a simple closure and formative assessment activity, the students en joyed the slope dance so much that this was used to regain the students ’ attention for days two and three. On day two, students viewed a video about their lo cal community museum that created real stained glass windows. The video was used to connect their researched on stained - glass windows to slopes. The anchor video engaged the students right away and was the first E phase, Engagement, for day 2. Once the video finished, a “ Job Offer ” intro duction letter was handed out to each student. The letter posed a problem faced by their community art museum that included background information to understand the problem. The letter explained that an existing stained - glass window was damaged during a storm due to the curved lines in the origi nal design that were structurally unsound. The stu dents were tasked to design a new stained - glass window using straight lines, which is more stable. To explore the second E, the students designed a “ blueprint ” for their own stained - glass window prototype for the community art museum. The “ Stained - Glass Equation ” worksheet, along with a sheet of graph paper and a ruler, was given to each student. The students requirements called for a

minimum of ten lines and the design had to include each type of slope (e.g. positive, negative). On the provided “ Stained - Glass Equation ” worksheet, stu dents indicated the y - intercept, slope, whether the slope was positive, negative, zero or undefined. The equation of each line is shown in Figure 1 and Figure 2. The students ’ imagination was limitless. It was captivating to see the diversity around the classroom. The students were so excited to experi ence a creative activity within a math class. More over, their creativity was so abundant that they had to be reminded to stay within the specifications of the original job offer in order to ensure their de signs met the customer ’ s requirements. By the end of the day, the students were expected to have their design on graph paper and the “ Stained - Glass Equation ” worksheet completed so on the last day, their prototype could be made for the public presentation. As the students completed their de signs and “ Stained - Glass Equation ” worksheet, they were instructed to “ trade and check. ” They swapped their design and “ Stained - Glass Equation ” worksheet with another student. They verified that the lines in the design matched the equations and requirements on the “ Stained - Glass Equation ” worksheet. Trading papers and checking each oth er ’ s work, addressed the explain E phase because the students were communicating by checking and explaining, as they verified their partners work.

Figure 1: Student in the beginning design phase

Figure 2: Student in the beginning design phase

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The last day, day three was fantastic! The day fo cused the elaboration E phase of the 5E lesson model. It allowed an opportunity students to move beyond typical calculations as they built their win dows. The students saw their learning and efforts transform into a practical application and a con crete product. The students were ready eager to build their designs. The building process was im plemented in one step increments as a whole class activity. First, the students were instructed to get out their designs. The plastic pocket paper protec tors were handed out. They inserted their design in to a plastic pocket protector (8.5x11) and set it aside. Next, they traced all lines from their graph paper on the exterior pocket protector using a black sharpie and a ruler. After the design was traced, the students colored the open space areas using colored sharpies. They placed their completed colored pocket protector in a safe place. Figure 3 shows de sign components preassembled. Next, the frames were passed out and the students carefully took the backing off of the frame. Pieces of aluminum foil, slightly larger than 8.5x11, were handed out. The students were asked to ball it up gently then straighten it out on a flat surface. The students wrapped the interior backing of the frame with the crinkled aluminum foil and set it aside. They turned their attention to the completed colored pocket protector and retraced any black lines on their design that may have faded. Next, they took the graph paper out of the pocket protector and re placed it with the aluminum foil wrapped frame backing. Then they placed their designs into frame and lock it into place. There were so many great designs and the students were so proud of what they accomplished that they wanted to show all their peers. The joy of math was in the eyes and it was magical. The final step was making a placard that described their prototype.

The final “ evaluation ” phase for the 5 E model was the public presentation of their products. The stu dents completed a 3 - 2 - 1 exit ticket: 3 - types of slopes, 2 - things they learned from the lesson that they did not know before, and 1 - thing they liked about the lesson. Assessment Students were assessed each day and throughout the lesson. The initial prior knowledge assessment was given at the beginning of day one with the “ Linear Equation Warm - up ” worksheet. T benefit of the PBL framework is anticipating and knowing the students were going to research, investigate, and discover a deeper meaning of linear equations and slopes throughout the process. Furthermore, each of the stations were paired with a worksheet to serve as formative assessments. Since the stu dents were working in small groups for the first three stations, it was easy to peruse the class and give guidance as needed. The class went outside for station 4 and were able to see slopes of lines in the real - world. As they were returning to the class room, they were asked to show each of the four types of slopes with their bodies or arms. The con nection to the real - world objects and their individu al bodies gave some students the coveted “ ah - ha ” moments. On day two, the students began the sum mative assessment, which was the design of their window. Using the “ Stained Glass Equation Work sheet ” and graph paper, the students had to create their own design and use a minimum of ten linear equations. This assessment was critical to the PBL lesson and to the students understanding, because they could connect algebra to a real - world situa tion, which strengthen their new understanding. By plotting y - intercepts and deciding on the slope in order to find the equation for the lines, the students interacted with the algebra rather than memorizing formulas and plugging in numbers. The last day of the lesson was building the prototype and the stu dents were given a small summative 3 - 2 - 1 assess ment at the end of the day. Since they were work ing individually on their design, discussions flour ished with students one - on - one where they were asked to explain their designs, reasoning and what if scenarios. The combination of the PBL lesson framework within the 5E model, resulted in a suc cessful student learning experience, where students deeply connected to the content. Evidence of Student Learning Student learning was apparent throughout the les son. It began on day one with the prior knowledge assessment “ Linear Equation Warm - Up ” worksheet

Figure 3: Student designs components before assembly

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tent to a real world application that further devel oped students ’ understanding about slopes.

Station 2, “ Find the Mistakes, ” gave the students the opportunity to recognize where the math could go wrong. Identifying common misconceptions and fixing the mistakes, helped the students to be aware common errors when working with basic linear equation and slope problems (See Figure 5). Stu dents worked individually at this station, but teach ers were closely monitoring student work and con versations to ensure misconceptions were ad dressed. The common misconceptions identified at this station included adding instead of subtracting, or vise versa, when moving terms across the equal to find the slope and using the distributive proper ty appropriately.

Figure 4: Student completed warm - up

(see Figure 4). Familiarity with the content led to a smooth lesson. The PBL framework linked the con

Figure 6: Station 1 “ Research Station ” Student Work

The “ Research Station ” (Station 1) and the “ Slopes in the Real World ” (Station 4) were excel lent examples of students deeper understanding of slopes. During these stations, students identified slopes in pictures and object that are not typically considered in the math classroom. Students made deeper connections to the content. Figures 6 and 7 show students finding slopes outside of the carte sian coordinate plane displayed on the paper.

Student learning was evident with their final prod ucts. Students connected their created designs on

Figure 5: “ Find the Mistakes ” student work

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Figure 10: Students display their work

References Bybee, R., Taylor, J., Gardner, A., Van Scotter, P., Powell, J., Westbrook, A., & Landes, N. (2006). The BSCS 5E Instructional Model: Ori gins and Effectiveness. Colorado Springs. Re trieved from https://bscs.org/resources/reports/ the - bscs - 5e - instructional - model - origins - and effectiveness/. Krajcik, J., McNeill, K., & Reiser, B. (2007). Learning ‐ goals ‐ driven design model: Develop ing curriculum materials that align with nation al standards and incorporate project ‐ based ped agogy." Sci. Ed, 92, 1 – 32. doi:10.1002/ sce.20240. Larmer, J. & Mergendoller, J. (2010). 7 Essentials for Project - Based Learning. Educational Lead ership , 34 – 37. Larmer, J., Ross, D., & Mergendoller, J. (2009). PBL Starter Kit : Novato. Buck I. for Ed.

Figure 7: “ Slopes in the Real World ” Student Work

Figure 8: Student beginning stained glass window design

the graph paper to the linear equation that they found and present their designs for public display. Figure 8 shows the beginning of the stained - glass design process followed by Figure 9, that shows the final product that was used for public display. The PBL format joined with the 5E model, helped students develop a deeper understanding of the content by connecting the seventh grade algebra content to a real - world context.

Brittany Houdashelt ODU Mathematics Education Graduate

Dr. Sarah Ferguson ODU College of Science

Deja Richardson ODU Mathematics Student

Kelly Johnson ODU Mathematics Education Graduate

Figure 9: Student completed project for public display

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Note from the Virginia Department of Education Tina Mazzacane

On March 13, 2020, schools across the Common wealth closed their doors amidst much uncertainty surrounding the COVID - 19 virus. Virginia public schools were required to quickly transition from a brick - and - mortar model of education to a technol ogy - enabled and remote learning model. Students, families, educators, and entire communities had to manage this unprecedented transition, while ensur ing that equitable access and support for a variety of student learning needs are met. Achieving equi ty in remote learning is more complex than simply providing equality in access to learning resources and technology. Providing ongoing support to stu dents and their families to ensure student success remained a top priority. The Virginia Department of Education (VDOE) is providing ongoing guidance to school divisions to ensure continuity for learning after schools closed and VDOE continues to support schools as they plan for the reopening of schools for the 2020 2021 school year. In addition, the VDOE Mathe matics Team provides ongoing support to mathe matics educators across the state. In this article, I will share some of the many resources that are available. The Virginia Learns Anywhere guidance docu ment, created by the Continuity of Learning (C4L) Task Force, was intended to address the immediate needs of school divisions to meet the ongoing learning needs of all students when school closed last spring. The document includes equity consid erations and key steps when choosing alternative options for virtual, online, and other instructional delivery models and strategies. Learning in Place Mathematics Resources were compiled by the VDOE Mathematics Team to as sist teachers, parents, and students in identifying quality resources to support ongoing learning at home since March 2020 school closures. Re sources include Virginia Standards of Learning Mathematics Tracking Logs for Kindergarten through Algebra II to identify the standards stu dents had sufficient exposure to and experience with prior to the COVID - 19 school closure, and to

support teachers decision - making regarding when and how experiences with new standards may oc cur moving forward. There are also online re sources, playlists, and collections of select eMedi aVA resources, and suggested offline activities. When Governor Northam ’ s June 9, 2020 News Release, announcing Virginia ’ s Phased Return to School Plan, simultaneously a guidance document was simultaneously released by the Virginia De partment of Education called, Recover, Redesign, Restart 2020. It is intended to help guide school divisions in making plans for the 2020 - 2021 reo pening of schools. This comprehensive guidance document includes the recommendations from the Return to School Recovery Task Force, the Ac creditation Task Force and the Continuity of Learning (C4L) Task Force. The document out lines the phased approach to reopen schools and it includes resources to support school divisions with instructional plan support. As mathematics teachers prepare for the return to school in the fall, equity and rigorous content should guide instructional decisions. It is estimat ed that content learning loss for some students may be as high as 50% due to the shut down in March 2020, which is dependent on the instruc

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vital. The plan is to have teachers implement a “ Just - in - time ” diagnosis of unfinished learning and reteach foundational content required for students to access the new content. These have been and continue to be unprecedented times for all of us. Your colleagues will need your support and collaboration as we tackle the chal lenges that are ahead together. Challenges also provide opportunities to advance our own knowledge and skills as learners and educators as it does for our students. It is our role to help them achieve success.

tional quality students received during school clo sures. This learning loss may be exacerbated for many of our marginalized students. It is important for teachers to move forward with the new grade level mathematics instruction in the fall of 2020 versus sending students backward to recover all of the unfinished learning that may exist from the pre vious grade level or course. Identifying the prereq uisite knowledge that students need to learn the new mathematics content in the new school year is

Tina Mazzacane Mathematics Coordinator Virginia Department of Education

Organization Membership Information

National Council of Teachers of Mathematics Membership Options:

Essential Membership: $94/year, full membership Premium Membership, plus research journal: $149/year Emeritus Membershp: $49/year, premium Membership Student Membership plus online research journal: $49/year

Virginia Council of Teachers of Mathematics Membership Options:

$20 Individual One - Year Membership, Student Membership $20 Institutional One - Year Membership $39 Individual Two - Year Membership $57 Individual Three - Year Membership $500 Individual Lifetime Membership

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How to Keep Students from Falling Off the Slope

Dawn D. Hakkenberg & Diana Fisher

and she asks her students to explain what the sign represents. In addition, she discusses the units of measure, their placement in the slope ratio, and the significance of this information (Teuscher and Reys, 2010). In the first two activities, equations are used to describe how a function behaves over time, within the context. The last activity uses line ar regression to give the students the opportunity to see how the past behavior of a function can be used to predict future behavior. Samples of her stu dents ’ work are shown below. Algebra students learn more deeply when they are actively engaged in problem solving activities through predicting, investigating, and drawing con clusions. In this way, they “ see and touch ” the math. In addition, group activities support students ’ development of mathematical language as they dis cuss their ideas and solutions with each other (Wolbert, 2017). For Ms. Hakkenberg, this repre sented a shift toward student - centered learning, which is preferred to traditional learning and pro vide students with the opportunity to develop their intuitive insights, which in turns helps them devel op a firm mathematical foundation (Kaput, 2018).

Introduction

The purpose of this paper is to introduce readers to a set of three hands - on, technology - based Algebra activities that use multiple representations that fo cus on slope - based mathematical content. The au thors used the Online Desmos Graphing calculator to provide students with a visual representation of function behavior over time. In addition, they pro vided repeated experiences with multiple represen tations for linear functions in a real - world context. The first activity provides an introduction to Vir ginia SOL A.7 (Virginia Department of Education, 2017) material using a real - world problem provid ed by Dan Meyer ’ s Water Tank task (Meyer, 2011). The students explore four ways the linear function can be represented. In the second activity, the students ’ focus is on A.6 concepts as they use walking patterns and a motion detector to create various distance vs. time graphs. In the third activ ity, the students toss a ball, make a people vs. time table, find a line of best fit and make predictions for outputs with given inputs, SOL A.9. Through out these three activities, the teacher (Ms. Hakken berg) frequently asks about the sign of the slope

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Virginia SOL 2016 A.6

The student will a) determine the slope of a line when given an equation of the line, the graph of the line, or two points on the line; write the equation of a line when given the graph of the line, two points on the line,

or the slope and a point on the line; and graph linear equations in two variables.

A.7

The student will investigate and analyze linear and quadratic function families and their characteristics both algebraically and graphically, including a) determining whether a relation is a function; domain and range; c) zeros; d) intercepts; e) values of a function for elements in its domain; and f) connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs. The student will collect and analyze data, determine the equation of the curve of best fit in order to make predictions, and solve practical problems, using mathematical models of linear and quadratic functions.

A.9

These instructional shifts, when coupled with a Math 360 and Smart Pal Curriculum , increasingly keep students on the slope. This is evident by her increasing SOL pass rates; up to 87% in 2019 for her urban high school students, in which strong math students typically enroll in Algebra I courses in middle school. Before her instructional shift, most of her students completed worksheets correct

her students ’ faces, and watched them become more engaged as their mathematical self confidence grows. Some of her students now ask if they can stay and work in her class, rather than go to their next class when the bell rings. We share these shifts with Virginia educators in the hope that all Virginia students deepen their understanding of Algebra. This activity is chosen first because the images of the water tank filling and emptying are great exam ples of positive slope and negative slope respec tively, and provide good reference points for dis cussions. The activity begins with the three acts of the Water Tank task (Meyer, 2011). The first and last act are whole class activities, while the second act is a partner activity. All the resources needed are available in the Links & Show Notes of a Vrain Waves podcast (Kalb, Peters and Meyer, 2018). Act One: The students view a short video clip of water flowing from a hose into a water tank. The class discusses the questions that arise in their minds and what they notice while watching the video. They conclude the discussion by agreeing on the question: how long will it take to fill up the water tank? Each student predicts a possible cor rect answer. Activity 1: Using Three Acts to Introduce Slope

ly, but they had difficulty relating their learning to the real world. They were unable to car ry out linear trans formations and they would demon strate conceptual misunderstandings. For example, be lieving a line with a negative slope is a line with a nega tive y - intercept. Her teaching prac tices continue to improve, so that

Figure 1: Photo by Dan Meyer (2011) used with permission

today most of her students can accurately identify sign, ratio, and units for a slope. These are skills that many Algebra I students find challenging to master. (Teuscher and Reys, 2010). In addition, she has seen the joy of intuitive understanding on

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Photo

Graph

Table

Figure 2: Four representations of the Water Tank. Photo by Dan Meyer (2011) used with permission

Act Two: The students request additional infor mation about filling the water tank as they work with their partners to answer the question. Subse quently, dimensions given in the video, and the time it took to fill the tank, are used to find the equation of a line representing the relationship be tween the volume of water in the tank versus time. Some students have time to complete the Sequel , which asks: How long will it take to empty the tank? In this lesson they are exposed to the concept of negative slope as the water tank is emptied. Act Three: The students see the answers and dis cuss and compare their solutions. This discussion includes Sequel values, even though some students did not discover these themselves. Once the stu dents have completed the Water Tank activity, they are introduced to the multiple representations of the linear function (see Figure 2) (Leinwand, 2009). This activity is followed by a Green Screen activity where the students work in pairs to explain various

tomed to problem solving, it is helpful to explain the process and goals of the three act learning mod el before the lesson begins. Students may need to be reminded that they can derive the formula for the area of a regular polygon by breaking the poly gon down into a set of n isosceles triangles, where n is the number of sides, and multiplying the area of one triangle by n . Students may also need to be reminded to keep their units consistent, either sec onds or minutes and either cubic centimeters or ounces.

Activity 2: Using a Motion Detector to Reinforce Slope

This is an activity using a motion detector device to help students connect the rate and type of physical movement to slope.

Supplies:

a) 1 motion detector (often available from the physics teacher in your school). b) 1 copy of the software (free with the motion detector or it can be downloaded from the com pany website). It displays the graph for an ob jects motion as it moves in front of the detector. c) 1 computer d) 1 computer projection unit a) Lesson handout for students to record motion graphs produced in class and to summarize the most important ideas learned from the lesson. The motion detector is connected to the computer and placed on the edge of a chair pointing outward toward an unobstructed path in the classroom (see Figure 3). Per student:

real world graphs by identifying and explaining the meaning of the la bels on the x - and y - axis, the x - and y - intercepts, the slope and a coordi nate pair of their choice. Common Difficul ties and Challenges If students have difficulties because they are unaccus

Figure 3: Student walking in front of a motion detector

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A student stands about 0.5 meters in front of the detector and is asked to walk backward at a slow, constant, steady pace. The teacher asks students to predict the shape of the graph, that shows the dis tance between the motion detector and the student. The teacher asks the students to sketch their predic tion on their lesson sheet. Next, the student is asked to walk when the motion detector is activated. The graph produced is a straight - line segment. The teacher points to the place where the graph inter cepts the y - axis and asks what part of the motion is described by that point on the graph. The teacher asks why the graph is tilted upward (Hakkenberg, 2017). The teacher asks for another volunteer and posi tions this student 0.5 meters in front of the motion detector (facing the detector). This time, the stu dent is asked to walk at a quick, steady, constant pace backwards from the detector. Students in the class are asked to draw a graph of their predictions. The teacher asks what characteristic of the graph should be the same as the previous scenario. Many students correctly predict that the tilt will still be upward because distance from the detector increas es with time. The teacher then asks if it will be the same tilt. Students often know that the tilt will be steeper because the rate of change has increased. What characteristic of the motion would produce a steeper graph? Will the graph still be a straight line? Why or why not? What words in the de scription of how the person is supposed to walk suggest it might be a straight line? While keeping the graph from the first experiment on the screen, the student is asked to walk quickly and at a steady, even pace backwards. The new graph is superim posed on the same grid as the first graph so the two can be compared and the students predictions veri fied. Students are asked for the mathematical term that represents the tilt of the lines that have been pro duced during the first two experiments. They are also asked for the mathematical term that identifies the location where the line touches or crosses the vertical axis. Sometimes the line does not rise im mediately, but has a small horizontal segment be fore it rises. Students are asked what caused that horizontal segment. Usually, a few students can explain that it was caused by the delayed reaction time between the start of the motion detector and the actual movement of the student. Finally, anoth er student is asked to stand about two meters away from the detector, facing the detector. This student is asked to walk at a slow, steady, and constant pace toward the detector. Again, students predict

the graph that will be produced and explain what will be similar and what will be different from the previous graphs and they will explain why. Next, students are shown a graph with a positive slope, followed by a zero slope, followed by a neg ative slope and asked to describe how a student would have to move to create each graph. The soft ware has some simple graphs that can be projected onto the screen so that students can take turns try ing to walk in ways that will match the graph. Their motion graphs are superimposed on the “ target ” graph that is displayed. This is quite en joyable for the students. Usually three or four stu dents, in sequence, are given opportunities to try to match the graph by walking. Homework and assessments give students descrip tions of movements and asks them to sketch graphs of the movements, or give students graphs and ask them to describe the movement represented by the graph. The main concepts reinforced in this lesson in clude: slope is related to velocity, and the distance from the motion detector is captured by the upward tilt or downward tilt of the line segment over time. The positive or negative slope has meaning with regard to the physical movement of the student, and the segment is straight because the movement

is “ steady and constant. ” The lesson also reinforces the meaning of the y intercept.

Common Difficulties and Challenges

Some student have difficul ties labeling the axis. They are not sure which should be time and which should be

Figure 4: Students stand three feet apart and pass the ball down the line.

distance. If this occurs, it provides an excellent opportunity to discuss dependent and independent variables. Some students will predict that a student walking backwards results in a negative slope. Why does the line go up? Noting the units on the graph is often helpful at this point. That is, as time increases, so does distance. Other students may have difficulty knowing how to start to make pre dictions. If this is the case, referring them back to the Green Screen activity is helpful. For example, what are the x - and y - intercepts at the beginning and end of each round? At the beginning of the Round 1, 0 seconds, how far were you from the motion detector? At the end of the Round 1, 5 sec

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