Virginia Mathematics Teacher Fall 2016

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

V IRGINIA M ATHEMATICS T EACHER Vol. 43, No. 1 Fall 2016

Spec i a l I s sue : Vi rg i ni a Ma thema t i c s and Sc i ence Par tner shi p Grant s

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Editorial Staff

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

Dr. Margie Mason Associate Editor of the Special Issue mmmaso@wm.edu The College of William and Mary

Brian Pratt Assistant Editor Radford University

Liam Downey Assistant Editor Radford University

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Virginia Council of Teachers in Mathematics

Many thanks to our Blind Peer Reviewers for Fall 2016

President: Jamey Lovin, Virginia Beach Public Schools

Alfreda Jernigan, VCTM Board Member

Past President: Cathy Shelton, Fairfax County Public Schools

Dr. Steve Corwin, Radford University

Secretary: Lisa Hall, Henrico County Public Schools

Dr. Robert Berry

Membership Chair: Ruth Harbin-Miles

Dr. Betti Kreye, Virginia Tech

Treasurer: Virginia Lewis, Longwood University

Mrs. Anita Lockett, Fairfax County Public Schools

Webmaster: Ian Shenk, Hanover County Public Schools

Jean Mistele, Radford University

NCTM Representative: Betsy Steadman, Hanover County Public Schools

Dr. Matthew Reames, University of Virginia

Elementary Representatives: Meghann Cope, Bedford County Public Schools; Eric Vicki Bohidar, Hanover County Public Schools

Dr. Wendy Hageman-Smith, Longwood University

Middle School Representatives: Melanie Pruett, Chesterfield County Public Schools; Skip Tyler, Henrico County Public Schools

Dr. Maria Timmerman, Longwood University

Secondary Representatives: Pat Gabriel; Samantha Martin, Powhatan Public Schools

Dr. Kateri Thunder, James Madison University

Math Specialist Representative: Spencer Jamieson, Fairfax County Public Schools

Dr. Ann Howard Wallace, James Madison University

2 Year College: Joe Joyner, Tidewater Community College

Pam Bailey, VCTM Board Member

4 Year College: Ann Wallace, James Madison University;

Joyce Xu, Virginia Tech

Robert Berry, University of Virginia

Karen Zwanch, Virginia Tech

Dr. Anthony Dove, Radford University

The acceptance rate for VCTM journals is 20%

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Table of Contents:

Awards Page

5

Message from the Editor

6

Note from the President

7

Virginia Department of Education

8

Technology-Enhanced Inquiry of Light and Optics Concepts 9

HEXA Challenge Fall 2016

14

Solutions to Spring 2016 HEXA Challenge Problems 16

Secondary Mathematics Professional Development Center 20

Key to the Spring 2016 Puzzlemaker

25

Technology Review

26

Enhancing Pedagogical Practices Through PD

27

Grant Opportunities

32

Good Reads

33

Upcoming Math Competitions

34

Interactive Mathematics Institute for Middle School Teachers 35

Math and Science Partnership Grants

40

Call for Manuscripts

44

VISTA ELIS Professional Development

45

Unsolved Mathematical Mysteries

51

Integrating Mandates for the Benefit of Professional Learning 53

Math Jokes

58

Evaluation of Lesson Study Based Teacher PD Model

59

Busting Blockbusters

64

The Puzzlemaker

67

Conferences of Interest

68

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Featured Awards Congratulations to the 2016 Winners of the William C. Lowry Mathematics Educator Award :

Middle School Awardee: Becky Pierce, Smyth County

High School Awardee: Amy Lamb, Northumberlamb County

College Awardee: Betti Kreye, Virginia Tech

Math Specialists Awardee: Corrine Magee, Arlington County

Congratulations to our Reigning Champion Hexa Challenge Winner Veronica Moldoveanu! Teaches AP Calculus AB and Geometry at Falls

Church High School. With two consecutive victories, in the Hexa Challenge, she is now our Reigning Champion! Be sure to attempt our Hexa Challenge problems featured in this issue!

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Sharing the Knowledge: AMessage from the Editor Dr. Agida Manizade

We are pleased to introduce this special issue of the Virginia Mathematics Teacher journal, focused on Mathematics and Science Partnership (MSP) grants

64, and the Hexachallenge, page 14. We also invite you to read the Technology Review and Good Reads columns. Please consider applying for our

sponsored by the Department of Education for the past six years. These awards have been granted to 25 institutions, from universities to public schools. More than 70 teams of educators have received this funding and have worked tirelessly to improve the state of mathematics education in the Commonwealth. As funding will cease in 2018, our editorial staff have decided that

featured grant on page 32, and join us at the conferences listed here. If you have never participated in the journal, consider acting as a reviewer for a future issue, or writing an article based on your experiences. We believe sharing experiences based on reflective practices is one of the most effective ways of

distributing professional knowledge.

If

you are interested, please see the call for

mathematics teachers should hear about these projects, and have access to the products created by the teams. This can allow teachers across the Commonwealth to access and effectively utilize the educational products in their own classrooms. We invited every Principal Investigator (PI) to submit an article briefly describing their project and any information which would be helpful to practicing teachers. You can find a list of the projects, PIs, and their contact information, provided by the Department of Education, on page 40. In addition to the articles related to the special issue, the normal recurring sections and challenges are included in this issue. Please participate and invite your friends to answer Busting Blockbusters, page

manuscripts on page 44. Thank for your continued interest in our publication, and please enjoy this special issue.

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

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Note from the President Jamey Lovin

Welcome Back! I hope everyone was able to enjoy time with families and friends this summer. Your VCTM Board did spend some of our summer together, meeting to talk about our plans for the organization in the

outstanding support for our teachers in its implementation. It is an exciting time and I am glad to be part of that as an educator in Virginia. So, I will end where I started. We your board, and me, your President, want to be a source of encouragement to you, the Virginia Mathematics Educator community . When speaking of our membership, one board member shared a most fitting t-shirt she saw worn by a 2015 conference attendee. It said, “I am a Math Teacher, what is your Super Power?” We all agreed Virginia teachers are super heroes! It shaped what we did that day and became the theme for our 2016 Conference: We Are a Community of Math Heroes! Please feel free to contact myself, or any board member, with ideas on how we can better serve you.

upcoming year. We examined our vision statement, goals, and objectives and renewed our commitment to them. We want our focus to be on YOU , our members , and creating opportunities for you to be actively engaged with the organization, not just at conference time, but also all year long. Teachers of mathematics in Virginia face a challenging year thinking about how we will transition to the new Standards of Learning. I am optimistic about the changes and believe our state leaders have developed an exceptional plan that provides a first rate education for students and

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

Organization Membership Information National Council of Teachers of Mathematics Membership Options:

Individual One-Year Membership : $93/year, full membership Individual One-Year Membership, plus research journal: $120/year Base Student E-Membership:$46/year Student E-Membership plus online research journal: $61/year Pre-K-8 Membership: $160/year with one journal Pre-K-8 E-Membership: $81/year with one digital journal -$10 for per additional teacher

Current National Council of Teachers of Mathematics Membership: 70,000 Members

Virginia Council of Teachers of Mathematics Membership Options: $20 Individual, One-Year Membership $20 Institutional One-Year Membership

Current Virginia Council of Teachers of Mathematics Membership: 750 Members

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Note from The Virginia Department of Education: The Mathematics and Science Partnership Michael Bolling

The Mathematics and Science Partnership (MSP) program is designed to improve the content knowledge of teachers and the performance of students in the areas of mathematics and science by encouraging states, institutions of higher education (IHE), local education agencies (LEA), and elementary and secondary schools to participate in programs that:  Improve and upgrade the status and stature of mathematics and science teaching by encouraging IHE to improve mathematics and science teacher education;  Focus on the education of mathematics and science teachers as a career-long process;  Bring mathematics and science teachers together with scientists, mathematicians, and engineers to improve their teaching skills; and  Provide summer institutes and ongoing professional development for teachers to improve their knowledge and teaching skills. Partnerships between high-need school districts and the science, technology, engineering and mathematics (STEM) faculty in institutions of higher education are the core of these improvement efforts. Other partners may include state education agencies, public charter schools or other public or private schools, businesses, and nonprofit or for- profit organizations concerned with mathematics and science education. The MSP program is a formula grant program to the states, with the size of individual state awards based on student population and poverty rates. Each state is responsible for administering a competitive grant competition, in which grants are made to partnerships to improve teacher content knowledge in mathematics and science. As a part of MSP grant requirements, IHE are required to provide public access to products and professional development resources developed through grant funding. The Virginia Department of Education (VDOE) provides access to these resources through the VDOE MSP website. http://www.doe.virginia.gov/federal_programs/ esea/title2/part_b/index.shtml Over 80 MSP grant projects have been

funded since 2003, providing growth opportunities to an estimated 7,500 mathematics and science teachers.

Michael Bolling Virginia Department of Education Michael.Bolling@vdoe.virginia.gov

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Technology-Enhanced Inquiry of Light and Optics Concepts: Teachers’ Professional Development Jennifer Maeng, Richard A. Lindgren, Jesse T Senechal

Abstract

PD Model

The Developing Science Teachers’ Understanding of Light and Optics Professional Development (PD) provided an integrated approach to teaching science through inquiry and educational technology for upper elementary, middle, and high school teachers with the goal of increasing their content and pedagogical knowledge for teaching physical science. Below, we describe the PD model employed as well as teacher and student outcomes. Results indicated teacher’s understandings of light and optics content and their pedagogical knowledge for teaching through inquiry and technology improved following participation in the PD. These results have implications for the implementation of PD that supports middle and high school teachers’ understanding of light and optics content. Products of the PD include teacher-generated lesson plans. The Developing Science Teachers’ Understanding of Light and Optics Professional Development (PD) provided an integrated approach to teaching science through inquiry and educational technology. This MSP project, led by Dr. Richard Lindgren, was a collaboration between the University of Virginia (UVa), Jefferson National Laboratory (JLab), the Virginia School University Partnership, Albemarle County Public Schools, Charlottesville City Schools, Newport News City Schools, and Hampton City Schools. Goals of the project were to (1) support upper elementary, middle, and high school teachers’ content knowledge and conceptual modeling instructional skills to effectively teach science content outlined in Virginia’s Science SOLs and (2) to support teachers in integrating technology-enhanced inquiry to improve student achievement in science. To accomplish these goals, the program held two summer institutes, one at UVa and one at JLab, during the summer of 2014. The summer institute focused on increasing teachers’ pedagogical knowledge for teaching science through technology-enhanced inquiry and light and optics content knowledge. Introduction

Previous research has identified five components of PD likely to influence teacher quality and student achievement (e.g. Loucks- Horsley et al., 2010). These components include (1) immersing teachers in inquiry, questioning, and experimentation to model inquiry teaching, (2) engaging teachers in concrete teaching tasks based on their experiences with students, and (3) focusing on subject-matter knowledge and deepening teachers' content knowledge. Further, effective PD should be (4) intensive, long term, and coherent and (5) be grounded in a common set of PD standards in order to show teachers how to connect their work to specific learning standards for student performance (Loucks-Horsley et al., 2010). These components informed the Developing Science Teachers’ Understanding of Light and Optics PD. POE Inquiry Model. Inquiry can be defined simply as “Students answering a research question through the analysis of data” (Bell, Smetana, & Binns, 2005). Several models of inquiry instruction exist. This PD taught light and optics content through a predict-observe-explain (POE) inquiry model (Haysom & Bowen, 2010). The POE model involves eliciting student ideas, discussing student predictions, students making observations and explaining their observations, and the teacher supporting students’ explanations with the scientific explanation. Technology-enhanced inquiry. Research indicates integrating computer-based models into inquiry instruction promotes students’ conceptual understanding of scientific phenomena (NRC, 2011). The role of computer simulations is not to replace inquiry investigations but to provide students with supplemental contact with the variables tested in a real experiment or to visualize the process that occurs at sub-atomic scale (Luft, Gess-Newsome, & Bell, 2008). Research also indicates that computer simulations are useful for simulating labs that are impractical, expensive, or too dangerous to conduct in a school-based setting, contribute to conceptual change, and provide tools

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for scientific inquiry and problem solving (e.g. Maeng, Mulvey, Smetana, & Bell, 2013; NRC, 2011; Windschitl, 2000).

refraction and wavelength. They summarized patterns in their data to explain their observations, which resulted in a formal statement of Snell’s Law (i.e. the Law of Refraction). PHET Simulations, developed by UC-Boulder, are a repository of free, downloadable simulations that address a variety of math and science concepts. In a parallel morning activity, teachers used a plastic block, a laser beam and a protractor to measure angles and verify Snell’s Law (Figure 1). This activity required some knowledge of sines and cosines, but could have also been completed just by making use of the Pythagorean Theorem. Knowledge of sines and cosines has the potential to help teachers in other mathematical applications of science (e.g. determining the work done when pulling a cart along a road with a handle inclined at a 45° angle from the road). Teachers discussed the affordances and limitations of simulations and generated resource

PD Context

Morning sessions during the summer institute involved teachers engaging in modeled hands-on activities to build their content knowledge and that they could easily modify to include in their own classroom instruction. Across the 10 days of the summer institute, teachers engaged in 43 different hands-on investigations of light and optics. Many of these investigations related to the reflection and refraction of light (e.g. VA SCIENCE SOL 5.3, 8.9, PH.8) and relied heavily on an understanding of angles (e.g. VA MATH SOL 3.15, 5.11, G.3, G.4). For example, teachers explored the ray model of light, in which light is represented as straight lines emanating from an object. Using a laser beam as a ray of light and a single plane mirror teachers investigated the Law of Reflection (the angle of incidence is equal to the angle of reflection). They also used multiple plane mirrors to track a ray of light over several reflections to locate the final image. Later, they used these same mirrors to understand how images are actually formed, which required additional math skills. Using protractors, they measured the angles of incidence and reflection and looked for patterns in how the angle of reflection varied with the angle of incidence (e.g. VA MATH SOL 3.19, 5.17, 7.13). Each activity teachers engaged in involved some combination of making measurements, performing calculations, creating and interpreting graphs, describing patterns, and made use of their knowledge of geometry and trigonometry. Afternoon sessions addressed practical aspects of classroom implementation of light and optics content using a technology-enhanced POE model. During this time, teachers learned strategies to effectively integrate simulations and animations to support students’ scientific investigations through modeled lessons designed to reinforce the content they explored during morning investigations. For example, in one afternoon activity, teachers used a PHET simulation ( https:// phet.colorado.edu/en/simulation/bending-light ) to explore the Law of Refraction. First, they predicted an answer to the question, “What happens to the speed of light as a light ray passes through different mediums (for example air into water)? Why? How might this affect what we see?” Teachers used the simulation to make observations by manipulating the media (i.e. water, air, glass) through which the light waves traveled and measured the angle of

banks of simulations and animations to support their teaching of light and optics content. They also developed and received feedback on lesson plans that incorporated these strategies that they could directly implement into their classroom science instruction. Finally, they discussed how they could integrate the POE inquiry model and educational technology in cross curricular ways. For example, because the PHET simulation included a protractor Figure 1. A laser beam is used to illustrate how a ray of light incident on a glass block is reflected from the block as a faint red ray and the main ray is refracted through the block. A protractor is used to measure the angles involved.

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Content Knowledge

to measure angles as the light rays passed through various media, teachers discussed ways in which they could reinforce students’ understanding of angles and protractor use to both science and mathematics content. For participating in the PD, teachers received physics graduate course credit, all materials needed to implement the modeled light and optics activities in their classrooms as well as generate new activities, feedback from peers and instructors through follow-up sessions, and the opportunity to attend and present lessons they developed through the project at the annual Virginia Association of Science Teachers Professional Development Institute. Research Design Participants included 24 teachers from 22 schools in 15 divisions. Twenty (80%) of the teachers taught in middle school, 28% of teachers had 5 or fewer years of experience and 64% held Master’s degrees in education. The PD was evaluated through a quasi- experimental pre-/post-test design in which teacher and student pre-assessments served as their own control. The design assessed changes in teachers’ content and pedagogical knowledge, and their perceptions of the PD as well as their students’ science achievement. Teachers’ content knowledge was assessed on two light and optics content- knowledge assessments pre- and post-summer institute as well as year-end. Teachers also completed surveys related to pedagogical knowledge. Changes in their students’ science content knowledge were assessed at the beginning of the year and following light and optics instruction via a researcher-developed instrument. Quantitative data were analyzed descriptively and inferentially and qualitative data were thematically analyzed. Two major limitations need to be considered when interpreting the results described below. First, all data related to pedagogical knowledge were self-reported by participants. Second, the research design did not employ a control group, therefore, causal inferences regarding the impact of the PD must be interpreted with caution. Results suggested the PD positively influenced teachers’ knowledge related to light and optics content, pedagogical knowledge for teaching light and optics, and their students’ light and optics content knowledge. In addition, teachers had positive perceptions of the PD. Results

Results indicated teachers’ content knowledge significantly improved from pre- to post-instruction on both Light and Optics content assessments; assessment 1 pre (M = 10.3), post (M = 15.9) (t = 5.883, p < .001), assessment 2 pre (M = 28.8), post (M = 32.1) (t = 3.776, p = .001). These results suggest the PD positively influenced teachers’ understanding of physics content. Themes in the qualitative data supported these findings. For example, teachers perceived their content knowledge to be limited prior to the PD and that the PD helped them develop a deeper understanding of the content. For example, one teacher noted, “I haven’t honestly had physics since college, so it was really good for me to refresh my memory of physics and my knowledge of physics and even go beyond what my students need.” They also perceived that the content of the PD went beyond what they needed to know to address grade-level physical science standards. For example, a teacher indicated, “There were some things that I won’t teach in eighth grade, like all the understanding of the distance of the lens from the focal point and all that. That’s high school and while I feel I need to know that, if it were professional development it goes beyond what I need.” Finally, developing their content knowledge through the PD appeared to provide teachers with confidence in their ability to explain physics ideas, answer questions, and design lesson plans. A teacher described how the PD supported her understanding the mathematics behind the physics involved in the content: “It taught me more of the physics and the mathematical part behind things so as I’m doing labs I’m not just following directions and getting through but I actually understand the reasoning behind some of the things that we’re doing.” Pedagogical Knowledge Teachers’ self-rated pedagogical knowledge was statistically significantly higher following participation in the PD for all assessed pedagogical skills (Tables 1 and 2). Qualitative data support these findings. As exemplified by the interview response below, many teachers noted the use of simulations was a novel instructional tool that they would use to support their physical science teaching. “I honestly have not used simulations in the past and so being able to find a wide variety of simulations as a result of that class was great.” Others noted the use of the POE model supported their thinking about how to integrate inquiry into their physical science instruction:

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When we talk about the predict, and then the observe and then the explain; when I do other lessons now I keep that in the back of my mind. Like, Ok, when they should predict something then they need to be the ones that actually observe it, and then we need to kind of regroup and explain and make sure we kind of revisit the predictions and make sure that they have the knowledge that they need before they more on and so that model has really been helpful with other lessons as well. A subset of teachers submitted pre/post student achievement data using a researcher- developed 14-item Light and Optics content instrument. The mean student score increased 17 points from pretest to posttest, from 53% (SD = 20%) correct to 70% (SD = 17%) correct. Paired samples t-test indicated this was a statistically significant gain, t (278) = -17.196, p < .001. In interviews, teachers indicated they perceived that the POE instructional model, combined with their enhanced content knowledge facilitated their students’ learning. For example, one teacher indicated: I think they seemed like they understood it, based on hearing their explanations of defending their answer with the initial prediction and trying to support it, and also just when they were explaining their reasoning for why they thought a certain way, they were able to support the answer. Other teachers responded similarly, that the investigative nature through which they were able to teach the content had a positive impact on student learning. Perceptions of the PD Overall, the majority of teachers (95%) indicated the PD program met or exceeded their expectations. Further, 87% of participants said they would recommend or highly recommend the program to other teachers. Most participants responded that the hands- on labs including the integration of technology, and the collaborative nature of the projects were the most effective components of the PD. For example, one teacher wrote, “Getting all the hands -on experience in doing the labs. It made all the concepts ‘real’ for me, and I'm glad that we got to discover them on our own.” The “talented and knowledgeable professors” were also identified as a program strength. When asked to make suggestions for improvements, a number of Student Achievement

teachers suggested more modeling prior to labs. For example, one teacher wrote, “Some of the lab activities were difficult to follow through just by reading the directions. Having the teacher model the procedure before more complicated activities would have been helpful. This would also model for the teachers how to model procedure for their students.” Conclusions and Implications Overall, results of this investigation suggest that PD that supports middle and high school teachers integrating technology through a POE model to teach light and optics content has the potential to positively influence middle and high school students’ understandings of these concepts. Teacher-generated lesson plans to teach light and optics, including alignment with Virginia SOLs, assessment plans, and associated student handouts, were the primary product generated through the UVa-JLab project. These lesson plans as well as PD materials from the summer institute (lab activities, PowerPoint TM slides, instructional videos and photos) are available at: http:// teacher_institute/2014-labs.html . These materials have the potential to support teachers in implementing light and optics content through a technology-enhanced POE model as well as facilitating teachers’ considering the potential to integrate science and mathematics content (e.g. patterns, data collection, angles). galileo.phys.virginia.edu/outreach/ ProfessionalDevelopment/UVa-JLab/ Bell, R., Smetana, L. K., & Binns, I (2005). Simplifying inquiry instruction. The Science Teacher, 72, 30–34. Haysom, J. and Bowen, M. (2010). Predict, observe, explain: Activities enhancing scientific understanding. NSTA Press: Arlington, VA. Loucks-Horsley, S., Stiles, K.E., Mundry, S., Love, N., & Hewson, P. (2010) Designing professional development for teachers of mathematics and science. (3 rd Ed.) Thousand Oaks, CA: Corwin Press. Luft, J., Gess-Newsome, J. & Bell, R.L. (eds). (2008). Technology in the secondary science classroom. NSTA Press: Arlington, VA. Maeng, J.L., Mulvey, B.K., Smetana, L.K., & Bell, R.L. (2013). Preservice teachers' TPACK: References

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hgtechnology inquiry instruction. Journal of Science Education and Technology, 22, 838-857 . DOI: 10.1007/s10956-013-9434-z National Research Council (NRC). (2011). Learning science through computer games and simulations. Washington, DC: National Academy Press. Windschitl, M. (2000). Supporting the development of science inquiry skills with special classes of software. Educational Technology Research and Development, 48 , 81-95. to support

Jennifer Maeng Curry School of Education University of Virginia jlc7d@virginia.edu

Richard A. Lindgren Department of Physics University of Virginia ral5q@eservices.virginia.edu

Jesse T Senechal MERC School of Education Virginia Commonwealth University senechaljt@vcu.edu

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HEXA Cha l l enge

Problems created by: Dr. Oscar Tagiyev

October Challenge: It is 2016. If we continue writing the digits, 2, 0, 1, 6, in this order N times, we’ll GET a different number K=201620162016...2016 where the digits 2, 0,1, 6 are repeated N times. Prove that K can not be a perfect square of any integer number.

November Challenge: There are N people that live in a city, where there are two main competing companies. Out of these people, n know each other, because they work for the same company. m people also know each other because they work in the same city, but in a rival company. Personal relationships between workers at rival companies are not allowed. What portion of the population of the city, does not work for either of these companies, but can knows exactly 1 person from each company.

December Challenge: Given an infinitely large set of different types of triangles, if one randomly selects a triangle, what are the chances of this triangle being obtuse?

Contest Alert!

Virginia Mathematics Teacher is conducting a contest for educators and students who can solve the greatest number of problems cor­ rectly by 2/29/2017 The winner will receive a prize and will be featured in the next issue of the VMT. Send your solutions to vmt@radford.edu with the email subject line: Hexa Challenge

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Please be sure to state your assumptions as you solve each problem. Answers to the Fall 2016 Hexa Challenge Problems will be featured in the Spring 2017 Issue of Virginia Mathematics Teacher.

January Challenge: Solve the following equation:

February Challenge: Prove the following for any integer number, n :

March Challenge: In an Isosceles triangle ABC, AB=BC, Angle B is 20 degrees, AC is five units Point D is on BC so that Angle BAD is 30degrees, and angle DAC is 50 degrees. Point E is on the side AB, so that Angle ECB is 60 degrees and angle ECA is 20 degrees. Find the length of DE. See Figure 1.

Figure 1

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So l ut i ons t o Spr i ng 2016 HEXA Cha l l enge Probl ems

April Challenge:

In a trapezoid ABCD. E is the midpoint of AB, F lies on CD, and EF is perpendicular to CD. Estimate the area of the trapezoid if EF= h and CD = b.

SOLUTION :

Draw line GE parallel to CD. G lies on line AD. H is the point of intersection between lines GE and BC. The area of the parallelogram GHCD is equal to the area of trapezoid ABCD, because the triangles EHB and AGE are equal to each other by angle-side-angle congruence condition. Thus the area of the trapezoid equals b times h.

May Challenge:

In a triangle ABC. AD and CF are medians, D is on BC and F is on AB. The medians intersect at O. What is the ratio of the triangular areas DOF to ABC?

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So l ut i ons t o Spr i ng 2016 HEXA Cha l l enge Probl ems

Construct the third median BG. The three medians AD, CF and BG split ABC into six triangles (AOF, AOG, GOC, DOC, BOD, and BOF) whose areas are equal to one sixth of the ABC area. The area of the triangle BDF is one fourth of ABC area, as those triangles are similar. Thus the area DOF equals area of DOFB - area of BDF = 2 x one sixth - one fourth = one twelfth of ABC. The area of DOF equals the area of DOFB minus the area of BDF. The area of DOFB equals 2 times 1/6 th the area of ABC, as it is the sum of the areas of the trian­ gles BOD and BOF. As the area of BDF equals 1/4 th the area of ABC, the area of DOF equals of the area of triangle ABC. June Challenge: AB is the diameter of a semicircle. AC is the diameter of smaller semicircle which completely inside the first semicircle. The line FG is tangent to the smaller semicircle and parallel to AB. The length of FG is 10 units. Find the ar­ ea of the bigger semicircle that does not include the area of the smaller semicircle.

SOLUTION :

Let’s translate the smaller semicircle to the right, so the center points of both semicircles will overlap. The shaded area will remain the same, as the area of the smaller semicircle is not changing: Therefore:

Shaded area equals:

July Challenge Mr Saver has $ A, spends nothing and saves all money he has been paid by the clients. If a client pays any­ thing, Mr Saver estimates the "significance" of the payment by calculating the ratio of this payment to the total amount he has saved, including the last payment. During this month, Mr Saver has had N clients, who paid him $ B in total. Assume R - the maximum "significance" ratio. Find the minimum value of R.

SOLUTION :

Assume so far Mr. Saver has had N clients who have paid him the amounts x1, x2, x3, respectively. If R is the "significance" of their payments, then:

...

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So l ut i ons t o Spr i ng 2016 HEXA Cha l l enge Probl ems The inequalities could be rewritten in the following way:

...

Let’s multiply all of the inequalities, then :

August Challenge: Four spheres are all tangent to a plane. Three spheres with the radii 2 units, 1 unit, and 1 unit respectively, are standing on the plane around the fourth sphere which is smaller. If all four of the spheres are tangent to each other, what is the radius of the smaller sphere? Note that If two spheres with radii R and r are on the same plane and touch each other, the distance between the points at which the spheres are touching the plane can be expressed through the Pythagorean theorem Then, all the arrangement of the four sphere centers O1, O2, O3 and O4 could be projected on a plane as points A, B, C, and D respectively. The projection of the centers on the plane is the figure below. Where dis­ tance AB is the distance between the projected centers. Below on the left is the side view

Consider the top view, of the four spheres with centers O1, O2, O3, and O4 (above right). We can see that

The solution to this equation can then be calculated as follows:

We have to choose the smaller root because x must be less than 1. Therefore

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So l ut i ons t o Spr i ng 2016 HEXA Cha l l enge Probl ems September Challenge:

Inside a sphere (the original print had a typo that said circle) there is a polyhedron where all n vertices are on the sphere. Prove that the number of points where the diagonals intersect each other, cannot exceed:

SOLUTION : The number of quadrilaterals that can be formed out of n points on the circle is:

Each of the quadrilaterals has two diagonals, therefore they can intersect at only one point. As a result, the total number of points of intersection for the polyhedron can never exceed this number.

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Secondary Mathematics Professional Development Center Dr. Agida Manizade, Dr. Laura Jacobsen, Christine Belcher, Robert Thien, Jamey Lovin, Stephanie Brady, Dee Baker

based on work related to our projects in state, national, and international professional outlets. The purpose of this paper is to present materials developed by secondary mathematics teachers, for secondary mathematics teachers. The SMPDC was originally funded in 2010 by a Math and Science Partnership grant from the VDOE and has received six consecutive years of funding through MSP grants. The individual projects’ foci varied each year, based on the goals as identified by VDOE, however, they shared the common themes of: 1. Improving teachers’ mathematical knowledge and pedagogical content knowledge; 2. Developing and improving an online PD model; and 3. Developing resources for mathematics classrooms in critical areas identified by the VDOE. In addition, the project Principal Investigators (PIs) published dozens of research papers (e.g., Corey, D.et. al., 2016; Manizade & Jacobsen, 2013; Manizade, & Martinovic, 2016) in national, international, and state outlets. Our purpose for this paper is not to focus on the research findings, but instead the outcomes most relevant and useful for practicing mathematics teachers. We Introduction

Abstract

The Secondary Mathematics Professional Development Center (SMPDC) has been funded by multiple Math and Science Partnership grants through the Virginia Department of Education (VDOE) since 2010 through the present. The SMPDC serves high school mathematics teachers as well as middle school teachers teaching high- school-level mathematics classes. The main goal of the Center is to provide a professional development program to teachers interested in improving their mathematical knowledge and pedagogical content knowledge (PCK). Our research has shown that teachers who participate in these programs have significantly improved their subject matter knowledge as well as their PCK in algebra, statistics, probability, geometry, and Algebra Functions and Data Analysis. The SMPDC has partnered with public school systems across the Commonwealth, several institutes of higher education in Virginia, the Virginia Math and Science Coalition and NASA. The SMPDC has served 352 mathematics teachers since its original funding in 2010, providing over 700 contact hours of graduate-level professional development

coursework to the teachers. The participants have created several

hundred products available

intend to use this paper to disseminate the knowledge gained by teachers during these projects and provide an opportunity to use the

to teachers across the commonwealth, including lesson plans, unit plans, performance- based assessments, and classroom videos. SMPDC faculty have produced 39 publications and presentations

materials produced

by

their peers. These include, but are not limited to

Figure 1 . Sample performance-based assessment task

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hundreds of unit and lesson plans, problem based assessments, classroom videos, and teacher interviews. All of these products are published on the project website and are freely available for use by teachers across the Commonwealth. Several examples will be discussed later in this paper. The online PD model for secondary teachers was developed through a collaborative effort with public schools, state universities, and other institutions across the Commonwealth such as Virginia Commonwealth University, the College of William and Mary, Longwood University; the Virginia Math and Science Coalition; and NASA’s Langley research facility. Courses are transferable between partnering institutions. Online classes were designed to be hands-on and interactive, see an example at http://lecture-play.radford.edu/ Mediasite/ Play/84c88e2983c44e4eb35189c4f6e4ee911d . The focus of the model was to improve secondary mathematics teachers’ subject matter knowledge in Algebra 1, Algebra 2, AFDA, Geometry, Modeling, and Probability and Statistics. Since 2010, six grant projects have been conducted, serving 352 teachers, and providing more than 700 hours of instruction. The number of partners each grant year varied between 25 and 52. The complete list of partners and details on each project are available on our website at http://www.radford.edu/ rumath-smpdc/PartnersMap.html . Teachers’ subject matter knowledge and pedagogical content knowledge have long been widely acknowledged as essential components of teaching expertise (e.g., Shulman, 1986). The Literature Review and Research Overview

Figure 3 . Sample of student work

Mathematical Education of Teachers II (AMS/ CBMS, 2012) report recommended continual improvement of mathematical knowledge and teaching skills, with such improvement promoted by regular interactions between teachers, mathematicians, and mathematics educators in the creation and analysis of mathematics lessons, texts,

Figure 2 . Rubric for student evaluations

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the American Mathematical Society (AMS) and the Conference Board of the Mathematical Sciences (CBMS) indicated, “Although high school mathematics teachers frequently major in mathematics, too often the mathematics courses they take emphasize preparation for graduate study or careers in business rather than advanced perspectives on the mathematics that is taught in high school” (p. 5). The high school curriculum prepares students for sophisticated and often abstract understandings, but may not prepare teachers to see the mathematics through students’ eyes (McCrory, Floden, Ferrini-Mundy, Reckase, & Senk, 2012). Teachers need both advanced mathematical knowledge and an understanding of how that advanced mathematical knowledge relates to the high school curriculum

Figure 4 . Example worksheet

(AMS & CBMS, 2012; McCrory et al., 2012). Our primary underlying assumption is that by providing teachers with quality educational experiences, the corresponding increase in their subject and pedagogical content knowledge will translate into changes in classroom practices that ultimately have a positive impact on students’ learning The SMPDC’s research examines the Center’s impact on measuring: (1) gains in teacher content knowledge, (2) impact on student achievement, and (3) progress towards meeting the assessed needs of partnering school divisions. Toward these ends, assessment measures include pre- and post-assessments of teachers’ mathematical content knowledge and mathematical knowledge for teaching Algebra I, Algebra II, and

and curriculum materials as well as examinations of the underlying mathematics. The SMPDC provides mathematics teachers with extensive opportunities such as online graduate level courses in mathematics designed for secondary teachers, summer institutes at NASA, and classroom observations and feedback. These programs help develop teachers’ knowledge bases and ultimately increase students’ mathematics achievement. Teachers’ mathematical knowledge for teaching contributes significantly to gains in students’ mathematics achievement (e.g., Hill, Rowan, & Ball, 2005; Sample McMeeking, Orsi, & Cobb, 2012; Telese, 2012). The Mathematical Education of Teachers II (2012) co-publication by

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These products then went through two to three rounds of blinded peer-review with external reviewers, and are currently available for use by teachers across the Commonwealth. The products implement best practices for teaching mathematics, and feature hands-on, interactive activities with various STEM applications. In this paper we will share two example products including a unit plan and a performance-based assessment. Our first example is a performance-based learning and assessment task that was designed by teachers Dee Baker, Jamey Lovin and Robert Thien 20Building%20a%20Recreation%20Center.pdf ). In this task, learners were prompted to explore constructions of triangle centers to determine the best location for a recreation center with respect to three communities in a geographic area (Figure 1). To solve the task, students used applications such as GeoGebra, Google Maps, as well as standard construction tools. The task included two assessment components and rubrics which assess students’ understanding of the content (Figure 2). Students were expected to select a site and justify their selection using credible research, geometric principles, and constructing triangles (Figure 3). Our next example is a unit plan by Christine Belcher and Stephanie Brady ( http:// www.radford.edu/rumath-smpdc/Units/src/A% 20Change%20in%20the%20Weather.pdf ). This unit, “A change in the weather”, guides students through data analysis, finding equations of the curve of best fit, mathematical modeling that includes polynomial, exponential, and logarithmic functions, and making predictions, as well as teaching them about climate as one of the real world applications of mathematics. Students create charts and models using provided data of 17 different locations across the world (Figure 4, Figure 5), and make scatterplots and trend lines of the data (Figure 5), as well as other indicators of climate change such as global mean sea level and temperature. We also have samples of classroom videos of our participants implementing different approaches to teaching mathematics. These approaches include the Structuralist, Integrated- Environmentalist, and Formative approaches. They also feature the teachers’ educational knowledge and insights gained from the program. They can be watched at the SMPDC website http:// www.radford.edu/rumath-smpdc/ VideoCourse.html (see Figure 6). Our project ( http://www.radford.edu/rumath-smpdc/ Performance/src/Dee%20Baker%20-%

Figure 5 . Provided temperature data

AFDA; multiple classroom observations of selected participating teachers using Instructional Quality Assessment (IQA) rubrics (Junker et al., 2006); teacher surveys addressing content or pedagogical content knowledge; assessments of gains in high school students’ understanding and knowledge of Algebra I, Algebra II, and AFDA; surveys of participating school administrators; participating teacher surveys to evaluate the strengths and to offer suggestions for the SMPDC; and course fidelity assessments to evaluate whether courses are designed and taught in ways that correspond with identified needs of partnering school divisions. Based on the SMPDC project evaluation reports from the past six years, teachers have shown a significant gain in their knowledge of algebra, geometry, and probability and statistics as well their ability to implement new educational technologies in their classrooms. Readers who wish to engage with the data and analysis portions of the projects can find full evaluation reports for each of the six projects at the SMPDC website. ( http:// www.radford.edu/~amanizade/projects.html ) One of the main outcomes of the MSP projects are the hundreds of products generated and shared by the participating teachers. These include lesson plans, unit plans, and performance-based assessments and have been made available in the Teacher Resources section of our project’s website, http://www.radford.edu/rumath-smpdc/ . Products Generated

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website includes hundreds of more products like the examples included in our papers. We encourage readers to examine and utilize these free resources.

address the major obstacle of online professional development courses, the lack of interaction and hands-on experiences. Finally, we suggest that all of the teacher-generated products created during the projects be available for use by teachers throughout the Commonwealth, as this is one of the most effective ways of distributing the knowledge generated in each program. Conclusions During the past six years of the MSP projects at Radford University, we were consistently able to show growth in the following areas: 1) Teachers’ content knowledge and pedagogical content knowledge in Algebra, Geometry, and probability and statistics. 2) Students’ mathematical knowledge in the aforementioned areas. We also met the needs of the participating partners, and addressed the goals identified by the Commonwealth. The major challenges of the MSP projects relate to the following: 1) The availability of reliable and valid assessment measures of teachers’ professionally situated knowledge at the high-school level. 2) The rate of turnover of teachers in the partner divisions. 3) The length of the award periods that prevent longitudinal studies. We would also like to invite readers to explore the products and resources created as a result of these projects, which can be found on our website, ( http://www.radford.edu/rumath-smpdc/ Resources.html ). American Mathematical Society & Conference Board of the Mathematical Sciences (2012). The Mathematical Education of Teachers II . Providence, RI: AMS and CBMS. Corey, D., Jacobsen, L., Manizade, A., Dove, A., Galeshi, R. & Younes, R. (2016). Best Practices: Lessons Learned From an Online Statewide Collaborative Master's in Mathematics Education Program. In Proceedings of Society for Information Technology & Teacher Education International Conference 2016 (pp. 2497- 2498). Chesapeake, VA: Association for the Advancement of Computing in Education (AACE). Hill, H., Rowan, B., & Ball, D. L. (2005). Effects of teachers’ mathematical knowledge for teaching on student achievement. American Educational Research Journal , 42 (2), 371- 406. References

Implications and Recommendations

Providing teachers with an online, ongoing professional development program that involves partnerships between multiple institutions of higher education, and which utilizes the strengths of these institutions, allows the creation of an effective, statewide program for the continuing improvement of mathematics education. An important element of this model is simple and easy credit transfers between partner institutions, allowing teachers flexibility in taking graduate-level classes. This also allows institutions and their faculty to focus on their strong subject areas, increasing the overall quality of the professional development. An important part of our professional development projects over the past six years has been our presence in participants’ classrooms following the courses. We observed and interviewed teachers and provided them with critical feedback related to the

Figure 6. Screenshot of an online video

quality of their mathematics instruction. This not only allows us to see if the theoretical knowledge learned during the course has been implemented in the participants’ classrooms, but also provides an opportunity for additional professional development in the field. We also encourage the continued development of online courses that feature hands- on, interactive instructional practices that allow teachers to engage with the ideas of the content area through exploration and discovery. We recommend the use of all available technological tools such as Adobe Connect, Geogebra, Cinderella, Mathematica, and others to aid in the creation of an interactive online environment. We also recommend providing participants with manipulatives and corresponding explorations to complete as part of their coursework. This helps

Junker, B., Weisberg, Y., Matsumura, L. C.,

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