Virtual Instructional Unit

Classifying Polygons and Quadrilaterals – A 5th Grade Mathematics Virtual Unit

Rachel NachmanEDTC650 – Teaching and Learning in K-12 Virtual Schools

November 8, 2014Dr. Allen Grant

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Unit Introduction

This unit focuses on classifying polygons and quadrilaterals. Students will first begin by learning about polygons; more specifically, they will sort shapes by those that are polygons and those that are not polygons. Students will also learn the difference between convex and concave polygons, and will have multiple opportunities to classify polygons into these two categories. Additionally, students will learn the definition of a quadrilateral, and will be exposed to the different types of quadrilaterals. A majority of this unit focuses on classifying quadrilaterals into the different types, as well as exploring the relationship between the different types of quadrilaterals. At the end of this unit, students will be asked to develop their own classification system of quadrilaterals that illustrates the hierarchal relationships between the different types.

Unit Title: Classifying Polygons and Quadrilaterals

Grade Level: 5th Grade

Content Area: Mathematics

Standards (Common Core State Standards):3.5.A.3 – Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles.

3.5.A.4 – Classify two-dimensional figures in a hierarchy based on properties.

Setting: Synchronous (1 hour session; 5 times a week)

Web Conferencing Platform Used: Adobe Connect

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Unit Structure

A fully developed lesson play for day #1 of this unit is included below.

Day #1:

Lesson Title: Classifying ShapesGrade Level/Subject Area: 5th Grade (Mathematics)Concept/Topic and Essential Questions:

What is a polygon? What is a concave polygon? What is a convex polygon?

Length of Lesson: 1 hourStandards Addressed:Common Core State Standards (Mathematics)3.5.A.4 – Classify two-dimensional figures in a hierarchy based on properties.Mastery Objectives:

Students will be able to classify two-dimensional shapes as polygons or non-polygons. Students will be able to classify polygons as convex or concave.

Lesson Materials: Adobe Connect 9 (web-conferencing software) “Polygon or Not?” (Appendix A) “Polygons and Non-Polygons T-Chart” (Appendix B) “Polygons” Presentation (See Attached File) Math Learning Center Virtual Geoboard

(http://www.mathlearningcenter.org/web-apps/geoboard/) “Sorting Logic Blocks” (http://nrich.maths.org/content/id/7192/JulySh.swf) “Polygon Assessment” (Appendix C)

Introduction:This is the first lesson of this geometry unit, and so to introduce the lesson, the teacher will explain to students that they will be learning about different shapes, as well as how to categorize and classify these different shapes. In this lesson, students will learn about polygons and will have the opportunity to use virtual geoboards to create their own polygons. Lesson Procedure (to be completed after the introduction):

Using Adobe Connect’s audio (or video) conferencing and desktop sharing features, show students the resource “Polygon or Not?” Ask students to first think about the characteristics that all polygons have in common. After providing students with time to think independently, ask students to volunteer to share their thoughts (by using the “raise-hand” feature). Project a blank t-chart labeled “Polygons” and “Non-Polygons” (using Adobe Connect’s white board feature) and record student responses in the “Polygons” category. Once students have shared their answers, summarize student responses and provide a definition of a polygon (“closed” 2-dimensional shapes with straight line segments).

Display the “Polygons or Not?” resource again. Ask students to think about the characteristics that all non-polygons have in common. After providing time for students

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to think independently, ask students to volunteer to share their thoughts (by using the “raise-hand” feature). Project the t-chart and record student responses in the “Non-Polygons” category. Once students have shared their answers, summarize student responses and explain that shapes with rounded sides, or shapes that are “open”, are not polygons.

Using the desktop sharing feature, display slide 1 of the “Polygons” presentation. Ask students to “raise their hands” and classify this shape as either a polygon or a non-polygon. Encourage students to explain their reasoning. Use the “marker” tool to trace the straight line segments that make up this polygon.

Display slide 2 of the “Polygons” presentation. Ask students to “raise their hands” and classify this shape as either a polygon or a non-polygon. Encourage students to explain their reasoning. Use the “marker” tool to trace the rounded figure of the shape that does not make it a polygon.

Display slide 3 of the “Polygons” presentation. Ask students to “raise their hands” and classify this shape as either a polygon or a non-polygon. Encourage students to explain their reasoning. Use the “marker” tool to trace the straight line segments that make up this polygon.

Display slide 4 of the “Polygons” presentation. Ask students to “raise their hands” and classify this shape as either a polygon or a non-polygon. Encourage students to explain their reasoning. Use the “marker” tool to trace the rounded figure of the shape that does not make it a polygon.

Project the completed “Polygons and Non-Polygons” t-chart and review the characteristics of polygons.

Display slide 5 of the “Polygons” presentation. Explain to students that polygons can either be convex or concave. Explain that convex polygons are polygons whose angles measure less than 180 degrees, while concave polygons are polygons with at least one angle that is greater than 180 degrees. Explain that there is a straight line on the blue concave figure to show how that angle measures greater than 180 degrees.

Display slide 6 of the “Polygons” presentation. Ask students to “raise their hands” and classify this shape as either a convex polygon or a concave polygon. Encourage students to explain their reasoning. Use the marker tool to identify the angles of this shape and explain how these angles are less than 180 degrees (and so this is a convex polygon). “Draw” a straight line on the angles to show students how they can use the line as a reference point when determining whether an angle is greater than or less than 180 degrees.

Display slide 7 of the “Polygons” presentation. Ask students to “raise their hands” and classify this shape as either a convex polygon or a concave polygon. Encourage students to explain their reasoning. Use the marker tool to identify the angles of this shape and identify the angle that is greater than 180 degrees (and so this is a concave polygon). “Draw” a straight line on the angle to show students how they can use the line as a reference point when determining whether an angle is greater than or less than 180 degrees.

Display slide 8 of the “Polygons” presentation. Ask students to “raise their hands” and classify this shape as either a convex polygon or a concave polygon. Encourage students to explain their reasoning. Use the marker tool to identify the angles of this shape and identify the angle that is greater than 180 degrees (and so this is a concave polygon).

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Using the marker tool, continue the straight line that is already on the polygon. Use this straight line to show students how the angle is greater than 180 degrees.

Display slide 9 of the “Polygons” presentation. Ask students to “raise their hands” and classify this shape as either a convex polygon or a concave polygon. Encourage students to explain their reasoning. Use the marker tool to identify the angles of this shape and explain how these angles are less than 180 degrees (and so this is a convex polygon). “Draw” a straight line on the angles to show students how they can use the line as a reference point when determining whether an angle is greater than or less than 180 degrees.

Using the “desktop sharing” feature, model how to use the Math Learning Center Virtual Geoboard. Model using the rubber bands to create different shapes. Explain to students that they will use the geoboards to create different shapes.

Using Adobe Connect’s text chat feature, send students the link to the virtual geoboard. Have students access the virtual geoboard (while still logged on to the class) and provide time for students to create different shapes.

After allowing students to create different shapes, ask students to create a concave polygon. Ask for students to volunteer to share their shape with the class, and allow students to share their shape using Adobe Connect’s desktop sharing feature. Ask students to explain how they know that their shape is a polygon and how they know that their shape is concave.

Ask students to create a convex polygon. Ask for students to volunteer to share their shape with the class, and allow students to share their shape using Adobe Connect’s desktop sharing feature. Ask students to explain how they know that their shape is a polygon and how they know that their shape is convex.

Explain to students that they will now be divided into groups to practice classifying shapes and polygons or non-polygons, as well as classifying polygons as convex or concave.

Using the file sharing feature, send students the “Polygons” presentation. Explain that students will navigate through the two remaining slides of the presentation (one student per group can serve as the “presenter” and share his or her computer screen with the rest of the group). Students will work together to complete the activity on each slide. Encourage students to review the entire presentation prior to completing the small-group activities. If students finish early, they can identify “real life” examples of polygons (and identify whether these are convex or concave).

Using Adobe Connect’s “breakout room” feature to divide students into small groups. These groups should be homogeneous based on students’ mathematical ability level.

o Note – Because this is the first lesson of the unit, there is no assessment data to use when dividing students into groups. Thus, the teacher should use previous assessment data, as well as professional judgment and observational data, to divide students into homogeneous groups.

Once students are in their breakout rooms, visit each breakout room to ensure that students are either voice-chatting or videoconferencing while working on the remainder of the presentation’s activities.

o Re-Teaching – For the group(s) of students who may need re-teaching in this concept:

Spend time reviewing the difference between a polygon and a non-

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polygon Review the “Polygons and Non-Polygons” t-chart to review the

characteristics of a polygon Review the difference between a convex and a concave polygon Spend time reviewing the “Polygons” presentation in order to reinforce the

mathematical content. Complete part of each activity with students, utilizing a scaffolding model

and providing guidance as needed.o Extension – For the group(s) or students who may need extension and

acceleration in this concept: Ensure that students completed the remaining activities correctly (ask

students to explain their reasoning for their answers; ask students to explain the difference between polygons and non-polygons; ask students to explain the difference between convex and concave polygons)

Once students have demonstrated proficiency with this lesson, introduce the “Sorting Logic Blocks” activity to students. Explain that each student will sort his or her shapes into two different categories. Then, once every member in the group has finished sorting, they will take turns sharing their sort with the rest of the group. The other group members will then guess the rule used to sort the shape. Encourage students to sort based on characteristics of shapes (as opposed to color or size).

When there is 10 minutes left in the session, end the breakout rooms so that all students are back in the “main” virtual classroom. Explain that students will now complete a “Polygons” assessment. Students will take this assessment independently. This assessment is comprised of six short answer questions that require students to not only provide an answer, but explain their reasoning as well.

o Note – This assessment is embedded within a PowerPoint presentation. The assessment was created using the “Adobe Presenter” plug-in for PowerPoint. This plug-in allows quizzes to be embedded into presentations used within Adobe Connect.

The Adobe Presenter tool costs money and was not actually used to create the quiz. However, should this unit actually be implemented in a virtual classroom, Adobe Presenter would be utilized. For the purpose of this assignment, the questions for the assessment can be found in Appendix C.

Assessment: Students will complete the “Polygons” assessment independently. This assessment

requires students to identify shapes as either polygons or non-polygons. Additionally, students are asked to identify polygons as either convex or concave. Students are also asked to explain their answer for each question. This assessment directly aligns to the objectives of this lesson and will allow me to see who met the lesson’s objectives and who did not.

Closure:Explain to students that the next day, they will be learning about quadrilaterals. Students will also be learning about shapes that are equilateral and shapes that are equiangular.

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A lesson outline for each of the remaining days of the unit is included below.

Day #2During the second day of this unit, students will practice classifying polygons based on properties of their sides and angles. Using the same whole-group presentation method that was used in the previous day’s lesson, students will first learn the definition of a quadrilateral (a polygon with exactly four sides). They will then learn the definition of a rhombus (a quadrilateral whose sides are all equal). Students will view different virtual pattern blocks (via the whole-group presentation) and will be asked to identify which of these pattern blocks are rhombuses. They will also be asked to compare the different rhombus pattern blocks. Just as with the previous day, students will be expected to share their reasoning. The teacher will record students’ answers on a virtual white board.

Next, students will explore the question of whether a square is a rhombus. They will have an opportunity to develop their own rule for classifying squares and rhombuses and will then share their rule with their classmates.

Students will then be introduced to the term equiangular and equilateral. The teacher will model how to identify shapes that are equiangular and/or equilateral. This will end the whole-group portion of this lesson.

Similar to the previous day, students will then meet in breakout rooms. However, the groups for this day’s lesson will be determined based on students’ performance on the previous day’s assessment. In their small groups, students will be asked to complete the remaining activities on the PowerPoint presentation. These activities will require students to sort polygons into “rhombuses” and “not rhombuses”, as well as sort shapes by those that are equiangular, equilateral, both, or none. The teacher will visit all break out rooms; however, will spend more time with the group(s) that may need re-teaching of the previous day’s concept. For those students who require enrichment, they will visit the “Shape Sorter” activity (http://illuminations.nctm.org/Activity.aspx?id=3581) once they have finished completing the required activities. This sorting activity allows students to sort shapes based on different rules and attributes.

This lesson will end with an independent assessment in which students will be asked to identify whether a shape is a rhombus or not a rhombus. Students will also be asked to identify whether a shape is equiangular, equilateral, both, or none. Students will be asked to explain their thinking for all answers. This assessment will be delivered via the Adobe Presenter tool, which allows quizzes to be embedded into presentations that are delivered through Adobe Connect.

Day #3On this day, students will again classify polygons based on their sides and angles. However, while the previous day’s lesson focused on rhombuses, this lesson will focus on trapezoids and parallelograms.

First, students will be asked to access a virtual geoboard (the same geoboards used on day #1) to create a quadrilateral with at least one pair of parallel sides. If necessary, the teacher will review

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the definition of quadrilateral and parallel (the term parallel was not covered in this unit; however, it has been covered in previous units).

Using the same whole-group presentation method that was used during the previous two days, students will learn the definition of a trapezoid. After seeing pictures of different trapezoids, students will be asked to think about and discuss the characteristics that all trapezoids have in common. The teacher will record these characteristics on a virtual white board. Students will then be asked to revisit their virtual geoboards and build quadrilaterals that are not trapezoids. Students will then have the opportunity to share their desktop with the class and explain why their shapes are not trapezoids.

After students have shared, the teacher will explain that these shapes are called parallelograms. Students will come up with the definition and attributes of a parallelogram, which will be recorded on a virtual white board. The teacher will then model how to identify whether a shape is a parallelogram, trapezoid, or neither. This will end the whole-group portion of the lesson.

Similar to the previous day, students will then meet in breakout rooms. However, the groups for this day’s lesson will be determined based on students’ performance on the previous day’s assessment. In their small groups, students will be asked to complete the remaining activities on the PowerPoint presentation. These activities will require students to sort polygons into “trapezoids” and “parallelograms”. The teacher will visit all break out rooms; however, will spend more time with the group(s) that may need re-teaching of the previous day’s concept.

This lesson will end with an independent assessment in which students will be asked to identify whether a shape is a parallelogram or a trapezoid (and explain their answers). This assessment will be delivered via the Adobe Presenter tool, which allows quizzes to be embedded into presentations that are delivered through Adobe Connect.

Day #4On day #4, students will be introduced to the hierarchy of quadrilaterals (the categories and subcategories of quadrilaterals). First, the teacher will ask students to provide the definition of parallelograms and rectangles. These definitions will be recorded on to a blank presentation slide (which will be projected on students’ computers). Examples of parallelograms and rectangles will be shown to students. Students will then be asked to think about and explain how rectangles and parallelograms are related.

Next, students will then provide the definition of a rhombus and a square, and these definitions will also be recorded. The teacher will show examples of rhombuses and squares. The teacher will then state how a rhombus, rectangle, parallelogram, and a square are all related; however, the relationship between these shapes is a hierarchy. No additional clarification or explanation as to how these shapes are related will be provided.

The teacher will then ask for the definition of a trapezoid, which will also be recorded on to the presentation slide. The teacher will show students examples of trapezoids. This presentation slide will then be sent to students for them to have future access to.

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The teacher will explain that for the unit assessment, students will be creating a web-based presentation that illustrates how these types of quadrilaterals are related. They will be creating their own classification chart of quadrilaterals that shows this hierarchal relationship. Students will be allowed to choose whichever “design” they would like to use to visually illustrate their classification method; however, their classification chart must clearly show all categories, subcategories, and must include all types of quadrilaterals. Additionally, students must provide examples of shapes for each category of their chart. Students will be asked to share their classification chart with the class on the following day.

The teacher will model how to use Zoho Show (the Web 2.0 tool that will be used for this assessment). Students will then have the opportunity to spend the remainder of the class time logged in and working on their presentation. This presentation is expected to be finished prior to the next day.

Day #5For the last day of the unit, students will be asked to present their quadrilateral classification chart with the class. Each student will have the opportunity to share his or her desktop, allowing the rest of the class to see the chart. Along with the visual chart, students will be asked to provide a verbal explanation of how they classified their quadrilaterals. They will also be asked to explain the hierarchal relationship between quadrilaterals. Students in the “audience” will also be encouraged to respond to the presenter and ask questions about his or her chart. This will serve as the assessment for this unit.

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Unit Assessment

For a unit assessment, students will be asked to create a web-based presentation using Zoho Show (a Web 2.0 tool). This presentation will illustrate a classification system of the different types of quadrilaterals. Students will be asked to develop their own classification system that also highlights the hierarchal relationship between the different types of quadrilaterals. Additionally, students will also provide examples of each type of quadrilateral. Students will be allowed to choose whichever “design” they would like to use to visually illustrate their classification method; however, their classification chart must clearly show all categories, subcategories, and must include all types of quadrilaterals.

Once students have created their classification system, they will then share their presentation with the rest of the class (using Adobe Connect’s “desktop sharing” feature). Students will be asked to provide a verbal explanation of how they classified their quadrilaterals. They will also be asked to explain the hierarchal relationship between quadrilaterals. Students in the “audience” will also be encouraged to respond to the presenter and ask questions about his or her chart.

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Media and Technologies Used

For the implementation of this unit, Adobe Connect was used. This is a videoconferencing software with various features that make it a valuable tool for distance education. These features were all utilized throughout the implementation of this unit (and within the fully developed lesson plan for day #1). These features include:

Audio Chat – Participants can communicate with each other via voice/audio

Video Chat – Participants can communicate with each other via video (with and without audio)

Desktop Sharing – The presenter can share his or her desktop with the other participants. The role of “presenter” can change throughout the session (for example, the teacher can be the presenter and share his or her desktop, or a student can be a presenter and share his or her desktop).

“Raise-Hand” – Participants can press the “raise-hand” button to signal to the teacher that they have something to say (either sharing an answer or asking a question).

“White Board” and Markers – Participants can see a virtual white board on which the teacher can use the markers to write notes and important information.

Text Chat – Participants can communicate with each other via text

Breakout Rooms – Participants can be divided into different “breakout rooms”, in which they meet with a smaller group of people. The host of the conference can then navigate through each room and communicate with students. This is an ideal feature when implementing small group instruction.

File Sharing – Files can be shared among participants in a conference

Quizzes – Participants can take quizzes through Adobe Connect. Quiz answers can then be sent to the teacher for grading (or, depending on the type of quiz, can be graded automatically).

In addition to Adobe Connect, virtual manipulatives were used throughout this unit. Because this unit is designed to be implemented in a completely virtual setting, students may not have access to the physical math manipulatives often found in traditional classrooms. Thus, students explored virtual geoboards and virtual pattern blocks throughout this unit. According to the National Council of Teachers of Mathematics (as cited in Drickey, 2006), “mathematics instructional programs should use technology to help all students understand mathematics and should prepare them to use mathematics in an increasingly technological world” (p. 109). The use of virtual manipulatives have had proven positive effects including an increase in student performance (Steen, Brooks, & Lyon, 2006), Johnson (2011), and Chang, Yuan, Lee, Chen, and Huang (2013) as well as an improvement in student attitudes (Steen et. al, 2006) and (Lee and Ferrucci, 2012).

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Furthermore, various PowerPoint presentations were used within this unit. These PowerPoints were presented to the entire class during the whole-group instruction block of four lessons, and students also explored these PowerPoints during the small-group block of three lessons.

In addition, on day #2, the students who required enrichment visited a virtual “shape sorter”, which allowed them to sort shapes based on different rules and attributes. This online activity allows students to apply their mathematical knowledge in an engaging online setting.

Lastly, Zoho Show was used as a tool for the unit assessment. Zoho Show is a Web 2.0 tool that allows students to create web-based multimedia presentations. Students used Zoho Show to create a quadrilateral classification presentation. A description of this assessment can be found in greater detail within the “Unit Assessment” section of this report.

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Pedagogical Approaches

Various pedagogical approaches were used throughout this unit.

First, a majority of the lessons (four out of five) included a whole-group component. During the whole-group section of each lesson, the teacher provided direct instruction and modeling. However, students still had the opportunity to share their thoughts, ideas, and questions. As previously discussed, PowerPoint presentations were used during whole-group instruction. These presentations were shared using the “desktop sharing” feature of Adobe Connect. Additionally, the teacher and students could communicate via audio or video chat.

A small-group component followed the whole-group component of a lesson. Students were divided into homogeneous groups (based on student performance and ability). Each group met in a separate “breakout room” and completed various collaborative activities. The teacher then visited each “breakout room”, provided re-teaching, and provided enrichment.

Students then had the opportunity to apply what they have learning in an independent setting. Three out of the five lessons included an independent assessment. This assessment, conducted at the end of the lesson, was administered via a PowerPoint presentation (using the Adobe Presenter tool). This tool allows users to embed quizzes into presentations being shown within Adobe Connect. Students’ responses to these quizzes were then graded and used as formative assessments.

The summative (unit) assessment of this unit employed a more constructivist approach. Rather than simply telling (or showing) students the traditional way of classifying quadrilaterals, they were asked to develop their own classification system that still illustrated the various hierarchal relationships. Students then presented their classification systems to their classmates and teacher, allowing them an opportunity to present and share their mathematical thinking and reasoning.

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Justification

When designing the structure of this unit, great thought was given to how to implement an effective learning model in a virtual setting.

As discussed in the pedagogical approaches section of this report, a scaffolding model was utilized throughout this unit. This model includes whole-group instruction, small-group instruction, and independent practice. Throughout the unit, students are provided with explicit, direct instruction, as well as an opportunity for guided, collaborative practice. Thus, students are given sufficient exposure to the content, as well as multiple practice opportunities, prior to completing an independent assessment.

This scaffolding model advances students’ learning of the mathematical content being covered in this unit. The direct instruction provided during a whole-group lesson models the correct understandings and skills needed to meet the objective of the lesson. Additionally, the collaborative small group activities allow students to apply their knowledge to new problem situations, while working together and receiving any additional needed support.

The final unit assessment also requires students to develop their own conclusions about quadrilaterals and their properties, and this assessment employs a constructivist approach. According to the Educational Broadcasting Corporation (2014), students develop a deeper understanding of the academic content when they are required to formulate their own conclusions. Students should not be the passive receivers of information – they should be actively engaging with the academic content and should be guiding their own learning.

Lastly, virtual manipulatives and tools were used throughout this unit. As previously discussed, virtual manipulatives have many positive effects on student learning, including an increase in student performance, as well as an improvement in student learning and motivation. Additionally, the mathematical concepts discussed in this unit may be too abstract for some fifth grade students, and thus, manipulatives can help make these abstract concepts more concrete. Because these students are enrolled in a virtual school and may not have physical manipulatives readily available, virtual manipulatives can help foster an understanding of polygons and quadrilaterals.

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Appendix A – “Polygon or Not?”

Polygon Non-Polygon

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Appendix B – “Polygons and Non-Polygons”

The T-Chart below is a sample chart. For the lesson, create this chart on a blank virtual white board screen.

Polygons Non-Polygons

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Appendix C – Polygons Assessment

Should this lesson be implemented, these questions would be embedded into a quiz using the Adobe Presenter tool within Microsoft PowerPoint. This tool allows quizzes to be administered within Adobe Connect. However, this tool costs money and was not purchased for this assignment. Thus, the questions that would be in the quiz can be found below.

Question 1: Is the shape below a polygon? Explain why or why not.

Question 2: Is the shape below a polygon? Explain why or why not.

Question 3: Is the shape below a polygon? Explain why or why not.

Question 4: Is the polygon below convex or concave? Explain your answer.

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Question 5: Is the polygon below convex or concave? Explain your answer.

Question 6: Is the polygon below convex or concave? Explain your answer.

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References

Education Broadcasting Company. (2014). Constructivism as a paradigm for teaching and

learning. Retrieved from

http://www.thirteen.org/edonline/concept2class/constructivism/index_sub6.html

National Governors Association Center for Best Practices & Council of Chief State School

Officers. (2010). Common Core State Standards (Mathematics). Retrieved from

http://www.corestandards.org

Chang, W. L., Yuan, Y., Lee, C. Y., Chen, M. H., & Huang, W. G. (2013). Using Magic Board

as a teaching aid in third graders learning of area concepts. Educational Technology &

Society, 16(2), 163-173.

Drickey, N. (2006). Learning technologies for enhancing student understanding of mathematics.

International Journal of Learning, 13(5), 109-116.

Johnson, K. R. (2011). Physical and virtual math manipulatives and their effectiveness in a

fourth grade classroom. In T. E. Stone (Ed.), Models of Applied Research in Educational

Technology (pp. 35-51). Adelphi, MD: UMUC Teachers Press.

Lee, N. H., & Ferrucci, B. (2012). Enhancing learning of fraction through the use of virtual

manipulatives. Electronic Journal of Mathematics & Technology, 6(2), 126-140.

Steen, K., Brooks, D., & Lyon, T. (2006). The impact of virtual manipulatives on first grade

geometry instruction and learning. Journal of Computers in Mathematics and Science

Teaching, 25(4), 373-391.

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