Informal Math Education

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INFORMAL MATH EDUCATION: USING ORIGAMI TO TEACH ELEMENTARY SCHOOL STUDENTS MATH CONCEPTS BY GINA FAWKS A Thesis Submitted to the Division of Natural Sciences New College of Florida in partial fulfillment of the requirements for the degree Ba chelor of Arts Under the sponsorship of Sandra Gilchrist Sarasota, Florida April 2012

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ii Acknowledgements I want to thank my sponsor Dr. Gilchrist for all of her help, not only on this project but also throughout my career at New College. I wan t to thank Dr. Cooper for his help on my results section. I want to thank my Bacc alaureate Committee including Professor Gilchrist, Professor Poimenidou, and Professor Henckell, for their time. I want to thank everyone at the Boys and Girls Club who help ed me and made this project possible. Yvonne Zaccone was very nice and helpful getting me into the right Boys and Girls Club for my project. Yvonne helped me get all of my volunteering paperwork in order so that I could start helping the children even be fore I started my the woman working at the front desk calling the students for me and all three of the teachers who kept the students who were not a part of my proje ct working on their homework. I also want to thank all of my friends and family who helped and supported me through my thesis year. Especially to my roommates especially Jasmine Zeki for helping me with my dogs, my fianc for giving me the time I needed, and my mother for ALWAYS being there for me.

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iii Preface I first looked at origami as an educational supplement in a project that I did for Florida Academy of Sciences Conference 2011 at Florida Institute of Technology in Melbourne. We proposed the use of origami to teach basic concepts of marine animals and promote conservation. We mainly focused on using models for a whale, fish and turtle, each with a different difficulty level. I have used these same models in a couple of subsequent workshops. D uring my 2011 January ISP I taught three 6 th grade classes various geometric properties, such as the properties of a quadrilateral and what a transversal line was, while folding a fish. This thesis is a way to research and document the results of this kin d of teaching tool to add to the anecdotal papers and books that describe how origami can be useful in the math classroom With this thesis I examined how applicable origami may be in teaching math concepts specifically to elementary school students. For students struggling to understand different basic math concepts origami can potentially be a very useful supplemental resource From the outcomes of this study I can begin to understand how students learn math informally and what their understanding of math as a concept is. I wish to take what I learn and apply it to the way that I teach in the future

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iv Table of Contents vi Chapter One: Introduction 1.1 Mathematics in the Classroom a nd Beyond................................................... 1 3 1.2 Informal Ed ucation 5 1.3 7 1.4 13 Chapter Two: Methods 2.1 Designing the Study 2.2 Set up and Consent Process 19 2.3 Study in Actio 21 Chapter Three: Results 27 34 Chapter Four: Conclusion and Discussion .35 40 4.1.1 Raw Data 4.1.2 Statistics 44

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v 4.3 46 50 59 Appendix A Location Re Location Approva 64 Appendix B 68 73 75 97 Appendix C

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vi Pre 101 113 Pre 116 Test: Conc 125 127 132 147 154 Selected S 175 179 191 Post 195 st 210 Post 213 222

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vii List of Illustrations Figures ....8 Tables Tab 19 Table 3.2: Attendance for Groups 1 Table 3.4 Attendance for Groups 5 Table 3.6: Participant Test Scores (in 27 Table 3.8: Score Breakdown per Test (including attendance) (in percentages)....29 Table 3

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viii Informal Math Education: using origami to teach elementary school students math concepts Gina Fawks New College of Florida 2012 ABSTRACT There is a disconnect that students have between what they learn in the math can be used to bridge that gap. A specific type of informal math education is the use of origami to teach math concepts. I worked with forty five elementary school students at the Roy McBean Boys and Girls Club of America. I did three origami projects covering a variety of math concepts that students in kindergarten through fifth grade sh ould know or learn based on Florida Sunshine Standards ([Anonymous] Florida 2010) with corresponding worksheets for the non origami group and nothing for a control group. Each group took two tests, a vocabulary based pre and post test and a test reviewin g math concepts such as addition, multiplication and fractions. T This is a pilot study. There were few statistically significant outcomes in the short term. The Origami experimental group did improve their c oncepts test scores more than the control group. Research should be conducted to further our understanding of the usefulness of informal education methods, and origami in particular. _________________________________ Sandra Gilchrist

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1 Chapter One: Introduction 1.1 Mathematics in the Classroom and Beyond Why do we need to know how to do math? What could we possib ly use it for after we graduate? Regardless of what profession we choose, math will always surround us. Whether you despise math or love it, whether it comes naturally to you, or takes you some time, you will need to know some of the basics. Today, tech nology allows us to calculate a tip at a restaurant on our phones and use a computer to double a recipe. However, to be successful we need to know that the change the cashier gave us is correct and how many packages of eight juices we need to buy for our thirty two students. The math you learn when you are younger is necessary for a multitude of reasons, including giving you the ability to function as a resident of a society (Ernest 2010). Success in mathematics is more than jus mindset and attitude play a role as well as knowing how to apply and identify basic math knowledge like formulas in problems (Garofalo 1989). There are many studies that have ut mathematics. Students have trouble associating what they learn in the classroom and what they do and see (Corte et al 2000, Mason 2003 Fuchs et.al. 2004) There are multiple beliefs about mathematics that are accepted by students which may impair their ability to perform and understand math problems that are presented to them (Mason 2003) It is important to know about these beliefs to be able to teach students so that they comprehend and can apply what they learn.

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2 In table 1.1 several noted beliefs that students have about mathematics are recorded. Belief 1, can hinder a students ability to solve novel problems and succeed on tests if they give up after they are unable to recognize how to do a problem or question quickly. Num ber Belief 1 A problem is insolvable and not worth the minutes to solve 2 Size and quantity of numbers in a problem correlates to its difficulty 3 Only one (or possibly two) operations should solve all math p roblems 4 Operation needed to solve the problem can be found in a few key words at the beginning or end of the question 5 Time available dictates whether the student will check his or her work Belief 2 is based on some realistic conditions. Personally, I have helped students who would think that 23 x 7 is easier to do than 1000 x 5. This makes many problems appear to be scarier and harder than they should be. This negative attitude can affec t the can result in students getting very confused and frustrated on a question that would be fairly simple if broken up into a few steps. Many students will not read the whole question becau se they believe that the operation to be used to solve the question will be indicated by a few key words at the beginning or end of the question (Belief 4) This thought process can be detrimental to a students success. Although the idea is based on the way that questions are often worded, when the students do not actually read the questions they can get the wrong answer. Children that I have worked with have been taught to take the F CATS, a Florida standardized test. They are taught to look for the nu mbers and key words in the problem so that they can complete the test in the alloted time. This often means that they

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3 do not actually read what the question is asking and can be tricked into the wrong answer by using the wrong operation or giving what the y assume would be asked for. For total number of apples instead of how many the y found Joe to have or some students might simply say Joe has four. Both of these anwers are incorrect and are gle a ned by students not taking the time to read the entire problem. Based on how the tests are written, how well the students do on these tests and thus how they are prepared for the tests do not necessarily reflect the mathematical knowledge that they have (Boaler 2003). Lastly, students check their work based on time available (Belief 5). If students do not check their work they could miss a mistake that they made that could have been easily caught by reviewing the question and thinking logically about their answer. For example, if the question asked how many more apples Sally has than Joe if Sally has four apples and Joe has three apples, i f the student answers seven then checks their work they would realize that that answer does not make sense and they made an operation error. It is hard for students to understand why they might be getting a problem wrong if they are driven by the assumpti ons just discussed. When these thoughts are disproved it makes math appear more difficult than if these beliefs had not been established (Mason 2003). 1.2 Learning Learning is human nature (Parker 2005) Learning is something that has been studied for many many years and yet we do not know a lot about many of its processes. Learning occurs across disciplines and is studied from those different perspectives. It is

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4 not just limited to education but is important in psychology, sociology and even eco nomy because nations function through the people that run them which are affected by what they learn and what they know. By having multiple angles to observe this complex topic, there is no generally accepted definition of learning (Illeris 2009). Numero us people have written articles on learning and have posited theories of learning. The book T heir Own W ords (Illeris 2009) has sixteen chapters each discussing a different learning theory written by a different author. This book contains a limited selection of all of the theories of learning that exist. Some learning theories overlap and some slightly contradict others. Howard n, are very well known and currently popular. Gardner theorized that there are seven intelligences consisting of musical intelligence, bodily kinesthetic intelligence, logical mathematical intelligence, linguistic intelligence, spatial intelligence, inter personal intelligence and intrapersonal intelligence (Gardner 1993). Any one person may have more than one of these but the idea of multiple intelligences challenges the standard testing of intelligences and IQ scores (Gardner 1993). Many theories, simil learning as being individualized. This has even been broken down to the mechanics of these theories focus on individuals and their le arning differences, Robert Slavin (1990) stressed the importance on cooperative learning. Cooperative learning is not commonly used in the classroom due to the fear of cheating and difficulty of assessing individuals in a group. This is unfortunate since fifty seven percent of studies 1 found significant results 1 36 0ut of 63 studies before 1990 when Slavin wrote the book Cooperati ve Learning: Student Tea ms

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5 for better learning and only one study ( 1.6%) found significant negative results (Slavin 1990). Jean Piaget a classic well known psychologist, theorized that children needed to be challenged al though they should not be pushed to learn something that is too high of a level for them ( Huitt and Hummel 2003). Piaget also emphasized that students need to participate in many different activities and engage in the world around them to learn (Boakes an d Stockton 2009). Origami is an activity different than the students typically see in a classroom which may help them better connect to real world mathematics. 1.3 Informal Education To discuss origami used in informal education, it is first important to try and define what informal education is. Informal education, such as the use of educational mathematics (Ritzhaupt et al. 2010). Informal education is loosely defined as lea rning that takes place informally. It is hard to define informal education strictly. This is partly because it often occurs spontaneously in any location but also because it overlaps with formal education. Some try to define informal education as not ha ving tests, grades and/or class rankings (Fenichel and Schweingruber 2010). This does not always create a clear boundary between formal and informal education. For instance, one might have a summer program that will test the children at the end of the pr ogram on what they have learned, possibly as a way to assess the program. This would be an informal education program but has a formal test. Similarly, I have attended courses at New College that have had no grades or tests as a way to assess learning.

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6 Another way to try to fit informal education into a box with clear boundaries is to state that informal education occurs outside of the classroom and formal education strictly occurs in the classroom (Ramey Gassert 1997). There is a lot of gray area that occurs with restricting these two types of learning in this way. Where would a school field trip to a museum fit into this definition of formal and informal education? Think about a student who, while in a math class solving a problem about the area of a window, sparks a conversation about the recent hurricane. This spontaneous conversation could result in the student learning valuable information about hurricanes, weather, architecture, or even repair work to a house. This would be classified as infor mal education that happened to occur inside of a school classroom. Additionally, where would home schooling fit in? Depending on the style of home schooling, it can replicate formal education but occur outside the formal setting Regardless of what way you choose to define informal education, there are certain inherent characteristics. Informal education engages students or participants in a variety of ways like physically and emotionally, encourage self motivation to have direct contact with what they are learning, provide dynamic portrayals of the subject and allow students or participants to have control over their learning experience (Fenichel and Schweingruber 2010). There are many factors that make studies involving or evaluating informal educat ion complicated. The multifaceted idea of learning is one obstacle of studying informal education. Learning is no longer narrowly described as what content you can recall for a test. Another hurdle for the study of informal education is the complicatio n of evaluating the informal learning or programs. By certain definitions informal education

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7 way of testing what students learn does not always apply in these diffe rent environments. Many of the journals and articles that are available are either more anecdotal (not a scientific study) or the results are not conclusive (Fenichel and Schweingruber 2010). Fenichel and Schweingruber (2010) wrote a textbook on in formal education which was sponsored by the National Science Foundation (NSF). NSF is a forerunner in informal education research. The National Science Foundation has started a program for pre Kindergarten children to enhance their mathematic, science an d literary skills ( Ferrante 2012), as well as many other programs. The NSF website even provides a list (Petti 2012) that has a project for folding different polyhedrons. 1. 4 Origami A specific form of informal education is using origami as a teaching tool. Origami is an example of informal education because it is not part of the typical textbook teaching style of formal education but also because the teacher is directly tea ching the students how to fold a unit and supplementing the academic knowledge into the activity. Simply put, origami is the act of folding paper to create something. The word origami comes from the Japanese words oru which means to fold and kami which m eans paper

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8 (Pearl 2008). Origami uses paper and does not require any cutting, tape, or other modifications. Origami, even when constrained to a single sheet of paper can offer unlimited possibilities (Hull 2010) ( Figure 1 al sheets of paper folded into units and assembled to create a new multifaceted structure (Chen 2005). a b Figure 1 .1 : E xample s of the possibilities available with origami. Each of the models shown above the swan (a) and the dragon (b), was created with a single sheet of paper (a: [Anonymous] 2011, b: Greenhough 2010) Teachers and students have access to books that use origami as a base for teaching origami is, as well as briefly stating what scientific experiments an d hypotheses are. In the book there is a diagram for a boat that can be used to teach about properties of water teach about gravity. Also, an origami box can be used to help children understand with the idea of teaching math concepts through origami. Two such books are Math in

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9 Motion: origami in the classroom by Barbra Pearl (2008) a nd Unfolding Mathematics with Unit Origami by Betsy Franco (1999). Both are written for the teacher with ideas written for elementary or middle school classrooms and F middle school or high school classrooms. When origami is used to teach m athematical concepts it is often employed in the high school level and higher to study complex geometric properties and ideas (Hull 1994 ; Dacorogna et al 2010 ; Wares 2010 ). Dacorogna and colleagues (2010) wrote a paper titled, Origami and Partial Differential Equations The y discussed using origami as a means to demonstrate solutions to some systems of partial differential equations. Students do not tend to learn partial differential equations until they are in college. This paper is another example of the ways that origam i can be used in upper level math classes to help students understand mathematical concepts. One of the most logical and simple reasons for the fact that origami is often more applicable to higher level math classes has to do with what origami can be used to teach. Origami is very useful in teaching geometry concepts due to the fact that when folding a piece of paper you are constantly working with shapes and angles. Two dimensional shapes that students learn in elementary school and some middle school cl asses can be understood simply with a drawing on paper. Three dimensional shapes, however, can be rather complicated to draw and grasp the concept, particularly the more obscure shapes you might work with in a college level calculus class ( Table 1.2 ). Wi th origami, students

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10 can create some of these three dimensional shapes and be able to observe them from different angles to truly get an idea of how the shape looks in three dimensional space. Geometric concepts, particularly pertaining to angles, can be shown to scale and manipulated in an origami model for the students to better understand. There are also many computer programs available to students that allow them to manipulate three dimensional images is a way that provides greater understanding than images in a textbook. However, there is not an agreement across studies that computer programs are beneficial to student learning ( Roschelle 2000). Also not all schools can afford to supply enough computers to use these types of programs effectively (Bec ker 1984). Most of the geometry concepts such as angles and complex three dimensional shapes are not studied until at least middle school but mostly found in high school ([Anonymous] Florida 2010). The use of origami has also on occasion been applied at t he middle school level (Pope 2002, Boakes and Stockton 2009). There is not as much research regarding the use of origami to teach math concepts to elementary school children. Children start learning geometry when they can talk, by learning their basic s hapes. They start learning how to fold things like paper airplanes and simple boats when they start kindergarten. So origami as a mathematics teaching tool can be used from kindergarten with basic shapes through high school and beyond with more complex g eometric theorems. Annette Purnell (2009) wrote Providing for Creativity through Origami Instruction S he discusses how she participated in an after school program called Learn & Lead in South Bronx, New York. She was teaching elementary school aged chi ldren origami and became interested in what they could learn through origami.

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11 Some of the concepts that the students began learning about was new vocabulary with which they were able to become a part of a larger group identity, how to read and write diagra ms, and math concepts (Purnell 2009) Purnell observed many positive social aspects that resulted from teaching a few students how to do origami. The use of specific vocabulary and the ability to teach others, allowed the students to exhibit leadership q ualities and make a connection with people they may or may not have talked with without the origami. This branching out fr o m what was taught by the teacher based on the excitement and enthusiasm for folding allowed the children to problem solve and discov firsthand that math is in the model and that learning is integrated all around them, not just in the textbook and the classroom (3) make the square paper that you need for origami by cutting a rectangle piece of paper which was more readily available Purnell does not delve much into the math concepts learned or whether any math concepts were explicitly taught to the students. Purnell is not the only person to discuss the social benefits of origami which can origami with deaf and hard of hearing students. The benefits that she describes however are u niversal. Similar to Purnell, Chen noticed an increase in math vocabulary skills with the use of origami as well as development of problem solving skills and learning to work better in groups. Although students learn through both active and passive learn ing, they mostly learn when actively involved in the process (Chen 2005).

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12 Shape Name Image of shape Grade level that these shapes may be learned Triangle Kindergarten Octagon Elementary school Hexahedron May see a cube or other polyhedrons in fifth grade. May not learn a hexahedron by name until later if at all. Irregular Shapes High school Tetrahedron High school or College (depending on whether it is con sidered a pyramid or tetrahedron Hyperboloid College: Calculus III Table 1.2: Chart of shapes ([Anonymous] Hexahedron 2010, Watkins 2011, Palais 2006)

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13 Boakes and Stockton ( 2009 ) performed a study with middle school aged childre n visualize two and three udy of geometry in middle school. Although this study focused on middle school aged students geometry is important in a school curriculum from kindergarten through twelve grades and beyond (Boakes and Stockton 2009). Sedanur akmak (2009) wrote a thesis for graduate school based on using origami with elementary school students to teach spatial ability. The results of that study showed that the origami had a positive effect on the cepts with elementary school children?

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14 Chapter 2: Methods 2.1 Designing the Study For my study I ended up in the Roy McBean Boys and Girls Club. The Boys and Girls Club is a place where children can go afterschool. After settling on the loc ation and student pool, I began to design my study. I looked up the Florida Sunshine standards ([Anonymous] 201 0 ) and cross referenced what students, kindergarten through fifth grade, should be learning and/or know with books, and websites containing orig ami diagrams ([Anonymous ] 2011 [Anonymous] 2004, Gross 1993). I elected to teach the vocabulary words: square, pentagon, triangle, quadrilateral, hexagon, line of symmetry, polygon, trapezoid, congruent, rhombus, regular and irregular shapes, polyhedron, hexahedron, geometry, origami, and the math concepts: multiplication and division, adding and subtracting, reasonableness, counting edges, faces and vertices, counting (triangles), pairs (triangles), fractions (1/2), area of square and triangle, fractions (using colors/ out of 3). I also decided on using the diagrams for an origami twist fish, whale, and hexahedron (Appendix B). Each diagram is a different difficulty level and encompasses a set of vocabulary words and concepts from those listed above. I created two pre tests and post tests (Appendix C) to correlate with the vocabulary words and concepts to be taught through the origami as well as worksheets (Appendix C) to correspond with each origami lesson for the non experimental group. I also made a survey for the experimental and non experimental groups to take at the conclusion of the project. (Appendix B shows the lesson plan)

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15 2.2 Set up and Consent Process I started volunteering at the Boys and Girls Club approximately eight weeks before I be gan my project so that I could become familiar with routines and have the children get to know me. After receiving IRB approval (IRB Number: 11 024) and getting the official go ahead from the Boys and Girls Club, I began the consenting process (Refer to A ppendix A for IRB information, location approval and Consent form). Owing to the way the Boys and Girls Club is set up, the best way to conduct the consenting process was to wait in the front office and talk to the parents about my project as they picked up their elementary school children. I went to the front office for one week. I began on a Tuesday after getting the final approval from the Boys and Girls Club. On that Tuesday, based on times suggested to me from the person in charge of the club, I wa s at the front desk from 3:50 6:05pm. As the parents came to the desk to sign out their children, I asked them if they had elementary aged students. If they said yes, I explained my project and asked if they would say yes or no to their child (ren) parti cipating in my study. Then I signed the consent form and handed them a copy to keep for their own records. The next day, Wednesday, I consented from 4:10 5:00 pm. On the following day, Thursday, I stayed from 4:20 6:20 pm. I noticed that since the hom ework help session in which I will be working with the children on the project, occurs from 4:00 6:00 pm, students who leave before 4:15 pm will not have the time to participate in my study and thus I began coming after 4:00 pm to do my consenting. On Fri day I was there from 4:45 6:00 pm. The following Monday was my last day of the week of the consenting process. I stayed from 4:10 5:00 pm.

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16 There were several factors that kept me from getting certain parents to see/ sign the consent form for their chil d (ren) to participate in my study. There were many occasions when the parents came in talking on cell phones which made it difficult to get their attention or try to talk to them about why I was there. Several parents did not bother to come into the off ice to sign their child (ren) out. They either waited outside for the front desk staff to see them, recognize them and call the students to leave or they walked walked b ack out. This did not allow me to talk with any of those parents. Additionally there was a set of students who are allowed to walk home, so I was never able to see or talk to those parents. There were very few adults with whom I talked that would consid er and say that they would talk to the mother, but never brought the form back. I also had of the students and thus were unable to sign the consent forms. The last factor that hindered my participant gathering was the tutoring program that the Boys and Girls Club has recently implemented. If the student was signed up to participate in the Si2 tutoring then they did not attend the homework room where I did the study. I may have unknowingly collected signatures from students who were not be able to come to the homework room to participate but I did try to be aware of those parents who I knew ha d signed up their students for Si2 tutoring and did not discuss my project with them because their students would be unable to participate. After the consenting week was over I looked at how many students I had available for my project. I received thirty three signed consent forms with fifty four students

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17 available to participate in my study. I immediately dropped eight students who were in a tutoring program that occurred at the same time as the time that I was given to work with the students. I also dr opped one student whose mother did not include their name or grade on the consent form. I split the students into groups ( Table 2.1 ) and created a time schedule for the project ( Table 2.3 ). I needed at least one control group, one experimental group and one non experimental group. If I divided all of the participants into the three groups I would have one group with sixteen and two with fifteen. Both space and practicality made it difficult to work with that many students at once so I decided to split up all forty five participants into six groups. The method to which I split up the participants was first to organize them by grade level. Not only is it easier to work with groups of children of similar ages versus a group with a large age range, it was also convenient based on location to break them up by ages. During the homework help sessions, which was the setting of my project, students from 6 8 years old come in from 4:00 5:00pm to receive help with their homework and 9 12 year olds come in from 5 :00 6:00 pm. I ended up with seven kindergarteners, eight first graders, eight second graders, eight third graders, five fourth graders, and nine fifth graders. I decided that Groups 1, 3, and 5 would be for kindergarteners through second graders and Gro ups 2, 4, and 6 would be for third through fifth graders. Each group would contain seven or eight students. First I put the three students whose parent said that they leave early into the control group which correlated to their age groups, which I sched uled to occur at around 3:40 pm, before the homework help room was open. I then worked on randomly assigning the rest of the students to the

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18 sixth groups, trying to keep as equal a gender and grade spread as possible with the students I was provided ( Tabl e 2.1 and 2.2 ). After the students were placed into manageable groups and divided into the Control Group, the Non Group, I coded them. Each student received a number that correlated to their name. This information was saved in a password protected excel sheet. The numbers were assigned to the students first based on group. Group One has students from 11 18 with all of the tens digits being one for Group One. The ones digit was assigned at rando m starting with the students in the younger grades first up through the students in the higher grade of that group. For example, in Group One 11, 12 and 13 are kindergarteners, 14, 15, and 16 correlates to first graders and 17 and 18 are second graders. Group # of students per Grade included Type Total # of students Assigned Numbers assigned to students Group 1 3 K 3 1 st 2 2 nd Experimental 8 11 18 Group 2 3 3 rd 2 4 th 3 5 th Experimental 8 21 28 Group 3 2 K 3 1 st 3 2 nd Control 8 31 38 Group 4 3 3 rd 1 4 th 3 5 th Control 7 41 47 Group 5 2 K 2 1 st 3 2 nd Non Experimental 7 51 57 Group 6 2 3 rd 2 4 th 3 5 th Non Experimental 7 61 67 Table 2.1: Details pertaining to how the participants were grouped

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19 Group Frequency Percent Group 1 8 17.78 Gr oup 2 7 15.56 Group 3 8 17.78 Group 4 9 20.00 Group 5 7 15.56 Group 6 6 13.33 Total 45 100.00 Females 27 60.00 Males 18 40.00 Total 45 100.00 Control 17 37.78 Origami 15 33.33 Worksheet 13 28.89 Total 45 100.00 Kindergarten 7 15.56 Fi rst Grade 8 17.78 Second Grade 8 17.78 Third Grade 8 17.78 Fourth Grade 5 11.11 Fifth Grade 9 20.00 Total 45 100.00 Table 2.2: Group Participants Breakdown Once I established my participant pool and when I would meet with them, I read each student t he child assent form when I first met with them and their group and asked them to sign it if they wished to participate in my study (Appendix A has a copy of the Child Assent Form). For those who agreed, I began the project. 2.3 Study in Action Ba sed on constraints of the location (See Appendix A for Location Re Creation), I worked on my project with the students for about the first fifteen minutes of their time slot and then helped them with their homework if they needed it. I worked with Groups

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20 1, 2, 3, and 4 for about two weeks and then I worked with Groups 5 and 6 for about two weeks ( Table 2.3 shows the schedule used ). Day Groups I worked with What we worked on 1 Group 3, Group 1, Group 2 Child Assent, Pre Test: Vocab 2 Group 3, Group1, Group 2 Pre Test: Concepts 3 Group 4 Group1, Group 2 Child Assent, Pre Test: Vocab, Pre Test: Concepts Whale Origami model: focus on vocab words 4 Mixed Group due to absences Group 1, Group 2 Child Assent, Pre Test: Vocab, Pre Test: Concepts Wh ale Origami model: focus on concepts 5 Group 1, Group 2 Hexahedron 6 Mixed Group due to absences Hexahedron 7 Group 3 Group 1, Group 2 Post Test: Concepts Twist Fish model: focus on vocab words and a concept 8 Group 3 Group1, Group 2 Post Test: V ocab, Survey Twist Fish project: focus on concepts 9 Group 4, Group 2, Group 1 Post Test: Concepts, Post Test: Vocab, Survey 10 Mixed Group due to absences Group 5, Group 6 Post Test: Concepts, Post Test: Vocab, Survey Child Assent, Pre Test: Vocab, Pre Test: Concepts 11 Mixed Group due to absences Child Assent, Pre Test: Vocab, Pre Test: Concepts 12 Group 5, Group 6 Worksheet A 13 Group 5, Group 6 Worksheet B 14 Group 5, Group 6 Worksheet C 15 Mixed Group due to absences Worksheets A,B, C 16 Group 5, Group 6 Post Test: Concepts, Post Test: Vocab, Survey 17 Mixed Group due to absences Post Test: Concepts, Post Test: Vocab, Survey Table 2.3: Project Schedule

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21 I worked with the children on the project for seventeen days. I graded their pre te sts and post Tests and recorded the results. I also included gender, whether they had done origami before and attendance. Then I conducted descriptive statistics.

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22 Chapter Three: Results (Peng 2009, 35) 3.1 Raw Data One student from Group 2 has been included in numbers for Group 4 for the statistic testing. This participant, initially included in Group 2, o nly completed the pre and post Tests so their test scores were included with the control group as they participated in the same conditions. A participant from Group 6 fell into the same circumstances and is also included in the numbers for Group 4 when t he data were analyzed. Group Completed all assignments Completed some of the assignments Completed no assignments Total Initial Number used for test comparisons 1 2 3 3 8 4 2 2 4 2 8 6 3 4 3 1 8 4 4 3 4 0 7 3 5 6 1 0 7 7 6 2 3 2 7 4 Total 19 18 8 4 5 29 Table 3.1: Students Completion of Project Tables 3.2 3.4 show the attendance of the students. Table 3.2 has the attendance for Groups 1, 2, and 3. Table 3.3 has the attendance for Group 4 and Table 3.4 has the attendance recorded for Groups 5 a nd 6. Not all participants were called each day. The subjects highlighted in yellow are those that were removed and I stopped calling, those highlighted in red are the students that I could definitely not use in the final results of this study, and the g reen subjects could definitely be used in the final results of this study. The attendances recorded in green means that they were present that day, the red means that the student was absent and the blue means that they were not called that day.

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23 Dates Number Nov. 15 2011 Nov. 16 2011 Nov. 17 2011 Nov. 21 2011 Nov. 22 2011 Nov. 28 2011 Nov. 29 2011 Nov. 30 2011 11 N N N R R R R R 12 N N N R R R R R 13 Y N Y Y Y Y Y Y 14 Y Y Y N N N Y Y 15 Y Y Y Y Y Y Y Y 16 Y Y Y N N N N Y 17 N N N R R R R R 18 Y Y N Y N Y Y N 21 Y Y Y Y Y Y N Y 22 Y Y Y Y Y Y Y Y 23 N N N R R R R R 24 Y Y Y Y Y Y N Y 25 Y Y Y N N N Y Y 26 Y Y Y Y Y Y Y Y 27 Y N N N N N N N 28 N Y N N N N N Y 31 Y Y N/A N/A N/A Y Y N/A 32 N Y N/A N/A N/A Y Y N/A 33 N N N/A N/A N/A N N N/A 34 Y Y N/A N/A N/A Y Y N/A 35 Y Y N/A N/A N/A N Y Y 36 Y Y N/A N/A N/A Y N N 37 Y N N/A N/A N/A N N N/A 38 Y Y N/A N/A N/A N N N/A KEY Y = Present N = Not Present R = Removed N/A = Not Called Table 3.2: Attendance for Groups 1 3 Date s Number Nov. 17 2011 Nov. 21 2011 Nov. 22 2011 Nov. 28 2011 Nov. 29 2011 Nov. 30 2011 Dec. 1 2011 Dec. 2 2011 Dec. 9 2011 41 Y N/A N/A N/A N/A Y N/A N/A N/A 42 N N N/A N/A N/A N N Y N/A 43 N Y N/A N/A N/A N Y N/A N/A 44 N N N/A N/A N/A N N Y N/A 45 Y N/A N/A N/A N/A N N N N 46 N N N/A N/A N/A N N Y N/A 47 Y N/A N/A N/A N/A Y N/A N/A N/A KEY Y = Present N = Not Present R = Removed N/A = Not Called Table 3.3: Attendance for Group 4

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24 Dates Number Dec. 1 2011 Dec. 2 2011 Dec. 5 2011 Dec. 6 2011 Dec. 7 2011 Dec. 8 2011 Dec. 9 2011 51 Y N/A Y Y Y Y Y 52 Y N/A Y Y Y Y N/A 53 Y N/A Y N N Y Y 54 Y N/A Y Y Y Y N/A 55 N Y Y Y Y N Y 56 Y N/A Y Y Y Y N/A 57 Y N/A Y Y Y Y N/A 61 N Y N N N N N 62 N N N N N N N/A 63 Y N/A Y N N Y Y 64 Y N/ A Y Y Y Y N/A 65 N Y N N N N Y 66 N Y Y Y N Y N/A 67 N N N N N N N/A Table 3.4: Attendance for Groups 5 and 6 for the students who participated in the control conditions, an O for the students that participated in the origami projects and the W represents the students who completed the worksheets in the non the student said that they had done origami before, a question asked on the pre and post vocabulary tests. The column exhibits if the students liked doing the origami if they were in Group 1 or Group 2. Those that have an N/A either did not take the tests or just did not answer that question.

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25 Subject Number Group Gender Origami Learned? Liked? 1 11 O F N/A N/A N/A 2 12 O F N/A N/A N/A 3 13 O M N/A N/A N/A 4 14 O F Y Y Y 5 15 O F Y Y Y 6 16 O F Y Y N/A 7 17 O M N/A N/A N/A 8 18 O F N Y Y 9 21 O M Y Y Y 10 22 O M Y Y Y 11 23 O F N/A N/A N/A 12 24 O F Y N Y 13 25 O M Y Y Y 14 26 O F Y Y Y 15 27 O F Y N/A N/A 16 28 C M Y N N 17 31 C M Y N N/A 18 32 C M Y N/A N/A 19 33 C F N/A N/A N/A 20 34 C F Y Y N/A 21 35 C F Y Y N/A 22 36 C F Y N/A N/A 23 37 C M Y N/A N/A 24 38 C M Y N/A N/A 25 41 C M Y Y N/A 26 42 C F N Y N/A 27 43 C F Y Y N/A 28 44 C M Y Y N/A 29 45 C M Y N/A N/A 30 46 C F Y Y N/A 31 47 C F Y Y N/A 32 51 W M Y Y N/A 33 52 W F N Y N/A 34 53 W F Y Y N/A 35 54 W F Y Y N/A 36 55 W F Y Y N/A 37 56 W F Y N/A N/A 38 57 W M N N/A N/A 39 61 W M Y N/A N/A 40 62 W M N/A N/A N/A 41 63 W F Y Y N/A 42 64 W M Y Y N/A 43 65 C F Y N N/A 44 66 W F Y Y N/A 45 67 W F N/A N/A N/A Table 3.5: Descriptive Information on Participants

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26 In the next table are all of the raw data recorded as percentages for the test scores and attendance. The represents the diff erences in the Vocabulary tests (PostV PreV = DiffV). The DiffC column represents the differences in the Concept tests (PostC PreC = DiffC). The DiffC or Diff V, a difference could not be calculated. The Attendance column lists the percentage of the days that they participated in of the days that they were called. Select Number Group PreV PostV Diff V PreC PostC DiffC Attendance 11 O / / / / / / 0 12 O / / / / / / 0 13 O 13 13 / / 26 / 88 14 O 63 80 17 57 70 13 63 15 O 50 53 +3 52 57 +5 100 16 O 56 67 +11 52 43 9 50 17 O / / / / / / 0 18 O 63 60 3 57 74 +17 63 21 O 44 33 11 30 43 +13 88 2 2 O 50 53 +3 74 74 0 100 23 O / / / / / / 0 24 O 44 47 +3 43 70 +27 88 25 O 88 87 1 83 87 +4 63 26 O 81 93 +12 96 100 +4 100 27 O 27 / / 52 / / 13 28 C 48 87 +39 83 65 18 25 31 C 56 33 23 26 17 9 100 32 C 56 53 3 13 26 +13 100 33 C / / / / / / 0 34 C 56 67 +11 52 39 13 100 35 C 59 60 +1 52 52 0 100 36 C 44 / / 35 35 0 75 37 C 63 / / / / / 25 38 C 50 / / 65 / / 50 41 C 44 67 +23 87 78 9 100

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27 Number Group PreV PostV DiffV PreC PostC DiffC Attendance 42 C / 87 / / 83 / 25 43 C 58 67 +9 87 70 17 100 44 C / 67 / / 74 / 50 45 C 69 / / 96 / / 50 46 C / 73 / / 83 / 50 47 C 72 83 +11 87 96 +9 100 51 W 63 70 +7 43 52 +9 100 52 W 25 33 +8 26 52 +26 100 53 W 77 60 17 52 61 +9 67 54 W 73 53 20 43 61 +18 100 55 W 38 53 +15 83 65 18 83 56 W 56 47 9 70 61 9 100 57 W 63 67 +4 70 65 5 100 61 W 56 / / 87 / / 17 62 W / / / / / / 0 63 W 94 100 +6 83 83 0 67 64 W 56 73 +17 83 65 18 100 65 C 75 87 +12 87 65 22 29 66 W 75 93 +18 83 91 +8 67 67 W / / / / / / 0 Table 3.6: Participant Test Scores (in percentages) 3.2 Statistical Results The first step I took to analyze the data was to find the mean scores and descriptive statistics ( Table 3.7 stands for the dif ference of pre and post concept test scores and the difference of pre and post vocabulary test scores respectively. Positive Diff scores imply that the students improved over the study and negative test scores imply that the students did worse on the post tests than the pre tests. The minimum and maximum tests scores are also included for each group to show the range in ability of the participants. Figure 3.1 shows the means for all groups DiffC and DiffV values. The next table shows the means of al l of the test scores for all of the participants as well as mean attendance.

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28 Group Number of participants Test Scores Mean (%) Standard Deviation (%) Minimum (%) Maximum (%) 1 4 DiffC +6.5 11.5 9 17 5 DiffV +5.6 8.2 3 17 2 5 DiffC +9.6 10.8 0 27 5 DiffV +1.2 8.3 11 12 3 5 DiffC 1.8 10.0 13 13 4 DiffV 3.5 14.3 23 11 4 5 DiffC 11.4 12.3 22 9 5 DiffV +18.8 12.5 9 39 5 7 DiffC +4.3 15.6 18 26 7 DiffV 1.7 13.6 20 15 6 3 DiffC 3.3 13.3 18 8 3 DiffV +13.7 6.7 6 18 Table 3.7: Te st Scores Breakdown per Group (in percentages) Figure 3.1 : Mean Difference Test Scores

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29 Variable # of participants Mean Standard Deviation Minimum Maximum Pre Test: Vocab 35 58.57% 16.39% 13% 94% Post Test: Vocab 32 64.56% 2 0.21% 13% 100% DiffV 29 +4.93% 13.14% 23% +39% Pre Test: Concepts 33 63.30% 22.90% 13% 96% Post Test: Concepts 33 63.12% 20.31% 17% 100% DiffC 29 +0.97% 13.53% 22% +27% Attend 38 73.58% 28.98% 13% 100% Table 3.8: Score Breakdown per Test (including attendance) After finding the mean scores, I did an ANOVA Procedure. This test is used to determine whether the mean scores of the different groups are statistically different (Peng 2009). The Null Hypothesis for the first ANOVA I did was that the mean DiffV scores for the Origami, Control and Worksheet groups were the same ( The Alternative Hypothesis states that they are not all equal The ANOVA re sulted in a p value of 0.5684 and was not statistically significant. Therefore the Null Hypothesis is not rejected and there is not a significant difference between the three groups for DiffV. To determine if the ANOVA was appropriate to use I ran a tes t for homogeneity of variance using the Levene HOV test (Zhang 1998) which resulted in a p value of 0.2367. The ANOVA was thus appropriate because the Levene HOV test was not significant which means the distributions for the different groups were similar in shape. This means that on average all of the students improved their vocabulary test scores by

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30 about 5% from the pre test to the post Test. The breakdown of how each o f the groups performed on average for the vocabulary tests consist of the Control groups (Group 3 and Group 4) having a mean score of 8.89% increase with a standard deviation of 17.12%, the Origami, experimental groups (Group 1 and Group 2) having a mean s core of 3.4% increase with a standard deviation of 8.14%, and the Worksheet, non experimental groups (Group 5 and Group 6) having a mean score of 2.9% increase with a standard deviation of 13.70%. I also did an ANOVA for the DiffC scores. The Null Hypoth esis for this ANOVA states that the mean DiffC scores for the Origami, Control and Worksheet groups were the same ( A test for homogeneity was also conducted to determine if the ANOVA was appropriate for this situation. Since the variance f or the three groups was not significant with a p value of 0.3988 using a Levene HOV test, the ANOVA is appropriate to use. The p value for this ANOVA test was 0.0496. Therefore the Null Hypothesis is rejected and the Alternative Hypothesis that the means are different is supported. This means that the different conditions had an effect on the DiffC scores. The effect size was this means that about twenty one percent of the variability can be explained by the different conditions. The me average all of the students improved their vocabulary test scores by about 1% from the pre test to the post Test. The breakdown of how each of the groups performed on average for the concept tests consist of the Control groups (Group 3 and Group 4) having a mean score of 6.6% decrease with a standard deviation of 11.75%, the Origami, experimental groups (Group 1 and Group 2) having a mean score of 8.2% increase with a

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31 standard dev iation of 10.52%, and the Worksheet, non experimental groups (Group 5 and Group 6) having a mean score of 2% increase with a standard deviation of 14.68%. Since the results for the DiffC ANOVA were significant I did a Post Hoc Test ignificant Difference (LSD) t Test ( Table 3.9 contains results ). For this test, the groups with the same letter are not significantly different. Based on the results the mean DiffC scores for the Origami group and the control group are significantly diff erent. The Worksheet group is not significantly different from the Origami or the Control groups. T Grouping Mean N Group A 8.2% 9 O B A 2.0% 10 W B 6.6% 10 C Table 3.9: Results of t Tests LSD for DiffC The ANOVA test resulted in a difference in m eans for the DiffC scores. The LSD t Test showed that there was a difference in the DiffC means between the Origami and Control groups. I also did a two variable t Test to confirm that there was a statistically significant difference between the mean Diff C scores for the Origami and Control groups. The Null Hypothesis for this test thus stated The difference between the mean for the Origami Group (+8.2%) and the Control Group ( 6.6%) was significant with a p value of 0.0103. The Null Hypo thesis is rejected and the Alternative Hypothesis that the means are not equal is supported. The t Test is acceptable because the Equality of Variances had a p value of 0.7662; this means that the distribution shapes are similar enough to use a t Test.

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32 I additionally did a t Test for the mean DiffV scores between the Control Group and the Origami Group. This resulted in a p value of 0.3765 which is not significant, however; the Equality of Variances had a p value of 0.0396. This means that the t Test is not appropriate to use since the distributions are not close. I did a Mann Whitney Test which does not assume a specific shape for the population distribution and is referred to as distribution free (Peng 2009). The p value for the Mann Whitney was simi lar to the t Test at 0.3670 so there is not a significant difference in the DiffV means between the Origami and Control Groups and the Null Hypothesis is supported. Although the LSD t Test did not show a significant difference in means between the Origami and Worksheet groups for the DiffC mean scores, I did a two sample t Test to confirm. The t Tests were used to assess whether two variables represented two populations with different means or if any difference between the two groups was based on random ch ance (Sheskin 1997). The p value for the t Test was 0.3082. This is not significant and confirms that the Null Hypothesis ( ) is supported. The t Test is applicable with an Equality of Variances p value of 0.3607. I then did another two sam ple T Test between the Origami Group and the Worksheet Group for mean DiffV scores. This was also not significant with a p value of 0.9221. The test is acceptable with a p value of 0.1369 for the Equality of Variances. Lastly, I investigated any differen ces in scores related to gender. I did a two sample t Test for both DiffV and DiffC mean scores. The Null Hypothesis for the t Test related to the DiffC mean scores was An Equality of Variance test was performed to confirm that the t Test is appropriate. The p value was 0.5238, meaning the variances between the two groups was not significantly different. A t Test was

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33 performed which resulted in a non significant p value of 0.4016. Therefore the Null Hypothesis is supported and there is no significant difference. Nineteen females and ten males were included in the t Test for the DiffC values. The mean difference in concept test scores for the females in my study was about +2.5% and the mean difference in concept test scores for the male s in my study was 2%. The Null Hypothesis for the t Test related to the DiffV mean scores was the same as for the last test ( ). An Equality of Variance test was performed to confirm that the t Test is appropriate. The p value was 0.1192, so the variances between the two groups were not significantly different. A t Test was performed which resulted in a non significant p value of 0.9829. Therefore the Null Hypothesis is preserved and there is no difference between male and female test sco res for the Concepts test. For the t Test for gender influence on the DiffV values only eighteen females and eleven males were included. The mean difference in vocabulary test scores for the females was about +4.9% and the mean difference in concept te st scores for the males was +5%. Table 3.10 is a summary of all aforementioned statistical tests and their results. The rows in red contain significant results. Statistical Test Groups Looked at DiffC or DiffV p value Significant? ANOVA O,W,C DiffV 0. 5684 No ANOVA O,W,C DiffC 0.0496 Yes 2 variable T Test O, C DiffC 0.0103 Yes 2 variable T Test O, C DiffV 0.3765 Test not reasonable Mann Whitney O, C DiffV 0.3670 No 2 variable T Test O, W DiffC 0.3082 No 2 variable T Test O, W DiffV 0.9221 No 2 va riable T Test F, M DiffV 0.9829 No 2 variable T Test F, M DiffC 0.4016 No

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34 Finally, I looked at the correlation between variables. The Pearson Correlation Coefficients are displayed in Figure 3.2 and Table 3.11. The correlations in Table 3.11 that are red are statistically significant. Figure 3.2: Correlation Coef ficients Variables Pearson Correlation Coefficient P Value Variables Pearson Correlation Coefficient P Value PostV: PreV 0.77646 <0.0001 DiffC: PostV 0.27928 0.1501 DiffV: PreV 0.10257 0.5958 DiffC: DiffV 0.38067 0.0457 DiffV: PostV 0.54705 0 .0021 DiffC: PreC 0.52384 0.0035 PreC: PreV 0.39038 0.0247 DiffC: PostC 0.07219 0.7098 PreC: PostV 0.73130 <0.0001 Attend: PreV 0.23734 0.1698 PreC: DiffV 0.57205 0.0015 Attend: PostV 0.47356 0.0062 PostC: PreV 0.55931 0.0013 Attend: DiffV 0.3 5671 0.0575 PostC: PostV 0.72532 <0.0001 Attend: PreC 0.26969 0.1291 PostC: DiffV 0.40607 0.0288 Attend: PostC 0.25456 0.1528 PostC: PreC 0.81178 <0.0001 Attend: DiffC 0.21496 0.2628 DiffC: PreV 0.00590 0.9757 Key: Correlations in red are statis tically significant Table 3.11: Pearson Correlation Coefficients

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35 Chapter Four : Conclusion and Discussion 4.1 Comments on Results 4.1.1 Raw Data The raw data were recorded as percentages for ease of comparison. The Vocabulary tests had a different number of points than the Concepts tests. Also the Pre Concepts test had different scoring than the Post Concepts test. Thus percentages allowed for easy comparisons. There are three types of missing data, an absence of information, invalid responses, or unus able responses (Peng 2009). A few students gave up on their tests before working on them to completion. Doing so would give a slightly unrealistic difference of scores between their pre tests and post Tests, especially since, as a whole, the students were better at completing the pre tests as the pre tests were a new and exciting experience from their normal routine. Also due to attendance, not all of the students in each group had the same experience. In the Post Test: Vocab I asked the students if they learned anything during my project, what they learned, and, if they were part of the origami groups (Group 1 and 2), I asked if they liked the origami and why. Twenty four of the twenty eight students that answered the question pertaining to whether they felt they had learned something with Several wrote that they learned abou whether they liked the origami stated that they did like the origami. When responding to

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36 responses similar to what Cakmak (2009) and Purnell (2009) received with their work. 4.1.2 Statistics When looking at the means for the Diff scores for different groups ( T able 3.7 ) I had a curious observation that the younger groups (3 and 5) for the Control and Worksheet non experimental groups improved more in their concepts than vocabulary. The older group s (4 and 6) for these same conditions improved more in their vocabulary. Based on the set up of the tests I may have expected the opposite due to the amount of reading and understanding required in the concepts tests over the multiple choice picture vocab ulary tests. One justification for this observation is that the younger students knew the vocabulary fairly well with the exception of the higher level words such as hexahedron and polygon and did not learn those higher level words in the Control or Works heet groups and thus showed no improvement in the vocabulary tests but the younger students were able to learn new concepts, improving their concept scores. The older students in these two groups may have had a similar situation where they knew most of th e concepts presented without the ability to learn many new concepts resulting in little visible improvement but they were able to learn the higher level vocabulary terms and show notable improvement in their vocabulary scores. There were not many statistic ally significant results from this pilot study. Gender with these findings while others contradict them. Boakes (2008) found that gender did play a role in middle schoo

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37 visualization skills. A recent article (Robelen 2012) on the gender gap in mathematics could be eliminated by removing the stereotype that boys are better in math than girls. This article (R obelen 2012) also noted that many studies show that there is no gender difference for math learnin g. The origami group did significantly better on the Post Test: Concepts compared to their Pre Test: Concepts than the control group (p value of 0.0103). Thi s supports the idea that the origami projects helped the students learn certain math concepts. Cakmak visualization and orientation. Boakes and Stockton (2009) howev er did not find any significant improvement in geometry skills due to origami based lessons. I however do not believe they incorporated the origami well as a lesson and the items for their tests al sense, but not to text may not have included the same information that was presented on the testing material. If the lessons and tests do not correlate well than t he tests may not accurately reflect the success of the lesson. I got several noteworthy and statistically significant results with the Pearson Correlation Coefficients. These can be useful in designing future experiments. There was a negative correlation between attendance and the Post Test: Vocab scores for all students. This could be significant because of the low numbers of participants. Although many of the other studies I have been looking at have similar or lower numbers of participants (Yuzawa an d Bart 2012; 24 participants, Boakes and Stockton 2009; 56 participants, Cakmak 2009: 38 participants). It could have also been affected by certain

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38 early before complet ing the post vocabulary test resulting in a lower score than he/she may have gotten. This student had an 88% attendance but a low attention span and often left mid lesson. Another factor that could have resulted in this unexpected significant result was that students in all groups, particularly in the Control groups, who missed days around the tests ended up taking both pre tests or both post tests in the same day. This could have resulted in the lower attendance producing higher Post Test: Vocab scores. For the pre tests the students took the vocabulary test first then the concepts test but for the post tests the students took the concepts test and then the vocabulary test. Both tests had some overlap in material and they could have used what was state d as fact on the concept test to assist them on the vocabulary test questions. For example, question number 16 on the Post with an image of a hexahedron included and question number 13 on Post T est: Vocab C for test examples). This means that if they did not know what a hexahedron was they still might have gotten the answer right on the vocabulary test by reme mbering what they just saw on the concepts test. I considered removing these students from the data for more accurate results, however, then I would also have to take a closer look at several other participants that may need to be removed as well which wo uld leave me with very few who went through the whole procedure correctly. For example, one of the students from the control group asked to do an origami fish after he was done with his homework. Without realizing he was in the control group and not the origami group I folded the fish with him. Although I did not teach the lesson that went along with the fish I should not

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39 have exposed him to the origami. There is a possibility that since both of his sisters were in the origami group they may have taught him the fish at home. Another student, who was in the origami group, was very interested in the origami and folded many more whales and fish after the lessons were over and the other students left. The lessons were not taught again but she may have had an advantage on the post tests than the students The other significant correlations included the positive correlation between doing well on the Pre Test: Vocab and the Post Test: Vocab as well as between Pre Test: Concepts scores and the Post: Test Concepts both with a p value of less than 0.0001. This means that the students who performed well on the vocabulary and concepts pre test tended to do well on the post Test and vice versa. This can be expected regardless of the conditions that the students were exposed to. There were seven other statistically significant positive correlations and two statistically significant negative correlations. There was a high positive correlation between the Post Test: Vocab scores and Post Test: C oncepts scores, and between the Pre Test: Concepts and Post Test: Vocab scores with a p value of less than 0.0001. The positive correlation between both types of post Test scores could be explained by the fact that a student who does well on tests in gen eral would likely do well on both tests. Students could have been in good frames of mind, well rested and well fed in the time frame that they took both post Tests. This same explanation can be applied to the statistically significant positive correlatio n between the Pre Test: Concepts scores and the Pre Test: Vocab Scores which had a p value of 0.0247. The positive correlation between the Pre Test Concepts and Post Test: Vocab could also be observed as a correlation due to good test takers in general. Similarly, there are

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40 positive correlations between the Post Test: Vocab scores and the difference between pre and post vocabulary test scores with a p value of 0.0021, between the Pre Test: Concepts scores and the difference between pre and post vocabul ary test scores with a p value of 0.0015, between the Post Test: Concepts scores and the Pre Test: Vocab scores with a p value of 0.0013, and between Post Test: Concepts scores and the difference between pre and post vocabulary test scores with a p valu e of 0.0288, can be inferred to exist due to those students being good overall test takers. The negative correlations that were statistically significant were more thought provoking. There was a negative correlation between the difference in pre and post vocabulary test scores and the difference between the pre and post concept test scores with a p value of 0.0457. Students who learned new vocabulary or concepts and had a positive improvement in test scores in one category tended to not do well or imp rove test scores in the other. The last statistically significant result from the Pearson Correlation Coefficients was the negative correlation between the Pre Test: Concepts and the difference between the Pre and Post Test: Concepts scores with a p valu e of 0.0035. Students who did well on the Pre Test: Concepts did not show improvement in their concept test scores. Based on the calculations used to find a difference score there tends to be a negative correlation with the pretest values (Linn and Slin de 1977). This strong negative correlation is likely a result of those students who did really well so they had no room to improve thus earning low DiffC scores. Those students who did not perform as well on the Pre Test: Concepts and learned new concept s earned higher DiffC values than the students who tested well in the first place.

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41 4.2 Comments on Study This project is a pilot study. There are not enough participants to get a lot of valuable significant statistical data. There were several factors that contributed to the low participant numbers. The timing of the research played into several of the participants not completing the whole project. I worked with the students in November and December which meant that I was working with the students ar ound the Thanksgiving break and their winter holi day break. Children were less likely to come to the homework help session and they had less focus for working on academic activities. Since I started volunteering in the Boys and Girls Club in September I got to observe that on Mondays when the students had homework due for the week the homework help room was very busy with students wanting to get their homework done. As the week progressed and the students got closer to the weekend attendance and enthusia sm decreased. A similar phenomenon occurred as both breaks that I was present for approached. The time was not a major issue because Groups 1, 2, and 3 were finished before t h e Thanksgiving break and Groups 4, 5, and 6 completed the project in the time s egment between the two breaks. Attendance was an issue for the success of the project. Some of the attendance was influenced by the time of year. Ten students were not able to be included in the pre test/pos t Test comparisons due to completing more tha n one but less than all four of the tests. Several students who were still included in the pre test/ pos t Test comparisons, did not participate in all of th e research activities due to poo r attendance. Some students did not bother to come even when they were present at the Boys and Girls Club. This could have been due to them not hearing the front desk call them to the room, them not feeling

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42 like coming, or they had another activity they would prefer to participate in instead of coming to the homework h elp room. Attention was another complication with the project. Some children would come late or not at all because they had something they wanted to do instead. Many students had different mindsets when approaching this project. I had one participant nervously ask me what happens if they do not do well, every time they took one of the tests. This stress of failing in a situation where not all students had the opportunity to succeed one hundred percent could have affected their attention to the projec t as well as their test scores. Other students were so happy to get the one on one attention that I was providing along with the feeling that they were special that they were on board for anything I had them do. There were even a few who wanted to partic ipate which was made evident by their getting upset when they were finished with the project. However, they felt it necessary to not give their full attention and participate. Still, it was the location that I had to work in that contributed to many of t he attention issues. Many of the students refused to try on concepts with which they were unfamiliar. It was unrealistic to try to teach kindergarteners multiplication and division in a sentence or two which made it complicated to try to work with the o lder ones in the same group who may have been able to learn something. I began the project by reading to the younger group the pre tests one question at a time so that I knew each student understood what was being asked, especially when many of the student s struggled with reading. I had to repeat most of the questions because all the students could not hear me over the other noises in the room (a layout of the location is located in Appendix A). Reading the questions aloud also drew the attention from the

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43 other students who wanted to understand what we were doing. Additionally it seemed to give the students a greater feeling that they were allowed to talk about the test questions aloud. The larger issue with reading aloud to the students was the time tha t it took. Those who finished early simply left and I was under strict time constraints so that they could also complete all their homework in the time allotted to us. I began asking the students if they had any trouble reading and to ask me if they need ed anything specific read to them. This appeared to work, my concern lay in whether each student had the same opportunity to succeed as well as if they truly understood the questions asked of them. Students who come from lower income families tend to ente r school without a good foundation for reading. This is mainly due to the fact that these students are not exposed to as many words in the home as middle class students (Walker 2010). I was told when I started volunteering with the Boys and Girls Club th at they could always use volunteers to help the students with their reading. This was because the teachers in the homework help room had too many children to help to take the time to read a story with a single child. To practice and improve their reading skills many students are assigned to read a book out loud with a parent, older sibling or teacher so that when they stumble on a word they can get help. Even when I tried to sit down and help a student read a story I had to stop several times to help ano ther student with a homework question. I observed that many students would not read the instructions for their homework and try to just guess what was needed to complete the assignment. This may or may not have been a result of the fact that many of the students had weak reading skills.

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44 One factor that should have been more controlled in my study was a better introduction to the project for the students so that they understood what was going to be asked of each of them individually as well as reading and explaining the testing. The loud chaotic noises in the homework help room made it difficult to talk with the students in my group. Something that I could have done to help participation and attention was the length of the project for each participan t. For the experimental and non experimental groups I did three projects between the pre tests and pos t Test s. Doing just one or maybe two projects instead may have resulted in more participants completing a higher percent of the project. For the whale model (Appendix B), which I folded twice with the students to teach them different lessons, the students wanted to fold the model faster than I could teach the new lessons because they already knew the folds. In the environment I was in to perform this stu dy family ties were very important. This meant older siblings felt it necessary to assist younger siblings. Origami encourages cooperation which in certain circumstances can provide benefits to those involved (Purnell 2009; Chen 2005). This did not hav e a huge effect on my project because I divided the students into two different age groups and tried to put siblings into different conditions but I did have a few problems with some of the older students trying to help the younger students on their tests. The collaborative and social nature of informal activities does make it difficult to access learning and success of these activities (Fenichel and Schweingruber 2009).

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45 4.3 Discussion Purnell (2009) observed positive feedback from doing the or igami with the children; however, there are numerous social learning aspects that I was not able to observe. To look at the true efficiency of this project it may help to stay longer and perform a secondary set of post tests a month or so later to observe how well the origami helped the students retain information. These social leanings that Purnell (2009) observed with her origami groups will likely also crop up if I had had more time with the students. She comments that a big thing that they learned wa s vocabulary. We looked at different types of vocabulary. Possibly my vocabulary should have been ingrained in the origami more effectively. Both Chen (2005) and Purnell (2009) discuss the idea that students will be self motivated to work together and c reate a sense of community with the origami (Levenson n.d.). I did not see this phenomenon in my groups. This is likely attributed to the location and the size of the group. With only ten students at most at a time I was available to assist them as they needed it so they had no need to turn to their peers. The location also did not lend to quietly and calmly work together on a shared goal. Even though I did not observe all of the same behaviors as Purnell (2009), I did notice students desire to be a part of the project although not all of them projected this desire positively. I had one student resist participating every time that they were called. Then after a few days the student began to ask when the project would be over. When I no longer ne eded his participation he was upset. Another student ran and gave me a hug every day that I was there and working with her. Both students, albeit in their own way, appreciated the benefits that my project had to offer them. These benefits not only

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46 inclu ded my one on one attention that was a rarity at the homework help sessions but also increased self esteem, pride in their work, and empowerment (Levenson n.d.). Another study ( Yuzawa 2010), which examined size comparison strategies in young children an d how origami might help, had a different type of teaching style when it came to the origami. They had a tendency to fold the paper wrong and ask the students if they thought it was appropriate. This integrated the size comparison learning that they want ed to teach the children easily. The way that origami was used to teach in this study was different than the method of incorporating the origami into a lesson that Yuzawa and Bart (2012) and I used. It would be a good follow up study to look at the diffe rence between origami assistance in learning simply as a hands on activity versus integrated in a lesson plan. There is a new concept of origami lessons that has developed in the last few years called storigami. This is when a story is incorporated with the academic origami lesson. Storytelling + Origami =Storigami Mathematics (Mastin 2007) is an article that describes this practice. I found the story that was given as an example interesting but it had too much going on and not enough involved in the l essons. This might, however, be well used in an elementary classroom particularly when trying to teach several subject lessons in one activity. It would also work very well in the informal setting. You could integrate just about any kind of lesson into the storigami. It might also be more interesting and lesson for a test but instead just to get them involved so that they might pick up some new knowledge, stori gami would be suitable.

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47 4.4 Implications of the Study Origami addresses all of the different learning types so that all students have the opportunity to be engaged and gain new knowledge. The most commonly identified styles of learning include auditory symbolic abstract, visual spatial, and kinesthetic (Denig 2004, Boakes and Stockton 2009). Origami addresses auditory learning through verbal step by step instruction and lecture. Symbolic abstract learning occurs through origami diagrams and any works heets that may accompany the origami. Visual spatial learning comes in when the students watch the teacher or instructor perform the origami steps to follow and with the different shapes that the students create. Lastly, when the students manipulate the paper kinesthetic learning takes place. (Boakes and Stockton 2009) This study, even though it is a pilot, shows that there can be positive outcomes for informal education, such as using origami, to teach mathematical concepts. If there are enough studies done on the different types of informal education that can be applied to make it into the school curriculum. Getting origami and other educational tools into the classro om will not only help the students understand the knowledge presented to them a little better it will in addition provide them with a well rounded learning environment. 4.5 Future Recommendations ience Foundation identifies the importance of helping young children incorporate math in their everyday life so that they can create a strong foundation for future math learning ( Ferrante 2012) It is important to continue researching what types of progra ms can be implemented to

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48 highly functioning citizens (Ernest 2010). As far as the design of the study is concerned a control origami group may have worked better t han the non experimental worksheet group. I wanted to see whether the students were capable of learning the concepts presented to the origami group and tried to present it to them in a way that they might come across it in a formal classroom setting. Or igami projects should be observed integrated in the classroom. This is an important aspect of the idea that informal education as a bridge between formal education and real world interactions. Having the project conducted in the classroom may assist with the attendance, attention, and participant number issues. Although not all students have perfect attendance and some skip classes there is a greater expectation for students to show up to classes than it is for them to show up to certain after school pro grams or extracurricular activities. Having a classroom or classroom like setting would also give the instructor the ability to write notes on a board to supplement the origami. For example when I tried to teach the younger students about fractions, eve n though the origami visuals may have helped them learn the concept, they had a hard time understanding how a fraction is supposed to be written simply through verbal explanation. In my setting bringing in any type of poster or large paper note board woul d have been slightly difficult. This would also have been novel and caused those not in my study to run over and disrupt the lessons. Although in other informal settings notes or worksheets could be used very effectively, the ability to integrate these n otes into a classroom lesson with the origami may result in the students gaining more from the lesson.

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49 A new location is necessary for this project to work better. A classroom does not have to be the next step for this research but the issue of attentio n in particular would be improved with a better location. The Boys and Girls Club could still be utilized if the project were to have its own room to take away the distractions (See Appendix A for Location Re Creation). Although using origami in the sett ing that I had could have its uses to study origami as an informal learning tool to assist formal classroom learning, the project should be conducted in a similar situation to the formal classroom. Population variability may need to be taken into account when more data are collected for this type of study. The population I had to work with consisted of students from different schools. However, they came from similar racial, economic and attitude backgrounds. Students from low income families are not as ready for school as students from middle class families (Janus and Duku 2007). This could mean that the population that I had is not representative of the general population of elementary school students. Another alteration to the pilot, when there are m ore participants, is to break down the project into the different concepts and vocabulary words (See Appendix B for example). This would allow me or another researcher to be able to pin point what concepts the origami helps with the most. This could show teachers when the best time Based on the statistically significant results for the negative correlation between the difference in test scores for the pre and post T ests for vocabulary and concepts it might be a good idea to look at the breakdown of the project. This significance may have resulted from the fact that the younger students tended to do better on the concepts tests

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50 and the older students tended to perfor m better on the vocabulary tests in the worksheet non experimental groups and the control group. Another recommendation for further research would be to look at how effective origami could be when used as a study tool similar to how I used it at Saint S tephens Episcopal School. While at Saint Stephens I was able to teach the students how to fold an origami fish while reviewing concepts that would be on their next test. This replaced the normal review the teacher would do. It would be interesting to se e if the origami would help the students retain information better for a test. Additionally a study to compare the use of origami with related mathematical computer programs could be a useful avenue for future research. 4.6 End ld emphasize activities that encourage students to learn in the classroom and what they need to do in the world, informal education techniques should be investigated. Or igami can be used to teach many concepts ( Pearl 2008 observe the potential of origam i for teaching math concepts to elementary school students.

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51 Bibliography Anonymous. Wonderful Whale! [Online]. Instructor reproducible. 2004 Available from: http://teacher.scholastic.com/products/instructor/images/April04/ Origami/whale_repro.pdf Accessed 2011 Sept 30. Anonymous. 2010. Florida Department of Education. [online]. Available from: http://www.floridastandards.org/Downloads.aspx Accessed 2011 Sept 30. Anonymous. 2011. Origami Resource Center [Online]. Available from: http://www .origami resource center.com/index.html. Accessed 20011 Sept 30 Becker, H. Computers in Schools Today: Some Basic Considerations. American Journal of Education [Internet]. 1984. [cited 2012 April]; 93(1): 22 39. Available from: http://www.jstor.org/discover/10.2307/1085088?uid=3739600&uid=2129&uid=2 &uid=70&uid=4&uid=3739256&sid=47698847810997 Boaler, J. When Learning No Longe r Matters: Standardized Testing and the Creation of Inequality. Phi Kappa Delta [Internet]. 2003 [cited 2012 Mar 10]; 84(7): 502 506. Available from: http://web.ebscohost.com.ezproxy.lib.usf.edu/ehost/pdfviewer /pdfviewer?vid=14&hid=112&sid=bd11a0a1 9795 4c76 a09f 75184b3c9c2d%40 sessionmgr112 Boakes, N. Origami Mathematics Lessons: Paper Foldin g as a Teaching Tool Mathitudes [Internet]. 2008 [cited 2011 March 13]; 1(1 ):1 9. Available from: http://www.coe fau.edu/CentersAndPrograms/mathitudes/20080901bMathitudes Oct08_revisionFinalVersionforpublicationOct.%2024,200 8.pdf Boakes, N., Stockton, R. Origami Instruction in the Middle School Mathematics Classroom: Its Impact on Spatial Visualization and Geometry Knowledge of

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52 Students. Research on Middle Level Education [Internet]. 2009 [cited 2011 Sept 28]; 32(7): 1 12. Available from: http://www.amle.org/portals/0/pdf/publications /RMLE/rmle_vol32 _no7.pdf akmak, S. 2009. An Investigation of the Effect of Origami Based instructi on on Elementary Students Spatial Ability in Mathematics. Middle East Technical University Library E Thesis Archive [Online] 1 133. Available from: http://etd.lib.metu.edu.tr/ upload/3/12610864/index.pdf Accessed 2011 Sept. 28. Chen, K. Math in Motion: Origami Math for Students who are Deaf or Hard of Hearing. Journal of Deaf Studies and Deaf Education [Internet]. 2006 [cited 2012 Mar 10 ]; 11(2): 262 266. Available from: http://jdsde.oxfordjournals.org/content/11/2/262.full.pdf+html?sid=e89b69fd 6ce2 43ac a821 a140074457f0 Corte, E., Verschaffel, L., Greer, B. Connecting Mathematics Problem solving to the Real World. Proceedings of the International Conference on Mathematics Education into the 21st Century: Mathematics for living [Internet]. 2000 [Cited 2011 April 17]; 66 73 Available from: http://math.unipa.it/~grim/Jdecorte.PDF Dacorogna, B. Marcellini, P. and Paolini, E. Origami and Partial Differential Equations. Notices of the American Mathematics Society [Inter net]. 2010 [cited 2011 Sept. 30]; 57(5):598 606 Available from: http://web.math.unifi.it/users/marcellini/lavori/ DMP_ Notices_AMS.pdf Denig, S. Multiple Intelligenc es and Learning Styles: Two Complementary Dimensions. Teachers College Record [Internet]. 2004 [cited 2012 April 4]; 106(1): 96 111. Available from:

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53 htt p://projects.cbe.ab.ca/central/altudl/files/multiple_intellegences_learning_styles .pdf Childhood Education in the Context of Mathematics, Science, and Literacy. National Science Foundation: Award Abstract [Internet] 2010. [cited 2012 April 7. Avail able from: http://www.nsf.gov/awardsearch/showAward.do?AwardNumber=1020118 Ernest, P. Add it up; Why Teach M athematics?. Professional Educator [Internet]. 2010 [cited 2011 March 30; 9(2): 44 47. Available from: http://docs.mohammadzadeh info/Publications/Math/Refrences/ Conversation%20with%20a%20Keynote.pdf .pdf. Fawks, G, Verbeek, M. Tressler, P. and Gilchrist S. 2011. Integrating scien ce outreach into the curriculum. Florida Academy of Sciences Conference. Available from: http://www.barry.edu/fas/abstracts/Default.htm Fenichel, M. and Schweingruber, H. A. Surrounded by Scie nce: Learning Science in Informal Environments. Washington, DC: The National Academic Press. 2010. Children Learn Math. National Science Foundation: Discovery. 2012. [cited 2012 Ap ril 7]. Available from: http://www.nsf.gov/discoveries/disc_summ.jsp?cntn_id=123183&org=NSF Fuchs, L. Fuchs, D. Finelli, R. Courey, S. and Hamlet, C. Mathematical Problems E xpanding Schema Based Transfer Instruction to Help Third Graders Solve Real Life. American Educational Research Journal [Internet]. 2004 [cited 2011 March

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54 31]; 41(2): 419 445. Available from: h ttp://aer.sagepub.com/content/ 41/2/419.short Gardner, H. Multiple intelligences: The Theory in Practice. 1 st Ed. New York: BasicBooks; 1993. Available from: http:/ /www.chaight.com/Wk%2011%20Gardner%20 %20Multiple%20Intelligences.pdf Garofalo, J. Beliefs and Their Influence on Mathematical Performance. Mathematics Teacher [Internet]. 1989 [cited 2012 Mar 7]; 82(7): 502 505. Available from: http://www.jstor.org/stable/27966379 Greenhough, C. 10 Unbelievable Origami and Papercraft Sculptures. The Inquisitr [Internet]. 2010 [cited 2011 Dec 5]; Available From: http://www.inquisitr.com/67232/10 unbelievable origami a nd papercraft sculptures/ Huitt, W. and Hummel, J. Piaget's theory of cognitive development. Educational Psychology Interactive Valdosta, GA: Valdosta State University. [Internet] 2003. [cited 2012 Feb 26] Available from: http://www.edpsycinteractive.org/topics/ cognition/piaget.html Hull, T. Congresses Numerantium [Internet]. 1994 [cited 2011 Sept 30]; 100: 215 224. Available from: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.28.2679 Illeris, K (Ed.). Contempor ar Words. 1 st Ed. New York: Routledge. 2009.

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5 5 Janus, M. and Duku, E. The School Entry Gap: Socioeconomic, Family and Health and Development [Internet]. 2007 [cited 2012 Mar 14]; 18(3): 375 403. Available from: http://www.tandfonline.com.ezproxy.lib.usf.edu/doi/pdf/10.1080/104092 80701610796a LaFosse, M. Making Origami Science Experiments. 1 st Ed. New York: The Rosen Publishing Group, Inc. 2004. Levenso n, G. The Educational Benefits of Origami [Internet]. N.D. [cited 2012 Mar 5] Available from: http://www.informeddemocracy.com/sadako/fold/edbens.html Linn. R. and Slinde. J. The Dete rmination of the Significance of Change Between Pre and Posttesting Periods. Review of Educational Research [Internet]. 1997 [cited 2012 Jan 28]; 47(1): 121 150. Available from: http://rer.sagepub.com/content/ 47/1/121.extract and Their Achievement in Maths: A cross sectional study. Educational Psychology [Internet]. 2003 [cited 2011 May 21]; 23(1): 73 85. Available from: http://www cimm.ucr.ac.cr/ciaemPortugues/articulos/educacion/ aprendizaje/High%20School%20student%C2%B4s%20beliefs%20about%20math s,%20mathematical%20problem%20solving,%20ande%20their%20achievement %20in%20maths*Mason%20Luc%C3%ADa.*Mans on%20L.%20High%20Schoo l%20Students%20Beliefs%20About%20...2003.pdf.

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56 Mastin, M. Storytelling + Origami = Storigami Mathematics. Teaching Children Mathematics [Internet]. 2007 [cited 2012 Mar 5]; 14(4): 206 212. Available from: http://www.nctm.org Parker, S. The Biology of Learning. Human Learning: A Holistic Approach [Book]. Ed. Parker, S and Jarvis, P. 2005 [cited 2012 April 5] New York: Routledge. 16 31 Pearl, B. Math in Motion: Origami in the classroom. 1 st ed. 2008. Peng. C. Data Analysis Using SAS. 2009. California: Sage Publications; 2009. Petti, W. Make Your Own Geometric Forms!. Math Cats [Internet]. 2012. [cited 2012 April 7]. Available from: http://www .mathcats.com/crafts/spaceforms.html Pope, S. The Use of Origami in the Teaching of Geometry. Proceedings of the British Society for Research into Learning Mathematics [Internet]. 2002 [cited 2011 Sept 30]; 22(3): 67 73. Available from: http://bsrlm.org.uk/IPs/ip22 3/BSRLM IP 22 3 12.pdf Purnell, A. Providing for Creativity through Origami Instruction. New York City Writing Project [Internet]. 2009. [cited 2011 March 31]; 1 7. Availabl e from: http://www.lehman.edu/deanedu /litstudies/ELP/PDF/AnnettePurnell_Origami%5 B1%5D.pdf. Ritzhaupt, A., Higgins, H. and Allred, B. Teacher Experiences on the Integration of Modern Educational Games in the Mid dle School Mathematics Classroom. International Society for Technology in Education [Internet]. 2010 [cited 2011 March 13]; 1 38. Available from: http://center.uoregon.edu/ISTE/uploads/ ISTE2010/KEY_48692743 /Ritzhaupt_ISTE_Teacher_RP.pdf Robelen, E. Study Questions Explanation for Gender Gap in Math. Education Week [Internet]. 2012 [cited 2012 March 12]; 31(18): 12. Available from:

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57 http://web.ebscohost.com/ehost/pdfviewer/pdfviewer?sid=28f217d0 decd 4d7c 9ae7 585cf43fd783%40sessionmgr112&vid=7&hid=10 Robinson, J Instructor Reproducible [online]. 2004 [cited 2011 Sept 25] Available from: http://www.scholastic.com/content/collateral_resources/pdf/ p/products_instructor _images_whale_repro.pdf Roschelle, J., Pea, R., Hoadley, C., Gordin, D., and Means, B. Changing How and What Children Learn in School with Computer Based Technologies. Children and Computer Technology [Internet]. 2000 [cited 2012 Apri l 5]; 10(2): 76 101. Available from: http://www.princeton.edu/futureofchildren/publications/journals/article/index.xml ?journalid=45&a rticleid=203 Sheskin. D. Handbook of Parametric and nonparametric statistical procedures. Florida: CRC Press Inc; 1997. Slavin, R. Cooperative Learning: Student Teams. 2 nd ed. 1990. Washington D.C.: National Education Association. Enhanced Lea Foundation: Award Abstract. 2001. [cited 2012 April 7]. Available from: http://www.nsf.gov/awardsearch/showAward.do?Awa rdNumber=9903409 Walker, J. Streamlining Literacy. Principle Leadership [Internet]. 2010 [cited 2012 Mar 14]; 11(3):48 51. Available from: http://web.ebscohost.com.ezproxy.lib.usf.edu/ehost/pdfviewer/pdfviewer?vid=15 &hid=112&sid=9b6fe66c fc37 4a74 b3bf 2795244d82f2%40sessionmgr112

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58 Wares, A. Using Origami to Explore Concepts of Geometry and Calculus. International Journal of Mathematical Education in Science and Technology [Internet]. 2011. [cited 2012 Feb 21]; 42(2): 264 272. Available from: http://www.tandfonline.com/doi/ab s/10.1080/0020739X.2010.519797 Effect or Origami Exercises. The Journal or Genetic Psychology [Internet]. 2002 [cited 2012 Feb 21]; 163(4): 459 478. Available from: http://web.ebscohost.com/ehost/detail?vid=4&hid=107&sid=94b1ab0b 6747 4b5b a3 f1 79a3c9163854%40sessionmgr14&bdata=JnNpdGU9ZWhvc3QtbGl2ZQ%3d%3d #db=psyh&AN=2005 07300 007 Zhang, S. Fourteen Homogeneity of Variance Tests: When and How To Use Them. [Online] 1998 [cited 2012 March 12]. Available from: http://eric.ed.gov/PDFS/ ED422392.pdf Useful Websites http://www.mathsisfun.com/geometry/polygons.htm http://www2.scholastic.com/browse/article.jsp?id=3597 Images from http://www.origami resource center.com/images/EasyOrigamiMontrollSwan.jpg http://www.oneinchpunch.net/wordpress/wp content/uploads/2007/04/origami1.jpg

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59 The Hexahedron model is designed by Molly Kahn and can be found on page 101 of The Art of Origami by Gay Merrill Gross Copyright 1993 by Michael Friedman Publishing Group, Inc.

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Appendix A 60 Appendix A IRB Approval Location Re Creation Location Approval Consent Form Child Assent Form

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Appendix A 61 IRB Approval I received IRB approval on 10/29/2011 which expires on 10/29/12. My IRB number is 11 024 Location Re Creation Site for study

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Appendix A 62 Location Approval Dear Jeanne Viviani: It is my understanding that Gina Fawks will be conducting a research study at the Roy McBean Boys & Girls Club on youth mathematics and origami. Ms. Fawks has informed our staff of the design of the study as well as the targeted population. I support this effort and will provide any assistance necessary for the successful implementation of this study. If you have any questions, please do not hesitate to con tact me. Best, Yvonne Zaccone Volunteer and Outcomes Specialist

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Appendix A 63 Consent Letter and Form 11/7/11 Dear Parent or Guardian: I am a student at New College of Florida. I am conducting a research project on using origami to teach math. I am wri ting to request permission for your child to participate. The study consists of each student taking a short pre test, then some doing worksheets and some doing origami, then each student will take a post test. The project will be explained in terms that your child can understand, and your child will participate only if he or she is willing to do so. Only I will have access to information from your child. At study will be made available in the thesis section of the New College of Florida library. Participation in this study is voluntary. Your decision whether or not to allow your child to participate will not affect the services normally provided to your child by the Roy loss of any benefits to which he or she is otherwise entitled. Even if you give your permission for your child to participate, your child is free to refuse to pa rticipate. If your child agrees to participate, he or she is free to end participation at any time. You and your participation in this research study. Should you have any questions or desire further information, please call me or email me at 941 302 0279 or Gina.Fawks@ncf.edu You may also contact my thesis sponsor, Dr. Gilchrist, at 941 487 4377 or Gilchrist@ncf.edu Keep a copy of this letter for your records. If you have any questions about your rights as a research subject, you may contact the New College I nstitutional Review Board (IRB) by mail at Research Programs & Services New College of Florida, COH 214, 5800 Bay Shore Road, Sarasota, FL 34243, by phone contact Jeanne Viviani at ( 941) 487 4649 or by e mail at irb@ncf.edu. This study (IRB 11 024 ) was approved by the IRB on 10/29/11. Sincerely, Gina Fawks New College of Florida

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Appendix A 64 Consent Form Please indicate whether or not you wish to allow your child to participate in this project by checking one of the statements below, signing your name and handing it back to me (Gina Fawks). Sign both copies and keep one for your records. _____ n Informal Math Education. _____ Informal Math Education __ ________________________ ______________________________ Signature of Parent/Guardia n Printed Parent/Guardian Name ____ ______________________ ________________ _______ _________ Printed Name of Child Date _____________________________ __ Gina F awks _________________ Si gnature of Principal Investigator Printed Name or Principal Investigator

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Appendix A 65 Child Assent You are being asked to participate in Gina asked to complete some worksheets and participate in a few les sons. This is completely voluntary and will not affect the help you get on your homework at the Boys and Girls Club. Feel free to ask Miss Gina if you have any questions or concerns. Please sign your name below if you would like to help with this projec t. ____________________ ___________ _________________________________ ____ Age ___________________ ______ _______ Gina Fawks_______ ______________

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Appendix B Lesson Plan Hexahedron Diagram Whale and Twist Fish Diagrams Example Whale Lesson Plan Example Concept Breakdown Chart Select Florida Sunshine Standards

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A ppendix B 67 Lesson Plan s Origami Consent forms Pre test Vocab: square, pentagon, triangle, quadrilateral, hexagon, line of symmetry, polygon, trapezoid, congruent, rhombus, regular and irregular shapes, polyhedron, hexahedron, geometry, o rigami, Concepts: multiplication and division, adding and subtracting, reasonableness, counting edges, faces and vertices, counting (triangles), pairs (triangles), fractions (1/2), area of square and triangle, fractions (using colors/ out of 3) Whale V ocab: quadrilateral, polygon, triangle, line of symmetry, congruent, regular and irregular shapes, pentagon Concepts: counting edges and vertices, counting (triangles), pairs (triangles), fractions (1/2) Hexahedron Vocab: hexahedron, rhombus, squar e, polyhedron, line of symmetry Concepts: multiplication and division, adding and subtracting, counting edges, faces and vertices, area of square and triangle, counting (triangles), fractions (using colors/ out of 3) Twist Fish Vocab: trapezoid, hexa gon, quadrilateral, pentagon Concepts: multiplication and division, adding and subtracting, reasonableness, counting edges and vertices Post Test (same vocab and concepts as pre test) Origami Post thesis Survey Control Consent forms Pre test Vocab: square, pentagon, triangle, quadrilateral, hexagon, line of symmetry, polygon, trapezoid, congruent, rhombus, regular and irregular shapes, polyhedron, hexahedron, geometry, origami Concepts: multiplication and division, adding and subtracting, reasonable ness, counting edges, faces and vertices, counting (triangles), pairs (triangles), fractions (1/2), area of square and triangle, fractions (using colors/ out of 3) Worksheet A Vocab: quadrilateral, polygon, triangle, line of symmetry, congruent, regular and irregular shapes Concepts: counting edges and vertices, counting (triangles), pairs (triangles), fractions (1/2)

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A ppendix B 68 Worksheet B Vocab: hexahedron, rhombus, square, polyhedron, line of symmetry Concepts: multiplication and division, adding and sub tracting, counting edges, faces and vertices, area of square and triangle, counting (triangles), fractions (using colors/ out of 3) Worksheet C Vocab: trapezoid, hexagon, quadrilateral, pentagon Concepts: multiplication and division, adding and subt racting, reasonableness, counting edges and vertices Post Test (same vocab and concepts as pre test) Post thesis Survey Alternate Origami Diagrams (if the children need something easier/harder or I feel like I need more: journal Vocab: rectangle quadrilateral Concepts: counting, fractions, line of symmetry, counting edges and vertices Cube Vocab: Cube, polyhedron, polygon Concepts: count edges and vertices, count faces, fractions (using colors/ out of 2, 3, 6) Vocab: rec tangle, quadrilateral, triangle, pentagon, regular and irregular shapes Concepts: line of symmetry, counting, area of a square and triangle Dog/cat Vocab: triangle, congruent, pentagon, hexagon, polygon concepts: Counting, count vertices and edges Sailboat Vocab: triangle, trapezoid, quadrilateral, square, congruent Concepts: counting, count edges and vertices

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A ppendix B 69 Diagrams

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A ppendix B 70

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A ppendix B 71

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A ppendix B 72

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A ppendix B 73

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A ppendix B 74 Example Concept Breakdown Chart Grade Concepts Student 11 12 13 14 K Represent numbers up to 20 Times seen Pre Test Score Post Test Score Appropriate Grade Level? Identify squares, triangles, circles, rectangles, hexagons, and trapezoid Times seen Pre Test Score Post Test Score Appropriate Grade Level? Counting Times seen Pre Test Score Post Test Score Appropriate Grade Level? 1 st Addition and subtraction Times seen Pre Test Score Post Test Score Appropriate Grade Level? Quadrilateral Time s seen Pre Test Score Post Test Score Appropriate Grade Level? 2nd Multiplication and division Times seen Pre Test Score Post Test Score Appropriate Grade Level? Congruence Times seen Pre Test Sco re Post Test Score Appropriate Grade Level? Fractions Times seen Pre Test Score Post Test Score Appropriate Grade

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A ppendix B 75 Level? 4 th Area of a rectangle/square Times seen Pre Test Score Post Test Score Appropriate Grade Level? 5th Reasonableness problems Times seen Pre Test Score Post Test Score Appropriate Grade Level? Polyhedron Times seen Pre Test Score Post Test Score Appropriate Grade Level? Counting Faces, Edges and Vertices Times seen Pre Test Score Post Test Score Appropriate Grade Level? Area of Triangle Times seen Pre Test Score Post Test Score Appropriate Grade Level? Other Irreg ular or Regular shapes and line of symmetry Times seen Pre Test Score Post Test Score Appropriate Grade Level?

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A ppendix B 76 Florida Sunshine Standards (K 6) Mathematics Standards GRADE: K Big Idea 1: BIG IDEA 1 Represent, compare, an d order whole numbers and join and separate sets. BENCHMARK CODE BENCHMARK MA.K.A.1.1 Represent quantities with numbers up to 20, verbally, in writing, and with manipulatives. Cognitive Complexity/Depth of Knowledge Rating: Moderate MA.K.A.1.2 Solve problems including those involving sets by counting, by using cardinal and ordinal numbers, by comparing, by ordering, and by creating sets up to 20. Cognitive Complexity/Depth of Knowledge Rating: Moderate MA.K.A.1.3 Solve word problems involving sim ple joining and separating situations. Cognitive Complexity/Depth of Knowledge Rating: High Big Idea 2: BIG IDEA 2 Describe shapes and space. BENCHMARK CODE BENCHMARK MA.K.G.2.1 Describe, sort and re sort objects using a variety of attributes such a s shape, size, and position. Cognitive Complexity/Depth of Knowledge Rating: Moderate MA.K.G.2.2 Identify, name, describe and sort basic two dimensional shapes such as squares, triangles, circles, rectangles, hexagons, and trapezoids.

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A ppendix B 77 Cognitive Compl exity/Depth of Knowledge Rating: Moderate MA.K.G.2.3 Identify, name, describe, and sort three dimensional shapes such as spheres, cubes and cylinders. Cognitive Complexity/Depth of Knowledge Rating: Moderate MA.K.G.2.4 Interpret the physical world wi th geometric shapes, and describe it with corresponding vocabulary. Cognitive Complexity/Depth of Knowledge Rating: Moderate MA.K.G.2.5 Use basic shapes, spatial reasoning, and manipulatives to model objects in the environment and to construct more co mplex shapes. Cognitive Complexity/Depth of Knowledge Rating: High Big Idea 3: BIG IDEA 3 Order objects by measurable attributes. BENCHMARK CODE BENCHMARK MA.K.G.3.1 Compare and order objects indirectly or directly using measurable attributes such a s length, height, and weight. Cognitive Complexity/Depth of Knowledge Rating: Moderate Supporting Idea 4: Algebra Algebra BENCHMARK CODE BENCHMARK MA.K.A.4.1 Identify and duplicate simple number and non numeric repeating and growing patterns. Cogn itive Complexity/Depth of Knowledge Rating: Moderate

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A ppendix B 78 Supporting Idea 5: Geometry and Measurement Geometry and Measurement BENCHMARK CODE BENCHMARK MA.K.G.5.1 Demonstrate an understanding of the concept of time using identifiers such as morning, aftern oon, day, week, month, year, before/after, shorter/longer. Cognitive Complexity/Depth of Knowledge Rating: Moderate GRADE: 1 Big Idea 1: BIG IDEA 1 Develop understandings of addition and subtraction strategies for basic addition facts and related sub traction facts. BENCHMARK CODE BENCHMARK MA.1.A.1.1 Model addition and subtraction situations using the concepts of "part whole," "adding to," "taking away from," "comparing," and missing addend." Cognitive Complexity/Depth of Knowledge Rating: Modera te MA.1.A.1.2 Identify, describe, and apply addition and subtraction as inverse operations. Cognitive Complexity/Depth of Knowledge Rating: Moderate MA.1.A.1.3 Create and use increasingly sophisticated strategies, and use properties such as Commutati ve, Associative and Additive Identity, to add whole numbers. Cognitive Complexity/Depth of Knowledge Rating: Moderate MA.1.A.1.4 Use counting strategies, number patterns, and models as a means for solving basic addition and subtraction fact problems. Cognitive Complexity/Depth of Knowledge Rating: High

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A ppendix B 79 Big Idea 2: BIG IDEA 2 Develop an understanding of whole number relationships, including grouping by tens and ones. BENCHMARK CODE BENCHMARK MA.1.A.2.1 Compare and order whole numbers at least to 1 00. Cognitive Complexity/Depth of Knowledge Rating: Moderate MA.1.A.2.2 Represent two digit numbers in terms of tens and ones. Cognitive Complexity/Depth of Knowledge Rating: Low MA.1.A.2.3 Order counting numbers, compare their relative magnitudes, and represent numbers on a number line. Cognitive Complexity/Depth of Knowledge Rating: Moderate Big Idea 3: BIG IDEA 3 Compose and decompose two dimensional and three dimensional geometric shapes. BENCHMARK CODE BENCHMARK MA.1.G.3.1 Use appropriat e vocabulary to compare shapes according to attributes and properties such as number and lengths of sides and number of vertices. Cognitive Complexity/Depth of Knowledge Rating: Moderate MA.1.G.3.2 Compose and decompose plane and solid figures, includ ing making predictions about them, to build an understanding of part whole relationships and properties of shapes. Cognitive Complexity/Depth of Knowledge Rating: High Supporting Idea 4: Algebra Algebra

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A ppendix B 80 BENCHMARK CODE BENCHMARK MA.1.A.4.1 Extend re peating and growing patterns, fill in missing terms, and justify reasoning. Cognitive Complexity/Depth of Knowledge Rating: High Supporting Idea 5: Geometry and Measurement Geometry and Measurement BENCHMARK CODE BENCHMARK MA.1.G.5.1 Measure by usin g iterations of a unit, and count the unit measures by grouping units. Cognitive Complexity/Depth of Knowledge Rating: Moderate MA.1.G.5.2 Compare and order objects according to descriptors of length, weight, and capacity. Cognitive Complexity/Dept h of Knowledge Rating: Moderate Supporting Idea 6: Number and Operations Number and Operations BENCHMARK CODE BENCHMARK MA.1.A.6.1 Use mathematical reasoning and beginning understanding of tens and ones, including the use of invented strategies, to so lve two digit addition and subtraction problems. Cognitive Complexity/Depth of Knowledge Rating: High MA.1.A.6.2 Solve routine and non routine problems by acting them out, using manipulatives, and drawing diagrams. Cognitive Complexity/Depth of Knowl edge Rating: High

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A ppendix B 81 GRADE: 2 Big Idea 1: BIG IDEA 1 Develop an understanding of base ten numerations system and place value concepts. BENCHMARK CODE BENCHMARK MA.2.A.1.1 Identify relationships between the digits and their place values through the thou sands, including counting by tens and hundreds. Cognitive Complexity/Depth of Knowledge Rating: Moderate MA.2.A.1.2 Identify and name numbers through thousands in terms of place value, and apply this knowledge to expanded notation. Cognitive Complex ity/Depth of Knowledge Rating: Low MA.2.A.1.3 Compare and order multi digit numbers through the thousands. Cognitive Complexity/Depth of Knowledge Rating: Moderate Big Idea 2: BIG IDEA 2 Develop quick recall of addition facts and related subtraction facts and fluency with multi digit addition and subtraction. BENCHMARK CODE BENCHMARK MA.2.A.2.1 Recall basic addition and related subtraction facts. Cognitive Complexity/Depth of Knowledge Rating: Low MA.2.A.2.2 Add and subtract multi digit whole n umbers through three digits with fluency by using a variety of strategies, including invented and standard algorithms and explanations of those procedures. Cognitive Complexity/Depth of Knowledge Rating: Moderate MA.2.A.2.3 Estimate solutions to multi digit addition and subtraction problems through three digits.

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A ppendix B 82 Cognitive Complexity/Depth of Knowledge Rating: Moderate MA.2.A.2.4 Solve addition and subtraction problems that involve measurement and geometry. Cognitive Complexity/Depth of Knowledge R ating: High Big Idea 3: BIG IDEA 3 Develop an understanding of linear measurement and facility in measuring lengths. BENCHMARK CODE BENCHMARK MA.2.G.3.1 Estimate and use standard units, including inches and centimeters, to partition and measure length s of objects. Cognitive Complexity/Depth of Knowledge Rating: Moderate MA.2.G.3.2 Describe the inverse relationship between the size of a unit and number of units needed to measure a given object. Cognitive Complexity/Depth of Knowledge Rating: Moder ate MA.2.G.3.3 Apply the Transitive Property when comparing lengths of objects. Cognitive Complexity/Depth of Knowledge Rating: Moderate MA.2.G.3.4 Estimate, select an appropriate tool, measure, and/or compute lengths to solve problems. Cognitive C omplexity/Depth of Knowledge Rating: High Supporting Idea 4: Algebra Algebra BENCHMARK CODE BENCHMARK MA.2.A.4.1 Extend number patterns to build a foundation for understanding multiples and factors for example, skip counting by 2's, 5's, 10's.

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A ppendix B 83 Cog nitive Complexity/Depth of Knowledge Rating: Moderate MA.2.A.4.2 Classify numbers as odd or even and explain why. Cognitive Complexity/Depth of Knowledge Rating: Moderate MA.2.A.4.3 Generalize numeric and non numeric patterns using words and tables. Cognitive Complexity/Depth of Knowledge Rating: High MA.2.A.4.4 Describe and apply equality to solve problems, such as in balancing situations. Cognitive Complexity/Depth of Knowledge Rating: High MA.2.A.4.5 Recognize and state rules for function s that use addition and subtraction. Cognitive Complexity/Depth of Knowledge Rating: High Supporting Idea 5: Geometry and Measurement Geometry and Measurement BENCHMARK CODE BENCHMARK MA.2.G.5.1 Use geometric models to demonstrate the relationships between wholes and their parts as a foundation to fractions. Cognitive Complexity/Depth of Knowledge Rating: Moderate MA.2.G.5.2 Identify time to the nearest hour and half hour. Cognitive Complexity/Depth of Knowledge Rating: Low MA.2.G.5.3 Identi fy, combine, and compare values of money in cents up to $1 and in dollars up to $100, working with a single unit of currency. Cognitive Complexity/Depth of Knowledge Rating: Moderate MA.2.G.5.4 Measure weight/mass and capacity/volume of objects. Includ e the use of the appropriate unit of measure and their abbreviations including cups, pints, quarts, gallons, ounces (oz), pounds (lbs), grams (g), kilograms (kg), milliliters (mL) and liters (L).

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A ppendix B 84 Cognitive Complexity/Depth of Knowledge Rating: Low Supp orting Idea 6: Number and Operations Number and Operations BENCHMARK CODE BENCHMARK MA.2.A.6.1 Solve problems that involve repeated addition. Cognitive Complexity/Depth of Knowledge Rating: Moderate GRADE: 3 Big Idea 1: BIG IDEA 1 Develop understa ndings of multiplication and division and strategies for basic multiplication facts and related division facts. BENCHMARK CODE BENCHMARK MA.3.A.1.1 Model multiplication and division including problems presented in context: repeated addition, multiplicat ive comparison, array, how many combinations, measurement, and partitioning. Cognitive Complexity/Depth of Knowledge Rating: Moderate MA.3.A.1.2 Solve multiplication and division fact problems by using strategies that result from applying number prope rties. Cognitive Complexity/Depth of Knowledge Rating: High MA.3.A.1.3 Identify, describe, and apply division and multiplication as inverse operations. Cognitive Complexity/Depth of Knowledge Rating: Moderate Big Idea 2: BIG IDEA 2

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A ppendix B 85 Develop an unde rstanding of fractions and fraction equivalence. BENCHMARK CODE BENCHMARK MA.3.A.2.1 Represent fractions, including fractions greater than one, using area, set, and linear models. Cognitive Complexity/Depth of Knowledge Rating: Moderate MA.3.A.2.2 D escribe how the size of the fractional part is related to the number of equal sized pieces in the whole. Cognitive Complexity/Depth of Knowledge Rating: Moderate MA.3.A.2.3 Compare and order fractions, including fractions greater than one, using models and strategies. Cognitive Complexity/Depth of Knowledge Rating: Moderate MA.3.A.2.4 Use models to represent equivalent fractions, including fractions greater than 1, and identify representations of equivalence. Cognitive Complexity/Depth of Knowled ge Rating: Moderate Big Idea 3: BIG IDEA 3 Describe and analyze properties of two dimensional shapes. BENCHMARK CODE BENCHMARK MA.3.G.3.1 Describe, analyze, compare, and classify two dimensional shapes using sides and angles including acute, obtuse, and right angles and connect these ideas to the definition of shapes. Cognitive Complexity/Depth of Knowledge Rating: Moderate MA.3.G.3.2 Compose, decompose, and transform polygons to make other polygons, including concave and convex polygons with three, four, five, six, eight, or ten sides. Cognitive Complexity/Depth of Knowledge Rating: High MA.3.G.3.3 Build, draw, and analyze two dimensional shapes from several orientations in order to examine and apply congruence and symmetry.

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A ppendix B 86 Cognitive Complexity/Depth of Knowledge Rating: Moderate Supporting Idea 4: Algebra Algebra BENCHMARK CODE BENCHMARK MA.3.A.4.1 Create, analyze, and represent patterns and relationships using words, variables, tables, and graphs. Cognitive Complexity/Depth o f Knowledge Rating: High Supporting Idea 5: Geometry and Measurement Geometry and Measurement BENCHMARK CODE BENCHMARK MA.3.G.5.1 Select appropriate units, strategies, and tools to solve problems involving perimeter. Cognitive Complexity/Depth of K nowledge Rating: High MA.3.G.5.2 Measure objects using fractional parts of linear units such as 1/2, 1/4, and 1/10. Cognitive Complexity/Depth of Knowledge Rating: Low MA.3.G.5.3 Tell time to the nearest minute and to the nearest quarter hour, and d etermine the amount of time elapsed. Cognitive Complexity/Depth of Knowledge Rating: Moderate Supporting Idea 6: Number and Operations Number and Operations BENCHMARK CODE BENCHMARK MA.3.A.6.1 Represent, compute, estimate, and solve problems using numbers through hundred thousands.

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A ppendix B 87 Cognitive Complexity/Depth of Knowledge Rating: High MA.3.A.6.2 Solve non routine problems by making a table, chart ,or list and searching for patterns. Cognitive Complexity/Depth of Knowledge Rating: High Suppo rting Idea 7: Data Analysis Data Analysis BENCHMARK CODE BENCHMARK MA.3.S.7.1 Construct and analyze frequency tables, bar graphs, pictographs, and line plots from data, including data collected through observations, surveys, and experiments. Cognitiv e Complexity/Depth of Knowledge Rating: High GRADE: 4 Big Idea 1: BIG IDEA 1 Develop quick recall of multiplication facts and related division facts and fluency with whole number multiplication. BENCHMARK CODE BENCHMARK MA.4.A.1.1 Use and describe v arious models for multiplication in problem solving situations, and demonstrate recall of basic multiplication and related division facts with ease. Cognitive Complexity/Depth of Knowledge Rating: Moderate MA.4.A.1.2 Multiply multi digit whole numbers through four digits fluently, demonstrating understanding of the standard algorithm, and checking for reasonableness of results, including solving real world problems. Cognitive Complexity/Depth of Knowledge Rating: High

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A ppendix B 88 Big Idea 2: BIG IDEA 2 Develop an understanding of decimals, including the connection between fractions and decimals. BENCHMARK CODE BENCHMARK MA.4.A.2.1 Use decimals through the thousandths place to name numbers between whole numbers. Cognitive Complexity/Depth of Knowledge Ratin g: Low MA.4.A.2.2 Describe decimals as an extension of the base ten number system. Cognitive Complexity/Depth of Knowledge Rating: High MA.4.A.2.3 Relate equivalent fractions and decimals with and without models, including locations on a number line. Cognitive Complexity/Depth of Knowledge Rating: Moderate MA.4.A.2.4 Compare and order decimals, and estimate fraction and decimal amounts in real world problems. Cognitive Complexity/Depth of Knowledge Rating: Moderate Big Idea 3: BIG IDEA 3 Dev elop an understanding of area and determine the area of two dimensional shapes. BENCHMARK CODE BENCHMARK MA.4.G.3.1 Describe and determine area as the number of same sized units that cover a region in the plane, recognizing that a unit square is the sta ndard unit for measuring area. Cognitive Complexity/Depth of Knowledge Rating: Moderate MA.4.G.3.2 Justify the formula for the area of the rectangle "area = base x height". Cognitive Complexity/Depth of Knowledge Rating: Moderate MA.4.G.3.3 Selec t and use appropriate units, both customary and metric, strategies, and measuring tools to estimate and solve real world area problems.

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A ppendix B 89 Cognitive Complexity/Depth of Knowledge Rating: Moderate Supporting Idea 4: Algebra Algebra BENCHMARK CODE BENCHMA RK MA.4.A.4.1 Generate algebraic rules and use all four operations to describe patterns, including nonnumeric growing or repeating patterns. Cognitive Complexity/Depth of Knowledge Rating: High MA.4.A.4.2 Describe mathematics relationships using exp ressions, equations, and visual representations. Cognitive Complexity/Depth of Knowledge Rating: High MA.4.A.4.3 Recognize and write algebraic expressions for functions with two operations. Cognitive Complexity/Depth of Knowledge Rating: High Sup porting Idea 5: Geometry and Measurement Geometry and Measurement BENCHMARK CODE BENCHMARK MA.4.G.5.1 Classify angles of two dimensional shapes using benchmark angles (45, 90, 180, and 360) Cognitive Complexity/Depth of Knowledge Rating: Low MA .4.G.5.2 Identify and describe the results of translations, reflections, and rotations of 45, 90, 180, 270, and 360 degrees, including figures with line and rotational symmetry. Cognitive Complexity/Depth of Knowledge Rating: Moderate MA.4.G.5.3 Ident ify and build a three dimensional object from a two dimensional representation of that object and vice versa.

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A ppendix B 90 Cognitive Complexity/Depth of Knowledge Rating: Moderate Supporting Idea 6: Number and Operations Number and Operations BENCHMARK CODE BENCH MARK MA.4.A.6.1 Use and represent numbers through millions in various contexts, including estimation of relative sizes of amounts or distances. Cognitive Complexity/Depth of Knowledge Rating: Moderate MA.4.A.6.2 Use models to represent division as: the inverse of multiplication as partitioning as successive subtraction Cognitive Complexity/Depth of Knowledge Rating: Moderate MA.4.A.6.3 Generate equivalent fractions and simplify fractions. Cognitive Complexity/Depth of Knowledge Rating: Mod erate MA.4.A.6.4 Determine factors and multiples for specified whole numbers. Cognitive Complexity/Depth of Knowledge Rating: Moderate MA.4.A.6.5 Relate halves, fourths, tenths, and hundredths to decimals and percents. Cognitive Complexity/Depth of Knowledge Rating: Moderate MA.4.A.6.6 Estimate and describe reasonableness of estimates; determine the appropriateness of an estimate versus an exact answer. Cognitive Complexity/Depth of Knowledge Rating: High

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A ppendix B 91 GRADE: 5 Big Idea 1: BIG IDEA 1 D evelop an understanding of and fluency with division of whole numbers. BENCHMARK CODE BENCHMARK MA.5.A.1.1 Describe the process of finding quotients involving multi digit dividends using models, place value, properties, and the relationship of division to multiplication. Cognitive Complexity/Depth of Knowledge Rating: Moderate MA.5.A.1.2 Estimate quotients or calculate them mentally depending on the context and numbers involved. Cognitive Complexity/Depth of Knowledge Rating: Moderate MA.5.A.1. 3 Interpret solutions to division situations including those with remainders depending on the context of the problem. Cognitive Complexity/Depth of Knowledge Rating: High MA.5.A.1.4 Divide multi digit whole numbers fluently, including solving real worl d problems, demonstrating understanding of the standard algorithm and checking the reasonableness of results. Cognitive Complexity/Depth of Knowledge Rating: High Big Idea 2: BIG IDEA 2 Develop an understanding of and fluency with addition and subtra ction of fractions and decimals. BENCHMARK CODE BENCHMARK MA.5.A.2.1 Represent addition and subtraction of decimals and fractions with like and unlike denominators using models, place value, or properties. Cognitive Complexity/Depth of Knowledge Rati ng: Moderate MA.5.A.2.2 Add and subtract fractions and decimals fluently, and verify the reasonableness of results, including in problem situations.

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A ppendix B 92 Cognitive Complexity/Depth of Knowledge Rating: Moderate MA.5.A.2.3 Make reasonable estimates of fra ction and decimal sums and differences, and use techniques for rounding. Cognitive Complexity/Depth of Knowledge Rating: Moderate MA.5.A.2.4 Determine the prime factorization of numbers. Cognitive Complexity/Depth of Knowledge Rating: Moderate Bi g Idea 3: BIG IDEA 3 Describe three dimensional shapes and analyze their properties, including volume and surface area. BENCHMARK CODE BENCHMARK MA.5.G.3.1 Analyze and compare the properties of two dimensional figures and three dimensional solids (polyh edra), including the number of edges, faces, vertices, and types of faces. Cognitive Complexity/Depth of Knowledge Rating: High MA.5.G.3.2 Describe, define, and determine surface area and volume of prisms by using appropriate units and selecting strat egies and tools. Cognitive Complexity/Depth of Knowledge Rating: High Supporting Idea 4: Algebra Algebra BENCHMARK CODE BENCHMARK MA.5.A.4.1 Use the properties of equality to solve numerical and real world situations. Cognitive Complexity/Depth of Knowledge Rating: Moderate MA.5.A.4.2 Construct and describe a graph showing continuous data, such as a graph of a quantity that changes over time.

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A ppendix B 93 Cognitive Complexity/Depth of Knowledge Rating: High Supporting Idea 5: Geometry and Measurement G eometry and Measurement BENCHMARK CODE BENCHMARK MA.5.G.5.1 Identify and plot ordered pairs on the first quadrant of the coordinate plane. Cognitive Complexity/Depth of Knowledge Rating: Low MA.5.G.5.2 Compare, contrast, and convert units of measur e within the same dimension (length, mass, or time) to solve problems. Cognitive Complexity/Depth of Knowledge Rating: Moderate MA.5.G.5.3 Solve problems requiring attention to approximation, selection of appropriate measuring tools, and precision of measurement. Cognitive Complexity/Depth of Knowledge Rating: High MA.5.G.5.4 Derive and apply formulas for areas of parallelograms, triangles, and trapezoids from the area of a rectangle. Cognitive Complexity/Depth of Knowledge Rating: High Suppo rting Idea 6: Number and Operations Number and Operations BENCHMARK CODE BENCHMARK MA.5.A.6.1 Identify and relate prime and composite numbers, factors, and multiples within the context of fractions. Cognitive Complexity/Depth of Knowledge Rating: Mod erate MA.5.A.6.2 Use the order of operations to simplify expressions which include exponents and parentheses.

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A ppendix B 94 Cognitive Complexity/Depth of Knowledge Rating: Moderate MA.5.A.6.3 Describe real world situations using positive and negative numbers. Cognitive Complexity/Depth of Knowledge Rating: Moderate MA.5.A.6.4 Compare, order, and graph integers, including integers shown on a number line. Cognitive Complexity/Depth of Knowledge Rating: Moderate MA.5.A.6.5 Solve non routine problems using v Cognitive Complexity/Depth of Knowledge Rating: High Supporting Idea 7: Data Analysis Data Analysis BENCHMARK CODE BENCHMARK MA.5.S.7.1 Construct and analyze li ne graphs and double bar graphs. Cognitive Complexity/Depth of Knowledge Rating: High MA.5.S.7.2 Differentiate between continuous and discrete data, and determine ways to represent those using graphs and diagrams. Cognitive Complexity/Depth of Know ledge Rating: Moderate GRADE: 6 Big Idea 1: BIG IDEA 1 Develop an understanding of and fluency with multiplication and division of fractions and decimals. BENCHMARK CODE BENCHMARK MA.6.A.1.1 Explain and justify procedures for multiplying and dividin g fractions and decimals.

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A ppendix B 95 Cognitive Complexity/Depth of Knowledge Rating: Moderate MA.6.A.1.2 Multiply and divide fractions and decimals efficiently. Cognitive Complexity/Depth of Knowledge Rating: Low MA.6.A.1.3 Solve real world problems involvin g multiplication and division of fractions and decimals. Cognitive Complexity/Depth of Knowledge Rating: High Big Idea 2: BIG IDEA 2 Connect ratio and rates to multiplication and division. BENCHMARK CODE BENCHMARK MA.6.A.2.1 Use reasoning about mul tiplication and division to solve ratio and rate problems. Cognitive Complexity/Depth of Knowledge Rating: High MA.6.A.2.2 Interpret and compare ratios and rates. Cognitive Complexity/Depth of Knowledge Rating: Moderate Big Idea 3: BIG IDEA 3 Wri te, interpret, and use mathematical expressions and equations. BENCHMARK CODE BENCHMARK MA.6.A.3.1 Write and evaluate mathematical expressions that correspond to given situations. Cognitive Complexity/Depth of Knowledge Rating: Moderate MA.6.A.3.2 Write, solve, and graph one and two step linear equations and inequalities. Cognitive Complexity/Depth of Knowledge Rating: Moderate MA.6.A.3.3 Work backward with two step function rules to undo expressions.

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A ppendix B 96 Cognitive Complexity/Depth of Knowledg e Rating: Moderate MA.6.A.3.4 Solve problems given a formula. Cognitive Complexity/Depth of Knowledge Rating: Moderate MA.6.A.3.5 Apply the Commutative, Associative, and Distributive Properties to show that two expressions are equivalent. Cognitiv e Complexity/Depth of Knowledge Rating: Moderate MA.6.A.3.6 Construct and analyze tables, graphs, and equations to describe linear functions and other simple relations using both common language and algebraic notation. Cognitive Complexity/Depth of Kn owledge Rating: High Supporting Idea 4: Geometry and Measurement Geometry and Measurement BENCHMARK CODE BENCHMARK MA.6.G.4.1 Understand the concept of Pi, know common estimates of Pi (3.14; 22/7) and use these values to estimate and calculate the cir cumference and the area of circles. Cognitive Complexity/Depth of Knowledge Rating: Moderate MA.6.G.4.2 Find the perimeters and areas of composite two dimensional figures, including non rectangular figures (such as semicircles) using various strategie s. Cognitive Complexity/Depth of Knowledge Rating: Moderate MA.6.G.4.3 Determine a missing dimension of a plane figure or prism given its area or volume and some of the dimensions, or determine the area or volume given the dimensions. Cognitive Com plexity/Depth of Knowledge Rating: Moderate Supporting Idea 5: Number and Operations Number and Operations

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A ppendix B 97 BENCHMARK CODE BENCHMARK MA.6.A.5.1 Use equivalent forms of fractions, decimals, and percents to solve problems. Cognitive Complexity/Depth o f Knowledge Rating: Moderate MA.6.A.5.2 Compare and order fractions, decimals, and percents, including finding their approximate location on a number line. Cognitive Complexity/Depth of Knowledge Rating: Moderate MA.6.A.5.3 Estimate the results of c omputations with fractions, decimals, and percents, and judge the reasonableness of the results. Cognitive Complexity/Depth of Knowledge Rating: Moderate Supporting Idea 6: Data Analysis Data Analysis BENCHMARK CODE BENCHMARK MA.6.S.6.1 Determine t he measures of central tendency (mean, median, mode) and variability (range) for a given set of data. Cognitive Complexity/Depth of Knowledge Rating: Low MA.6.S.6.2 Select and analyze the measures of central tendency or variability to represent, descr ibe, analyze, and/or summarize a data set for the purposes of answering questions appropriately. Cognitive Complexity/Depth of Knowledge Rating: High This report was generated by CPALMS www.floridastan dards.org

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Appendix C Appendix C Pre Test: Vocabulary Answer Key Test: Vocabulary Pre Test: Concepts Answer Key Test: Concepts Copy of Worksheet A Selected Copy of Worksheet B Copy of Worksheet C Post Test: Vocabulary Answer Key Test: Vocabulary Post T est: Concepts Answer Key Test: Concepts

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A ppendix C 128 Worksheet A A Triangle is a shape with 3 sides and 3 corners. 1) Draw an d color three different triangles in the space below. 2) Circle the triangle Count how many of each shape there are below 3) __________________ 4) __________________

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A ppendix C 129 A quadrilateral is a shape with 4 sides and 4 corners. 5) Draw and color three different quadrilaterals. 6) Circle the quadrilateral A poly gon is a flat shape with straight lines and has no openings (it is closed). 7) Draw and color three different polygons 8) Circle the polygon This is a polygon. It is flat It has straight lines It is closed This is NOT a polygon It has a curved edge This is NOT a polygon It is not closed It has a an opening

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A ppendix C 130 A line of symmetry is a line that you can make so that if you fold the image or object it is the same on both sides. It is like a reflection. Examples of lines of symmetry: 9) Draw a line of symmetry through the shapes below Regular shapes are when all of the sides are equal. Irregular shapes have some sides that are longer than others and some sides that are shorter than other. Examples: Regular Hexagon Irregular hexagon Regular Pentagon Irregular Pentagon 10) color the regular shapes purple and the irregular shapes red

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A ppendix C 131 A pair is when you group two objects together that have something in common or are similar. 11) Draw a line between the two shapes that make a pair. When two shapes are congruent they are equal. This means that they are both the same shape and size. 12) Draw two shapes that are congruent 13) Circle the shapes that are congruent

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A ppendix C 132 Edges are the si des of a shape. Vertices are the corners of a shape. So this triangle has 3 edges and 3 vertices 14) How many edges and vertices does the rectangle have? _______________ edges ________________ vertices 15) How many edges and vertices d oes the pentagon have? _______________ edges ________________ vertices A fraction is a part of a whole. When you write a fraction you write how many parts you have over the whole number of parts. Examples: 1 piece out of the two pieces that make up the pizza is pepperoni so, of the pizza is pepperoni What fraction of the shape below is colored in? 16) Edge Vertices

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A ppendix C 148 Worksheet B A square has 4 sides that are all equal in length. That means that all the sides are the same size. 1) Draw and color a square 2) Circle the square A rhombus is like a square turned on its side. It has four sides that are equal in length. 3) Draw and color a rhombus 4) Circle the rhombus Cou nt how many of each shape there are below 5) __________________

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A ppendix C 149 6) ____________ ______ When you add you bring two or more numbers (or things) together to make a total number (or amount of things). Example: 2 + 1 = 3 + = Do the following addition problems. Draw pictures if you need to. 7) 5 + 3 = _______________ 8) 4 + 2 = ________________ When you subtract you take one number away from the other. Example: 6 2 = 4 = Do the following sub traction problems. Draw pictures if you need to. 9) 5 1 = _______________ 10) 9 3 = ________________ Edges are the sides of a shape. Vertices are the corners of a shape. So this triangle has 3 edges and 3 vertices Edge Vertices

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A ppendix C 150 11) How many edges and verti ces does the rhombus have? _______________ edges ________________ vertices 12) How many edges and vertices does the arrow have? _____________ edges ________________ vertices Multiplication is grouping numbers in a way that is f aster than addition. Learn your multiplication facts! They are important. Examples: 2 x 8 = 16 (if you know addition add 2, 8 times: 2+2+2+2+2+2+2+2 =16) = Do the following multiplication probl ems. Draw pictures if you need to. 13) 1 x 9 = _______________ 14) 5 x 2 = ________________ Division is the opposite of multiplication. Division is much easier when you know your multiplication facts.

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A ppendix C 151 Example: 9 3 = 3 ( If you know multiplication: x 3 = 9 ?) = Do the following division problems. Draw pictures if you need to. 15) 10 5 =_______________ 16) 8 2 = __________________ A line of symmetry is a line that you can make so that if you fold the image or object it is the same on both sides. It is like a reflection. Examples of lines of symmetry: 17) Draw a line of symmetry through the shapes below A hexahedron is a 3D (or solid) shape that has 6 sides or faces. A cube is an exampl e of a hexahedron. 18) Circle the hexahedron

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A ppendix C 152 A polyhedron is a closed 3D (or solid) shape that has straight edges. 19) Circle the polyhedron The area of a square is base x height or the length of the side times the length of another side. Example: The area of this square is 2 x 2= 4 20) What is the area of the square? Area = _________________ This is a polyhedron. It is 3D (or solid) It has straight lines It is closed This is NOT a polyhedron It has a curved edge This is NOT a polyhedron It is flat 2 2 3 3

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A ppendix C 153 The area of a triangle is half the area of a square. The area of a triangle is x the base x the height. Example: 2 triangles can make a square so if the area of this square is 2 x 2= 4, then the area of the blue triangle is half of that or 4 2= 2. Area of blue triangle= 2 x 2 = 42 = 2 21) What is the area of the triangle? Area of triangle = _______________ A fraction is a part of a whole. When you write a fraction you write how many parts you have over the whole number of parts. Examples: 1 piece out of the four pie ces that make up the pizza is cheese so, of the pizza is cheese What fraction of the shape below is colored in? 22) 2 2 2 2 4 2

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A ppendix C 176 Worksheet C A quadrilateral is a shape with 4 sides and 4 corners. 17) Draw and color three different quadrilaterals. 18) Circle the quadrilateral A t rapezoid has four sides. Two of its sides are parallel. 19) Circle the trapezoid A pentagon is a closed shape with 5 sides 20) Draw and color two different pentagons 21) Circle the pentagon

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A ppendix C 177 A hexagon is a closed shape with 6 sides 22) Draw and col or two different hexagons 23) Circle the hexagon When you add you bring two or more numbers (or things) together to make a total number (or amount of things. Example: 1 + 2 = 3 + = Do the following addition problems. Draw pictures if you need to. 24) 2 + 3 = _______________ 25) 4 + 1 = ________________ When you subtract you take one number away from the other. Example: 5 3 = 2 = Do the following subtraction problems. Draw pictures if you need to. 26) 6 2 = _______________

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A ppendix C 178 27) 3 1 = ________________ Edges are the sides of a shape. Vertices are the corners of a shape. So this triangle has 3 edge s and 3 vertices 28) How many edges and vertices does the circle have? ____________ edges ___________ vertices 29) How many edges and vertices does the square have? ____________ edges ___________ vertices Multiplication is grouping numbers in a way that is faster than addition. Learn your multiplication facts! They are important. Examples: 3 x 5 = 15 (if you know addition, you can add 3, 5 times: 3+3+3+3+3=15) = = Vertices Edge

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A ppendix C 179 Do the following multiplication p roblems. Draw pictures if you need to. 30) 2 x 4 = _______________ 31) 3 x 2 = ________________ Division is the opposite of multiplication. Division is much easier when you know your multiplication facts. Example: 10 2 = 5 (If you know multiplica tion: x 2 = 10 ?) = Do the following division problems. Draw pictures if you need to. 32) 8 2 =_______________ 33) 12 4 = __________________ Answer the following question and make sure that your answer is re asonable. This means make sure that your answer makes sense. Use the space below to draw pictures or show work. 34) Tom has 13 toys. He is trying to move his toys from one room to the other with his wagon. His wagon will hold 3 toys each trip. How many t rips will he have to make to bring all of his toys into the other room. ____________ trips

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