Geometry And Spatial Sense, Grades 4 To 6 - EWorkshop

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Geometry andSpatial Sense,Grades 4 to 6A Guide to Effective Instructionin Mathematics,Kindergarten to Grade 62008

Geometry andSpatial Sense,Grades 4 to 6A Guide to Effective Instructionin Mathematics,Kindergarten to Grade 6

Every effort has been made in this publication to identify mathematics resources andtools (e.g., manipulatives) in generic terms. In cases where a particular product is usedby teachers in schools across Ontario, that product is identified by its trade name, in theinterests of clarity. Reference to particular products in no way implies an endorsement ofthose products by the Ministry of Education.

ContentsIntroduction5Working Towards Equitable Outcomes for Diverse Students.5Accommodations and Modifications.7The Mathematical Processes. 10Addressing the Needs of Junior Learners. 12The Big Ideas in Geometry and Spatial Sense14Overview. 14General Principles of Instruction. 16Levels of Geometric Thought. 18Properties of Two-Dimensional Shapes and Three-Dimensional Figures. 19Geometric Relationships. 25Location and Movement. 31Relating Mathematics Topics to the Big Ideas. 37Learning About Two-Dimensional Shapes in the Junior Grades38Introduction. 38Investigating Angle Properties. 40Investigating Congruence. 44Investigating Polygon Properties. 45Investigating Triangles and Quadrilaterals. 48Learning About Three-Dimensional Figures in the Junior Grades53Introduction. 53Properties of Prisms and Pyramids. 55Representing Three-Dimensional Figures in Two Dimensions. 58Learning About Location and Movement in the Junior Grades61Introduction. 61Grid and Coordinate Systems. 62Relationships in Transformational Geometry. 66Congruence, Orientation, and Distance. 69Relationships Between Transformational and Coordinate Geometry. 70

References73Learning Activities75Introduction to the Learning Activities. 77Grade 4 Learning Activities79Two-Dimensional Shapes: Comparing Angles. 79Three-Dimensional Figures: Construction Challenge. 91Location: Check Mate.108Movement: Hit the Target. 119Grade 5 Learning Activities129Two-Dimensional Shapes: Triangle Sort.129Three-Dimensional Figures: Package Possibilities.140Location: City Treasure Hunt.156Movement: Drawing Designs.168Grade 6 Learning Activities178Two-Dimensional Shapes: Connect the Dots. 178Three-Dimensional Figures: Sketching Climbing Structures. 191Location: Name My Shapes.213Movement: Logo Search and Design .223 Appendix: Guidelines for Assessment239Glossary243

IntroductionGeometry and Spatial Sense, Grades 4 to 6 is a practical guide that teachers will find useful inhelping students to achieve the curriculum expectations outlined for Grades 4 to 6 in theGeometry and Spatial Sense strand of The Ontario Curriculum, Grades 1–8: Mathematics, 2005.This guide provides teachers with practical applications of the principles and theories thatare elaborated in A Guide to Effective Instruction in Mathematics, Kindergarten to Grade 6, 2006.This guide provides: an overview of each of the three “big ideas”, or major mathematical themes, in theGeometry and Spatial Sense strand. The overview stresses the importance of focusing onthe big ideas in mathematical instruction to achieve the goal of helping students gain adeeper understanding of the mathematical concepts; three sections that focus on the important curriculum topics of two-dimensional shapes,three-dimensional figures, and location and movement. Each of these sections provides adiscussion of mathematical models and instructional strategies that have proved effectivein helping students understand the mathematical concepts related to the topics; sample learning activities for Grades 4, 5, and 6. These learning activities illustrate how alearning activity can be designed to:– focus on an important curriculum topic;– involve students in applying the seven mathematical processes described in themathematics curriculum document and reproduced on pages 10–11 of this document;– develop understanding of the big ideas in Geometry and Spatial Sense.This guide also contains a list of the references cited throughout the guide. At the end of theguide is an appendix that discusses assessment strategies for teachers. There is also a glossarythat includes mathematical and pedagogical terms used throughout the guide.Working Towards Equitable Outcomes for Diverse StudentsAll students, whatever their socio-economic, ethnocultural, or linguistic background, musthave opportunities to learn and to grow, both cognitively and socially. When students can

make personal connections to their learning, and when they feel secure in their learningenvironment, their true capacity will be realized in their achievement. A commitment toequity and inclusive instruction in Ontario classrooms is therefore critical to enabling allstudents to succeed in school and, consequently, to become productive and contributingmembers of society.To create effective conditions for learning, teachers must take care to avoid all forms ofbias and stereotyping in resources and learning activities, which can quickly alienatestudents and limit their learning. Teachers should be aware of the need to provide a varietyof experiences and to encourage multiple perspectives, so that the diversity of the class isrecognized and all students feel respected and valued. Learning activities and resources forteaching mathematics should be inclusive, providing examples and illustrations and usingapproaches that recognize the range of experiences of students with diverse backgrounds,knowledge, skills, interests, and learning styles.The following are some strategies for creating a learning environment that acknowledges andvalues the diversity of students and enables them to participate fully in the learning experience: providing mathematics problems with situations and contexts that are meaningful to allstudents (e.g., problems that reflect students’ interests, home-life experiences, and culturalbackgrounds and that arouse their curiosity and spirit of enquiry); using mathematics examples drawn from diverse cultures, including those ofAboriginal peoples; using children’s literature that reflects various cultures and customs as a source of mathematical examples and situations; understanding and acknowledging customs and adjusting teaching strategies as necessary.For example, a student may come from a culture in which it is considered inappropriate fora child to ask for help, express opinions openly, or make direct eye contact with an adult; considering the appropriateness of references to holidays, celebrations, and traditions; providing clarification if the context of a learning activity is unfamiliar to students (e.g.,describing or showing a food item that may be new to some students); evaluating the content of mathematics textbooks, children’s literature, and supplementarymaterials for cultural or gender bias; designing learning and assessment activities that allow students with various learningstyles (e.g., auditory, visual, tactile/kinaesthetic) to participate meaningfully; providing opportunities for students to work both independently and interdependentlywith others; Geometry and Spatial Sense, Grades 4 to 6

providing opportunities for students to communicate orally and in writing in theirhome language (e.g., pairing English language learners with a first-language peer whoalso speaks English); using diagrams, pictures, manipulatives, sounds, and gestures to clarify mathematicalvocabulary that may be new to English language learners.For a full discussion of equity and diversity in the classroom, as well as a detailed checklistfor providing inclusive mathematics instruction, see pages 34–40 in Volume 1 of A Guide toEffective Instruction in Mathematics, Kindergarten to Grade 6, 2006.An important aspect of inclusive instruction is accommodating students with special education needs. The following section discusses accommodations and modifications as theyrelate to mathematics instruction.Accommodations and ModificationsThe learning activities in this guide have been designed for students witha range of learning needs. Instructional and assessment tasks are openended, allowing most students to participate fully in learning experiences.In some cases, individual students may require accommodations and/ormodifications, in accordance with their Individual Education Plan (IEP), tosupport their participation in learning activities.PROVIDING ACCOMMODATIONSStudents may require accommodations, including special strategies,support, and/or equipment to allow them to participate in learning activities. There are three types of accommodations: Instructional accommodations are adjustments in teaching strategies,including styles of presentation, methods of organization, or the use oftechnology or multimedia. Environmental accommodations are supports or changes that the studentmay require in the physical environment of the classroom and/or theschool, such as preferential seating or special lighting. Assessment accommodations are adjustments in assessment activitiesand methods that enable the student to demonstrate learning, such asallowing additional time to complete tasks or permitting oral responses totest questions.The term accommodationsis used to refer to thespecial teaching andassessment strategies,human supports, and/orindividualized equipmentrequired to enable a studentto learn and to demonstratelearning. Accommodationsdo not alter the provincialcurriculum expectations forthe grade.Modifications are changesmade in the ageappropriate grade-levelexpectations for a subject. . . in order to meet astudent’s learning needs.These changes may involvedeveloping expectationsthat reflect knowledgeand skills required in thecurriculum for a differentgrade level and/orincreasing or decreasing thenumber and/or complexityof the regular grade-levelcurriculum expectations.(Ontario Ministry ofEducation, 2004,pp. 25–26)Introduction

Some of the ways in which teachers can provide accommodations with respect to mathematicslearning activities are listed in the following chart.Instructional Accommodations Vary instructional strategies, using different manipulatives, examples, and visuals (e.g., concretematerials, pictures, diagrams) as necessary to aid understanding. Rephrase information and instructions to make them simpler and clearer. Use non-verbal signals and gesture cues to convey information. Teach mathematical vocabulary explicitly. Have students work with a peer. Structure activities by breaking them into smaller steps. Model concepts using concrete materials, and encourage students to use them when learningconcepts or working on problems. Have students use calculators and/or addition and multiplication grids for computations. Format worksheets so that they are easy to understand (e.g., use large-size font; an unclutteredlayout; spatial cues, such as arrows; colour cues). Encourage students to use graphic organizers and graph paper to organize ideas and written work. Provide augmentative and alternative communications systems. Provide assistive technology, such as text-to-speech software. Provide time-management aids (e.g., checklists). Encourage students to verbalize as they work on mathematics problems. Provide access to computers. Reduce the number of tasks to be completed. Provide extra time to complete tasks.Environmental Accommodations Provide an alternative workspace. Seat students strategically (e.g., near the front of the room; close to the teacher in group settings; witha classmate who can help them). Reduce visual distractions. Minimize background noise. Provide a quiet setting. Provide headphones to reduce audio distractions. Provide special lighting. Provide assistive devices or adaptive equipment.Assessment Accommodations Have students demonstrate understanding using concrete materials or orally rather than in writtenform. Have students record oral responses on audiotape. Have students’ responses on written tasks recorded by a scribe. Provide assistive technology, such as speech-to-text software. Provide an alternative setting. Provide assistive devices or adaptive equipment. Provide augmentative and alternative communications systems. Geometry and Spatial Sense, Grades 4 to 6

Assessment Accommodations Format tests so that they are easy to understand (e.g., use large-size font; an uncluttered layout;spatial cues, such as arrows; colour cues). Provide access to computers. Provide access to calculators and/or addition and multiplication grids. Provide visual cues (e.g., posters). Provide extra time to complete problems or tasks or answer questions. Reduce the number of tasks used to assess a concept or skill.MODIFYING CURRICULUM EXPECTATIONSStudents who have an IEP may require modified expectations, which differ from the regulargrade-level curriculum expectations. When developing modified expectations, teachersmake important decisions regarding the concepts and skills that students need to learn.Most of the learning activities in this document can be adapted for students who requiremodified expectations. The following chart provides examples of how a teacher coulddeliver learning activities that incorporate individual students’ modified expectations.Modified ProgramWhat It MeansExampleModified learning expectations,same activity, same materialsThe student with modifiedexpectations works on thesame or a similar activity, usingthe same materials.The learning activity involvessorting and classifying quadrilaterals (regular and irregular)by geometric properties relatedto symmetry, angles, andsides using a variety of tools(e.g., geoboards, protractors).Students with modified expectations identify and comparequadrilaterals (e.g., square,rectangle, rhombus, trapezoid)and sort and classify them bygeometric properties (e.g.,sides of equal length, parallelsides, right angles), using avariety of tools.Modified learning expectations,same activity, different materialsThe student with modifiedexpectations engages in thesame activity, but uses differentmaterials that enable him/herto remain an equal participantin the activity.The activity involves sketchingdifferent perspectives andviews of three-dimensionalfigures. Students with modified expectations may buildthree-dimensional figuresfrom a picture or model, usinginterlocking cubes.(continued)Introduction

Modified ProgramWhat It MeansExampleModified learning expectations,different activity, differentmaterialsStudents with modified expectations participate in differentactivities.Students with modified expectations work on angle activitiesthat reflect their learningexpectations, using a variety ofconcrete materials.(Adapted from Education for All: The Report of the Expert Panel on Literacy and Numeracy Instruction forStudents With Special Education Needs, Kindergarten to Grade 6, p. 119.)It is important to note that some students may require both accommodations andmodified expectations.The Mathematical ProcessesThe Ontario Curriculum, Grades 1–8: Mathematics, 2005 identifies seven mathematical processes through which students acquire and apply mathematical knowledge and skills. Themathematical processes that support effective learning in mathematics are as follows: problem solving connecting reasoning and proving representing reflecting communicating selecting tools andcomputational strategiesThe learning activities described in this guide demonstrate how the mathematical processeshelp students develop mathematical understanding. Opportunities to solve problems, toreason mathematically, to reflect on new ideas, and so on, make mathematics meaningfulfor students. The learning activities also demonstrate that the mathematical processes areinterconnected – for example, problem-solving tasks encourage students to represent mathematical ideas, to select appropriate tools and strategies, to communicate and reflect onstrategies and solutions, and to make connections between mathematical concepts.Problem Solving: Each of the learning activities is structured around a problem or an inquiry.As students solve problems or conduct investigations, they make connections between newmathematical concepts and ideas that they already understand. The focus on problem solvingand inquiry in the learning activities also provides opportunities for students to: find enjoyment in mathematics; develop confidence in learning and using mathematics; work collaboratively and talk about mathematics; communicate ideas an

Geometry and Spatial Sense, Grades 4 to 6 is a practical guide that teachers will find useful in helping students to achieve the curriculum expectations outlined for Grades 4 to 6 in the Geometry and Spatial Sense strand of The Ontario Curriculum, Grades 1–8: Mathematics, 2005.

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