WIXLIB

IntroductionPurpose of This DocumentGrade 7 Mathematics: Support Document for Teachers provides various suggestions forinstruction, assessment strategies, and learning resources that promote the meaningfulengagement of mathematics learners in Grade 7. The document is intended to beused by teachers as they work with students in achieving the learning outcomes andachievement indicators identified in Kindergarten to Grade 8 Mathematics: ManitobaCurriculum Framework of Outcomes (2013) (Manitoba Education).BackgroundKindergarten to Grade 8 Mathematics: Manitoba Curriculum Framework of Outcomes is basedon The Common Curriculum Framework for K–9 Mathematics, which resulted from ongoingcollaboration with the Western and Northern Canadian Protocol (WNCP). In its work,WNCP emphasizesQQcommon educational goalsQQthe ability to collaborate and achieve common goalsQQhigh standards in educationQQplanning an array of educational activitiesQQremoving obstacles to accessibility for individual learnersQQoptimum use of limited educational resourcesThe growing effects of technology and the need for technology-related skills havebecome more apparent in the last half century. Mathematics and problem-solvingskills are becoming more valued as we move from an industrial to an informationalsociety. As a result of this trend, mathematics literacy has become increasinglyimportant. Making connections between mathematical study and daily life, business,industry, government, and environmental thinking is imperative. The Kindergartento Grade 12 mathematics curriculum is designed to support and promote theunderstanding that mathematics isQQa way of learning about our worldQQpart of our daily livesQQboth quantitative and geometric in natureIntroduction1

OverviewBeliefs about Students and Mathematics LearningThe Kindergarten to Grade 8 mathematics curriculum is designed with theunderstanding that students have unique interests, abilities, and needs. As a result, itis imperative to make connections to all students’ prior knowledge, experiences, andbackgrounds.Students are curious, active learners with individual interests, abilities, and needs.They come to classrooms with unique knowledge, life experiences, and backgrounds.A key component in successfully developing numeracy is making connections to thesebackgrounds and experiences.Students learn by attaching meaning to what they do, and they need to constructtheir own meaning of mathematics. This meaning is best developed when learnersencounter mathematical experiences that proceed from the simple to the complex andfrom the concrete to the abstract. The use of manipulatives and a variety of pedagogicalapproaches can address the diversity of learning styles and developmental stages ofstudents, and enhance the formation of sound, transferable mathematical concepts.At all levels, students benefit from working with a variety of materials, tools, andcontexts when constructing meaning about new mathematical ideas. Meaningfulstudent discussions can provide essential links among concrete, pictorial, and symbolicrepresentations of mathematics.Students need frequent opportunities to developand reinforce their conceptual understanding,procedural thinking, and problem-solvingabilities. By addressing these three interrelatedcomponents, students will strengthen theirability to apply mathematical learning to theirdaily lives.The learning environment should value andrespect all students’ experiences and waysof thinking, so that learners are comfortabletaking intellectual risks, asking questions, andposing conjectures. Students need to exploreproblem-solving situations in order to developpersonal strategies and become mathematicallyliterate. Learners must realize that it isacceptable to solve problems in different waysand that solutions may vary.2Conceptual understanding:comprehending mathematicalconcepts, relations, and operationsto build new knowledge. (Kilpatrick,Swafford, and Findell 5)Procedural thinking: carrying outprocedures flexibly, accurately,efficiently, and appropriately.Problem solving: engaging inunderstanding and resolvingproblem situations where a methodor solution is not immediatelyobvious. (OECD 12)G r a d e 7 M a t h e m a t i c s: S u p p o r t D o c u m e n t f o r Te a c h e r s

First Nations, Métis, and Inuit PerspectivesFirst Nations, Métis, and Inuit students in Manitoba come from diverse geographic areaswith varied cultural and linguistic backgrounds. Students attend schools in a variety ofsettings, including urban, rural, and isolated communities. Teachers need to recognizeand understand the diversity of cultures within schools and the diverse experiences ofstudents.First Nations, Métis, and Inuit students often have a whole-world view of theenvironment; as a result, many of these students live and learn best in a holistic way.This means that students look for connections in learning, and learn mathematics bestwhen it is contextualized and not taught as discrete content.Many First Nations, Métis, and Inuit students come from cultural environments wherelearning takes place through active participation. Traditionally, little emphasis wasplaced upon the written word. Oral communication along with practical applicationsand experiences are important to student learning and understanding.A variety of teaching and assessment strategies are required to build upon the diverseknowledge, cultures, communication styles, skills, attitudes, experiences, and learningstyles of students. The strategies used must go beyond the incidental inclusion oftopics and objects unique to a culture or region, and strive to achieve higher levels ofmulticultural education (Banks and Banks).Affective DomainA positive attitude is an important aspect of the affective domain that has a profoundeffect on learning. Environments that create a sense of belonging, encourage risktaking, and provide opportunities for success help students develop and maintainpositive attitudes and self-confidence. Students with positive attitudes toward learningmathematics are likely to be motivated and prepared to learn, participate willingly inclassroom learning activities, persist in challenging situations, and engage in reflectivepractices.Teachers, students, and parents* need to recognize the relationship between the affectiveand cognitive domains, and attempt to nurture those aspects of the affective domainthat contribute to positive attitudes. To experience success, students must be taught to setachievable goals and assess themselves as they work toward reaching these goals.Striving toward success and becoming autonomous and responsible learners areongoing, reflective processes that involve revisiting the setting and assessment ofpersonal goals.*In this document, the term parents refers to both parents and guardians and is used with the recognition that insome cases only one parent may be involved in a child’s education.Introduction3

Middle Years EducationMiddle Years education is defined as the education provided for young adolescents inGrades 5, 6, 7, and 8. Middle Years learners are in a period of rapid physical, emotional,social, moral, and cognitive development.Socialization is very important to Middle Years students, and collaborative learning,positive role models, approval of significant adults in their lives, and a sense ofcommunity and belonging greatly enhance adolescents’ engagement in learning andcommitment to school. It is important to provide students with an engaging and socialenvironment within which to explore mathematics and to construct meaning.Adolescence is a time of rapid brain development when concrete thinking progressesto abstract thinking. Although higher-order thinking and problem-solving abilitiesdevelop during the Middle Years, concrete, exploratory, and experiential learning ismost engaging to adolescents.Middle Years students seek to establish their independence and are most engaged whentheir learning experiences provide them with a voice and choice. Personal goal setting,co-construction of assessment criteria, and participation in assessment, evaluation, andreporting help adolescents take ownership of their learning. Clear, descriptive, andtimely feedback can provide important information to the mathematics student. Askingopen-ended questions, accepting multiple solutions, and having students developpersonal strategies will help students to develop their mathematical independence.Adolescents who see the connections between themselves and their learning, andbetween the learning inside the classroom and life outside the classroom, are moremotivated and engaged in their learning than those who do not observe theseconnections.Adolescents thrive on challenges in their learning, but their sensitivity at this age makesthem prone to discouragement if the challenges seem unattainable. Differentiatedinstruction allows teachers to tailor learning challenges to adolescents’ individual needs,strengths, and interests. It is important to focus instruction on where students are and tosee every contribution as valuable.The energy, enthusiasm, and unfolding potential of young adolescents provide bothchallenges and rewards to educators. Those educators who have a sense of humour andwho see the wonderful potential and possibilities of each young adolescent will findteaching in the Middle Years exciting and fulfilling.4G r a d e 7 M a t h e m a t i c s: S u p p o r t D o c u m e n t f o r Te a c h e r s

Mathematics Education Goals for StudentsThe main goals of mathematics education are to prepare students toQQQQQQQQQQQQcommunicate and reason mathematicallyuse mathematics confidently, accurately, andefficiently to solve problemsappreciate and value mathematicsmake connections between mathematical knowledgeand skills and their applicationsMathematics educationmust prepare studentsto use mathematics tothink critically aboutthe world.commit themselves to lifelong learningbecome mathematically literate citizens, using mathematics to contribute to societyand to think critically about the worldStudents who have met these goals willQQgain understanding and appreciation of the contributions of mathematics as ascience, a philosophy, and an artQQexhibit a positive attitude toward mathematicsQQengage and persevere in mathematical tasks and projectsQQcontribute to mathematical discussionsQQtake risks in performing mathematical tasksQQexhibit curiosityIntroduction5

Conceptual Framework for Kindergarten to Grade 9 MathematicsC O N TheC E PchartT U A LbelowF R AprovidesMEWORKO R K - 9of MATHEMAT ICSan Foverviewhowmathematicalprocesses and the nature ofmathematics influence learning outcomes.The chart below provides an overview of how mathematical processesand the nature of mathematics influence learning ATICSPatterns and Relations Patterns Variables and EquationsCHANGEShape and SpaceCONSTANCY, NUMBER SENSE, PATTERNS,RELATIONSHIPS,SPATIAL SENSE,UNCERTAINTY Measurement3-D Objects and 2-DShapesGENERAL LEARNING OUTCOMES,SPECIFIC LEARNING OUTCOMES,AND ACHIEVEMENT INDICATORSTransformationsStatistics and Probability Data Analysis Chance and UncertaintyMATHEMATICAL PROCESSES:COMMUNICATION, CONNECTIONS, MENTALMATHEMATICS AND ESTIMATION, PROBLEMSOLVING, REASONING, TECHNOLOGY,VISUALIZATIONMathematical ProcessesThere are critical components that students must encounter in mathematics to achieve7the goals of mathematics education and encourage lifelong learning in mathematics.Conceptual Framework for K-9 MathematicsStudents are expected toQQQQconnect mathematical ideas to other concepts in mathematics, to everydayexperiences, and to other disciplinesQQdemonstrate fluency with mental mathematics and estimationQQdevelop and apply new mathematical knowledge through problem solvingQQdevelop mathematical reasoningQQselect and use technologies as tools for learning and solving problemsQQ6communicate in order to learn and express their understandingdevelop visualization skills to assist in processing information, making connections,and solving problemsG r a d e 7 M a t h e m a t i c s: S u p p o r t D o c u m e n t f o r Te a c h e r s

The common curriculum framework incorporates these seven interrelated mathematicalprocesses, which are intended to permeate teaching and learning:QQQQQQQQQQQQQQCommunication [C]: Students communicate daily (orally, through diagrams andpictures, and by writing) about their mathematics learning. They need opportunitiesto read about, represent, view, write about, listen to, and discuss mathematical ideas.This enables them to reflect, to validate, and to clarify their thinking. Journals andlearning logs can be used as a record of student interpretations of mathematicalmeanings and ideas.Connections [CN]: Mathematics should be viewed as an integrated whole, ratherthan as the study of separate strands or units. Connections must also be madebetween and among the different representational modes—concrete, pictorial, andsymbolic (the symbolic mode consists of oral and written word symbols as wellas mathematical symbols). The process of making connections, in turn, facilitateslearning. Concepts and skills should also be connected to everyday situations andother curricular areas.Mental Mathematics and Estimation [ME]: The skill of estimation requires asound knowledge of mental mathematics. Both are necessary to many everydayexperiences, and students should be provided with frequent opportunities topractise these skills. Mental mathematics and estimation is a combination ofcognitive strategies that enhances flexible thinking and number sense.Problem Solving [PS]: Students are exposed to a wide variety of problems in allareas of mathematics. They explore a variety of methods for solving and verifyingproblems. In addition, they are challenged to find multiple solutions for problemsand to create their own problems.Reasoning [R]: Mathematics reasoning involves informal thinking, conjecturing,and validating—these help students understand that mathematics makes sense.Students are encouraged to justify, in a variety of ways, their solutions, thinkingprocesses, and hypotheses. In fact, good reasoning is as important as finding correctanswers.Technology [T]: The use of calculators is recommended to enhance problem solving,to encourage discovery of number patterns, and to reinforce conceptual developmentand numerical relationships. They do not, however, replace the development ofnumber concepts and skills. Carefully chosen computer software can provideinteresting problem-solving situations and applications.Visualization [V]: Mental images help students to develop concepts and tounderstand procedures. Students clarify their understanding of mathematical ideasthrough images and explanations.These processes are outlined in detail in Kindergarten to Grade 8 Mathematics: ManitobaCurriculum Framework of Outcomes (2013).Introduction7

StrandsThe learning outcomes in the Manitoba curriculum framework are organized intofour strands across Kindergarten to Grade 9. Some strands are further subdivided intosubstrands. There is one general learning outcome per substrand across Kindergarten toGrade 9.The strands and substrands, including the general learning outcome for each, follow.NumberQQDevelop number sense.Patterns and RelationsQQPatternsQQQQUse patterns to describe the world and solve problems.Variables and EquationsQQRepresent algebraic expressions in multiple ways.Shape and SpaceQQMeasurementQQQQ3-D Objects and 2-D ShapesQQQQUse direct and indirect measure to solve problems.Describe the characteristics of 3-D objects and 2-D shapes, and analyze therelationships among them.TransformationsQQDescribe and analyze position and motion of objects and shapes.Statistics and ProbabilityQQData AnalysisQQQQChance and UncertaintyQQ8Collect, display, and analyze data to solve problems.Use experimental or theoretical probabilities to represent and solve problemsinvolving uncertainty.G r a d e 7 M a t h e m a t i c s: S u p p o r t D o c u m e n t f o r Te a c h e r s

Learning Outcomes and Achievement IndicatorsThe Manitoba curriculum framework is stated in terms of general learning outcomes,specific learning outcomes, and achievement indicators:QQQQQQGeneral learning outcomes are overarching statements about what students areexpected to learn in each strand/substrand. The general learning outcome for eachstrand/substrand is the same throughout the grades from Kindergarten to Grade 9.Specific learning outcomes are statements that identify the specific skills,understanding, and knowledge students are required to attain by the end of a givengrade.Achievement indicators are samples of how students may demonstrate theirachievement of the goals of a specific learning outcome. The range of samplesprovided is meant to reflect the depth, breadth, and expectations of the specificlearning outcome. While they provide some examples of student achievement, theyare not meant to reflect the sole indicators of success.In this document, the word including indicates that any ensuing items must beaddressed to meet the learning outcome fully. The phrase such as indicates that theensuing items are provided for illustrative purposes or clarification, and are notrequirements that must be addressed to meet the learning outcome fully.SummaryThe conceptual framework for Kindergarten to Grade 9 mathematics describes thenature of mathematics, the mathematical processes, and the mathematical concepts tobe addressed in Kindergarten to Grade 9 mathematics. The components are not meantto stand alone. Learning activities that take place in the mathematics classroom shouldstem from a problem-solving approach, be based on mathematical processes, and leadstudents to an understanding of the nature of mathematics through specific knowledge,skills, and attitudes among and between strands. Grade 7 Mathematics: Support Documentfor Teachers is meant to support teachers to create meaningful learning activities thatfocus on formative assessment and student engagement.Introduction9

AssessmentAuthentic assessment and feedback are a driving force for the suggestions forassessment in this document. The purposes of the

Grade 7 Mathematics Blackline Masters v Grades 5 to 8 Mathematics Blackline Masters viii Acknowledgements ix Introduction 1 Overview 2 Conceptual Framework for Kindergarten to Grade 9 Mathematics 6 Assessment 10 Instructional Focus 12 Document Organization and Format 13 Number 1 Numbe