Key Ideas of the Ontario Curriculum, Mathematics, Revised 2005, Kindergarten Grade 8. Developed by the TDSB for the TDSB.


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1 Key Ideas of the Ontario Curriculum, Mathematics, Revised 2005, Kindergarten Grade 8 Developed by the TDSB for the TDSB. 2009
2 Key Ideas of the Ontario Curriculum, Mathematics, Revised 2005, Kindergarten Grade Toronto District School Board Reproduction of this document for use by schools within the Toronto District School Board is encouraged. Members of the Ontario Coordinators Mathematics Association (OMCA) are permitted to print their own personal copy of this resource. For anyone other than Toronto District School Board staff, no part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any other means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the Toronto District School Board. This permission must be requested and obtained in writing from: Toronto District School Board Library and Learning Resources 3 Tippett Road Toronto, ON M3H 2V1 Tel: Fax: Every reasonable precaution has been taken to trace the owners of copyrighted material included in this document and to make due acknowledgement. Any omission will be gladly rectified in future printings. This document has been reviewed for equity.
3 Key Ideas of the Ontario Curriculum, Mathematics, Revised 2005, Kindergarten Grade 8 Developed by the TDSB for the TDSB. Key Ideas are based on the Ontario Curriculum and organized by strand. The purpose of the Key Ideas is to highlight the important mathematical ideas and to support teachers in seeing trends across the grades. The content of this document is based entirely on Ontario s The Kindergarten Program, 2006 (revised) and The Ontario Curriculum, Grades 1 8: Mathematics, 2005 (revised). It is intended to support planning around the key ideas of the curriculum, informing decisions made when planning for evaluation, and teaching in combinedgrade classrooms. While the Key Ideas identified are based on the content expectations of each strand of the mathematics curriculum, it is crucial to highlight the process expectations that support effective learning in mathematics: problem solving reasoning and proving reflecting selecting tools and computational strategies connecting representing communicating The process expectations occur in Grades 1 through 12. Contents Uses of the Key Ideas... 2 Planning for Instruction, Assessment, and Evaluation... 2 Use of Key Ideas in Planning for Instruction, Assessment, and Evaluation... 2 Use of Key Ideas in Combined Grades... 2 The Organization of the Key Ideas... 3 Key Ideas in Mathematics Number Sense and Numeration Measurement Geometry and Spatial Sense Patterning and Algebra Data Management and Probability Toronto District School Board Page 1 Key Ideas of the Ontario Curriculum, Mathematics, Revised 2005, Kindergarten Grade 8
4 Uses of the Key Ideas Planning for Instruction, Assessment, and Evaluation In The Report of the Expert Panel on Early Math in Ontario, one criterion identified for an effective learning environment is that teaching and learning focuses on big mathematical concepts: An effective learning environment is an environment that [ ] focuses on important mathematical concepts or big ideas [ ] Early Math Strategy, The Report of the Expert Panel on Early Math in Ontario, 2003, p. 12. Teaching with a focus on the Key Ideas provides students the opportunity to learn with coherence and a depth of understanding, while developing the ability to: connect with the mathematics apply the mathematics think using the mathematics Use of Key Ideas in Planning for Instruction, Assessment, and Evaluation All curriculum expectations must be accounted for in instruction, but evaluation focuses on students achievement of the overall expectations. A student s achievement of the overall expectations is evaluated on the basis of his or her achievement of related specific expectations (including the mathematical process expectations). The overall expectations are broad in nature, and the specific expectations define the particular content or scope of the knowledge and skills referred to in the overall expectations. Teachers will use their professional judgement to determine which specific expectations should be used to evaluate achievement of the overall expectations, and which ones will be covered in instruction and assessment (e.g., through direct observation) but not necessarily evaluated. The Ontario Curriculum, Grades 1 8: Mathematics, 2005 (revised), p. 19. In applying their professional judgment to determine which specific expectations to use for evaluation, teachers will find that using the Key Ideas emphasizes important mathematics that students encounter across the grades. When teachers plan instruction, focusing on the Key Ideas will allow students time to develop deeply concepts and skills that will support their success throughout the grades. Use of Key Ideas in Combined Grades Combinedgrade mathematics teaching enables teachers to program for students at varying levels of achievement within their grade. Many of the ideas of mathematics are introduced and developed from Kindergarten through Grade 8. Teaching combined grades will take different formats, depending on curricular content and students prior knowledge. The Key Ideas are especially useful for identifying the common concepts students learn across the grades and allows for planning based on student needs. To provide the necessary level of detail for each grade, teachers should use the Key Ideas in conjunction with the curriculum documents. Gradebased teaching Where the curricular content of one grade is substantially different from that of the other grade, teachers may choose to work with one grade, while providing independent work to the other. Teachers may then teach the other grade, while providing independent work to the first group. Smallgroup teaching Where gradetograde expectations are based on similar concepts, organizing students in small groups based on prior knowledge and skills enables differentiating for student needs and curricular content. Wholeclass teaching Where gradetograde expectations are highly similar, teaching the combined grade as a whole class and differentiating for the product supports all students in their learning. For example, the concept of equality begins in the Grade 1 Patterning and Algebra strand, in which students use the balance model to find how many more or less is needed to balance both sides. Then, in the Grades 2, 3, and 4 Patterning and Algebra strand, students find missing numbers in equations. When a teacher has a combined Grade 2 and Grade 3 class, the concept of equality and equations is vital in both grades. In Grade 2, the specific expectations focus on addition and subtraction to 18, while in Grade 3, the specific expectations focus on addition both onedigit and twodigit numbers. The prior knowledge from Grade 1 is foundational for both Grades 2 and 3. In this case, a teacher may choose to engage the class in wholeclass discussions and have students then work in small homogeneous groups. The Key Ideas relationships between quantities and understanding equality are pivotal to understanding equations Toronto District School Board Page 2 Key Ideas of the Ontario Curriculum, Mathematics, Revised 2005, Kindergarten Grade 8
5 The Organization of the Key Ideas Key Ideas are written for every strand, which is identified at the top of the section. The K 8 focus of each strand is summarized. Worth Noting provides a focus for thinking within each strand. One Overall Expectation from each grade is linked to one or two Key Ideas. Key Ideas provide the glue that links the gradetograde continuum with one Overall Expectation from each of the grades. Trends in the curriculum link to the Key Ideas and and are based on the Specific Expectations. Note: some trends begin in the early years and disappear in later grades. Other trends begin in later grades and continue. Examples In Grades 4 through 8, students are expected to relate the term with the term number. In Grades 3 through 6, students investigate repeating patterns involving transformational geometry Toronto District School Board Page 3 Key Ideas of the Ontario Curriculum, Mathematics, Revised 2005, Kindergarten Grade 8
6 Key Ideas in Mathematics Number Sense and Numeration Kindergarten Grade 8 The Number Sense and Numeration strand supports students in understanding number operations and number sense, and in gaining proficiency in computation. Number permeates all other areas of the mathematics discipline. Understanding what numbers are, how they are represented, and how numbers are related to each other defines the core meaning of number sense and numeration. Through exploration, investigation, and the use of a variety of concrete materials and tools, students develop and build their number. Worth Noting: In the early years, counting experiences help students develop onetoone correspondence and patterns. Students assign a name to a quantity and then move to manytoone groupings, leading to proportional reasoning in the later years. The use of symbols, strategies, and formalized standard notation to represent mathematical thinking, operations, and number relationships increases along the continuum. Students use estimation throughout the grades to check the reasonableness of answers. They develop conceptual number and operations by using concrete materials in relevant and authentic contexts. Developmentally appropriate experiences help students to develop flexibility in thinking about numbers and to build a repertoire of strategies. Key Idea Kindergarten Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Grade 6 Grade 7 Grade 8 Students develop an understanding of numbers through patterns, relationships, and proportional reasoning. Developing number sense is based on understanding the relationships between numbers. Continued... number, using concrete materials to explore and investigate counting, quantity, and number relationships Read, represent, compare, and order whole numbers to 50, and use concrete materials to investigate fractions and money amounts magnitude by counting forward to 100 and backwards from 20 Read, represent, compare, and order whole numbers to 100, and use concrete materials to represent fractions and money amounts to 100 cents magnitude by counting forward to 200 and backwards from 50, using multiples of various numbers as starting points Read, represent, compare, and order whole numbers to 1000, and use concrete materials to represent fractions and money amounts to $10 magnitude by counting forward and backwards by various numbers and from various starting points Read, represent, compare, and order whole numbers to , decimal numbers to tenths, and simple fractions, and represent money amounts to $100 magnitude by counting forward and backwards by 0.1 and by fractional amounts proportional reasoning by investigating wholenumber unit rates Read, represent, compare, and order whole numbers to , decimal numbers to hundredths, proper and improper fractions, and mixed numbers magnitude by counting forward and backwards by 0.01 proportional reasoning by investigating wholenumber rates Read, represent, compare, and order whole numbers to , decimal numbers to thousandths, proper and improper fractions, and mixed numbers relationships involving percent, ratio, and unit rate Represent, compare, and order numbers, including integers proportional relationships using percent, ratio, and rate Represent, compare, and order equivalent representations of numbers, including those involving positive exponents by using proportional reasoning in a variety of meaningful contexts 2009 Toronto District School Board Page 4 Key Ideas of the Ontario Curriculum, Mathematics, Revised 2005, Kindergarten Grade 8
7 Key Idea Kindergarten Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Grade 6 Grade 7 Grade 8 Students develop an understanding of numbers through patterns, relationships, and proportional reasoning. Developing number sense is based on understanding the relationships between numbers. Students begin to use place value to explore numbers in terms of part and whole relationships. Students count by rote to learn the counting sequence. Students think and build their quantity in terms of whole numbers and use, read, and write numerals to 10. Students compose and decompose numbers in a variety of ways. Students use referents to estimate quantities. Students count and skipcount to help identify counting patterns. Students recognize and use patterns to count and skipcount whole numbers from different starting points. Students begin to describe and demonstrate wholenumber multiplicative relationships, including simple unit rates. Students use concrete materials to understand and represent quantities that are parts of a whole. Students think about quantity in terms of money and fractions. Students read and print number words as they build their quantity and place value. Students use place value to compare numbers and to identify relationships among numbers. Students recognize and use patterns to count and skipcount fractions and decimals from different starting points. Students describe and demonstrate, using simple fractions and decimals, understanding of multiplicative relationships, including simple unit rates. Students use concrete materials to understand and represent quantities that are parts of a whole or a set. Students understand and use standard fractional notation to represent/compare/ order quantities. Students think about quantity in terms of decimals. Students represent numbers using expanded notation. Students represent, using fractions, decimals, and percents, multiplicative relationships, including rates and ratios. Students think about quantity in terms of integers. Students solve problems involving unit rates. Students compose and decompose numbers using 100s, 10s, and 1s. Students find multiples and factors for numbers. Students compare numbers to the nearest decade number. Students round numbers to the nearest 10. Students round numbers to the nearest 10, 100, and Students work with benchmark fractions of 0, ½, and 1. Students round decimal numbers to the nearest 10th. Students solve problems involving rates and proportions. Students estimate quantities using percentage benchmarks Toronto District School Board Page 5 Key Ideas of the Ontario Curriculum, Mathematics, Revised 2005, Kindergarten Grade 8
8 Key Idea Kindergarten Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Grade 6 Grade 7 Grade 8 There are meanings for the operations and they are related. There are many computational strategies for performing the operations. singledigit whole numbers, using a variety of strategies Students solve joining and separating problems to find how many in all, how many more, how many left, and the total or the difference. singledigit whole numbers, using a variety of strategies Students use concrete materials and drawings to solve problems that involve addition and subtraction. one and twodigit whole numbers, using a variety of strategies, and investigate multiplication and division Students understand multiplication as the combining of equal groups and division as sharing equally. one and twodigit whole numbers, using a variety of strategies, and investigate multiplication and division single and multidigit whole numbers, using a variety of strategies, and demonstrate an multiplication Students relate multiplication to repeated addition and division to repeated subtraction. single and multidigit whole numbers, using a variety of strategies, and demonstrate an multiplication Students use a variety of mentalmath strategies, studentgenerated algorithms, and standard algorithms to solve problems that involve addition whole numbers., subtraction, multiplication, and division of single and multidigit whole numbers, and decimal numbers to tenths and money amounts, using a variety of strategies, subtraction, multiplication, and division of single and multidigit whole numbers, and decimal numbers to tenths and money amounts, using a variety of strategies involving the multiplication and division of multidigit whole numbers, and decimal numbers to hundredths, using a variety of strategies involving the multiplication and division of multidigit whole numbers, and decimal numbers to hundredths, using a variety of strategies involving the multiplication and division of whole numbers, and the addition and subtraction of decimal numbers to thousandths, using a variety of strategies involving the multiplication and division of whole numbers, and the addition and subtraction of decimal numbers to thousandths, using a variety of strategies Students use a variety of mentalmath strategies, studentgenerated algorithms, and standard algorithms to solve problems that involve addition, subtraction, multiplication, and division of whole numbers and decimal numbers. addition and subtraction of fractions and integers, and apply a variety of computational strategies to solve problems involving whole numbers and decimal numbers involving whole numbers, decimal numbers, fractions, and integers, using a variety of computational strategies Students represent perfect squares and square roots. Students use exponential notation to represent repeated multiplication. addition and subtraction of fractions and integers, and apply a variety of computational strategies to solve problems involving whole numbers and decimal numbers involving whole numbers, decimal numbers, fractions, and integers, using a variety of computational strategies Students use a variety of mentalmath strategies, studentgenerated algorithms, and standard algorithms to solve problems that involve addition, subtraction, multiplication, and division of whole numbers, integers, and rational numbers Toronto District School Board Page 6 Key Ideas of the Ontario Curriculum, Mathematics, Revised 2005, Kindergarten Grade 8
9 Key Ideas in Mathematics Measurement Kindergarten Grade 8 The Measurement strand provides students with the opportunity to acquire the knowledge and practical skills of measurement that will serve them in everyday life. Through numerous experiences, students learn about measurable attributes, work with different units, and use a variety of measurement tools and strategies. Worth Noting: The number of attributes considered and the processes and tools students use to measure increase as they move from grade to grade. At first, students learn to measure using nonstandard units and comparison (for example, How many toothpickslong is my desk? How many pennies will it take to balance my baseball on a balance?). This leads to standard units and, eventually, to formal measuring tools. In the early years, students estimate in order to understand measuring and how units are used. As they gain experience, students estimate measurements by comparing with benchmarks (for example, doorknobs are usually 1 m or 100 cm from the floor). Early measurement experiences focus on understanding measurable attributes, the mechanics of measuring, and the units used to measure different attributes. Later, students apply these concepts and skills as they investigate the measurable attributes of twodimensional shapes and threedimensional figures. Students develop the formulas for calculating measurements through investigation and problem solving. Key Idea Kindergarten Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Grade 6 Grade 7 Grade 8 Some attributes can be measured using different strategies, tools, and units. Measure and compare length, mass, capacity, area, temperature of objects and/or materials, and the passage of time, using nonstandard units, through free exploration, focused exploration, and guided activity Estimate, measure, and describe length, area, mass, capacity, time, and temperature, using nonstandard units of the same size Students estimate to understand the concept of measurement. Estimate, measure, and record length, perimeter, area, mass, capacity, time, and temperature, using nonstandard units and standard units Estimate, measure, and record length, perimeter, area, mass, capacity, time, and temperature, using standard units Estimate, measure, and record length, perimeter, area, mass, capacity, volume, and elapsed time, using a variety of strategies Estimate, measure, and record perimeter, area, temperature change, and elapsed time, using a variety of strategies Estimate, measure, and record quantities, using the metric system Students estimate and measure to understand measurement units, by applying a variety of strategies and tools. Students compare the same attribute of an Students select and measure different attributes, using different standard units and different tools. object, using different nonstandard units. Students measure by comparing, counting, and choosing appropriate benchmarks. Report on research into reallife applications of area measurements Research, describe, and report on applications of volume and capacity measurements Students identify the applications of area, volume, and capacity of twodimensional and threedimensional shapes. Students apply measurement to reallife situations Toronto District School Board Page 7 Key Ideas of the Ontario Curriculum, Mathematics, Revised 2005, Kindergarten Grade 8
10 Key Idea Kindergarten Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Grade 6 Grade 7 Grade 8 Different measurements can be used to describe, compare, and understand objects and concepts. Measure and compare length, mass, capacity, area, temperature of objects and/or materials, and the passage of time, using nonstandard units, through free exploration, focused exploration, and guided activity Compare, describe, and order objects, using attributes measured in nonstandard units Compare, describe, and order objects, using attributes measured in nonstandard units and standard units Compare, describe, and order objects, using attributes measured in standard units Students order and compare objects with a benchmark object and with each other. Determine relationships among units and measurable attributes, including the area and perimeter of rectangles Determine relationships among units and measurable attributes, including the area of a rectangle and the volume of a rectangular prism Determine relationships among units and measurable attributes, including the area of a parallelogram, the area of a triangle, and the volume of a triangular prism Determine relationships among units and measurable attributes, including the area of a trapezoid and the volume of a right prism Students select the appropriate metric unit with which to measure and justify their selection. Determine relationships among units and measurable attributes, including the area of a circle and the volume of a cylinder Students consider each measurable attribute of an object independently. Students consider the relationship between measurable attributes to develop formulas. Students consider the size of the unit and the number needed to measure. Students select the appropriate standard units to measure different attributes and justify their choices. Students select, justify, and convert standard units. Students construct lines with specific lengths. Students construct twodimensional and threedimensional shapes with specific measurements. Students decompose twodimensional and threedimensional shapes with specific measurements Toronto District School Board Page 8 Key Ideas of the Ontario Curriculum, Mathematics, Revised 2005, Kindergarten Grade 8
11 Key Ideas in Mathematics Geometry and Spatial Sense Kindergarten Grade 8 In the Geometry and Spatial Sense strand, students learn to analyze the characteristics and properties of twodimensional and threedimensional geometric objects and to describe them mathematically in terms of spatial relationships and transformations. Throughout the strand, students learn through geometric modeling while solving problems. Worth Noting: Formal and informal experiences in geometry give students the opportunity to identify the properties and relationships among and between twodimensional and threedimensional geometric objects. Throughout the grades, students use their geometric properties and relationships to construct and draw/sketch shapes and figures. Students begin the description of location and movement of objects by using positional language, which extends to the use of maps and grids and, later, to coordinate systems. The emphasis in Grades 1 through 6 is on transformations of objects resulting in congruent shapes, which extends in Grade 7 to transformations resulting in similar noncongruent shapes, and then in Grade 8, to transformations of coordinate points. Key Idea Kindergarten Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Grade 6 Grade 7 Grade 8 Geometric objects have properties that allow them to be classified and described in a variety of different ways. Describe, sort, classify, and compare twodimensional shapes and threedimensional figures, and describe the location and movement of objects through investigation Identify common twodimensional shapes and threedimensional figures, and sort and classify them by their attributes Identify twodimensional shapes and threedimensional figures, and sort and classify them by their geometric properties Compare twodimensional shapes and threedimensional figures, and sort them by their geometric properties Identify quadrilaterals and threedimensional figures and classify them by their geometric properties, and compare various angles to benchmarks Identify and classify twodimensional shapes by side and angle properties, and compare and sort threedimensional figures Classify and construct polygons and angles Construct related lines, and classify triangles, quadrilaterals, and prisms the geometric properties of quadrilaterals and circles and the applications of geometric properties in the real world Students sort, compare, and describe to identify regular and irregular shapes and figures. Students sort, compare, and describe geometric objects to identify attributes (both geometric and nongeometric). Students sort, compare, and classify twodimensional shapes based on geometric properties ranging from symmetry, angles, congruence, number of vertices, number of sides, and other properties. Students investigate the properties of similar and congruent shapes. Students sort and classify quadrilaterals based on a wide variety of geometric properties. Students identify and describe examples of symmetry. Students investigate symmetry within shapes and resulting from transformations Toronto District School Board Page 9 Key Ideas of the Ontario Curriculum, Mathematics, Revised 2005, Kindergarten Grade 8
12 Key Idea Kindergarten Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Grade 6 Grade 7 Grade 8 Understanding relationships between geometric objects allows us to create any geometric object by composing and decomposing other geometric objects. The location and transformation of objects can be identified and described mathematically. Describe, sort, classify, and compare twodimensional shapes and threedimensional figures, and describe the location and movement of objects through investigation Compose and decompose common twodimensional shapes and threedimensional figures Compose and decompose twodimensional shapes and threedimensional figures Describe relationships between twodimensional shapes, and between twodimensional shapes and threedimensional figures Students identify relationships among and between twodimensional shapes and threedimensional figures. Describe, sort, classify, and compare twodimensional shapes and threedimensional figures, and describe the location and movement of objects through investigation Describe the relative locations of objects, using positional language Students describe relative location, using positional language. Describe and represent the relative locations of objects, and represent objects on a map Students describe the relative locations and movements on a map. Identify and describe the locations and movements of shapes and objects Students describe movement using a grid. Students investigate, identify, and name transformations. Construct threedimensional figures, using twodimensional shapes Identify and construct nets of prisms and pyramids Students construct a variety of representations of threedimensional prisms and pyramids (including regular objects in Grade 4), using nets. Identify and describe the location of an object, using a grid map, and reflect twodimensional shapes Students describe location, using a grid system. Students describe and perform transformations, with a focus on reflection. Students identify the symmetry and/or congruence of twodimensional shapes resulting from a transformation. Identify and describe the location of an object, using the cardinal directions, and translate twodimensional shapes Students use cardinal directions and coordinate systems. Students describe and perform transformations, with a focus on translations. Sketch threedimensional figures, and construct threedimensional figures from drawings Students construct models and sketch threedimensional figures from different perspectives. Describe location in the first quadrant of a coordinate system, and rotate twodimensional shapes Students plot points in the first quadrant of the Cartesian plane. Students describe and perform transformations, with a focus on rotations. Students identify how transformations can map one congruent figure onto another. Develop an similarity, and distinguish similarity and congruence Students relate angles and sides to compose and decompose shapes. Describe location in the four quadrants of a coordinate system, dilate twodimensional shapes, and apply transformations to create and analyze designs Develop geometric relationships involving lines, triangles, and polyhedra, and solve problems involving lines and triangles Students relate area, perimeter, and angle relationships. Students investigate the Pythagorean relationship. Represent transformations using the Cartesian coordinate plane, and make connections between transformations and the real world Students plot points in all four quadrants on a Cartesian coordinate system. Students describe and perform transformations, with a focus on dilations. Students identify how transformations can result in similar and congruent shapes. Students investigate and identify realworld movements that are transformations. Students transform points or sets of points rather than shapes Toronto District School Board Page 10 Key Ideas of the Ontario Curriculum, Mathematics, Revised 2005, Kindergarten Grade 8
13 Key Ideas in Mathematics Patterning and Algebra Kindergarten Grade 8 In the Patterning and Algebra strand, students develop an patterns and relationships. The strand builds toward an emphasis on the quantitative relationships between and among patterns and the use of algebraic symbols to represent mathematical relationships in a variety of changing contexts. Worth Noting: From Kindergarten through Grade 8, students identify, explore, describe, extend, and create a variety of geometric and numeric patterns via investigations, using a wide range of tools and materials appropriate to the level of development. The types of expressions and equations students use become more sophisticated as their knowledge, understanding, and skills in number develop. In the primary grades (K Grade 3), students begin to develop the concept of equality, using the balance model, concrete materials, and addition and subtraction. Later, they use multiplication and division to show equality between pairs of expressions. In the later junior grades (Grades 4 6), students use variables in equations and represent relationships graphically and algebraically. In Grade 8, students employ a variety of strategies to solve algebraic equations and verify their solutions. Key Idea Kindergarten Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Grade 6 Grade 7 Grade 8 Patterns represent identified regularities based on rules that describe the pattern s elements. Patterns can be modeled and represented in a variety of ways. Explore, recognize, describe, and create patterns, using a variety of materials in different contexts Identify, describe, extend, and create repeating patterns Identify, describe, extend, and create repeating patterns, growing patterns, and shrinking patterns Describe, extend, and create a variety of numeric patterns and geometric patterns Students investigate growing and shrinking patterns based on a variety of attributes, including repeated addition and subtraction. Describe, extend, and create a variety of numeric patterns and geometric patterns, make predictions related to the patterns, and investigate repeating patterns involving reflections Determine, through investigation using a table of values, relationships in growing and shrinking patterns, and investigate repeating patterns involving translations Describe and represent relationships in growing and shrinking patterns (where the terms are whole numbers), and investigate repeating patterns involving rotations Represent linear growing patterns (where the terms are whole numbers), using concrete materials, graphs, and algebraic expressions Represent linear growing patterns (where the terms are whole numbers), using graphs, algebraic expressions, and equations Students investigate growing and shrinking patterns based on a variety of attributes, including repeated addition, subtraction, and multiplication. In Grades 7 and 8, students work with linear growing patterns. Students connect the term with the term number. Students represent patterns in a table of values to investigate relationships. Students represent patterns, using graphs. Students investigate repeating patterns based on a variety of attributes. Students investigate repeating patterns involving transformational geometry. Students identify, describe, and create patterns. Students use words to express pattern rules for the subsequent term (using constant addition/subtraction/multiplication), and also use words to describe the general term. Students represent patterns algebraically. Students use words and algebraic notation to express pattern rules for the nth term Toronto District School Board Page 11 Key Ideas of the Ontario Curriculum, Mathematics, Revised 2005, Kindergarten Grade 8
14 Key Idea Kindergarten Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Grade 6 Grade 7 Grade 8 Relationships between quantities can be expressed symbolically. the concept of equality, using concrete materials and addition and subtraction to 10 the concept of equality between pairs of expressions, using concrete materials, symbols, and addition and subtraction to 18 equality between pairs of expressions, using addition and subtraction of oneand twodigit numbers equality between pairs of expressions, using addition, subtraction, and multiplication Demonstrate, through investigation, an the use of variables in equations Use variables in simple algebraic expressions and equations to describe relationships Model reallife linear relationships graphically and algebraically, and solve simple algebraic equations, using a variety of strategies, including inspection and guess and check Model linear relationships graphically and algebraically, and solve and verify algebraic equations, using a variety of strategies, including inspection, guess and check, and using a balance model Understanding equality is pivotal to understanding equations. Although kindergarten students consider equality in the Number Sense and Numeration strand, this key idea does not appear in the Patterning and Algebra strand of the Ontario curriculum. Students compose and decompose numbers in different ways, using concrete materials and tools to establish equality. Students use the balance model to find how many more or less are needed to balance both sides. Students represent two number expressions that are equal. Students investigate the inverse relationship of addition and subtraction. Students find missing numbers in equations. Students investigate the inverse relationship of multiplication and division. Students investigate variables as changing quantities. Students investigate variables as changing quantities and as unknown quantities as distinct from a constant in an equation. Starting in Grade 6, students make connections with the variables in measurement formulae. Students solve singlevariable linear equations, using a variety of strategies, including the use of an algebraic balance approach in Grade 8. Students consider equality through operations of addition and subtraction. Students use pattern to investigate the properties of numbers Toronto District School Board Page 12 Key Ideas of the Ontario Curriculum, Mathematics, Revised 2005, Kindergarten Grade 8
15 Key Ideas in Mathematics Data Management and Probability Kindergarten Grade 8 The Data Management and Probability strand facilitates students in becoming judicious about data, asking and answering questions, identifying trends, and making informed predictions and decisions. Worth Noting: As students move from grade to grade, they continuously collect data through surveys, experiments, and research. In the primary grades, the questions and investigations students conduct are close to their personal experiences. In later elementary grades, students use of secondary data includes crosscurricular and global data sets. Students construct various representations of data and explore trends through developmentally appropriate tasks. Analysis of data extends to the differences between measures of central tendency by Grade 6, and the effects of single data points by Grades 7 and 8. When studying probability, students progress from considering the likelihood of everyday events to predicting the outcome of simple games and experiments based on both experimental and theoretical probability. The data management and probability aspects of this strand are strongly intertwined by the use of data to enable students to make informed predictions and decisions. Key Idea Kindergarten Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Grade 6 Grade 7 Grade 8 Data can be collected, organized, and displayed in a variety of ways. Sort, classify, and display a variety of concrete objects, collect data, begin to read and describe displays of data, and begin to explore the concept of probability Collect and organize categorical primary data and display the data, using concrete graphs and pictographs, without regard to the order of labels on the horizontal axis Students sort objects according to one attribute, then display and organize them on templates provided by the teacher. In Grade 1, students record using prepared templates. Students participate in surveys and experiments to collect data. Collect and organize categorical or discrete primary data and display the data, using tally charts, concrete graphs, pictographs, line plots, simple bar graphs, and other graphic organizers, with labels ordered appropriately along horizontal axes, as needed Students sort and classify objects at the same time, using two attributes. They display data, using onetoone correspondence. Collect and organize categorical or discrete primary data and display the data, using charts and graphs, including vertical and horizontal bar graphs, with labels ordered appropriately along horizontal axes, as needed Collect and organize discrete primary data and display the data, using charts and graphs, including stemandleaf plots and doublebar graphs Collect and organize discrete or continuous primary data and secondary data and display the data, using charts and graphs, including brokenline graphs Collect and organize discrete or continuous primary data and secondary data and display the data, using charts and graphs, including continuousline graphs Students consider the data and choose the appropriate graph and suitable titles, labels, and scales. Displays use not only onetoone, but also manytoone, correspondence. Students plan and conduct surveys and experiments to collect primary data. Students use technology as one way of organizing and displaying data. Collect and organize categorical, discrete, or continuous primary data and secondary data and display the data, using charts and graphs, including relativefrequency tables and circle graphs Collect and organize categorical, discrete, or continuous primary data and secondary data and display the data, using charts and graphs, including frequency tables with intervals, histograms, and scatter plots 2009 Toronto District School Board Page 13 Key Ideas of the Ontario Curriculum, Mathematics, Revised 2005, Kindergarten Grade 8
16 Key Idea Kindergarten Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Grade 6 Grade 7 Grade 8 Sort, classify, and display a variety of concrete objects. Collect data, begin to read and describe displays of data Read and describe primary data presented in concrete graphs and pictographs Read and describe primary data represented in tally charts, concrete graphs, pictographs, line plots, simple bar graphs, and other graphic organizers Read, describe, and interpret primary data presented in charts and graphs, including vertical and horizontal bar graphs Read, describe, and interpret primary data presented in charts and graphs, including stemandleaf plots and doublebar graphs Read, describe, and interpret primary data and secondary data presented in charts and graphs, including brokenline graphs Read, describe, and interpret data, and explain relationships between sets of data Make and evaluate convincing arguments, based on the analysis of data Organized and displayed data can be analyzed to identify trends, make decisions and predictions, to solve problems, and make convincing arguments. The likelihood of an event occurring can be predicted through the analysis of data, the identification of trends, and the use of mathematics. Students use comparative language ( more, most ) to describe a set of data. Students read and describe primary data represented concretely in graphs. Students use mathematical language to describe a set of data. Students read and describe primary data represented concretely and nonconcretely in graphs. Students compare the parts of a single set of data. Students ask questions and make statements Students ask questions about data displayed using concrete graphs and and make statements graphic organizers. about data displayed using both concrete and nonconcrete graphs and graphic organizers. Begin to explore the concept of probability Describe the likelihood that everyday events will happen Describe probability in everyday situations and simple games Students use probability vocabulary to describe the likelihood of an event occurring. Students use the mode to describe a set of data. Students read and describe primary data represented nonconcretely in graphs. Students use median Students calculate to describe a set of mean to describe a set data. of data. Students read, interpret, and draw conclusions. Students use mean and demonstrate how it is different from median and mode. Students describe the effect on measures of central tendency of adding or removing data. Apply a variety of data management tools and strategies to make convincing arguments about data Students select the appropriate measure of central tendency. Students look for similarities and differences between two or more sets of data. Students consider the shape of data presented in a variety of ways in drawing conclusions, making Students critically examine graphs describing decisions, and making inferences. trends, making inferences, and identifying appropriate uses. Predict and investigate the frequency of a specific outcome in a simple probability experiment Students consider fairness through analysis of investigations. Predict the results of a Represent as a simple probability fraction the probability experiment, then that a specific outcome conduct the experiment will occur in a simple and compare the probability experiment, predictions to the using systematic lists results and area models Students informally consider sample size in drawing conclusions. Students use probability vocabulary to predict the frequency of an outcome, and then compare with the experimental results. Determine the theoretical probability of an outcome in a probability experiment, and use it to predict the frequency of the outcome Compare experimental probabilities with the theoretical probability of an outcome involving two independent events Use probability models to make predictions about reallife events Students use theoretical probability to make predictions based on probable outcomes. In Grade 7, students work with two independent events. In Grade 8, students focus on the relationships between theoretical and experimental probabilities, including consideration of sample size. Students conduct and analyze probability experiments. Students may conduct probability experiments, relating them to theoretical probability Toronto District School Board Page 14 Key Ideas of the Ontario Curriculum, Mathematics, Revised 2005, Kindergarten Grade 8
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