The Ontario Curriculum
Grades 1-8
Ministry of Education
2 0 0 5
Mathematics
R E V I S E D
ISBN 0-7794-8121-6 (Print)
ISBN 0-7794-8122-4 (Internet)

Introduction
This document replaces The Ontario Curriculum, Grades 1–8: Mathematics, 1997. Beginning in September 2005, all mathematics programs for Grades 1 to 8 will be based on the expectations outlined in this document.
The Importance of Mathematics
An information- and technology-based society requires individuals who are able to think critically about complex issues, analyse and adapt to new situations, solve problems of various kinds, and communicate their thinking effectively. The study of mathematics equips students with knowledge, skills, and habits of mind that are essential for successful and rewarding participation in such a  society. To learn mathematics in a way that will serve them well throughout
their lives, students need classroom experiences that help them develop mathematical understanding; learn important facts, skills, and procedures; develop the ability to apply the processes of mathematics; and acquire a positive attitude towards mathematics. The Ontario mathematics curriculum for Grades 1 to 8 provides the framework needed to meet these goals.
Learning mathematics results in more than a mastery of basic skills. It equips students with a concise and powerful means of communication. Mathematical structures, operations, processes, and language provide students with a framework and tools for reasoning, justifying conclusions, and expressing ideas clearly.Through mathematical activities that are practical and relevant to their lives, students develop mathematical understanding, problem-solving  skills, and related technological skills that they can apply in their daily lives and, eventually, in the workplace.
Mathematics is a powerful learning tool. As students identify relationships between mathematical concepts and everyday situations and make connections between mathematics and other subjects, they develop the ability to use mathematics to extend and apply their knowledge in other curriculum areas, including science,music, and language.

Principles Underlying the Ontario Mathematics Curriculum
This curriculum recognizes the diversity that exists among students who study mathematics. It is based on the belief that all students can learn mathematics and deserve the opportunity to do so. It recognizes that all students do not necessarily learn mathematics in the same way, using the same resources, and within the same time frames. It supports equity by promoting the active participation of all students and by clearly identifying the knowledge and skills students are expected to demonstrate in every grade. It recognizes different learning styles and sets expectations that call for the use of a variety of instructional and assessment tools and strategies. It aims to challenge all students by including expectations that require them to use higher-order thinking skills and to make connections between related  mathematical concepts and between mathematics, other disciplines, and the real world.

This curriculum is designed to help students build the solid conceptual foundation in mathematics that will enable them to apply their knowledge and further their learning successfully. It is based on the belief that students learn mathematics most effectively when they are given opportunities to investigate ideas and concepts through problem solving and are then guided carefully into an understanding of the mathematical principles involved. At the same time, it promotes a balanced program in mathematics. The acquisition of operational skills remains an important focus of the curriculum.
Attention to the processes that support effective learning of mathematics is also considered to be essential to a balanced mathematics program. Seven mathematical processes are identified in this curriculum document: problem solving, reasoning and proving, reflecting, selecting tools and computational strategies, connecting, representing, and communicating.The curriculum for each grade outlined in this document includes a set of “mathematical process expectations” that describe the practices students need to learn and apply in all areas of their study of mathematics. This curriculum recognizes the benefits that current technologies can bring to the learning and doing of mathematics. It therefore integrates the use of appropriate technologies, while recognizing the continuing importance of students’ mastering essential arithmetic skills.
The development of mathematical knowledge is a gradual process. A continuous, cohesive progam throughout the grades is necessary to help students develop an understanding of the “big ideas” of mathematics – that is, the interrelated concepts that form a framework for learning mathematics in a coherent way.The fundamentals of important concepts, processes, skills,
and attitudes are introduced in the primary grades and fostered through the junior and intermediate grades. The program is continuous, as well, from the elementary to the secondary level.
The transition from elementary school mathematics to secondary school mathematics is very important for students’ development of confidence and competence. The Grade 9 courses in the Ontario mathematics curriculum build on the knowledge of concepts and the skills that students are expected to have by the end of Grade 8. The strands used are similar to those used
in the elementary program,with adjustments made to reflect the more abstract nature of mathematics at the secondary level. Finally, the mathematics courses offered in secondary school are based on principles that are consistent with those that underpin the elementary program, a feature that is essential in facilitating the transition.

Roles and Responsibilities in Mathematics Education
Students. Students have many responsibilities with regard to their learning, and these increase as they advance through elementary and secondary school. Students who are willing to make the effort required and who are able to apply themselves will soon discover that there is a direct relationship between this effort and their achievement in mathematics. There will be some students, however, who will find it more difficult to take responsibility for their learning because of special challenges they face. For these students, the attention, patience, and encouragement of teachers and family can be extremely important factors for success. However, taking responsibility for their own progress and learning is an important part of education for all students.




Understanding mathematical concepts and developing skills in mathematics require a sincere commitment to learning.Younger students must bring a willingness to engage in learning activities and to reflect on their experiences. For older students, the commitment to learning requires an appropriate degree of work and study. Students are expected to learn and apply
strategies and processes that promote understanding of concepts and facilitate the application of important skills. Students are also encouraged to pursue opportunities outside the classroom to extend and enrich their understanding of mathematics.
Parents. Parents have an important role to play in supporting student learning. Studies show that students perform better in school if their parents or guardians are involved in their education.
By becoming familiar with the curriculum, parents can find out what is being taught in each grade and what their child is expected to learn. This awareness will enhance parents’ ability to discuss schoolwork with their child, to communicate with teachers, and to ask relevant questions about their child’s progress. Knowledge of the expectations in the various grades also helps parents to interpret their child’s report card and to work with teachers to improve their child’s learning.
There are other effective ways in which parents can support students’ learning. Attending parent-teacher interviews, participating in parent workshops and school council activities (including becoming a school council member), and encouraging students to complete their assignments at home are just a few examples. The mathematics curriculum has the potential to stimulate interest in lifelong learning not only for students but also for their parents and all those with an interest in education.
Teachers. 
Teachers and students have complementary responsibilities.Teachers are responsible for developing appropriate instructional strategies to help students achieve the curriculum expectations, and for developing appropriate methods for assessing and evaluating student learning.Teachers also support students in developing the reading, writing, and oral communication
skills needed for success in learning mathematics.Teachers bring enthusiasm and varied teaching and assessment approaches to the classroom, addressing different student needs and ensuring sound learning opportunities for every student.
Recognizing that students need a solid conceptual foundation in mathematics in order to further develop and apply their knowledge effectively, teachers endeavour to create a classroom environment that engages students’ interest and helps them arrive at the understanding of mathematics that is critical to further learning.
It is important for teachers to use a variety of instructional, assessment, and evaluation strategies, in order to provide numerous opportunities for students to develop their ability to solve problems, reason mathematically, and connect the mathematics they are learning to the real world around them. Opportunities to relate knowledge and skills to wider contexts will motivate students to learn and to become lifelong learners.
Principals. 
The principal works in partnership with teachers and parents to ensure that each student has access to the best possible educational experience.To support student learning, principals ensure that the Ontario curriculum is being properly implemented in all classrooms through the use of a variety of instructional approaches, and that appropriate resources are 
made available for teachers and students.To enhance teaching and student learning in all subjects, including mathematics, principals promote learning teams and work with teachers to facilitate teacher participation in professional development activities. Principals are also responsible for ensuring that every student who has an Individual Education Plan (IEP) is receiving the modifications and/or accommodations described in his or her plan – in other words, for ensuring that the IEP is properly developed, implemented, and monitored.

The Program in Mathematics

Curriculum Expectations
The Ontario Curriculum, Grades 1 to 8: Mathematics, 2005 identifies the expectations for each grade and describes the knowledge and skills that students are expected to acquire, demonstrate, and apply in their class work and investigations, on tests, and in various other activities on which their achievement is assessed and evaluated.
Two sets of expectations are listed for each grade in each strand, or broad curriculum area, of mathematics:
• The overall expectations describe in general terms the knowledge and skills that students are expected to demonstrate by the end of each grade.
• The specific expectations describe the expected knowledge and skills in greater detail. The specific expectations are grouped under subheadings that reflect particular aspects of the required knowledge and skills and that may serve as a guide for teachers as they plan learning activities for their students. (These groupings often reflect the “big ideas” of mathematics that are addressed in the strand.) The organization of expectations in subgroups is not meant to imply that the expectations in any one group are achieved independently of the expectations in the other groups. The subheadings are used merely to help teachers focus on particular aspects of knowledge and skills as they develop and present various lessons and learning activities for their students.
In addition to the expectations outlined within each strand, a list of seven “mathematical process expectations” precedes the strands in each grade. These specific expectations describe the key processes essential to the effective study of mathematics, which students need to learn and apply throughout the year, regardless of the strand being studied.Teachers should ensure that students develop their ability to apply these processes in appropriate ways as they work towards meeting the expectations outlined in all the strands.
When developing their mathematics program and units of study from this document, teachers are expected to weave together related expectations from different strands, as well as the relevant mathematical process expectations, in order to create an overall program that integrates and balances concept development, skill acquisition, the use of processes, and applications.
Many of the expectations are accompanied by examples and/or sample problems, given in parentheses. These examples and sample problems are meant to illustrate the specific area of learning, the kind of skill, the depth of learning, and/or the level of complexity that the expectation entails. The examples are intended as a guide for teachers rather than as an exhaustive or
mandatory list.Teachers do not have to address the full list of examples; rather, they may select one or two examples from the list and focus also on closely related areas of their own choosing.
Similarly, teachers are not required to use the sample problems supplied. They may incorporate the sample problems into their lessons, or they may use other problems that are relevant to the expectation.Teachers will notice that some of the sample problems not only address the requirements of the expectation at hand but also incorporate knowledge or skills described in
expectations in other strands of the same grade.

The Program in Mathematics

Some of the examples provided appear in quotation marks. These are examples of “student talk”, and are offered to provide further clarification of what is expected of students. They illustrate how students might articulate observations or explain results related to the knowledge and skills outlined in the expectation. These examples are included to emphasize the
importance of encouraging students to talk about the mathematics they are doing, as well as to provide some guidance for teachers in how to model mathematical language and reasoning for their students. As a result, they may not always reflect the exact level of language used by students in the particular grade.
Strands in the Mathematics Curriculum
Overall and specific expectations in mathematics are organized into five strands, which are the five major areas of knowledge and skills in the mathematics curriculum. 
The program in all grades is designed to ensure that students build a solid foundation in mathematics by connecting and applying mathematical concepts in a variety of ways.To support this process, teachers will, whenever possible, integrate concepts from across the five strands and apply the mathematics to real-life situations.
The five strands are Number Sense and Numeration, Measurement, Geometry and Spatial Sense, Patterning and Algebra, and Data Management and Probability.
Number Sense and Numeration. Number sense refers to a general understanding of number and operations as well as the ability to apply this understanding in flexible ways to make mathematical judgements and to develop useful strategies for solving problems. In this strand, students develop their understanding of number by learning about different ways of representing
numbers and about the relationships among numbers. They learn how to count in various ways, developing a sense of magnitude.They also develop a solid understanding of the four basic operations and learn to compute fluently, using a variety of tools and strategies.
A well-developed understanding of number includes a grasp of more-and-less relationships, part-whole relationships, the role of special numbers such as five and ten, connections between numbers and real quantities and measures in the environment, and much more.
Experience suggests that students do not grasp all of these relationships automatically. A broad range of activities and investigations, along with guidance by the teacher,will help students construct an understanding of number that allows them to make sense of mathematics and to know how and when to apply relevant concepts, strategies, and operations as they solve
problems.
Measurement. 
Measurement concepts and skills are directly applicable to the world in which students live. Many of these concepts are also developed in other subject areas, such as science, social studies, and physical education.
In this strand, students learn about the measurable attributes of objects and about the units and processes involved in measurement. Students begin to learn how to measure by working with non-standard units, and then progress to using the basic metric units to measure quantities such as length, area, volume, capacity, mass, and temperature.They identify benchmarks to help them recognize the magnitude of units such as the kilogram, the litre, and the metre. Skills associated with telling time and computing elapsed time are also developed. Students learn about important relationships among measurement units and about relationships involved in calculating the perimeters, areas, and volumes of a variety of shapes and figures.

Concrete experience in solving measurement problems gives students the foundation necessary for using measurement tools and applying their understanding of measurement relationships. Estimation activities help students to gain an awareness of the size of different units and to become familiar with the process of measuring. As students’ skills in numeration develop,
they can be challenged to undertake increasingly complex measurement problems, thereby strengthening their facility in both areas of mathematics.
Geometry and Spatial Sense. Spatial sense is the intuitive awareness of one’s surroundings and the objects in them. Geometry helps us represent and describe objects and their interrelationships in space. A strong sense of spatial relationships and competence in using the concepts and language of geometry also support students’ understanding of number and measurement.
Spatial sense is necessary for understanding and appreciating the many geometric aspects of our world. Insights and intuitions about the characteristics of two-dimensional shapes and three-dimensional figures, the interrelationships of shapes, and the effects of changes to shapes are important aspects of spatial sense. Students develop their spatial sense by visualizing, drawing, and comparing shapes and figures in various positions.
In this strand, students learn to recognize basic shapes and figures, to distinguish between the attributes of an object that are geometric properties and those that are not, and to investigate the shared properties of classes of shapes and figures. Mathematical concepts and skills related to location and movement are also addressed in this strand.
Patterning and Algebra. One of the central themes in mathematics is the study of patterns and relationships. This study requires students to recognize, describe, and generalize patterns and to build mathematical models to simulate the behaviour of real-world phenomena that exhibit observable patterns.
Young students identify patterns in shapes, designs, and movement, as well as in sets of numbers. They study both repeating patterns and growing and shrinking patterns and develop ways to extend them. Concrete materials and pictorial displays help students create patterns and recognize relationships. Through the observation of different representations of a pattern, students begin to identify some of the properties of the pattern.
In the junior grades, students use graphs, tables, and verbal descriptions to represent relationships that generate patterns.Through activities and investigations, students examine how patterns change, in order to develop an understanding of variables as changing quantities. In the intermediate grades, students represent patterns algebraically and use these representations to make predictions.
A second focus of this strand is on the concept of equality. Students look at different ways of using numbers to represent equal quantities.Variables are introduced as “unknowns”, and techniques for solving equations are developed. Problem solving provides students with opportunities to develop their ability to make generalizations and to deepen their understanding of the relationship between patterning and algebra.
Data Management and Probability. The related topics of data management and probability are highly relevant to everyday life. Graphs and statistics bombard the public in advertising, opinion polls, population trends, reliability estimates, descriptions of discoveries by scientists, and estimates of health risks, to name just a few.

In this strand, students learn about different ways to gather, organize, and display data. They learn about different types of data and develop techniques for analysing the data that include determining measures of central tendency and examining the distribution of the data. Students also actively explore probability by conducting probability experiments and using probability models to simulate situations. The topic of probability offers a wealth of interesting problems that can fascinate students and that provide a bridge to other topics, such as ratios, fractions, percents, and decimals. Connecting probability and data management to real-world problems helps make the learning relevant to students.

The Mathematical Processes

Presented at the start of every grade outlined in this curriculum document is a set of seven expectations that describe the mathematical processes students need to learn and apply as they work to achieve the expectations outlined within the five strands. The need to highlight these process expectations arose from the recognition that students should be actively engaged in applying these processes throughout the program, rather than in connection with particular strands.
The mathematical processes that support effective learning in mathematics are as follows:
• problem solving
• reasoning and proving
• reflecting
• selecting tools and computational strategies
• connecting
• representing
• communicating
The mathematical processes can be seen as the processes through which students acquire and apply mathematical knowledge and skills. These processes are interconnected. Problem solving and communicating have strong links to all the other processes. A problem-solving approach encourages students to reason their way to a solution or a new understanding. As students engage in reasoning, teachers further encourage them to make conjectures and justify solutions, orally and in writing.The communication and reflection that occur during and after the process of problem solving help students not only to articulate and refine their thinking but also to see the problem they are solving from different perspectives. This opens the door to recognizing
the range of strategies that can be used to arrive at a solution. By seeing how others solve a problem, students can begin to reflect on their own thinking (a process known as “metacognition”) and the thinking of others, and to consciously adjust their own strategies in order to make their solutions as efficient and accurate as possible.
The mathematical processes cannot be separated from the knowledge and skills that students acquire throughout the year. Students must problem solve, communicate, reason, reflect, and so on, as they develop the knowledge, the understanding of concepts, and the skills required in all the strands in every grade.
Problem Solving
Problem solving is central to learning mathematics. By learning to solve problems and by learning through problem solving, students are given numerous opportunities to connect mathematical ideas and to develop conceptual understanding. Problem solving forms the basis of effective mathematics programs and should be the mainstay of mathematical instruction.
The Mathematical Processes

It is considered an essential process through which students are able to achieve the expectations in mathematics, and is an integral part of the mathematics curriculum in Ontario, for the following reasons. Problem solving: 
• is the primary focus and goal of mathematics in the real world;
• helps students become more confident in their ability to do mathematics;
• allows students to use the knowledge they bring to school and helps them connect mathematics with situations outside the classroom;
• helps students develop mathematical understanding and gives meaning to skills and concepts in all strands;
• allows students to reason, communicate ideas, make connections, and apply knowledge and skills;
• offers excellent opportunities for assessing students’ understanding of concepts, ability to
solve problems, ability to apply concepts and procedures, and ability to communicate ideas;
• promotes the collaborative sharing of ideas and strategies, and promotes talking about mathematics; 
• helps students find enjoyment in mathematics;
• increases opportunities for the use of critical-thinking skills (estimating, evaluating, classifying, assuming, recognizing relationships, hypothesizing, offering opinions with reasons, and making judgements).
Not all mathematics instruction, however, can take place in a problem-solving context. Certain aspects of mathematics need to be taught explicitly. Mathematical conventions, including the use of mathematical symbols and terms, are one such aspect, and they should be introduced to students as needed, to enable them to use the symbolic language of mathematics.
Selecting Problem-Solving Strategies. Problem-solving strategies are methods that can be used to solve problems of various types.Teachers who use relevant and meaningful problemsolving experiences as the focus of their mathematics class help students to develop and extend a repertoire of strategies and methods that they can apply when solving various kinds of problems – instructional problems, routine problems, and non-routine problems. Students develop this repertoire over time, as they become more mature in their problem-solving skills. 
Eventually, students will have learned many problem-solving strategies that they can flexibly use and integrate when faced with new problem-solving situations, or to learn or reinforce mathematical concepts. Common problem-solving strategies include the following: making a model, picture, or diagram; looking for a pattern; guessing and checking; making an organized
list; making a table or chart; making a simpler problem; working backwards; using logical reasoning.
The Four-Step Problem-Solving Model. Students who have a good understanding of mathematical concepts may still have difficulty applying their knowledge in problem-solving activities, because they have not yet internalized a model that can guide them through the process. 
The most commonly used problem-solving model is George Polya’s four-step model: understand the problem; make a plan; carry out the plan; and look back to check the results.1 (These four steps are now reflected in the Thinking category of the achievement chart.)
1. First published in Polya’s How to Solve It, 1945.

The four-step model is generally not taught directly before Grade 3, because young students tend to become too focused on the model and pay less attention to the mathematical concepts involved and to making sense of the problem. However, a teacher who is aware of the model and who uses it to guide his or her questioning and prompting during the problem-solving process will help students internalize a valuable approach that can be generalized to other problem-solving situations, not only in mathematics but in other subjects as well.
The four-step model provides a framework for helping students to think about a question before, during, and after the problem-solving experience. By Grade 3, the teacher can present the problem-solving model more explicitly, building on students’ experiences in the earlier grades.The four-step model can then be displayed in the classroom and referred to often
during the mathematics lesson.
The stages of the four-step model are described in Figure 1. Students should be made aware that, although the four steps are presented sequentially, it may sometimes be necessary in the course of solving problems to go back and revisit earlier steps.

Figure 1: A Problem-Solving Model 
Understand the Problem(the exploratory stage)
. reread and restate the problem
. identify the information given and the information that needs to be determined 
Communication: talk about the problem to understand it better
Make a Plan
. relate the problem to similar problems solved in the past
. consider possible strategies
. select a strategy or a combination of strategies
Communication: discuss ideas with others to clarify which strategy or strategies would work best
Carry Out the Plan
. execute the chosen strategy
. do the necessary calculations
. monitor success
. revise or apply different strategies as necessary
Communication:
. draw pictures; use manipulatives to represent interim results
. use words and symbols to represent the steps in carrying out the plan or doing the calculations
. share results of computer or calculator operations
Look Back at the Solution
. check the reasonableness of the answer
. review the method used: Did it make sense? Is there a better way to approach the problem?
. consider extensions or variations
Communication: describe how the solution was reached, using the most suitable format, and explain the solution

Reasoning and Proving
The reasoning process supports a deeper understanding of mathematics by enabling students to make sense of the mathematics they are learning. The process involves exploring phenomena, developing ideas, making mathematical conjectures, and justifying results.Teachers draw on students’ natural ability to reason to help them learn to reason mathematically. Initially, students may rely on the viewpoints of others to justify a choice or an approach. Students should be
encouraged to reason from the evidence they find in their explorations and investigations or from what they already know to be true, and to recognize the characteristics of an acceptable argument in the mathematics classroom.Teachers help students revisit conjectures that they have found to be true in one context to see if they are always true. For example, when teaching
students in the junior grades about decimals, teachers may guide students to revisit the conjecture that multiplication always makes things bigger.

Reflecting
Good problem solvers regularly and consciously reflect on and monitor their own thought processes. By doing so, they are able to recognize when the technique they are using is not fruitful, and to make a conscious decision to switch to a different strategy, rethink the problem, search for related content knowledge that may be helpful, and so forth. Students’ problemsolving skills are enhanced when they reflect on alternative ways to perform a task, even if they have successfully completed it. Reflecting on the reasonableness of an answer by considering the original question or problem is another way in which students can improve their ability to make sense of problems. Even very young students should be taught to examine
their own thought processes in this way.
One of the best opportunities for students to reflect is immediately after they have completed an investigation, when the teacher brings students together to share and analyse their solutions. Students then share strategies, defend the procedures they used, justify their answers, and clarify any misunderstandings they may have had. This is the time that students can reflect on what made the problem difficult or easy (e.g., there were too many details to consider; they were not familiar with the mathematical terms used) and think about how they might tackle the problem differently. Reflecting on their own thinking and the thinking of others helps students make important connections and internalize a deeper understanding of the mathematical concepts involved.
Selecting Tools and Computational Strategies
Students need to develop the ability to select the appropriate electronic tools, manipulatives, and computational strategies to perform particular mathematical tasks, to investigate mathematicalideas, and to solve problems.
Calculators, Computers, Communications Technology.Various types of technology are useful in learning and doing mathematics. Although students must develop basic operational skills, calculators and computers can help them extend their capacity to investigate and analyse mathematical concepts and reduce the time they might otherwise spend on purely mechanical
activities.
Students can use calculators or computers to perform operations, make graphs, and organize and display data that are lengthier and more complex than those that might be addressed using only pencil-and-paper. Students can also use calculators and computers in various ways to investigate number and graphing patterns, geometric relationships, and different representations; to simulate situations; and to extend problem solving. When students use calculators and computers in mathematics, they need to know when it is appropriate to apply their mental computation, reasoning, and estimation skills to predict and check answers.
The computer and the calculator should be seen as important problem-solving tools to be used for many purposes. Computers and calculators are tools of mathematicians, and students should be given opportunities to select and use the particular applications that may be helpful to them as they search for their own solutions to problems. 
Students may not be familiar with the use of some of the technologies suggested in the curriculum. When this is the case, it is important that teachers introduce their use in ways that build students’ confidence and contribute to their understanding of the concepts being investigated. Students also need to understand the situations in which the new technology would be
an appropriate choice of tool. Students’ use of the tools should not be laborious or restricted to inputting or following a set of procedures. For example, when using spreadsheets and dynamic statistical software (e.g., TinkerPlots), teachers could supply students with prepared data sets, and when using dynamic geometry software (e.g., The Geometer’s Sketchpad), they
could use pre-made sketches so that students’work with the software would be focused on the mathematics related to the data or on the manipulation of the sketch, not on the inputting of data or the designing of the sketch.
Computer programs can help students to collect, organize, and sort the data they gather, and to write, edit, and present reports on their findings. Whenever appropriate, students should be encouraged to select and use the communications technology that would best support and communicate their learning. Students,working individually or in groups, can use computers, CD-ROM technology, and/or Internet websites to gain access to Statistics Canada, mathematics organizations, and other valuable sources of mathematical information around the world.
Manipulatives.2 Students should be encouraged to select and use concrete learning tools to make models of mathematical ideas. Students need to understand that making their own models is a powerful means of building understanding and explaining their thinking to others. 
Using manipulatives to construct representations helps students to:
• see patterns and relationships;
• make connections between the concrete and the abstract;
• test, revise, and confirm their reasoning;
• remember how they solved a problem;
• communicate their reasoning to others.
Computational Strategies. Problem solving often requires students to select an appropriate computational strategy.They may need to apply the written procedures (or algorithms) for addition, subtraction,multiplication, or division or use technology for computation. They may also need to select strategies related to mental computation and estimation. Developing the
ability to perform mental computations and to estimate is consequently an important aspect of student learning in mathematics.

2. See the Teaching Approaches section, on pages 24–26 of this document, for additional information about the use of
manipulatives in mathematics instruction. 

Mental computation involves calculations done in the mind, with little or no use of paper and pencil. Students who have developed the ability to calculate mentally can select from and use a variety of procedures that take advantage of their knowledge and understanding of numbers, the operations, and their properties. Using their knowledge of the distributive property, for example, students can mentally compute 70% of 22 by first considering 70% of 20 and then adding 70% of 2. Used effectively,mental computation can encourage students to think more deeply about numbers and number relationships.
Knowing how to estimate, and knowing when it is useful to estimate and when it is necessary to have an exact answer, are important mathematical skills. Estimation is a useful tool for judging the reasonableness of a solution and for guiding students in their use of calculators. The ability to estimate depends on a well-developed sense of number and an understanding of place value. It can be a complex skill that requires decomposing numbers, rounding, using compatible numbers, and perhaps even restructuring the problem. Estimation should not be taught as an isolated skill or a set of isolated rules and techniques. Knowing about calculations that are easy to perform and developing fluency in performing basic operations contribute to successful estimation.
Connecting
Experiences that allow students to make connections – to see, for example, how concepts and skills from one strand of mathematics are related to those from another – will help them to grasp general mathematical principles. As they continue to make such connections, students begin to see that mathematics is more than a series of isolated skills and concepts and that they can use their learning in one area of mathematics to understand another. Seeing the relationships among procedures and concepts also helps develop mathematical understanding. The more connections students make, the deeper their understanding. In addition, making connections between the mathematics they learn at school and its applications in their everyday lives
not only helps students understand mathematics but also allows them to see how useful and relevant it is in the world beyond the classroom.
Representing
In elementary school mathematics, students represent mathematical ideas and relationships and model situations using concrete materials, pictures, diagrams, graphs, tables, numbers,words, and symbols. Learning the various forms of representation helps students to understand mathematical concepts and relationships; communicate their thinking, arguments, and understandings;
recognize connections among related mathematical concepts; and use mathematics to model and interpret realistic problem situations. 

Students should be able to go from one representation to another, recognize the connections between representations, and use the different representations appropriately and as needed to solve problems. For example, a student in the primary grades should know how to represent four groups of two by means of repeated addition, counting by 2’s, or using an array of objects.
The array representation can help students begin to understand the commutative property (e.g., 2 x 4 = 4 x 2), a concept that can help them simplify their computations. In the junior grades, students model and solve problems using representations that include pictures, tables, graphs,words, and symbols. Students in Grades 7 and 8 begin to use algebraic representations
to model and interpret mathematical, physical, and social phenomena.

When students are able to represent concepts in various ways, they develop flexibility in their thinking about those concepts. They are not inclined to perceive any single representation as “the math”; rather, they understand that it is just one of many representations that help them understand a concept.

Communicating
Communication is the process of expressing mathematical ideas and understanding orally, visually, and in writing, using numbers, symbols, pictures, graphs, diagrams, and words. Students communicate for various purposes and for different audiences, such as the teacher, a peer, a group of students, or the whole class. Communication is an essential process in learning mathematics. Through communication, students are able to reflect upon and clarify their ideas, their understanding of mathematical relationships, and their mathematical arguments.

Teachers need to be aware of the various opportunities that exist in the classroom for helping students to communicate. For example, teachers can:
• model mathematical reasoning by thinking aloud, and encourage students to think aloud;
• model proper use of symbols, vocabulary, and notations in oral, visual, and written form;
• ensure that students begin to use new mathematical vocabulary as it is introduced (e.g., with the aid of a word wall; by providing opportunities to read, question, and discuss);
• provide feedback to students on their use of terminology and conventions;
• encourage talk at each stage of the problem-solving process;
• ask clarifying and extending questions and encourage students to ask themselves similar kinds of questions;
• ask students open-ended questions relating to specific topics or information (e.g.,“How do you know?” “Why?” “What if …?”,“What pattern do you see?”,“Is this always true?”);
• model ways in which various kinds of questions can be answered;
• encourage students to seek clarification when they are unsure or do not understand something. 

Effective classroom communication requires a supportive and respectful environment that makes all members of the class feel comfortable when they speak and when they question, react to, and elaborate on the statements of their classmates and the teacher.
The ability to provide effective explanations, and the understanding and application of correct mathematical notation in the development and presentation of mathematical ideas and solutions, are key aspects of effective communication in mathematics. 

Assessment and Evaluation of Student Achievement
Basic Considerations
The primary purpose of assessment and evaluation is to improve student learning. Information gathered through assessment helps teachers to determine students’ strengths and weaknesses in their achievement of the curriculum expectations in each subject in each grade. This information also serves to guide teachers in adapting curriculum and instructional approaches to students’ needs and in assessing the overall effectiveness of programs and classroom practices.

Assessment is the process of gathering information from a variety of sources (including assignments, day-to-day observations and conversations/conferences, demonstrations, projects, performances, and tests) that accurately reflects how well a student is achieving the curriculum expectations in a subject. As part of assessment, teachers provide students with descriptive
feedback that guides their efforts towards improvement. Evaluation refers to the process of judging the quality of student work on the basis of established criteria, and assigning a value to represent that quality. In Ontario elementary schools, the value assigned will be in the form of a letter grade for Grades 1 to 6 and a percentage grade for Grades 7 and 8.
Assessment and evaluation will be based on the provincial curriculum expectations and the achievement levels outlined in this document.
In order to ensure that assessment and evaluation are valid and reliable, and that they lead to the improvement of student learning, teachers must use assessment and evaluation strategies that:
• address both what students learn and how well they learn;
• are based both on the categories of knowledge and skills and on the achievement level descriptions given in the achievement chart on pages 22–23;
• are varied in nature, administered over a period of time, and designed to provide opportunities for students to demonstrate the full range of their learning;
• are appropriate for the learning activities used, the purposes of instruction, and the needs and experiences of the students;
• are fair to all students; 
• accommodate the needs of exceptional students, consistent with the strategies outlined in their Individual Education Plan;
• accommodate the needs of students who are learning the language of instruction (English or French);
• ensure that each student is given clear directions for improvement;
• promote students’ ability to assess their own learning and to set specific goals;
• include the use of samples of students’work that provide evidence of their achievement;
• are communicated clearly to students and parents at the beginning of the school year and at other appropriate points throughout the year.

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 characteristics given in the achievement chart (pages 22–23) for level 3 represent the “provincial standard” for achievement of the expectations. A complete picture of overall achievement at level 3 in mathematics can be constructed by reading from top to bottom in the shaded column of the achievement chart, headed “Level 3”. Parents of students achieving
at level 3 can be confident that their children will be prepared for work in the next grade.
Level 1 identifies achievement that falls much below the provincial standard, while still reflecting a passing grade. Level 2 identifies achievement that approaches the standard. Level 4 identifies achievement that surpasses the standard. It should be noted that achievement at level 4 does not mean that the student has achieved expectations beyond those specified for a particular grade. It indicates that the student has achieved all or almost all of the expectations for that grade, and that he or she demonstrates the ability to use the knowledge and skills specified for that grade in more sophisticated ways than a student achieving at level 3.
The Ministry of Education provides teachers with materials that will assist them in improving their assessment methods and strategies and, hence, their assessment of student achievement. These materials include samples of student work (exemplars) that illustrate achievement at each of the four levels.

The Achievement Chart for Mathematics
The achievement chart that follows identifies four categories of knowledge and skills in mathematics. The achievement chart is a standard province-wide guide to be used by teachers. It enables teachers to make judgements about student work that are based on clear performance standards and on a body of evidence collected over time.
The purpose of the achievement chart is to:
• provide a framework that encompasses all curriculum expectations for all grades and subjects represented in this document;
• guide the development of assessment tasks and tools (including rubrics);
• help teachers to plan instruction for learning;
• assist teachers in providing meaningful feedback to students;
• provide various categories and criteria with which to assess and evaluate student learning.
Categories of knowledge and skills. The categories, defined by clear criteria, represent four broad areas of knowledge and skills within which the subject expectations for any given grade are organized. The four categories should be considered as interrelated, reflecting the wholeness and interconnectedness of learning.

The categories of knowledge and skills are described as follows:
Knowledge and Understanding. Subject-specific content acquired in each grade (knowledge), and the comprehension of its meaning and significance (understanding). 
Thinking. The use of critical and creative thinking skills and/or processes,3 as follows: 
– planning skills (e.g., understanding the problem, making a plan for solving the problem) 
– processing skills (e.g., carrying out a plan, looking back at the solution)
– critical/creative thinking processes (e.g., inquiry, problem solving)

Communication. The conveying of meaning through various oral, written, and visual forms (e.g., providing explanations of reasoning or justification of results orally or in writing; communicating mathematical ideas and solutions in writing, using numbers and algebraic symbols, and visually, using pictures, diagrams, charts, tables, graphs, and concrete materials).

Application. The use of knowledge and skills to make connections within and between various contexts.

Teachers will ensure that student work is assessed and/or evaluated in a balanced manner with respect to the four categories, and that achievement of particular expectations is considered within the appropriate categories.

Criteria.Within each category in the achievement chart, criteria are provided, which are subsets of the knowledge and skills that define each category. For example, in Knowledge and Understanding, the criteria are “knowledge of content (e.g., facts, terms, procedural skills, use of tools)” and “understanding of mathematical concepts”. The criteria identify the aspects of
student performance that are assessed and/or evaluated, and serve as guides to what to look for.

Descriptors. A “descriptor” indicates the characteristic of the student’s performance, with respect to a particular criterion, on which assessment or evaluation is focused. In the achievement chart, effectiveness is the descriptor used for each criterion in the Thinking, Communication, and Application categories. What constitutes effectiveness in any given performance task will vary with the particular criterion being considered. Assessment of effectiveness may therefore focus on a quality such as appropriateness, clarity, accuracy, precision, logic, relevance, significance, fluency, flexibility, depth, or breadth, as appropriate for the particular criterion. For example, in the Thinking category, assessment of effectiveness might focus on the degree of relevance or depth apparent in an analysis; in the Communication category, on clarity of
expression or logical organization of information and ideas; or in the Application category, on appropriateness or breadth in the making of connections. Similarly, in the Knowledge and Understanding category, assessment of knowledge might focus on accuracy, and assessment of understanding might focus on the depth of an explanation. Descriptors help teachers to focus
their assessment and evaluation on specific knowledge and skills for each category and criterion, and help students to better understand exactly what is being assessed and evaluated.

Qualifiers. A specific “qualifier” is used to define each of the four levels of achievement – that is, limited for level 1, some for level 2, considerable for level 3, and a high degree or thorough for level 4. A qualifier is used along with a descriptor to produce a description of performance at a particular level. For example, the description of a student’s performance at level 3 with respect to the first criterion in the Thinking category would be: “The student uses planning
skills with considerable effectiveness”.

3. See the footnote on page 22, pertaining to the mathematical processes.

The descriptions of the levels of achievement given in the chart should be used to identify the level at which the student has achieved the expectations. Students should be provided with numerous and varied opportunities to demonstrate the full extent of their achievement of the curriculum expectations, across all four categories of knowledge and skills.

[Page 22 chart omitted]



Some Considerations for Program Planning in Mathematics

When planning a program in mathematics, teachers must take into account considerations in a number of important areas, including those discussed below.
The Ministry of Education has produced or supported the production of a variety of resource documents that teachers may find helpful as they plan programs based on the expectations outlined in this curriculum document. They include the following:
• A Guide to Effective Instruction in Mathematics,Kindergarten to Grade 6, 2005 (forthcoming; replaces the 2003 edition for Kindergarten to Grade 3), along with companion documents focusing on individual strands
• Early Math Strategy: The Report of the Expert Panel on Early Math in Ontario, 2003
• Teaching and Learning Mathematics: The Report of the Expert Panel on Mathematics in Grades 4 to 6 in Ontario, 2004
• Leading Math Success: Mathematical Literacy,Grades 7–12 – The Report of the Expert Panel on Student Success in Ontario, 2004
• Think Literacy: Cross-Curricular Approaches, Grades 7–12 – Mathematics: Subject-Specific Examples,Grades 7–9, 2004
• Targeted Implementation & Planning Supports (TIPS): Grades 7, 8, and 9 Applied Mathematics, 2003

Teaching Approaches
Students in a mathematics class typically demonstrate diversity in the ways they learn best. It is important, therefore, that students have opportunities to learn in a variety of ways – individually, cooperatively, independently, with teacher direction, through hands-on experience, through examples followed by practice. In addition, mathematics requires students to
learn concepts and procedures, acquire skills, and learn and apply mathematical processes. These different areas of learning may involve different teaching and learning strategies. It is assumed, therefore, that the strategies teachers employ will vary according to both the object of the learning and the needs of the students.
In order to learn mathematics and to apply their knowledge effectively, students must develop a solid understanding of mathematical concepts. Research and successful classroom practice have shown that an investigative approach, with an emphasis on learning through problem solving and reasoning, best enables students to develop the conceptual foundation they need.
When planning mathematics programs, teachers will provide activities and assignments that encourage students to search for patterns and relationships and engage in logical inquiry.

Teachers need to use rich problems and present situations that provide a variety of opportunities
for students to develop mathematical understanding through problem solving.
Some Considerations for Program
Planning in Mathematics

All learning, especially new learning, should be embedded in well-chosen contexts for learning – that is, contexts that are broad enough to allow students to investigate initial understandings, identify and develop relevant supporting skills, and gain experience with varied and interesting applications of the new knowledge. Such rich contexts for learning open the door for students to see the “big ideas”, or key principles, of mathematics, such as pattern or relationship. This
understanding of key principles will enable and encourage students to use mathematical reasoning
throughout their lives.
Effective instructional approaches and learning activities draw on students’ prior knowledge, capture their interest, and encourage meaningful practice both inside and outside the classroom. Students’ interest will be engaged when they are able to see the connections between the mathematical concepts they are learning and their application in the world around them
and in real-life situations. 
Students will investigate mathematical concepts using a variety of tools and strategies, both manual and technological. Manipulatives are necessary tools for supporting the effective learning of mathematics by all students. These concrete learning tools invite students to explore and represent abstract mathematical ideas in varied, concrete, tactile, and visually rich ways. Moreover, using a variety of manipulatives helps deepen and extend students’ understanding of mathematical concepts. For example, students who have used only base ten materials to represent two-digit numbers may not have as strong a conceptual understanding of place value as students who have also bundled craft sticks into tens and hundreds and used an abacus.
Manipulatives are also a valuable aid to teachers. By analysing students’ concrete representations of mathematical concepts and listening carefully to their reasoning, teachers can gain useful insights into students’ thinking and provide supports to help enhance their thinking.4
 
Fostering students’ communication skills is an important part of the teacher’s role in the mathematics classroom. Through skilfully led classroom discussions, students build understanding and consolidate their learning. Discussions provide students with the opportunity  to ask questions, make conjectures, share and clarify ideas, suggest and compare strategies, and explain their reasoning. As they discuss ideas with their peers, students learn to discriminate between effective and ineffective strategies for problem solving.
Students’ understanding is revealed through both oral communication and writing, but it is not necessary for all mathematics learning to involve a written communication component. Young students need opportunities to focus on their oral communication without the additional responsibility of writing.

Whether students are talking or writing about their mathematical learning, teachers can prompt them to explain their thinking and the mathematical reasoning behind a solution or the use of a particular strategy by asking the question “How do you know?”. And because mathematical reasoning must be the primary focus of students’ communication, it is important for teachers
to select instructional strategies that elicit mathematical reasoning from their students.

4. Lists of manipulatives appropriate for use in elementary classrooms are provided in the expert panel reports on mathematics, as follows: Early Math Strategy: The Report of the Expert Panel on Early Mathematics in Ontario, 2003, pp 21–24;
Teaching and Learning Mathematics: The Report of the Expert Panel on Mathematics in Grades 4 to 6 in Ontario, 2004,
pp. 61–63; Leading Math Success: Mathematical Literacy, Grades 7–12 – The Report of the Expert Panel on Student Success in
Ontario, 2004, pp. 48–49.

Promoting Positive Attitudes Towards Mathematics. Students’ attitudes have a significant effect on how they approach problem solving and how well they succeed in mathematics. Teachers can help students develop the confidence they need by demonstrating a positive disposition towards mathematics.5 Students need to understand that, for some mathematics problems,
there may be several ways to arrive at the correct answer. They also need to believe that they are capable of finding solutions. It is common for people to think that if they cannot solve problems quickly and easily, they must be inadequate.Teachers can help students understand that problem solving of almost any kind often requires a considerable expenditure of time and energy and a good deal of perseverance. Once students have this understanding, teachers can encourage them to develop the willingness to persist, to investigate, to reason and explore alternative solutions, and to take the risks necessary to become successful problem solvers.

Cross-Curricular and Integrated Learning
The development of skills and knowledge in mathematics is often enhanced by learning in other subject areas.Teachers should ensure that all students have ample opportunities to explore a subject from multiple perspectives by emphasizing cross-curricular learning and integrated learning, as follows:
a) In cross-curricular learning, students are provided with opportunities to learn and use related content and/or skills in two or more subjects. Students can use the concepts and skills of mathematics in their science or social studies lessons. Similarly, students can use what they have learned in science to illustrate or develop mathematical understanding. For
example, in Grade 6, concepts associated with the fulcrum of a lever can be used to develop a better understanding of the impact that changing a set of data can have on the mean.
b) In integrated learning, students are provided with opportunities to work towards meeting expectations from two or more subjects within a single unit, lesson, or activity. By linking expectations from different subject areas, teachers can provide students with multiple opportunities to reinforce and demonstrate their knowledge and skills in a range of settings. Also,  the mathematical process expectation that focuses on connecting encourages students to make connections between mathematics and other subject areas. For example, students in Grade 2 could be given the opportunity to relate the study of location and movement in the Geometry and Spatial Sense strand of mathematics to the study of movement in the Structures and Mechanisms strand in science and technology. Similarly, the same students could link their study of the characteristics of symmetrical shapes in Visual Arts to the creation of symmetrical shapes in their work in Geometry and Spatial Sense.

Planning Mathematics Programs for Exceptional Students 

In planning mathematics programs for exceptional students, teachers should begin by examining both the curriculum expectations for the appropriate grade level and the needs of the individual student to determine which of the following options is appropriate for the student:
• no accommodations6 or modifications; or
• accommodations only; or
• modified expectations, with the possibility of accommodations.

5. Leading Math Success, p. 42.
6. “Accommodations” refers to individualized teaching and assessment strategies, human supports, and/or individualized
equipment.

If the student requires either accommodations or modified expectations, or both, the relevant information, as described in the following paragraphs,must be recorded in his or her Individual Education Plan (IEP). For a detailed discussion of the ministry’s requirements for IEPs,  see Individual Education Plans: Standards for Development, Program Planning, and Implementation, 2000 (referred to hereafter as IEP Standards, 2000). More detailed information about planning programs for exceptional students can be found in The Individual Education Plan (IEP): A Resource Guide, 2004. (Both documents are available at http://www.edu.gov.on.ca.)

Students Requiring Accommodations Only.With the aid of accommodations alone, some exceptional students are able to participate in the regular grade-level curriculum and to demonstrate learning independently. (Accommodations do not alter the provincial curriculum expectations for the grade level.)
The accommodations required to facilitate the student’s learning must be identified in his or her IEP (see IEP Standards, 2000, page 11). A student’s IEP is likely to reflect the same accommodations for many, or all, subject areas.

There are three types of accommodations. Instructional accommodations are changes in teaching strategies, including styles of presentation, methods of organization, or use of technology and multimedia. Environmental accommodations are changes that the student may require in the classroom and/or school environment, such as preferential seating or special lighting. Assessment
accommodations are changes in assessment procedures that enable the student to demonstrate his or her learning, such as allowing additional time to complete tests or assignments or permitting oral responses to test questions (see page 29 of The Individual Education Plan (IEP): A Resource Guide, 2004, for more examples).

If a student requires “accommodations only” in mathematics, assessment and evaluation of his or her achievement will be based on the appropriate grade-level curriculum expectations and the achievement levels outlined in this document. 

Students Requiring Modified Expectations. Some exceptional students will require modified expectations, which differ from the regular grade-level expectations. In mathematics, modified expectations will usually be based on the knowledge and skills outlined in curriculum expectations for a different grade level. Modified expectations must indicate the knowledge and/or skills the student is expected to demonstrate and have assessed in each reporting period (IEP Standards, 2000, pages 10 and 11). Students requiring modified expectations need to develop knowledge and skills in all five strands of the mathematics curriculum. Modified expectations must represent specific, realistic, observable, and measurable achievements and must describe specific knowledge and/or skills that the student can demonstrate independently, given the appropriate assessment accommodations. They should be expressed in such a way that the student and parents can understand exactly what the student is expected to know or be able to do, on the basis of which his or her performance will be evaluated and a grade or mark recorded on the Provincial Report Card. The grade level of the learning expectations must be identified in the student’s IEP. The student’s learning expectations must be reviewed in relation to the student’s progress at least once every reporting period, and must be updated as necessary (IEP Standards, 2000, page 11).

If a student requires modified expectations in mathematics, assessment and evaluation of his or her achievement will be based on the learning expectations identified in the IEP and on the achievement levels outlined in this document. On the Provincial Report Card, the IEP box must be checked for any subject in which the student requires modified expectations, and the 
appropriate statement from the Guide to the Provincial Report Card, Grades 1–8, 1998 (page 8) must be inserted. The teacher’s comments should include relevant information on the student’s demonstrated learning of the modified expectations, as well as next steps for the student’s learning in the subject.

English As a Second Language and English Literacy Development (ESL/ELD)
Young people whose first language is not English enter Ontario elementary schools with diverse linguistic and cultural backgrounds. Some may have experience of highly sophisticated educational systems while others may have had limited formal schooling. All of these students bring a rich array of background knowledge and experience to the classroom, and all teachers
must share in the responsibility for their English-language development.
Teachers of mathematics need to incorporate appropriate instructional and assessment strategies to help ESL and ELD students succeed in their classrooms. These strategies include:
• modification of some or all of the curriculum expectations, based on the student’s level of English proficiency;
• use of a variety of instructional strategies (e.g., extensive use of visual cues, manipulatives, pictures, diagrams, graphic organizers; attention to the clarity of instructions; modelling of preferred ways of working in mathematics; previewing of textbooks; pre-teaching of key specialized vocabulary; encouragement of peer tutoring and class discussion; strategic use of students’ first languages);
• use of a variety of learning resources (e.g., visual material, simplified text, bilingual dictionaries, culturally diverse materials);
• use of assessment accommodations (e.g., granting of extra time; use of alternative forms of assessment, such as oral interviews, learning logs, or portfolios; simplification of language used in problems and instructions).

See The Ontario Curriculum, Grades 1–8: English As a Second Language and English Literacy Development – A Resource Guide, 2001 (available at www.edu.gov.on.ca) for detailed information about modifying expectations for ESL/ELD students and about assessing, evaluating, and reporting on student achievement. 

Antidiscrimination Education in Mathematics
To ensure that all students in the province have an equal opportunity to achieve their full potential, the curriculum must be free from bias and all students must be provided with a safe and secure environment, characterized by respect for others, that allows them to participate fully and responsibly in the educational experience.

Learning activities and resources used to implement the curriculum should be inclusive in nature, reflecting the range of experiences of students with varying backgrounds, abilities, interests, and learning styles. They should enable students to become more sensitive to the diverse cultures and perceptions of others, including Aboriginal peoples. For example, activities can be designed to relate concepts in geometry or patterning to the arches and tile work often found in Asian architecture or to the patterns used in Aboriginal basketry design. By discussing  aspects of the history of mathematics, teachers can help make students aware of the various cultural groups that have contributed to the evolution of mathematics over the centuries.  Finally, students need to recognize that ordinary people use mathematics in a variety of everyday
contexts, both at work and in their daily lives.

Connecting mathematical ideas to real-world situations through learning activities can enhance students’ appreciation of the role of mathematics in human affairs, in areas including health, science, and the environment. Students can be made aware of the use of mathematics in contexts such as sampling and surveying and the use of statistics to analyse trends. Recognizing
the importance of mathematics in such areas helps motivate students to learn and also provides a foundation for informed, responsible citizenship. Teachers should have high expectations for all students.To achieve their mathematical potential,
however, different students may need different kinds of support. Some boys, for example, may need additional support in developing their literacy skills in order to complete mathematical tasks effectively. For some girls, additional encouragement to envision themselves in careers involving mathematics may be beneficial. For example, teachers might consider providing strong role models in the form of female guest speakers who are mathematicians or who use mathematics in their careers.

Literacy and Inquiry/Research Skills
Literacy skills can play an important role in student success in mathematics. Many of the activities and tasks students undertake in mathematics involve the use of written, oral, and visual communication skills. For example, students use language to record their observations, to explain their reasoning when solving problems, to describe their inquiries in both informal and formal contexts, and to justify their results in small-group conversations, oral presentations, and written reports. The language of mathematics includes special terminology. The study of mathematics consequently encourages students to use language with greater care and precision and enhances their ability to communicate effectively.

Some of the literacy strategies that can be helpful to students as they learn mathematics include the following: reading strategies that help students build vocabulary and improve their ability to navigate textbooks; writing strategies that help students sort ideas and information in order to make connections, identify relationships, and determine possible directions for their thinking and writing; and oral communication strategies that help students communicate in smallgroup and whole-class discussions. Further advice for integrating literacy instruction into mathematics instruction in the intermediate grades may be found in the following resource documents:
• Think Literacy: Cross-Curricular Approaches, Grades 7–12, 2003
• Think Literacy: Cross-Curricular Approaches, Grades 7–12 – Mathematics: Subject-Specific Examples,Grades 7–9, 2004


As they solve problems, students develop their ability to ask questions and to plan investigations to answer those questions and to solve related problems. In their work in the Data Management and Probability strand, students learn to apply a variety of inquiry and research methods as they solve statistical problems. Students also learn how to locate relevant information
from a variety of sources such as statistical databases, newspapers, and reports.

The Role of Technology in Mathematics
Information and communication technologies (ICT) provide a range of tools that can significantly extend and enrich teachers’ instructional strategies and support students’ learning in mathematics.Teachers can use ICT tools and resources both for whole class instruction and to design programs that meet diverse student needs.Technology can help to reduce the time
spent on routine mathematical tasks and to promote thinking and concept development.

Technology can influence both what is taught in mathematics courses and how it is taught. Powerful assistive and enabling computer and handheld technologies can be used seamlessly in teaching, learning, and assessment. These tools include simulations,multimedia resources, databases, access to large amounts of statistical data, and computer-assisted learning modules. Information and communications technologies can also be used in the classroom to connect students to other schools, at home and abroad, and to bring the global community into the local classroom.

Guidance and Mathematics
The guidance and career education program should be aligned with the mathematics curriculum. Teachers need to ensure that the classroom learning across all grades and subjects provides sample opportunity for students to learn how to work independently (e.g., complete homework independently), cooperate with others, resolve conflicts, participate in class, solve problems,
and set goals to improve their work.
Teachers can help students to think of mathematics as a career option by pointing out the role of mathematics in the careers of people whom students observe in the community or in careers that students might be considering with their own future in mind. The mathematics program can also offer career exploration activities that include visits from guest speakers, contacts
with career mentors, involvement in simulation programs (e.g., Junior Achievement programs), and attendance at career conferences.

Health and Safety in Mathematics
Although health and safety issues are not usually associated with mathematics, they may be important when investigation involves fieldwork. Out-of-school fieldwork, for example measuring playground or park dimensions, can provide an exciting and authentic dimension to students’ learning experiences.Teachers must preview and plan these activities carefully to protect students’ health and safety. 

Grade 1
The following are highlights of student learning in Grade 1. They are provided to give teachers and parents a quick overview of the mathematical knowledge and skills that students are expected to acquire in each strand in this grade. The expectations on the pages that follow outline the required knowledge and skills in detail and provide information about the ways in which students are expected to demonstrate their learning, how deeply they will explore concepts and at what level of complexity they will perform procedures, and the mathematical processes they will learn and apply throughout the grade.

Number Sense and Numeration: representing and ordering whole numbers to 50; establishing the conservation of number; representing money amounts to 20¢; decomposing and composing numbers to 20; establishing a one-to-one correspondence when counting the elements in a set; counting by 1’s, 2’s, 5’s, and 10’s; adding and subtracting numbers to 20 
Measurement: measuring using non-standard units; telling time to the nearest half-hour; developing a sense of area; comparing objects using measurable attributes; comparing objects using non-standard units; investigating the relationship between the size of a unit and the number of units needed to measure the length of an object
Geometry and Spatial Sense: sorting and classifying two-dimensional shapes and threedimensional figures by attributes; recognizing symmetry; relating shapes to other shapes, to designs, and to figures; describing location using positional language
Patterning and Algebra: creating and extending repeating patterns involving one attribute; introducing the concept of equality using only concrete materials 

Data Management and Probability: organizing objects into categories using one attribute; collecting and organizing categorical data; reading and displaying data using concrete graphs and pictographs; describing the likelihood that an event will occur

Grade 1: Mathematical Process Expectations
The mathematical process expectations are to be integrated into student learning associated with all the strands.
Throughout Grade 1, students will:

PROBLEM SOLVING
• apply developing problem-solving strategies as they pose and solve problems and conduct
investigations, to help deepen their mathematical understanding;

REASONING AND PROVING
• apply developing reasoning skills (e.g., pattern recognition, classification) to make and
investigate conjectures (e.g., through discussion with others);

REFLECTING
• demonstrate that they are reflecting on and monitoring their thinking to help clarify their
understanding as they complete an investigation or solve a problem (e.g., by explaining to
others why they think their solution is correct);

SELECTING TOOLS AND COMPUTATIONAL STRATEGIES
• select and use a variety of concrete, visual, and electronic learning tools and appropriate
computational strategies to investigate mathematical ideas and to solve problems;


CONNECTING
• make connections among simple mathematical concepts and procedures, and relate mathematical
ideas to situations drawn from everyday contexts;

REPRESENTING
• create basic representations of simple mathematical ideas (e.g., using concrete materials;
physical actions, such as hopping or clapping; pictures; numbers; diagrams; invented
symbols), make connections among them, and apply them to solve problems;

COMMUNICATING
• communicate mathematical thinking orally, visually, and in writing, using everyday
language, a developing mathematical vocabulary, and a variety of representations.

Grade 1: Number Sense and Numeration
Overall Expectations
By the end of Grade 1, students will:
• read, represent, compare, and order whole numbers to 50, and use concrete materials to
investigate fractions and money amounts;
• demonstrate an understanding of magnitude by counting forward to 100 and backwards
from 20;
• solve problems involving the addition and subtraction of single-digit whole numbers, using a
variety of strategies.

Specific Expectations
Quantity Relationships
By the end of Grade 1, students will:
– represent, compare, and order whole numbers to 50, using a variety of tools (e.g., connecting cubes, ten frames, base ten
materials, number lines, hundreds charts) and contexts (e.g., real-life experiences, number stories);
– read and print in words whole numbers to ten, using meaningful contexts (e.g., storybooks, posters);
– demonstrate, using concrete materials, the concept of conservation of number (e.g., 5 counters represent the number 5, regardless whether they are close together or far apart);
– relate numbers to the anchors of 5 and 10 (e.g., 7 is 2 more than 5 and 3 less than 10);
– identify and describe various coins (i.e., penny, nickel, dime, quarter, $1 coin, $2 coin), using coin manipulatives or drawings, and state their value (e.g., the value of a penny is one cent; the value of a toonie is two dollars);
– represent money amounts to 20¢, through investigation using coin manipulatives;
– estimate the number of objects in a set, and check by counting (e.g.,“I guessed that there were 20 cubes in the pile.
I counted them and there were only 17 cubes. 17 is close to 20.”);
– compose and decompose numbers up to 20 in a variety of ways, using concrete materials (e.g., 7 can be decomposed using
connecting cubes into 6 and 1, or 5 and 2, or 4 and 3);
– divide whole objects into parts and identify and describe, through investigation, equal-sized parts of the whole, using
fractional names (e.g., halves; fourths or quarters).

Counting
By the end of Grade 1, students will:
– demonstrate, using concrete materials, the concept of one-to-one correspondence between number and objects when counting;
– count forward by 1’s, 2’s, 5’s, and 10’s to 100, using a variety of tools and strategies (e.g., move with steps; skip count on a number line; place counters on a hundreds chart; connect cubes to show equal groups; count groups of pennies, nickels,
or dimes);
– count backwards by 1’s from 20 and any number less than 20 (e.g., count backwards from 18 to 11), with and without the use of concrete materials and number lines;
– count backwards from 20 by 2’s and 5’s, using a variety of tools (e.g., number lines, hundreds charts);
– use ordinal numbers to thirty-first in meaningful contexts (e.g., identify the days of the month on a calendar). 

Operational Sense
By the end of Grade 1, students will:
– solve a variety of problems involving the addition and subtraction of whole numbers to 20, using concrete materials and
drawings (e.g., pictures, number lines) (Sample problem: Miguel has 12 cookies. Seven cookies are chocolate. Use counters
to determine how many cookies are not chocolate.);
– solve problems involving the addition and subtraction of single-digit whole numbers, using a variety of mental strategies (e.g., one more than, one less than, counting on, counting back, doubles);
– add and subtract money amounts to 10¢, using coin manipulatives and drawings.


Grade 1: Measurement
Overall Expectations
By the end of Grade 1, students will:
• estimate, measure, and describe length, area, mass, capacity, time, and temperature, using non-standard units of the same size;
• compare, describe, and order objects, using attributes measured in non-standard units.
Specific Expectations
Attributes, Units, and Measurement Sense
By the end of Grade 1, students will:
– demonstrate an understanding of the use of non-standard units of the same size (e.g., straws, index cards) for measuring
(Sample problem: Measure the length of your desk in different ways; for example, by using several different non-standard
units or by starting measurements from opposite ends of the desk. Discuss your findings.);
– estimate, measure (i.e., by placing nonstandard units repeatedly,without overlaps or gaps), and record lengths, heights, and distances (e.g., a book is about 10 paper clips wide; a pencil is about 3 toothpicks long);
– construct, using a variety of strategies, tools for measuring lengths, heights, and distances in non-standard units (e.g., footprints on cash register tape or on connecting cubes);
– estimate, measure (i.e., by minimizing overlaps and gaps), and describe area, through investigation using non-standard
units (e.g.,“It took about 15 index cards to cover my desk, with only a little bit of space left over.”);
– estimate, measure, and describe the capacity and/or mass of an object, through investigation using non-standard units
(e.g.,“My journal has the same mass as 13 pencils.” “The juice can has the same capacity as 4 pop cans.”);
– estimate, measure, and describe the passage of time, through investigation using nonstandard units (e.g., number of sleeps;
number of claps; number of flips of a sand timer);

– read demonstration digital and analogue clocks, and use them to identify benchmark times (e.g., times for breakfast, lunch, dinner; the start and end of school; bedtime) and to tell and write time to the hour and half-hour in everyday settings;
– name the months of the year in order, and read the date on a calendar;
– relate temperature to experiences of the seasons (e.g.,“In winter,we can skate because it’s cold enough for there to be ice.”).
Measurement Relationships
By the end of Grade 1, students will:
– compare two or three objects using measurable attributes (e.g., length, height, width, area, temperature, mass, capacity),
and describe the objects using relative terms (e.g., taller, heavier, faster, bigger, warmer; “If I put an eraser, a pencil, and a metre stick beside each other, I can see that the eraser is shortest and the metre stick is longest.”);
– compare and order objects by their linear measurements, using the same non-standard unit (Sample problem: Using a length of
string equal to the length of your forearm, work with a partner to find other objects that are about the same length.);
– use the metre as a benchmark for measuring length, and compare the metre with non-standard units (Sample problem: In
the classroom, use a metre stick to find objects that are taller than one metre and objects that are shorter than one metre.);
– describe, through investigation using concrete materials, the relationship between the size of a unit and the number of units needed to measure length (Sample problem: Compare the numbers of paper clips and pencils needed to measure the length of the same table.).

Grade 1: Geometry and Spatial Sense
Overall Expectations
By the end of Grade 1, students will:
• identify common two-dimensional shapes and three-dimensional figures and sort and classify them by their attributes;*
• compose and decompose common two-dimensional shapes and three-dimensional figures;
• describe the relative locations of objects using positional language.
Specific Expectations

Geometric Properties
By the end of Grade 1, students will:
– identify and describe common twodimensional shapes (e.g., circles, triangles, rectangles, squares) and sort and classify
them by their attributes (e.g., colour; size; texture; number of sides), using concrete materials and pictorial representations (e.g.,“I put all the triangles in one group. Some are long and skinny, and some are short and fat, but they all have three sides.”);
– trace and identify the two-dimensional faces of three-dimensional figures, using concrete models (e.g.,“I can see squares
on the cube.”);
– identify and describe common threedimensional figures (e.g., cubes, cones, cylinders, spheres, rectangular prisms) and
sort and classify them by their attributes (e.g., colour; size; texture; number and shape of faces), using concrete materials
and pictorial representations (e.g.,“I put the cones and the cylinders in the same group because they all have circles on
them.”);
– describe similarities and differences between an everyday object and a threedimensional figure (e.g.,“A water bottle
looks like a cylinder, except the bottle gets thinner at the top.”);
– locate shapes in the environment that have symmetry, and describe the symmetry.
Geometric Relationships
By the end of Grade 1, students will:
– compose patterns, pictures, and designs, using common two-dimensional shapes (Sample problem: Create a picture of a
flower using pattern blocks.);
– identify and describe shapes within other shapes (e.g., shapes within a geometric design);
– build three-dimensional structures using concrete materials, and describe the twodimensional shapes the structures contain;
– cover outline puzzles with two-dimensional shapes (e.g., pattern blocks, tangrams) (Sample problem: Fill in the outline of a boat with tangram pieces.).
Location and Movement
By the end of Grade 1, students will:
– describe the relative locations of objects or people using positional language (e.g., over, under, above, below, in front of, behind, inside, outside, beside, between, along);

* For the purposes of student learning in Grade 1, “attributes” refers to the various characteristics of twodimensional
shapes and three-dimensional figures, including geometric properties. (See glossary entries for
“attribute” and “property (geometric)”.) Students learn to distinguish attributes that are geometric properties
from attributes that are not geometric properties in Grade 2.

– describe the relative locations of objects on concrete maps created in the classroom (Sample problem:Work with your group
to create a map of the classroom in the sand table, using smaller objects to represent the classroom objects. Describe
where the teacher’s desk and the bookshelves are located.);
– create symmetrical designs and pictures, using concrete materials (e.g., pattern blocks, connecting cubes, paper for folding), and describe the relative locations of the parts.
Grade 1: Patterning and Algebra
Overall Expectations
By the end of Grade 1, students will:
• identify, describe, extend, and create repeating patterns;
• demonstrate an understanding of the concept of equality, using concrete materials and addition and subtraction to 10.
Specific Expectations
Patterns and Relationships
By the end of Grade 1, students will:

– identify, describe, and extend, through investigation, geometric repeating patterns involving one attribute (e.g., colour, size, shape, thickness, orientation);
– identify and extend, through investigation, numeric repeating patterns (e.g., 1, 2, 3, 1, 2, 3, 1, 2, 3, …);
– describe numeric repeating patterns in a hundreds chart;
– identify a rule for a repeating pattern (e.g., “We’re lining up boy, girl, boy, girl, boy, girl.”);
– create a repeating pattern involving one attribute (e.g., colour, size, shape, sound) (Sample problem: Use beads to make a
string that shows a repeating pattern involving one attribute.);
– represent a given repeating pattern in a variety of ways (e.g., pictures, actions, colours, sounds, numbers, letters) (Sample problem: Make an ABA,ABA,ABA pattern using actions like clapping or tapping.).
Expressions and Equality
By the end of Grade 1, students will:
– create a set in which the number of objects is greater than, less than, or equal to the number of objects in a given set;
– demonstrate examples of equality, through investigation, using a “balance” model (Sample problem: Demonstrate, using
a pan balance, that a train of 7 attached cubes on one side balances a train of 3 cubes and a train of 4 cubes on the
other side.);
– determine, through investigation using a “balance” model and whole numbers to 10, the number of identical objects that
must be added or subtracted to establish equality (Sample problem: On a pan balance, 5 cubes are placed on the left side
and 8 cubes are placed on the right side. How many cubes should you take off the right side so that both sides balance?).

Grade 1: Data Management and Probability
Overall Expectations
By the end of Grade 1, students will:
• 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;
• read and describe primary data presented in concrete graphs and pictographs;
• describe the likelihood that everyday events will happen.

Specific Expectations
Collection and Organization of Data
By the end of Grade 1, students will:
– demonstrate an ability to organize objects into categories by sorting and classifying objects using one attribute (e.g., colour, size), and by describing informal sorting experiences (e.g., helping to put away groceries) (Sample problem: Sort a collection of attribute blocks by colour. Re-sort the same collection by shape.);
– collect and organize primary data (e.g., data collected by the class) that is categorical (i.e., that can be organized into categories based on qualities such as colour or hobby), and display the data using one-to-one correspondence, prepared templates of concrete graphs and pictographs (with titles and labels), and a variety of recording methods (e.g., arranging objects, placing  stickers, drawing pictures, making tally marks) (Sample problem: Collect and organize data about the favourite fruit that students in your class like to eat.).
Data Relationships
By the end of Grade 1, students will:
– read primary data presented in concrete graphs and pictographs, and describe the data using comparative language (e.g.,
more students chose summer than winter as their single favourite season);
– pose and answer questions about collected data (Sample problem: What was the most popular fruit chosen by the students in
your class?).
Probability
By the end of Grade 1, students will:
– describe the likelihood that everyday events will occur, using mathematical language (i.e., impossible, unlikely, less likely, more likely, certain) (e.g.,“It’s unlikely that I will win the contest shown on the cereal box.”).

Grade 2
The following are highlights of student learning in Grade 2. They are provided to give teachers and parents a quick overview of the mathematical knowledge and skills that students are expected to acquire in each strand in this grade. The expectations on the pages that follow outline the required knowledge and skills in detail and provide information about the ways in which students are expected to demonstrate their learning, how deeply they will explore concepts and at what level of complexity they will perform procedures, and the mathematical processes they will learn and apply throughout the grade.

Number Sense and Numeration: representing and ordering numbers to 100; representing money amounts to 100¢; decomposing and composing two-digit numbers; investigating fractions of a whole; counting by 1’s, 2’s, 5’s, 10’s, and 25’s; adding and subtracting two-digit numbers in a variety of ways; relating equal-sized groups to multiplication and relating sharing
equally to division 

Measurement: measuring length using centimetres and metres; telling time to the nearest quarter-hour; measuring perimeter, area, mass, and capacity using non-standard units; describing and establishing temperature change; choosing personal referents for the centimetre and the metre; comparing the mass and capacity of objects using non-standard units; relating days to weeks and months to years


Geometry and Spatial Sense: distinguishing between attributes that are geometric properties and attributes that are not geometric properties; classifying two-dimensional shapes by geometric properties (number of sides and vertices); classifying three-dimensional figures by geometric properties (number and shape of faces); locating a line of symmetry; composing and decomposing shapes; describing relative locations and paths of motion Patterning and Algebra: identifying and describing repeating patterns and growing and shrinking patterns; developing the concept of equality using the addition and subtraction of numbers to 18 and the equal sign; using the commutative property and the property of zero in addition to facilitate computation

Data Management and Probability: organizing objects into categories using two attributes; collecting and organizing categorical and discrete data; reading and displaying data using line plots and simple bar graphs; describing probability, in simple games and experiments, as the likelihood that an event will occur

Grade 2: Mathematical Process Expectations
The mathematical process expectations are to be integrated into student learning associated with all the strands.
Throughout Grade 2, students will:
PROBLEM SOLVING
• apply developing problem-solving strategies as they pose and solve problems and conduct investigations, to help deepen their mathematical understanding;
REASONING AND PROVING
• apply developing reasoning skills (e.g., pattern recognition, classification) to make and investigate conjectures (e.g., through discussion with others);
REFLECTING
• demonstrate that they are reflecting on and monitoring their thinking to help clarify their understanding as they complete an investigation or solve a problem (e.g., by explaining to others why they think their solution is correct);
SELECTING TOOLS AND COMPUTATIONAL STRATEGIES
• select and use a variety of concrete, visual, and electronic learning tools and appropriate computational strategies to investigate mathematical ideas and to solve problems;
CONNECTING
• make connections among simple mathematical concepts and procedures, and relate mathematical ideas to situations drawn from everyday contexts;
REPRESENTING
• create basic representations of simple mathematical ideas (e.g., using concrete materials; physical actions, such as hopping or clapping; pictures; numbers; diagrams; invented symbols), make connections among them, and apply them to solve problems;
COMMUNICATING
• communicate mathematical thinking orally, visually, and in writing, using everyday language, a developing mathematical vocabulary, and a variety of representations.

Grade 2: Number Sense and Numeration
Overall Expectations
By the end of Grade 2, students will:
• read, represent, compare, and order whole numbers to 100, and use concrete materials to represent fractions and money amounts to 100¢;
• demonstrate an understanding of magnitude by counting forward to 200 and backwards from 50, using multiples of various numbers as starting points;
• solve problems involving the addition and subtraction of one- and two-digit whole numbers, using a variety of strategies, and investigate multiplication and division.
Specific Expectations
Quantity Relationships
By the end of Grade 2, students will:
– represent, compare, and order whole numbers to 100, including money amounts to 100¢, using a variety of tools (e.g., ten
frames, base ten materials, coin manipulatives, number lines, hundreds charts and hundreds carpets);
– read and print in words whole numbers to twenty, using meaningful contexts (e.g., storybooks, posters, signs);
– compose and decompose two-digit numbers in a variety of ways, using concrete materials (e.g., place 42 counters on ten
frames to show 4 tens and 2 ones; compose 37¢ using one quarter, one dime, and two pennies) (Sample problem: Use base
ten blocks to show 60 in different ways.);
– determine, using concrete materials, the ten that is nearest to a given two-digit number, and justify the answer (e.g., use
counters on ten frames to determine that 47 is closer to 50 than to 40);
– determine, through investigation using concrete materials, the relationship between the number of fractional parts of
a whole and the size of the fractional parts (e.g., a paper plate divided into fourths has larger parts than a paper plate divided into eighths) (Sample problem: Use paper squares to show which is bigger, one half of a square or one fourth of a square.);
– regroup fractional parts into wholes, using concrete materials (e.g., combine nine fourths to form two wholes and one
fourth);
– compare fractions using concrete materials, without using standard fractional notation (e.g., use fraction pieces to show that three fourths are bigger than one half, but smaller than one whole);
– estimate, count, and represent (using the ¢ symbol) the value of a collection of coins with a maximum value of one dollar.
Counting
By the end of Grade 2, students will:
– count forward by 1’s, 2’s, 5’s, 10’s, and 25’s to 200, using number lines and hundreds charts, starting from multiples of 1, 2, 5, and 10 (e.g., count by 5’s from 15; count by 25’s from 125);
– count backwards by 1’s from 50 and any number less than 50, and count backwards by 10’s from 100 and any number less than
100, using number lines and hundreds charts (Sample problem: Count backwards from 87 on a hundreds carpet, and describe any patterns you see.);
– locate whole numbers to 100 on a number line and on a partial number line (e.g., locate 37 on a partial number line that
goes from 34 to 41).
Operational Sense
By the end of Grade 2, students will:
– solve problems involving the addition and subtraction of whole numbers to 18, using a variety of mental strategies  (e.g.,“To add 6 + 8, I could double 6 and get 12 and then add 2 more to get 14.”);
– describe relationships between quantities by using whole-number addition and subtraction (e.g.,“If you ate 7 grapes and I
ate 12 grapes, I can say that I ate 5 more grapes than you did, or you ate 5 fewer grapes than I did.”);
– represent and explain, through investigation using concrete materials and drawings, multiplication as the combining of
equal groups (e.g., use counters to show that 3 groups of 2 is equal to 2 + 2 + 2 and to 3 x 2);
– represent and explain, through investigation using concrete materials and drawings, division as the sharing of a quantity
equally (e.g.,“I can share 12 carrot sticks equally among 4 friends by giving each person 3 carrot sticks.”);
– solve problems involving the addition and subtraction of two-digit numbers, with and without regrouping, using concrete
materials (e.g., base ten materials, counters), student-generated algorithms, and standard algorithms;
– add and subtract money amounts to 100¢, using a variety of tools (e.g., concrete materials, drawings) and strategies (e.g.,
counting on, estimating, representing using symbols).

Grade 2: Measurement
Overall Expectations
By the end of Grade 2, students will:
• estimate, measure, and record length, perimeter, area, mass, capacity, time, and temperature, using non-standard units and standard units;
• compare, describe, and order objects, using attributes measured in non-standard units and standard units.
Specific Expectations
Attributes, Units, and Measurement Sense
By the end of Grade 2, students will:
– choose benchmarks – in this case, personal referents – for a centimetre and a metre (e.g.,“My little finger is about as wide as one centimetre. A really big step is about one metre.”) to help them perform measurement tasks;
– estimate and measure length, height, and distance, using standard units (i.e., centimetre, metre) and non-standard units;
– record and represent measurements of length, height, and distance in a variety of ways (e.g., written, pictorial, concrete)
(Sample problem: Investigate how the steepness of a ramp affects the distance an object travels. Use cash-register tape for
recording distances.);
– select and justify the choice of a standard unit (i.e., centimetre or metre) or a nonstandard unit to measure length (e.g.,
“I needed a fast way to check that the two teams would race the same distance, so I used paces.”);
– estimate, measure, and record the distance around objects, using non-standard units (Sample problem: Measure around several
different doll beds using string, to see which bed is the longest around.);
– estimate, measure, and record area, through investigation using a variety of non-standard units (e.g., determine the number of yellow pattern blocks it takes to cover an outlined shape) (Sample problem: Cover your desk with index cards in more than one way. See if the number of index cards needed stays the same each time.);
– estimate, measure, and record the capacity and/or mass of an object, using a variety of non-standard units (e.g.,“I used the pan balance and found that the stapler has the same mass as my pencil case.”);
– tell and write time to the quarter-hour, using demonstration digital and analogue clocks (e.g.,“My clock shows the time
recess will start [10:00], and my friend’s clock shows the time recess will end [10:15].”);
– construct tools for measuring time intervals in non-standard units (e.g., a particular bottle of water takes about five seconds to empty);
– describe how changes in temperature affect everyday experiences (e.g., the choice of clothing to wear);
– use a standard thermometer to determine whether temperature is rising or falling (e.g., the temperature of water, air).

Measurement Relationships
By the end of Grade 2, students will:
– describe, through investigation, the relationship between the size of a unit of area and the number of units needed to cover a surface (Sample problem: Compare the numbers of hexagon pattern blocks and triangle pattern blocks needed to cover
the same book.);
– compare and order a collection of objects by mass and/or capacity, using non-standard units (e.g.,“The coffee can holds more sand than the soup can, but the same amount as the small pail.”);
– determine, through investigation, the relationship between days and weeks and between months and years.

Grade 2: Geometry and Spatial Sense
Overall Expectations
By the end of Grade 2, students will:
• identify two-dimensional shapes and three-dimensional figures and sort and classify them by their geometric properties;

• compose and decompose two-dimensional shapes and three-dimensional figures;
• describe and represent the relative locations of objects, and represent objects on a map.
Specific Expectations

Geometric Properties
By the end of Grade 2, students will:
– distinguish between the attributes of an object that are geometric properties (e.g., number of sides, number of faces) and the attributes that are not geometric properties (e.g., colour, size, texture), using a variety of tools (e.g., attribute blocks, geometric solids, connecting cubes);
– identify and describe various polygons (i.e., triangles, quadrilaterals, pentagons,hexagons, heptagons, octagons) and sort
and classify them by their geometric properties (i.e., number of sides or number of vertices), using concrete materials and pictorial representations (e.g.,“I put all the figures with five or more vertices in one group, and all the figures with fewer than five vertices in another group.”);
– identify and describe various threedimensional figures (i.e., cubes, prisms, pyramids) and sort and classify them by
their geometric properties (i.e., number and shape of faces), using concrete materials (e.g.,“I separated the figures that have square faces from the ones that don’t.”);
– create models and skeletons of prisms and pyramids, using concrete materials (e.g., cardboard; straws and modelling clay), and describe their geometric properties (i.e., number and shape of faces, number of edges);
– locate the line of symmetry in a twodimensional shape (e.g., by paper folding; by using a Mira).

Geometric Relationships
By the end of Grade 2, students will:
– compose and describe pictures, designs, and patterns by combining two-dimensional shapes (e.g.,“I made a picture of a flower from one hexagon and six equilateral triangles.”);
– compose and decompose two-dimensional shapes (Sample problem: Use Power Polygons to show if you can compose a rectangle from two triangles of different sizes.);
– cover an outline puzzle with twodimensional shapes in more than one way;
– build a structure using three-dimensional figures, and describe the two-dimensional shapes and three-dimensional figures in
the structure (e.g.,“I used a box that looks like a triangular prism to build the roof of my house.”).
Location and Movement
By the end of Grade 2, students will:
– describe the relative locations (e.g., beside, two steps to the right of ) and the movements of objects on a map (e.g.,“The path shows that he walked around the desk, down the aisle, and over to the window.”);
– draw simple maps of familiar settings, and describe the relative locations of objects on the maps (Sample problem: Draw a
map of the classroom, showing the locations of the different pieces of furniture.);
– create and describe symmetrical designs using a variety of tools (e.g., pattern blocks, tangrams, paper and pencil).
Grade 2: Patterning and Algebra
Overall Expectations
By the end of Grade 2, students will:
• identify, describe, extend, and create repeating patterns, growing patterns, and shrinking patterns;
• demonstrate an understanding of the concept of equality between pairs of expressions, using concrete materials, symbols, and addition and subtraction to 18.
Specific Expectations
Patterns and Relationships
By the end of Grade 2, students will:
– identify and describe, through investigation, growing patterns and shrinking patterns generated by the repeated addition
or subtraction of 1’s, 2’s, 5’s, 10’s, and 25’s on a number line and on a hundreds chart (e.g., the numbers 90, 80, 70, 60, 50, 40, 30, 20, 10 are in a straight line on a hundreds chart);
– identify, describe, and create, through investigation, growing patterns and shrinking patterns involving addition and subtraction, with and without the use of calculators (e.g., 3 + 1 = 4, 3 + 2 = 5, 3 + 3 = 6, …);
– identify repeating, growing, and shrinking patterns found in real-life contexts (e.g., a geometric pattern on wallpaper, a rhythm pattern in music, a number pattern when counting dimes);
– represent a given growing or shrinking pattern in a variety of ways (e.g., using pictures, actions, colours, sounds, numbers, letters, number lines, bar graphs) (Sample problem: Show the letter pattern A,AA, AAA,AAAA, … by clapping or hopping.);
– create growing or shrinking patterns (Sample problem: Create a shrinking pattern using cut-outs of pennies and/or
nickels, starting with 20 cents.);
– create a repeating pattern by combining two attributes (e.g., colour and shape; colour and size) (Sample problem: Use
attribute blocks to make a train that shows a repeating pattern involving two attributes.);
– demonstrate, through investigation, an understanding that a pattern results from repeating an operation (e.g., addition, subtraction) or making a repeated change to an attribute (e.g., colour, orientation).

Expressions and Equality
By the end of Grade 2, students will:
– demonstrate an understanding of the concept of equality by partitioning whole numbers to 18 in a variety of ways, using
concrete materials (e.g., starting with 9 tiles and adding 6 more tiles gives the same result as starting with 10 tiles and
adding 5 more tiles);
– represent, through investigation with concrete materials and pictures, two number expressions that are equal, using the equal sign (e.g.,“I can break a train of 10 cubes into 4 cubes and 6 cubes. I can also break 10 cubes into 7 cubes and 3 cubes. This means 4 + 6 = 7 + 3.”);
– determine the missing number in equations involving addition and subtraction to 18, using a variety of tools and strategies
(e.g., modelling with concrete materials, using guess and check with and without the aid of a calculator) (Sample problem:
Use counters to determine the missing number in the equation 6 + 7 =  + 5.);
– identify, through investigation, and use the commutative property of addition (e.g., create a train of 10 cubes by joining 4 red cubes to 6 blue cubes, or by joining 6 blue cubes to 4 red cubes) to facilitate computation with whole numbers (e.g.,
“I know that 9 + 8 + 1 = 9 + 1 + 8. Adding becomes easier because that gives 10 + 8 = 18.”);
– identify, through investigation, the properties of zero in addition and subtraction (i.e., when you add zero to a number, the number does not change; when you subtract zero from a number, the number does not change).


Grade 2: Data Management and Probability
Overall Expectations
By the end of Grade 2, students will:
• 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;
• read and describe primary data presented in tally charts, concrete graphs, pictographs, line plots, simple bar graphs, and other graphic organizers;
• describe probability in everyday situations and simple games.
Specific Expectations
Collection and Organization of Data
By the end of Grade 2, students will:
– demonstrate an ability to organize objects into categories, by sorting and classifying objects using two attributes simultaneously (e.g., sort attribute blocks by colour and shape at the same time);
– gather data to answer a question, using a simple survey with a limited number of responses (e.g.,What is your favourite
season?; How many letters are in your first name?);
– collect and organize primary data (e.g., data collected by the class) that is categorical or discrete (i.e., that can be counted, such as the number of students absent), and display the data using one-to-one correspondence in concrete graphs, pictographs, line plots, simple bar graphs, and other graphic organizers (e.g., tally charts, diagrams), with appropriate titles and labels and with labels ordered appropriately along horizontal axes, as needed (Sample problem: Record the number of times that specific words are used in a simple rhyme or poem.).
Data Relationships
By the end of Grade 2, students will:
– read primary data presented in concrete graphs, pictographs, line plots, simple bar graphs, and other graphic organizers
(e.g., tally charts, diagrams), and describe the data using mathematical language (e.g.,“Our bar graph shows that 4 more
students walk to school than take the bus.”);
– pose and answer questions about classgenerated data in concrete graphs, pictographs, line plots, simple bar graphs, and
tally charts (e.g.,Which is the least favourite season?);
– distinguish between numbers that represent data values (e.g.,“I have 4 people in my family.”) and numbers that represent
the frequency of an event (e.g.,“There are 10 children in my class who have 4 people in their family.”);
– demonstrate an understanding of data displayed in a graph (e.g., by telling a story, by drawing a picture), by comparing different parts of the data and by making statements about the data as a whole (e.g., “I looked at the graph that shows how
many students were absent each month. More students were away in January than in September.”).
Probability
By the end of Grade 2, students will:
– describe probability as a measure of the likelihood that an event will occur, using mathematical language (i.e., impossible, unlikely, less likely, equally likely, more likely, certain) (e.g.,“If I take a new shoe out of a box without looking, it’s equally likely that I will pick the left shoe or the right shoe.”);
– describe the probability that an event will occur (e.g., getting heads when tossing a coin, landing on red when spinning a
spinner), through investigation with simple games and probability experiments and using mathematical language (e.g., “I tossed 2 coins at the same time, to see how often I would get 2 heads. I found that getting a head and a tail was more
likely than getting 2 heads.”) (Sample problem: Describe the probability of spinning red when you spin a spinner that has
one half shaded yellow, one fourth shaded blue, and one fourth shaded red. Experiment with the spinner to see if the results
are what you expected.).

Grade 3
The following are highlights of student learning in Grade 3. They are provided to give teachers and parents a quick overview of the mathematical knowledge and skills that students are expected to acquire in each strand in this grade. The expectations on the pages that follow outline the required knowledge and skills in detail and provide information about the ways in which students are expected to demonstrate their learning, how deeply they will explore concepts and at what level of complexity they will perform procedures, and the mathematical processes they will learn and apply throughout the grade.

Number Sense and Numeration: representing and ordering numbers to 1000; representing money amounts to $10; decomposing and composing three-digit numbers; investigating fractions of a set; counting by 1’s, 2’s, 5’s, 10’s, 25’s, and 100’s; adding and subtracting three-digit numbers in a variety of ways; relating one-digit multiplication, and division by one-digit divisors,
to real-life situations 

Measurement: measuring distance using kilometres; telling time to the nearest 5 minutes; identifying temperature benchmarks; measuring perimeter using standard units; measuring mass in kilograms and capacity in litres; measuring area using grid paper; comparing the length, mass, and capacity of objects using standard units; relating minutes to hours, hours to days, days to weeks, and weeks to years 

Geometry and Spatial Sense: using a reference tool to identify right angles and to compare angles with a right angle; classifying two-dimensional shapes by geometric properties (number of sides and angles); classifying three-dimensional figures by geometric properties (number of faces, edges, and vertices); relating different types of quadrilaterals; naming prisms and pyramids; identifying congruent shapes; describing movement on a grid map; recognizing transformations 

Patterning and Algebra: creating and extending growing and shrinking patterns; representing geometric patterns with a number sequence, a number line, and a bar graph; determining the missing numbers in equations involving addition and subtraction of one- and two-digit numbers; investigating the properties of zero and one in multiplication

Data Management and Probability: organizing objects into categories using two or more attributes; collecting and organizing categorical and discrete data; reading and displaying data using vertical and horizontal bar graphs; understanding mode; predicting the frequency of an outcome; relating fair games to equally likely events

Grade 3: Mathematical Process Expectations
The mathematical process expectations are to be integrated into student learning associated with all the strands.

Throughout Grade 3, students will:

PROBLEM SOLVING
• apply developing problem-solving strategies as they pose and solve problems and conduct investigations, to help deepen their mathematical understanding;

REASONING AND PROVING
• apply developing reasoning skills (e.g., pattern recognition, classification) to make and investigate conjectures (e.g., through discussion with others);

REFLECTING
• demonstrate that they are reflecting on and monitoring their thinking to help clarify their understanding as they complete an investigation or solve a problem (e.g., by explaining to others why they think their solution is correct);

SELECTING TOOLS AND COMPUTATIONAL STRATEGIES
• select and use a variety of concrete, visual, and electronic learning tools and appropriate computational strategies to investigate mathematical ideas and to solve problems;

CONNECTING
• make connections among simple mathematical concepts and procedures, and relate mathematical ideas to situations drawn from everyday contexts;

REPRESENTING 
• create basic representations of simple mathematical ideas (e.g., using concrete materials; physical actions, such as hopping or clapping; pictures; numbers; diagrams; invented symbols), make connections among them, and apply them to solve problems;

COMMUNICATING
• communicate mathematical thinking orally, visually, and in writing, using everyday language, a developing mathematical vocabulary, and a variety of representations.

Grade 3: Number Sense and Numeration
Overall Expectations
By the end of Grade 3, students will:
• read, represent, compare, and order whole numbers to 1000, and use concrete materials to represent fractions and money amounts to $10;
• demonstrate an understanding of magnitude by counting forward and backwards by various numbers and from various starting points;
• solve problems involving the addition and subtraction of single- and multi-digit whole numbers, using a variety of strategies, and demonstrate an understanding of multiplication and division.
Specific Expectations
Quantity Relationships
By the end of Grade 3, students will:
– represent, compare, and order whole numbers to 1000, using a variety of tools (e.g., base ten materials or drawings of them, number lines with increments of 100 or other appropriate amounts);
– read and print in words whole numbers to one hundred, using meaningful contexts (e.g., books, speed limit signs);
– identify and represent the value of a digit in a number according to its position in the number (e.g., use base ten materials to show that the 3 in 324 represents 3 hundreds);
– compose and decompose three-digit numbers into hundreds, tens, and ones in a variety of ways, using concrete materials
(e.g., use base ten materials to decompose 327 into 3 hundreds, 2 tens, and 7 ones, or into 2 hundreds, 12 tens, and 7 ones);
– round two-digit numbers to the nearest ten, in problems arising from real-life situations;
– represent and explain, using concrete materials, the relationship among the numbers 1, 10, 100, and 1000, (e.g., use
base ten materials to represent the relationship between a decade and a century, or a century and a millennium);
– divide whole objects and sets of objects into equal parts, and identify the parts using fractional names (e.g., one half;
three thirds; two fourths or two quarters), without using numbers in standard fractional notation;
– represent and describe the relationships between coins and bills up to $10 (e.g., “There are eight quarters in a toonie and
ten dimes in a loonie.”);
– estimate, count, and represent (using the $ symbol) the value of a collection of coins and bills with a maximum value of
$10; 
– solve problems that arise from real-life situations and that relate to the magnitude of whole numbers up to 1000 (Sample
problem: Do you know anyone who has lived for close to 1000 days? Explain your reasoning.).
Counting
By the end of Grade 3, students will:
– count forward by 1’s, 2’s, 5’s, 10’s, and 100’s to 1000 from various starting points, and by 25’s to 1000 starting from multiples of 25, using a variety of tools and strategies (e.g., skip count with and without the aid of a calculator; skip count by 10’s using dimes);

– count backwards by 2’s, 5’s, and 10’s from 100 using multiples of 2, 5, and 10 as starting points, and count backwards by 100’s from 1000 and any number less than 1000, using a variety of tools (e.g., number lines, calculators, coins) and strategies.
Operational Sense
By the end of Grade 3, students will:
– solve problems involving the addition and subtraction of two-digit numbers, using a variety of mental strategies (e.g., to add 37 + 26, add the tens, add the ones, then combine the tens and ones, like this: 30 + 20 = 50, 7 + 6 = 13, 50 + 13 = 63);
– add and subtract three-digit numbers, using concrete materials, studentgenerated algorithms, and standard algorithms;
– use estimation when solving problems involving addition and subtraction, to help judge the reasonableness of a solution;
– add and subtract money amounts, using a variety of tools (e.g., currency manipulatives, drawings), to make simulated purchases and change for amounts up to $10 (Sample problem:You spent 5 dollars and 75 cents on one item and 10 cents on
another item. How much did you spend in total?);
– relate multiplication of one-digit numbers and division by one-digit divisors to reallife situations, using a variety of tools and strategies (e.g., place objects in equal groups, use arrays,write repeated addition or subtraction sentences) (Sample problem: Give a real-life example of when you might need to know that 3 groups of 2 is 3 x 2.);
– multiply to 7 x 7 and divide to 49 ÷ 7, using a variety of mental strategies (e.g., doubles, doubles plus another set, skip
counting).

Grade 3: Measurement
Overall Expectations
By the end of Grade 3, students will:
• estimate, measure, and record length, perimeter, area, mass, capacity, time, and temperature, using standard units;
• compare, describe, and order objects, using attributes measured in standard units. Specific Expectations
Attributes, Units, and Measurement Sense
By the end of Grade 3, students will:
– estimate, measure, and record length, height, and distance, using standard units(i.e., centimetre, metre, kilometre) (Sample problem: While walking with your class, stop when you think you have travelled one kilometre.);
– draw items using a ruler, given specific lengths in centimetres (Sample problem: Draw a pencil that is 5 cm long);
– read time using analogue clocks, to the nearest five minutes, and using digital clocks (e.g., 1:23 means twenty-three
minutes after one o’clock), and represent time in 12-hour notation; 
– estimate, read (i.e., using a thermometer), and record positive temperatures to the nearest degree Celsius (i.e., using a number line; using appropriate notation) (Sample problem: Record the temperature outside each day using a thermometer, and
compare your measurements with those reported in the daily news.);
– identify benchmarks for freezing, cold, cool,warm, hot, and boiling temperatures as they relate to water and for cold, cool, warm, and hot temperatures as they relate to air (e.g.,water freezes at 0°C; the air temperature on a warm day is about
20°C, but water at 20°C feels cool);
– estimate, measure, and record the perimeter of two-dimensional shapes, through investigation using standard units (Sample
problem: Estimate, measure, and record the perimeter of your notebook.);
– estimate, measure (i.e., using centimetre grid paper, arrays), and record area (e.g., if a row of 10 connecting cubes is approximately the width of a book, skip counting down the cover of the book with the row of cubes [i.e., counting 10, 20, 30, ...] is one way to determine the area of the book cover);
– choose benchmarks for a kilogram and a litre to help them perform measurement tasks;
– estimate, measure, and record the mass of objects (e.g., can of apple juice, bag of oranges, bag of sand), using the standard unit of the kilogram or parts of a kilogram (e.g., half, quarter);
– estimate, measure, and record the capacity of containers (e.g., juice can, milk bag), using the standard unit of the litre or parts of a litre (e.g., half, quarter).
Measurement Relationships
By the end of Grade 3, students will:
– compare standard units of length (i.e., centimetre, metre, kilometre) (e.g., centimetres are smaller than metres), and
select and justify the most appropriate standard unit to measure length;
– compare and order objects on the basis of linear measurements in centimetres and/or metres (e.g., compare a 3 cm object with a 5 cm object; compare a 50 cm object with a 1 m object) in problem-solving contexts;
– compare and order various shapes by area, using congruent shapes (e.g., from a set of pattern blocks or Power Polygons) and grid paper for measuring (Sample problem: Does the order of the shapes change when you change the size of the pattern blocks
you measure with?);
– describe, through investigation using grid paper, the relationship between the size of a unit of area and the number of units needed to cover a surface (Sample problem: What is the difference between the numbers of squares needed to cover the front of a book, using centimetre grid paper and using two-centimetre grid paper?); 
– compare and order a collection of objects, using standard units of mass (i.e., kilogram) and/or capacity (i.e., litre);
– solve problems involving the relationships between minutes and hours, hours and days, days and weeks, and weeks and years,
using a variety of tools (e.g., clocks, calendars, calculators).

Grade 3: Geometry and Spatial Sense
Overall Expectations
By the end of Grade 3, students will:
• compare two-dimensional shapes and three-dimensional figures and sort them by their geometric properties;
• describe relationships between two-dimensional shapes, and between two-dimensional shapes and three-dimensional figures;
• identify and describe the locations and movements of shapes and objects.
Specific Expectations
Geometric Properties
By the end of Grade 3, students will:
– use a reference tool (e.g., paper corner, pattern block, carpenter’s square) to identify right angles and to describe angles as greater than, equal to, or less than a right angle (Sample problem: Which pattern blocks have angles bigger than a right angle?);
– identify and compare various polygons (i.e., triangles, quadrilaterals, pentagons, hexagons, heptagons, octagons) and sort
them by their geometric properties (i.e., number of sides; side lengths; number of interior angles; number of right angles);
– compare various angles, using concrete materials and pictorial representations, and describe angles as bigger than, smaller than, or about the same as other angles (e.g., “Two of the angles on the red pattern block are bigger than all the angles on the green pattern block.”);
– compare and sort prisms and pyramids by geometric properties (i.e., number and shape of faces, number of edges, number
of vertices), using concrete materials; 

– construct rectangular prisms (e.g., using given paper nets; using Polydrons), and describe geometric properties (i.e., number and shape of faces, number of edges, number of vertices) of the prisms. 
Geometric Relat