This curriculum policy replaces The Ontario Curriculum, Grades 1–8: Mathematics, 2005. Beginning in September 2020, all mathematics programs for Grades 1 to 8 will be based on the expectations outlined in this curriculum policy.

elementary

# Mathematics (2020)

## The Strands in the Mathematics Curriculum Link

The expectations in the mathematics curriculum are organized into six distinct but related strands: A. Social-Emotional Learning (SEL) Skills in Mathematics and the Mathematical Processes; B. Number; C. Algebra; D. Data; E. Spatial Sense; and F. Financial Literacy.

The program in all grades is designed to ensure that students build a solid foundation in mathematics and develop a positive mathematical identity by connecting and applying mathematical concepts in a variety of ways. To support this process, teachers capitalize on students’ prior knowledge, skills, and experiences; integrate concepts from across the strands; and often apply the mathematics that students are learning to types of situations that might occur outside the classroom.

The following chart shows the flow of the learning through the curriculum and the interrelationships among its various components.

 Strand B. Number Strand C. Algebra Strand D. Data Strand E. Spatial Sense Strand F. Financial Literacy B1. Number Sense whole numbers rational and irrational numbers fractions, decimals, and percents B2. Operations properties and relationships math facts mental math addition and subtraction multiplication and division C1. Patterns and Relations patterns C2. Equations and Inequalities variables and expressions equalities and inequalities C3. Coding coding skills C4. Mathematical Modelling D1. Data Literacy data collection and organization data visualization data analysis D2. Probability E1. Geometric and Spatial Reasoning geometric reasoning location and movement E2. Measurement attributes length mass, capacity and volume area and surface area angles time the metric system Grades 1 to 8: F1. Money money concepts Grades 4 to 8: F1.  Finances financial management consumer and civic awareness

There is strong evidence that developing social-emotional learning skills at school contributes to all students’ overall health and well-being and to successful academic performance. It also supports positive mental health, as well as students’ ability to learn, build resilience, and thrive. The development of social-emotional learning skills throughout their school years will support all students in becoming healthier and more successful in their daily lives and as contributing members of society. In all grades, learning related to the expectations in this strand occurs in the context of learning related to the other five strands and is assessed and evaluated within these contexts.

Social-emotional learning skills can be developed across all subjects of the curriculum – including mathematics – as well as during various school activities, at home, and in the community. These skills support students in understanding mathematical concepts and in applying the mathematical processes that are key to learning and doing mathematics. They help all students – and indeed all learners, including educators and parents – develop confidence, cope with challenges, and think critically. This in turn enables them to improve and demonstrate mathematics knowledge, concepts, and skills in a variety of situations. Social-emotional learning skills help every student develop a positive identity as a capable “math learner”.

In all grades, Strand A comprises a single overall expectation and a chart listing the social-emotional learning skills, the mathematical processes, and the expected outcomes when students use these skills and processes to show their understanding and application of the mathematical content. The progression of learning from grade to grade is indicated in the examples, which are linked to each social-emotional learning skill in each grade and which highlight how the skills can be integrated with learning across the other five strands. The content and application of the learning changes as students develop and mature. Students’ application of the social-emotional learning skills and mathematical processes must be assessed and evaluated as a part of their achievement of the overall expectations in each of the strands for every grade.

The chart in Strand A outlines the social-emotional learning skills, the mathematical processes, and the expected outcomes when students apply both as they learn and do mathematics. The interaction of skills and processes is variable: Different social-emotional learning skills may be applied at different times in connection with different mathematical processes to achieve the outcomes.

#### Social-Emotional Learning Skills: Key Components and Sample Strategies

The following chart provides detailed information about each of the skills, including key ideas and sample strategies.

 Skills What are the skills? How do they help? What do they look like in mathematics? Key Components and Sample Strategies Identification and Management of Emotions Students often experience a range of emotions over the course of their day at school. They may feel happy, sad, angry, frustrated, or excited, or any number of emotions in combination. Students, and especially younger children, may struggle to identify and appropriately express their feelings. Learning to recognize different emotions, and to manage them effectively, can help students function and interact more successfully. When students understand the influence of thoughts and emotions on behaviour, they can improve the quality of their interactions. In mathematics, as they learn new mathematics concepts and interact with others while problem solving, students have many opportunities to develop awareness of their emotions and to use communication skills to express their feelings and to respond constructively when they recognize emotions in others. Recognizing a range of emotions in self and others Gauging the intensity and/or the level of emotion Understanding connections between thoughts, feelings, and actions Recognizing that new or challenging learning may involve a sense of excitement or an initial sense of discomfort Managing strong emotions and using strategies to self-regulate Applying strategies such as: using a “feelings chart” to learn words to express feelings using a “feelings thermometer” or pictures to gauge intensity of emotion Stress Management and Coping Every day, students are exposed to a range of challenges that can contribute to feelings of stress. As they learn stress management and coping skills, they come to recognize that stress is a part of life and that it can be managed. We can learn ways to respond to challenges that enable us to “bounce back” and, in this way, build resilience in the face of life’s obstacles. Over time, with support, practice, feedback, reflection, and experience, students begin to build a set of personal coping strategies that they can carry with them through life. In mathematics, students work through challenging problems, understanding that their resourcefulness in using coping strategies is helping them build personal resilience. Problem solving Seeking support from peers, teachers, family, or their extended community Managing stress through physical activity Applying strategies such as: breaking a task or problem down into pieces and tackling one piece at a time thinking of a similar problem deep breathing guided imagery stretching pausing and reflecting Positive Motivation and Perseverance Positive motivation and perseverance skills help students to “take a long view” and remain hopeful even when their personal and/or immediate circumstances are difficult. With regular use, practices and habits of mind that promote positive motivation help students approach challenges in life with an optimistic and positive mindset and an understanding that there is struggle in most successes and that repeated effort can lead to success. These practices include noticing strengths and positive aspects of experiences, reframing negative thoughts, expressing gratitude, practising optimism, and practising perseverance – appreciating the value of practice, of making mistakes, and of the learning process. In mathematics students have regular opportunities to apply these practices as they solve problems and develop an appreciation for learning from mistakes as a part of the learning process. Reframing negative thoughts and experiences Practising perseverance Embracing mistakes as a necessary and helpful part of learning Reflecting on things to be grateful for and expressing gratitude Practising optimism Applying strategies such as: using an iterative approach by trying out different methods, including estimating and guessing and checking, to support problem solving supporting peers by encouraging them to keep trying if they make a mistake using personal affirmations like “I can do this.” Healthy Relationship Skills When students interact in positive and meaningful ways with others, mutually respecting diversity of thought and expression, their sense of belonging within the school and community is enhanced. Learning healthy relationship skills helps students establish positive patterns of communication and inspires healthy, cooperative relationships. These skills include the ability to understand and appreciate another person’s perspective, to empathize with others, to listen attentively, to be assertive, and to apply conflict-resolution skills. In mathematics, students have opportunities to develop and practise skills that support positive interaction with others as they work together in small groups or in pairs to solve math problems and confront challenges. Developing these skills helps students to communicate with teachers, peers, and family about mathematics with an appreciation of the beauty and wonder of mathematics. Being cooperative and collaborative Using conflict-resolution skills Listening attentively Being respectful Considering other ideas and perspectives Practising kindness and empathy Applying strategies such as: seeking opportunities to help others taking turns playing different roles (e.g., leader, scribe or illustrator, data collector, observer) when working in groups Self-Awareness and Sense of Identity Knowing who we are and having a sense of purpose and meaning in our lives enables us to function in the world as self-aware individuals. Our sense of identity enables us to make choices that support our well-being and allows us to connect with and have a sense of belonging in various cultural and social communities. Educators should note that for First Nations, Métis, and Inuit students, the term “sense of identity and belonging” may also mean belonging to and identifying with a particular community and/or nation. Self-awareness and identity skills help students explore who they are – their strengths, difficulties, preferences, interests, values, and ambitions – and how their social and cultural contexts have influenced them. In mathematics, as they learn new skills, students use self-awareness skills to monitor their progress and identify their strengths; in the process, they build their identity as capable math learners. Educators play a key role in reinforcing that everyone – students, educators, and parents – is a math learner and in sharing an appreciation of the beauty and wonder of mathematics. Knowing oneself Caring for oneself Having a sense of mattering and of purpose Identifying personal strengths Having a sense of belonging and community Communicating their thinking, positive emotions, and excitement about mathematics Applying strategies such as: building their identity as a math learner as they learn independently as a result of their efforts and challenges monitoring progress in skill development reflecting on strengths and accomplishments and sharing these with peers or caring adults Critical and Creative Thinking Critical and creative thinking skills enable us to make informed judgements and decisions on the basis of a clear and full understanding of ideas and situations, and their implications, in a variety of settings and contexts. Students learn to question, interpret, predict, analyse, synthesize, detect bias, and distinguish between alternatives. They practise making connections, setting goals, creating plans, making and evaluating decisions, and analysing and solving problems for which there may be no clearly defined answers. Executive functioning skills – the skills and processes that allow us to take initiative, focus, plan, retain and transfer learning, and determine priorities – also support critical and creative thinking. In all aspects of the mathematics curriculum, students have opportunities to develop critical and creative thinking skills. Students have opportunities to build on prior learning, go deeper, and make personal connections through real-life applications. Making connections Making decisions Evaluating choices, reflecting on and assessing strategies Communicating effectively Managing time Setting goals Applying organizational skills Applying strategies such as: determining what is known and what needs to be found using various webs, charts, diagrams, and representations to help identify connections and interrelationships using organizational strategies and tools, such as planners, trackers, and goal-setting frameworks

Understanding how numbers work is foundational to many aspects of mathematics. Recognizing and understanding number properties is foundational to developing an understanding of branches of mathematics such as arithmetic and algebra. In the Number strand, as students progress through Grades 1 to 8, they learn about different types of numbers and how those numbers behave when various operations are applied to them.

A vital aspect of number work in elementary grades is to build what is often called number sense, where students develop the ability to flexibly relate numbers and relate computations. Students who have developed number sense regularly use number relationships to make sense of calculations and to assess the reasonableness of numbers used to describe situations, for example, in the media.

Students learn to count effectively and then become fluent with math facts in order to perform calculations efficiently and accurately, whether mentally or by using algorithms on paper. This strand is built on the belief that it is important to develop automaticity, which is the ability to use mathematical skills or perform mathematical procedures with little or no mental effort. Automaticity with math facts enables students to engage in critical thinking and problem solving.

Most students learn math facts gradually over a number of years, connecting to prior knowledge, using tools and calculators. Mastery comes with practice, and practice helps to build fluency and depth. Students draw on their ability to apply math facts as they manipulate algebraic expressions, equations, and inequalities. Mental math skills involve the ability to perform mathematical calculations without relying on pencil and paper. They enable students to estimate answers to calculations, and so to work accurately and efficiently on everyday problems and judge the reasonableness of answers that they have arrived at through calculation. In order to develop effective mental math strategies, all students need to have strong skills in number sense and a solid conceptual understanding of the operations.

Though individual students may progress at different rates, generally speaking, addition/subtraction facts should be mastered by the end of Grade 3, and multiplication/division facts should be mastered by the end of Grade 5. However, all students should continue to learn about effective strategies and to practice and extend their proficiency in the operations throughout the grades and in the context of learning in all the strands of the mathematics curriculum.

In this strand, students develop algebraic reasoning through working with patterns, variables, expressions, equations, inequalities, coding, and the process of mathematical modelling.

As students progress through the grades, they study a variety of patterns, including those found in real life. Students learn to identify regularities in numeric and non-numeric patterns and classify them based on the characteristics of those regularities. They create and translate patterns using various representations. Students determine pattern rules for various patterns in order to extend them, make near and far predictions, and determine their missing elements. They develop recursive and functional thinking as well as additive and multiplicative thinking as they work with linear patterns, and use this thinking to develop the general terms of the patterns to solve for unknown values. Understanding patterns and determining the relationship between two variables has many connections to science and is foundational to further mathematics. In the primary grades, students focus on understanding which quantities remain the same and which can change in everyday contexts, and on how to establish equality between numerical expressions. In the junior and intermediate grades, students work with variables in algebraic expressions, equations, and inequalities in various contexts.

As students progress through the grades, their coding experiences also progress, from representing movements on a grid, to solving problems involving optimization, to manipulating models to find which one best fits the data they are working with in order to make predictions. Coding can be incorporated across all strands and provides students with opportunities to apply and extend their math thinking, reasoning, and communicating.

Students in all grades also engage in the process of mathematical modelling.

#### The Mathematical Modelling Process

Mathematical modelling provides authentic connections to real-life situations. The process starts with ill-defined, often messy real-life problems that may have several different solutions that are all correct. Mathematical modelling requires the modeller to be critical and creative and make choices, assumptions, and decisions. Through this process, they create a mathematical model that describes a situation using mathematical concepts and language, and that can be used to solve a problem or make decisions and can be used to deepen understanding of mathematical concepts.

The process of mathematical modelling has four key components that are interconnected and applied in an iterative way, where students may move between and across, as well as return to, each of the four components as they change conditions to observe new outcomes until the model is ready to be shared and acted upon. While moving through these components, social-emotional learning skills and mathematical processes are applied as needed.

1. Understand the problem

• What information is needed?

2. Analyse the situation

• What assumptions do I make about the situation?
• What changes, what remains the same?

3. Create a mathematical model

• What representations, tools, technologies, and strategies will help build the model?
• What mathematical knowledge, concepts, and skills might be involved?

4. Analyse and assess the model

• Can this model provide a solution?
• What are alternative models?

The related topics of statistics and probability, which are addressed in this strand, are highly relevant to real life. The public is bombarded with data through advertising, opinion polls, politics, population trends, and scientific discoveries, to name just a few. Thus, one of the key focuses in this strand is to support students in developing critical thinking skills throughout their development of data literacy, so that they can analyse, synthesize, understand, generate, and use data, both as consumers and producers.

The main purpose for collecting and organizing data is to gather information in order to answer questions. When questions stimulate students’ curiosity, they become engaged in collecting, organizing, and interpreting the data that provide answers to their questions. Relevant questions often arise from class discussions; classroom events, issues, and thematic activities; and topics in various subject areas. When students collect and organize data, they have an opportunity to learn more about themselves, their environment, issues in their school or community, and so on. Learning activities should help students understand the processes that are involved in formulating questions, seeking relevant information, and organizing that information in meaningful ways. Involving students in collecting and organizing data allows them to participate in the decision making that is required at different steps of the process.

As students progress through the grades, they develop an understanding of qualitative data and both discrete and continuous quantitative data, and use that understanding to select appropriate ways to organize and display data. Students learn the fundamentals of statistics and develop the skills to visualize and critically analyse data, including identifying any possible biases within the data. Starting in the junior grades, students make intentional choices in creating infographics in order to represent key information about a set of data for a particular audience and to engage in the critical interpretation of data. In addition, students learn how to use data to make compelling arguments about questions of interest.

The learning in this strand also supports students in developing probabilistic reasoning. As students progress through the grades, they begin to understand the relationship between probability and data, and how data is used to make predictions about populations. Students’ intuitive understanding of probability is nurtured in the early grades to help them make connections to their prior experience with probability in everyday life, beginning with simply understanding that some events are likely to happen while others are not likely. Eventually, students begin to understand and represent these probabilities as fractions, decimals and percents. From Grades 5 to 8, students compare experimental probabilities involving independent and dependent events with their theoretical probabilities, and use these measures to make predictions about events.

This strand combines the areas of geometry and measurement in order to emphasize the relationship between the two areas and to highlight the role of spatial reasoning in underpinning the development of both. Study in this strand provides students with the language and tools to analyse, compare, describe, and navigate the world around them. It is a gateway to professions in other STEM (science, technology, engineering, and mathematics) disciplines, and builds foundational skills needed for construction, architecture, engineering, research, and design.

In this strand, students analyse the properties of shapes – the elements that define a shape and make it unique – and use these properties to define, compare, and construct shapes and objects, as well as to explore relationships among properties. Students begin with an intuition about their surroundings and the objects in them, and learn to visualize objects from different perspectives. Over time, students develop an increasingly sophisticated understanding of size, shape, location, movement, and change, in both two and three dimensions. They understand and choose appropriate units to estimate, measure, and compare attributes, and they use appropriate tools to make measurements. They apply their understanding of the relationships between shapes and measurement to develop formulas to calculate length, area, volume, and more.

All Ontario students need the skills and knowledge to take responsibility for managing their personal financial well-being with confidence, competence, a critical and compassionate awareness of the world around them.

Financial Literacy is a dedicated strand throughout the elementary math curriculum. Financial literacy is more than just knowing about money and financial matters and having the skills to work with this knowledge. Students develop the confidence and capacity to successfully apply the necessary knowledge, concepts, and skills in a range of relevant real-life contexts and for a range of purposes. They also develop the ability to make informed decisions as consumers and citizens while taking into account the ethical, societal, environmental, and personal aspects of those decisions.

In Grades 1 to 3, students demonstrate an understanding of the value and use of money by recognizing Canadian coins and bills, representing various amounts, and calculating change in simple transactions. In Grades 4 to 8, students extend their learning to the knowledge, concepts, and skills required to make informed financial decisions relevant to their lived experiences and plan simple sample  budgets. Students begin to develop consumer and civic awareness in the junior and intermediate grades. Making connections to what they are learning in the Media Literacy strand of the language curriculum as well as the social studies, history and geography curriculum, students become informed consumers and learn about the broader economic systems in their local communities, communities in other global contexts that their families are connected to, and beyond. Educators consider and respond to the range of equity issues related to the diverse circumstances and lived experiences of students and their families.

This strand connects with other mathematics strands in many ways, such as applying knowledge, concepts, and skills related to:

• numbers and operations to calculate change;
• percents to calculate sales tax and interest;
• mathematical modelling to understand real-life financial situations, including the financial applications of linear rates;
• unit rates to compare goods and services, and mental math to quickly determine those with the best value;
• social-emotional learning to become confident and critical consumers, and to persevere in managing financial well-being.