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### Mathematics Kindergarten to Grade 9 (2007, Updated 2016)

 Rationale & Philosophy
 Outcomes

 Number (K) Patterns & Relations - Patterns (K) Shape & Space - Measurement (K) Shape & Space - 3-D Objects & 2-D Shapes (K)

 Number (Gr. 1) Patterns & Relations - Patterns (Gr. 1) Patterns & Relations - Variables & Equations (Gr. 1) Shape & Space - Measurement (Gr. 1) Shape & Space - 3-D Objects & 2-D Shapes (Gr. 1)

 Number (Gr. 2) Patterns & Relations - Patterns (Gr. 2) Patterns & Relations - Variables & Equations (Gr. 2) Shape & Space - Measurement (Gr. 2) Shape & Space - 3-D Objects & 2-D Shapes (Gr. 2) Statistics & Probability - Data Analysis (Gr. 2)

 Number (Gr. 3) Patterns & Relations - Patterns (Gr. 3) Patterns & Relations - Variables & Equations (Gr. 3) Shape & Space - Measurement (Gr. 3) Shape & Space - 3-D Objects & 2-D Shapes (Gr. 3) Statistics & Probability - Data Analysis (Gr. 3)

 Number (Gr. 4) Patterns & Relations - Patterns (Gr. 4) Patterns & Relations - Variables & Equations (Gr. 4) Shape & Space - Measurement (Gr. 4) Shape & Space - 3-D Objects & 2-D Shapes (Gr. 4) Shape & Space - Transformations (Gr. 4) Statistics & Probability - Data Analysis (Gr. 4)

 Number (Gr. 5) Patterns & Relations - Patterns (Gr. 5) Patterns & Relations - Variables & Equations (Gr. 5) Shape & Space - Measurement (Gr. 5) Shape & Space - 3-D Objects & 2-D Shapes (Gr. 5) Shape & Space - Transformations (Gr. 5) Statistics & Probability - Data Analysis (Gr. 5) Statistics & Probability - Chance & Uncertainty (Gr. 5)

 Number (Gr. 6) Patterns & Relations - Patterns (Gr. 6) Patterns & Relations - Variables & Equations (Gr. 6) Shape & Space - Measurement (Gr. 6) Shape & Space - 3-D Objects & 2-D Shapes (Gr. 6) Shape & Space - Transformations (Gr. 6) Statistics & Probability - Data Analysis (Gr. 6) Statistics & Probability - Chance & Uncertainty (Gr. 6)

 Number (Gr. 7) Patterns & Relations - Patterns (Gr. 7) Patterns & Relations - Variables & Equations (Gr. 7) Shape & Space - Measurement (Gr. 7) Shape & Space - 3-D Objects & 2-D Shapes (Gr. 7) Shape & Space - Transformations (Gr. 7) Statistics & Probability - Data Analysis (Gr. 7) Statistics & Probability - Chance & Uncertainty (Gr. 7)

 Number (Gr. 8) Patterns & Relations - Patterns (Gr. 8) Patterns & Relations - Variables & Equations (Gr. 8) Shape & Space - Measurement (Gr. 8) Shape & Space - 3-D Objects & 2-D Shapes (Gr. 8) Shape & Space - Transformations (Gr. 8) Statistics & Probability - Data Analysis (Gr. 8) Statistics & Probability - Chance & Uncertainty (Gr. 8)

 Number (Gr. 9) Patterns & Relations - Patterns (Gr. 9) Patterns & Relations - Variables & Equations (Gr. 9) Shape & Space - Measurement (Gr. 9) Shape & Space - 3-D Objects & 2-D Shapes (Gr. 9) Shape & Space - Transformations (Gr. 9) Statistics & Probability - Data Analysis (Gr. 9) Statistics & Probability - Chance & Uncertainty (Gr. 9)
 Appendices

 There are 3 Teacher Resources related to the entire Program of Studies
Introduction
Mathematics Kindergarten to Grade 9 (2007, Updated 2016)

The Mathematics Kindergarten to Grade 9 Program of Studies has been derived from The Common Curriculum Framework for K - 9 Mathematics: Western and Northern Canadian Protocol, May 2006 (the Common Curriculum Framework). The program of studies incorporates the conceptual framework for Kindergarten to Grade 9 Mathematics and the general outcomes and specific outcomes that were established in the Common Curriculum Framework.

Background

The Common Curriculum Framework was developed by the seven ministries of education (Alberta, British Columbia, Manitoba, Northwest Territories, Nunavut, Saskatchewan and Yukon Territory) in collaboration with teachers, administrators, parents, business representatives, post-secondary educators and others. The framework identifies beliefs about mathematics, general and specific student outcomes, and achievement indicators agreed upon by the seven jurisdictions.

Beliefs About Students & Mathematics Learning

Students are curious, active learners with individual interests, abilities and needs. They come to classrooms with varying knowledge, life experiences and backgrounds. A key component in successfully developing numeracy is making connections to these backgrounds and experiences.

Students learn by attaching meaning to what they do, and they need to construct their own meaning of mathematics. This meaning is best developed when learners encounter mathematical experiences that proceed from the simple to the complex and from the concrete to the abstract. Through the use of manipulatives and a variety of pedagogical approaches, teachers can address the diverse learning styles, cultural backgrounds and developmental stages of students, and enhance within them the formation of sound, transferable mathematical understandings. At all levels, students benefit from working with a variety of materials, tools and contexts when constructing meaning about new mathematical ideas. Meaningful student discussions provide essential links among concrete, pictorial and symbolic representations of mathematical concepts.

The learning environment should value and respect the diversity of students' experiences and ways of thinking, so that students are comfortable taking intellectual risks, asking questions and posing conjectures. Students need to explore problem-solving situations in order to develop personal strategies and become mathematically literate. They must realize that it is acceptable to solve problems in a variety of ways and that a variety of solutions may be acceptable.

First Nations, Métis & Inuit Perspectives

First Nations, Métis and Inuit students in northern and western Canada come from diverse geographic areas with varied cultural and linguistic backgrounds. Students attend schools in a variety of settings, including urban, rural and isolated communities. Teachers need to understand the diversity of students' cultures and experiences.

First Nations, Métis and Inuit students often have a holistic view of the environment - they look for connections in learning and learn best when mathematics is contextualized. They may come from cultures where learning takes place through active participation. Traditionally, little emphasis was placed upon the written word, so oral communication and practical applications and experiences are important to student learning and understanding. By understanding and responding to nonverbal cues, teachers can optimize student learning and mathematical understanding.

A variety of teaching and assessment strategies help build upon the diverse knowledge, cultures, communication styles, skills, attitudes, experiences and learning styles of students.

Research indicates that when strategies go beyond the incidental inclusion of topics and objects unique to a culture or region, greater levels of understanding can be achieved (Banks and Banks, 1993).

Affective Domain

A positive attitude is an important aspect of the affective domain and has a profound impact on learning. Environments that create a sense of belonging, encourage risk taking and provide opportunities for success help develop and maintain positive attitudes and self-confidence within students. Students with positive attitudes toward learning mathematics are likely to be motivated and prepared to learn, participate willingly in classroom activities, persist in challenging situations and engage in reflective practices.

Teachers, students and parents need to recognize the relationship between the affective and cognitive domains, and attempt to nurture those aspects of the affective domain that contribute to positive attitudes. To experience success, students must be taught to set achievable goals and assess themselves as they work toward these goals.

Striving toward success and becoming autonomous and responsible learners are ongoing, reflective processes that involve revisiting the setting and assessing of personal goals.

Early Childhood

Young children are naturally curious and develop a variety of mathematical ideas before they enter Kindergarten. Children make sense of their environment through observations and interactions at home, in daycares, in preschools and in the community. Mathematics learning is embedded in everyday activities, such as playing, reading, beading, baking, storytelling and helping around the home.

Activities can contribute to the development of number and spatial sense in children. Curiosity about mathematics is fostered when children are engaged in, and talking about, such activities as comparing quantities, searching for patterns, sorting objects, ordering objects, creating designs and building with blocks.

Positive early experiences in mathematics are as critical to child development as are early literacy experiences.

Goals For Students

The main goals of mathematics education are to prepare students to:

• use mathematics confidently to solve problems
• communicate and reason mathematically
• appreciate and value mathematics
• make connections between mathematics and its applications
• commit themselves to lifelong learning
• become mathematically literate adults, using mathematics to contribute to society.

Students who have met these goals will:

• gain understanding and appreciation of the contributions of mathematics as a science, philosophy and art
• exhibit a positive attitude toward mathematics
• engage and persevere in mathematical tasks and projects
• contribute to mathematical discussions
• take risks in performing mathematical tasks
• exhibit curiosity.
Conceptual Framework For K–9 Mathematics

The chart below provides an overview of how mathematical processes and the nature of mathematics influence learning outcomes.

★ Achievement indicators for the prescribed program of studies outcomes are provided in the companion documentAlberta K–9 Mathematics Achievement Indicators, 2016

Mathematical Processes

There are critical components that students must encounter in a mathematics program in order to achieve the goals of mathematics education and embrace lifelong learning in mathematics.

Students are expected to:

Communication

communicate in order to learn and express their understanding

Connections

connect mathematical ideas to other concepts in mathematics, to everyday experiences and to other disciplines

Mental Mathematics and Estimation

demonstrate fluency with mental mathematics and estimation

Problem Solving

develop and apply new mathematical knowledge through problem solving

Reasoning

develop mathematical reasoning

Technology

select and use technologies as tools for learning and for solving problems

Visualization

develop visualization skills to assist in processing information, making connections and solving problems.

The program of studies incorporates these seven interrelated mathematical processes that are intended to permeate teaching and learning.

Communication

Students need opportunities to read about, represent, view, write about, listen to and discuss mathematical ideas. These opportunities allow students to create links between their own language and ideas, and the formal language and symbols of mathematics.

Communication is important in clarifying, reinforcing and modifying ideas, attitudes and beliefs about mathematics. Students should be encouraged to use a variety of forms of communication while learning mathematics. Students also need to communicate their learning using mathematical terminology.

Communication helps students make connections among concrete, pictorial, symbolic, oral, written and mental representations of mathematical ideas.

Connections

Contextualization and making connections to the experiences of learners are powerful processes in developing mathematical understanding. This can be particularly true for First Nations, Métis and Inuit learners. When mathematical ideas are connected to each other or to real-world phenomena, students begin to view mathematics as useful, relevant and integrated.

Learning mathematics within contexts and making connections relevant to learners can validate past experiences and increase student willingness to participate and be actively engaged.

The brain is constantly looking for and making connections. "Because the learner is constantly searching for connections on many levels, educators need to orchestrate the experiences from which learners extract understanding... Brain research establishes and confirms that multiple complex and concrete experiences are essential for meaningful learning and teaching" (Caine and Caine, 1991, p. 5).

Mental Mathematics and Estimation

Mental mathematics is a combination of cognitive strategies that enhance flexible thinking and number sense. It is calculating mentally without the use of external memory aids.

Mental mathematics enables students to determine answers without paper and pencil. It improves computational fluency by developing efficiency, accuracy and flexibility.

"Even more important than performing computational procedures or using calculators is the greater facility that students need - more than ever before - with estimation and mental math" (National Council of Teachers of Mathematics, May 2005).

Students proficient with mental mathematics "become liberated from calculator dependence, build confidence in doing mathematics, become more flexible thinkers and are more able to use multiple approaches to problem solving" (Rubenstein, 2001, p. 442).

Mental mathematics "provides the cornerstone for all estimation processes, offering a variety of alternative algorithms and nonstandard techniques for finding answers" (Hope, 1988, p. v).

Estimation is used for determining approximate values or quantities or for determining the reasonableness of calculated values. It often uses benchmarks or referents. Students need to know when to estimate, how to estimate and what strategy to use.

Estimation assists individuals in making mathematical judgements and in developing useful, efficient strategies for dealing with situations in daily life.

Problem Solving

Learning through problem solving should be the focus of mathematics at all grade levels. When students encounter new situations and respond to questions of the type How would you...? or How could you...?, the problem-solving approach is being modelled. Students develop their own problem-solving strategies by listening to, discussing and trying different strategies.

A problem-solving activity must ask students to determine a way to get from what is known to what is sought. If students have already been given ways to solve the problem, it is not a problem, but practice. A true problem requires students to use prior learnings in new ways and contexts. Problem solving requires and builds depth of conceptual understanding and student engagement.

Problem solving is a powerful teaching tool that fosters multiple, creative and innovative solutions. Creating an environment where students openly look for, and engage in, finding a variety of strategies for solving problems empowers students to explore alternatives and develops confident, cognitive mathematical risk takers.

Reasoning

Mathematical reasoning helps students think logically and make sense of mathematics. Students need to develop confidence in their abilities to reason and justify their mathematical thinking. High-order questions challenge students to think and develop a sense of wonder about mathematics.

Mathematical experiences in and out of the classroom provide opportunities for students to develop their ability to reason. Students can explore and record results, analyze observations, make and test generalizations from patterns, and reach new conclusions by building upon what is already known or assumed to be true.

Reasoning skills allow students to use a logical process to analyze a problem, reach a conclusion and justify or defend that conclusion.

Technology

Technology contributes to the learning of a wide range of mathematical outcomes and enables students to explore and create patterns, examine relationships, test conjectures and solve problems.

Calculators and computers can be used to:

• explore and demonstrate mathematical relationships and patterns
• organize and display data
• extrapolate and interpolate
• assist with calculation procedures as part of solving problems
• decrease the time spent on computations when other mathematical learning is the focus
• reinforce the learning of basic facts
• develop personal procedures for mathematical operations
• create geometric patterns
• simulate situations
• develop number sense.

Technology contributes to a learning environment in which the growing curiosity of students can lead to rich mathematical discoveries at all grade levels.

Visualization

Visualization "involves thinking in pictures and images, and the ability to perceive, transform and recreate different aspects of the visual-spatial world" (Armstrong, 1993, p. 10). The use of visualization in the study of mathematics provides students with opportunities to understand mathematical concepts and make connections among them.

Visual images and visual reasoning are important components of number, spatial and measurement sense. Number visualization occurs when students create mental representations of numbers.

Being able to create, interpret and describe a visual representation is part of spatial sense and spatial reasoning. Spatial visualization and reasoning enable students to describe the relationships among and between 3-D objects and 2-D shapes.

Measurement visualization goes beyond the acquisition of specific measurement skills. Measurement sense includes the ability to determine when to measure, when to estimate and which estimation strategies to use (Shaw and Cliatt, 1989).

Visualization is fostered through the use of concrete materials, technology and a variety of visual representations.

Nature of Mathematics

Mathematics is one way of trying to understand, interpret and describe our world. There are a number of components that define the nature of mathematics and these are woven throughout this program of studies. The components are change, constancy, number sense, patterns, relationships, spatial sense and uncertainty.

Change

It is important for students to understand that mathematics is dynamic and not static. As a result, recognizing change is a key component in understanding and developing mathematics.

Within mathematics, students encounter conditions of change and are required to search for explanations of that change. To make predictions, students need to describe and quantify their observations, look for patterns, and describe those quantities that remain fixed and those that change. For example, the sequence 4, 6, 8, 10, 12, ... can be described as:

• the number of a specific colour of beads in each row of a beaded design
• skip counting by 2s, starting from 4
• an arithmetic sequence, with first term 4 and a common difference of 2
• a linear function with a discrete domain (Steen, 1990, p. 184).
Constancy

Different aspects of constancy are described by the terms stability, conservation, equilibrium, steady state and symmetry (AAAS - Benchmarks, 1993, p. 270). Many important properties in mathematics and science relate to properties that do not change when outside conditions change. Examples of constancy include the following:

• The ratio of the circumference of a teepee to its diameter is the same regardless of the length of the teepee poles.
• The sum of the interior angles of any triangle is 180°.
• The theoretical probability of flipping a coin and getting heads is 0.5.

Some problems in mathematics require students to focus on properties that remain constant. The recognition of constancy enables students to solve problems involving constant rates of change, lines with constant slope, direct variation situations or the angle sums of polygons.

Number Sense

Number sense is an intuition about numbers. Number sense develops when students connect numbers to their own real-life experiences and when students use benchmarks and referents. This results in students who are computationally fluent and flexible with numbers.

A true sense of number includes and goes beyond the skills of counting, memorizing facts and the situational rote use of algorithms. Mastery of number facts occurs when students understand and recall facts and is expected to be attained by students as they develop their number sense. This mastery allows for application of number facts and facility with more complex computations.

Number sense can be developed by providing rich mathematical tasks that allow students to make connections to their own experiences and their previous learning.

Patterns

Mathematics is about recognizing, describing and working with numerical and non-numerical patterns. Patterns exist in all strands of this program of studies.

Working with patterns enables students to make connections within and beyond mathematics. These skills contribute to students' interaction with, and understanding of, their environment.

Patterns may be represented in concrete, visual or symbolic form. Students should develop fluency in moving from one representation to another.

Students must learn to recognize, extend, create and use mathematical patterns. Patterns allow students to make predictions and justify their reasoning when solving routine and nonroutine problems.

Learning to work with patterns in the early grades helps students develop algebraic thinking, which is foundational for working with more abstract mathematics in higher grades.

Relationships

Mathematics is one way to describe interconnectedness in a holistic worldview. Mathematics is used to describe and explain relationships. As part of the study of mathematics, students look for relationships among numbers, sets, shapes, objects and concepts. The search for possible relationships involves collecting and analyzing data and describing relationships visually, symbolically, orally or in written form.

Spatial Sense

Spatial sense involves visualization, mental imagery and spatial reasoning. These skills are central to the understanding of mathematics.

Spatial sense is developed through a variety of experiences and interactions within the environment. The development of spatial sense enables students to solve problems involving 3-D objects and 2-D shapes and to interpret and reflect on the physical environment and its 3-D or 2-D representations.

Some problems involve attaching numerals and appropriate units (measurement) to dimensions of shapes and objects. Spatial sense allows students to make predictions about the results of changing these dimensions; e.g., doubling the length of the side of a square increases the area by a factor of four. Ultimately, spatial sense enables students to communicate about shapes and objects and to create their own representations.

Uncertainty

In mathematics, interpretations of data and the predictions made from data may lack certainty.

Events and experiments generate statistical data that can be used to make predictions. It is important to recognize that these predictions (interpolations and extrapolations) are based upon patterns that have a degree of uncertainty.

The quality of the interpretation is directly related to the quality of the data. An awareness of uncertainty allows students to assess the reliability of data and data interpretation.

Chance addresses the predictability of the occurrence of an outcome. As students develop their understanding of probability, the language of mathematics becomes more specific and describes the degree of uncertainty more accurately.

Strands

The learning outcomes in the program of studies are organized into four strands across the grades K–9. Some strands are subdivided into substrands. There is one general outcome per substrand across the grades K–9.

The strands and substrands, including the general outcome for each, follow.

Number

• Develop number sense.

Patterns and Relations

Patterns

• Use patterns to describe the world and to solve problems.

Variables and Equations

• Represent algebraic expressions in multiple ways.

Shape and Space

Measurement

• Use direct and indirect measurement to solve problems.

3-D Objects and 2-D Shapes

• Describe the characteristics of 3-D objects and 2-D shapes, and analyze the relationships among them.

Transformations

• Describe and analyze position and motion of objects and shapes.

Statistics And Probability

Data Analysis

• Collect, display and analyze data to solve problems.

Chance and Uncertainty

• Use experimental or theoretical probabilities to represent and solve problems involving uncertainty.

An across-the-grades listing of outcomes by strand is provided in Appendix 1.

Outcomes

The program of studies is stated in terms of general outcomes and specific outcomes.

General outcomes are overarching statements about what students are expected to learn in each strand/substrand. The general outcome for each strand/substrand is the same throughout the grades.

Specific outcomes are statements that identify the specific skills, understanding and knowledge that students are required to attain by the end of a given grade.

In the specific outcomes, the word including indicates that any ensuing items must be addressed to fully meet the learning outcome. The phrase such as indicates that the ensuing items are provided for illustrative purposes or clarification and are not requirements that must be addressed to fully meet the learning outcome. Students investigate a variety of strategies, including standard/traditional algorithms, to become proficient in at least one appropriate and efficient strategy that they understand. The teaching professional has the flexibility and responsibility to meet the learning needs of each of his or her students. Over time, students refine their strategies to increase their accuracy and efficiency.

Notes are statements that clarify the intent of a learning outcome. Notes guide the teaching professional in making judgements about teaching and learning.

Notes in some Number outcomes for grades 2–5 highlight opportunities for students to investigate standard/traditional algorithms as a strategy for operations with whole numbers. The intent of these notes is to ensure that standard/traditional algorithms are explicitly included in students’ learning experiences. Students would then use their preferred strategy to demonstrate understanding of each outcome.

Notes in some outcomes for grades 4–9 highlight opportunities for students to maintain and refine previous learnings related to number facts and operations with whole numbers, fractions and integers. The intent of these notes is to indicate that through these outcomes, previous knowledge can be maintained. There may be other outcomes that provide similar opportunities for maintaining previous learning throughout the year.

Links to Information and Communication Technology (ICT) Outcomes

Some curriculum outcomes from Alberta Education's Information and Communication Technology (ICT) Program of Studies can be linked to outcomes in the mathematics program so that students will develop a broad perspective on the nature of technology, learn how to use and apply a variety of technologies, and consider the impact of ICT on individuals and society. The connection to ICT outcomes supports and reinforces the understandings and abilities that students are expected to develop through the general and specific outcomes of the mathematics program. Effective, efficient and ethical application of ICT outcomes contributes to the mathematics program vision.

Links to the ICT outcomes have been identified for some specific outcomes. These links appear in square brackets below the mathematical processes for an outcome, where appropriate. The complete wording of the relevant outcomes for ICT is provided in Appendix 2.

Summary

The conceptual framework for K–9 mathematics describes the nature of mathematics, mathematical processes and the mathematical concepts to be addressed in Kindergarten to Grade 9 mathematics. The components are not meant to stand alone. Activities that take place in the mathematics classroom should stem from a problem-solving approach, be based on mathematical processes and lead students to an understanding of the nature of mathematics through specific knowledge, skills and attitudes among and between strands.

Instructional Focus

The program of studies is arranged into four strands. These strands are not intended to be discrete units of instruction. The integration of outcomes across strands makes mathematical experiences meaningful. Students should make the connection between concepts both within and across strands.

Consider the following when planning for instruction:

• Integration of the mathematical processes within each strand is expected.
• Learning mathematics includes a balance between understanding, recalling and applying mathematical concepts.
• Problem solving, reasoning and connections are vital to increasing mathematical fluency and must be integrated throughout the program.
• There is to be a balance among mental mathematics and estimation, paper and pencil exercises, and the use of technology, including calculators and computers. Concepts should be introduced using manipulatives and be developed concretely, pictorially and symbolically.
• Students bring a diversity of learning styles and cultural backgrounds to the classroom. They will be at varying developmental stages.
Kindergarten
Number (K)
General Outcome: Develop number sense.

Specific Outcome 1

Say the number sequence 1 to 10 by 1s, starting anywhere from 1 to 10 and from 10 to 1.

Specific Outcome 2

Subitize (recognize at a glance) and name familiar arrangements of 1 to 5 objects or dots.

Specific Outcome 3

Relate a numeral, 1 to 10, to its respective quantity.

Specific Outcome 4

Represent and describe numbers 2 to 10, concretely and pictorially.

Specific Outcome 5

Compare quantities 1 to 10, using one-to-one correspondence.

Patterns & Relations - Patterns (K)
General Outcome: Use patterns to describe the world and to solve problems.

Specific Outcome 1

Demonstrate an understanding of repeating patterns (two or three elements) by:

• identifying
• reproducing
• extending
• creating
patterns using manipulatives, sounds and actions.

[ICT: P2–1.1]

Specific Outcome 2

Sort a set of objects based on a single attribute, and explain the sorting rule.

Shape & Space - Measurement (K)
General Outcome: Use direct and indirect measurement to solve problems.

Specific Outcome 1

Use direct comparison to compare two objects based on a single attribute, such as length (height), mass (weight) and volume (capacity).

Shape & Space - 3-D Objects & 2-D Shapes (K)
General Outcome: Describe the characteristics of 3-D objects and 2-D shapes, and analyze the relationships among them.

Specific Outcome 2

Sort 3-D objects, using a single attribute.

Specific Outcome 3

Build and describe 3-D objects.

Number (Gr. 1)
General Outcome: Develop number sense.

Specific Outcome 1

Say the number sequence 0 to 100 by:

• 1s forward between any two given numbers
• 1s backward from 20 to 0
• 2s forward from 0 to 20
• 5s and 10s forward from 0 to 100.

Specific Outcome 2

Subitize (recognize at a glance) and name familiar arrangements of 1 to 10 objects or dots.

Specific Outcome 3

Demonstrate an understanding of counting by:

• indicating that the last number said identifies "how many"
• showing that any set has only one count
• using counting-on
• using parts or equal groups to count sets.

Specific Outcome 4

Represent and describe numbers to 20, concretely, pictorially and symbolically.

Specific Outcome 5

Compare sets containing up to 20 elements, using:

• referents
• one-to-one correspondence
to solve problems.

Specific Outcome 6

Estimate quantities to 20 by using referents.

Specific Outcome 7

Demonstrate an understanding of conservation of number.

Specific Outcome 8

Identify the number, up to 20, that is:

• one more
• two more
• one less
• two less
than a given number.

Specific Outcome 9

Demonstrate an understanding of addition of numbers with answers to 20 and their corresponding subtraction facts, concretely, pictorially and symbolically, by:

• using familiar mathematical language to describe additive and subtractive actions
• creating and solving problems in context that involve addition and subtraction
• modelling addition and subtraction, using a variety of concrete and visual representations, and recording the process symbolically.

Specific Outcome 10

Describe and use mental mathematics strategies for basic addition facts and related subtraction facts to 18.

 Understand and apply strategies for addition facts up to and including 9 + 9 and related subtraction facts. Recall addition facts to a sum of 5 and related subtraction facts.

Patterns & Relations - Patterns (Gr. 1)
General Outcome: Use patterns to describe the world and to solve problems.

Specific Outcome 1

Demonstrate an understanding of repeating patterns (two to four elements) by:

• describing
• reproducing
• extending
• creating
patterns using manipulatives, diagrams, sounds and actions.

[ICT: P2–1.1]

Specific Outcome 2

Translate repeating patterns from one representation to another.

Specific Outcome 3

Sort objects, using one attribute, and explain the sorting rule.

Patterns & Relations - Variables & Equations (Gr. 1)
General Outcome: Represent algebraic expressions in multiple ways.

Specific Outcome 4

Describe equality as a balance and inequality as an imbalance, concretely and pictorially (0 to 20).

Specific Outcome 5

Record equalities, using the equal symbol.

Shape & Space - Measurement (Gr. 1)
General Outcome: Use direct and indirect measurement to solve problems.

Specific Outcome 1

Demonstrate an understanding of measurement as a process of comparing by:

• identifying attributes that can be compared
• ordering objects
• making statements of comparison
• filling, covering or matching.

Shape & Space - 3-D Objects & 2-D Shapes (Gr. 1)
General Outcome: Describe the characteristics of 3-D objects and 2-D shapes, and analyze the relationships among them.

Specific Outcome 2

Sort 3-D objects and 2-D shapes, using one attribute, and explain the sorting rule.

Specific Outcome 3

Replicate composite 2-D shapes and 3-D objects.

Specific Outcome 4

Compare 2-D shapes to parts of 3-D objects in the environment.

Number (Gr. 2)
General Outcome: Develop number sense.

Specific Outcome 1

Say the number sequence 0 to 100 by:

• 2s, 5s and 10s, forward and backward, using starting points that are multiples of 2, 5 and 10 respectively
• 10s, using starting points from 1 to 9
• 2s, starting from 1.

Specific Outcome 2

Demonstrate if a number (up to 100) is even or odd.

Specific Outcome 3

Describe order or relative position, using ordinal numbers (up to tenth).

Specific Outcome 4

Represent and describe numbers to 100, concretely, pictorially and symbolically.

Specific Outcome 5

Compare and order numbers up to 100.

Specific Outcome 6

Estimate quantities to 100, using referents.

Specific Outcome 7

Illustrate, concretely and pictorially, the meaning of place value for numerals to 100.

Specific Outcome 8

Demonstrate and explain the effect of adding zero to, or subtracting zero from, any number.

Specific Outcome 9

Demonstrate an understanding of addition (limited to 1- and 2-digit numerals) with answers to 100 and the corresponding subtraction by:

• using personal strategies for adding and subtracting with and without the support of manipulatives
• creating and solving problems that involve addition and subtraction
• using the commutative property of addition (the order in which numbers are added does not affect the sum)
• using the associative property of addition (grouping a set of numbers in different ways does not affect the sum)
• explaining that the order in which numbers are subtracted may affect the difference.

Note: Students investigate a variety of strategies, including standard/traditional algorithms, to become proficient in at least one appropriate and efficient strategy that they understand.

Specific Outcome 10

Apply mental mathematics strategies for basic addition facts and related subtraction facts to 18.

 Understand and apply strategies for addition facts up to and including 9 + 9 and related subtraction facts. Recall addition facts up to and including 5 + 5 and related subtraction facts.

Patterns & Relations - Patterns (Gr. 2)
General Outcome: Use patterns to describe the world and to solve problems.

Specific Outcome 1

Demonstrate an understanding of repeating patterns (three to five elements) by:

• describing
• extending
• comparing
• creating
patterns using manipulatives, diagrams, sounds and actions.

Specific Outcome 2

Demonstrate an understanding of increasing patterns by:

• describing
• reproducing
• extending
• creating
numerical (numbers to 100) and non-numerical patterns using manipulatives, diagrams, sounds and actions.

Specific Outcome 3

Sort a set of objects, using two attributes, and explain the sorting rule.

Patterns & Relations - Variables & Equations (Gr. 2)
General Outcome: Represent algebraic expressions in multiple ways.

Specific Outcome 4

Demonstrate and explain the meaning of equality and inequality, concretely and pictorially.

Specific Outcome 5

Record equalities and inequalities symbolically, using the equal symbol or the not equal symbol.

Shape & Space - Measurement (Gr. 2)
General Outcome: Use direct and indirect measurement to solve problems.

Specific Outcome 1

Relate the number of days to a week and the number of months to a year in a problem-solving context.

Specific Outcome 2

Relate the size of a unit of measure to the number of units (limited to nonstandard units) used to measure length and mass (weight).

Specific Outcome 3

Compare and order objects by length, height, distance around and mass (weight), using nonstandard units, and make statements of comparison.

Specific Outcome 4

Measure length to the nearest nonstandard unit by:

• using multiple copies of a unit
• using a single copy of a unit (iteration process).

Specific Outcome 5

Demonstrate that changing the orientation of an object does not alter the measurements of its attributes.

Shape & Space - 3-D Objects & 2-D Shapes (Gr. 2)
General Outcome: Describe the characteristics of 3-D objects and 2-D shapes, and analyze the relationships among them.

Specific Outcome 6

Sort 2-D shapes and 3-D objects, using two attributes, and explain the sorting rule.

Specific Outcome 7

Describe, compare and construct 3-D objects, including:

• cubes
• spheres
• cones
• cylinders
• pyramids.

Specific Outcome 8

Describe, compare and construct 2-D shapes, including:

• triangles
• squares
• rectangles
• circles.

Specific Outcome 9

Identify 2-D shapes as parts of 3-D objects in the environment.

Statistics & Probability - Data Analysis (Gr. 2)
General Outcome: Collect, display and analyze data to solve problems.

Specific Outcome 1

Gather and record data about self and others to answer questions.

[ICT: C4–1.3, C7–1.1]

Specific Outcome 2

Construct and interpret concrete graphs and pictographs to solve problems.

[ICT: C7–1.3]

Number (Gr. 3)
General Outcome: Develop number sense.

Specific Outcome 1

Say the number sequence 0 to 1000 forward and backward by:

• 5s, 10s or 100s, using any starting point
• 3s, using starting points that are multiples of 3
• 4s, using starting points that are multiples of 4
• 25s, using starting points that are multiples of 25.

Specific Outcome 2

Represent and describe numbers to 1000, concretely, pictorially and symbolically.

Specific Outcome 3

Compare and order numbers to 1000.

Specific Outcome 4

Estimate quantities less than 1000, using referents.

Specific Outcome 5

Illustrate, concretely and pictorially, the meaning of place value for numerals to 1000.

Specific Outcome 6

Describe and apply mental mathematics strategies for adding two 2-digit numerals.

Specific Outcome 7

Describe and apply mental mathematics strategies for subtracting two 2-digit numerals.

Specific Outcome 8

Apply estimation strategies to predict sums and differences of two 2-digit numerals in a problem-solving context.

Specific Outcome 9

Demonstrate an understanding of addition and subtraction of numbers with answers to 1000 (limited to 1-, 2- and 3-digit numerals), concretely, pictorially and symbolically, by:

• using personal strategies for adding and subtracting with and without the support of manipulatives
• creating and solving problems in context that involve addition and subtraction of numbers.

Note: Students investigate a variety of strategies, including standard/traditional algorithms, to become proficient in at least one appropriate and efficient strategy that they understand.

Specific Outcome 10

Apply mental mathematics strategies and number properties in order to understand and recall basic addition facts and related subtraction facts to 18.

 Understand, recall and apply addition facts up to and including 9 + 9 and related subtraction facts.

Specific Outcome 11

Demonstrate an understanding of multiplication to 5 x 5 by:

• representing and explaining multiplication using equal grouping and arrays
• creating and solving problems in context that involve multiplication
• modelling multiplication using concrete and visual representations, and recording the process symbolically
• relating multiplication to repeated addition
• relating multiplication to division.

 Understand and recall multiplication facts to 5 x 5.

Specific Outcome 12

Demonstrate an understanding of division (limited to division related to multiplication facts up to 5 x 5) by:

• representing and explaining division using equal sharing and equal grouping
• creating and solving problems in context that involve equal sharing and equal grouping
• modelling equal sharing and equal grouping using concrete and visual representations, and recording the process symbolically
• relating division to repeated subtraction
• relating division to multiplication.

 Understand and recall division facts related to multiplication facts to 5 x 5.

Specific Outcome 13

Demonstrate an understanding of fractions by:

• explaining that a fraction represents a part of a whole
• describing situations in which fractions are used
• comparing fractions of the same whole that have like denominators.

Patterns & Relations - Patterns (Gr. 3)
General Outcome: Use patterns to describe the world and to solve problems.

Specific Outcome 1

Demonstrate an understanding of increasing patterns by:

• describing
• extending
• comparing
• creating
numerical (numbers to 1000) and non-numerical patterns using manipulatives, diagrams, sounds and actions.

Specific Outcome 2

Demonstrate an understanding of decreasing patterns by:

• describing
• extending
• comparing
• creating
numerical (numbers to 1000) and non-numerical patterns using manipulatives, diagrams, sounds and actions.

Specific Outcome 3

Sort objects or numbers, using one or more than one attribute.

Patterns & Relations - Variables & Equations (Gr. 3)
General Outcome: Represent algebraic expressions in multiple ways.

Specific Outcome 4

Solve one-step addition and subtraction equations involving a symbol to represent an unknown number.

Shape & Space - Measurement (Gr. 3)
General Outcome: Use direct and indirect measurement to solve problems.

Specific Outcome 1

Relate the passage of time to common activities, using nonstandard and standard units (minutes, hours, days, weeks, months, years).

Specific Outcome 2

Relate the number of seconds to a minute, the number of minutes to an hour and the number of days to a month in a problem-solving context.

Specific Outcome 3

Demonstrate an understanding of measuring length (cm, m) by:

• selecting and justifying referents for the units cm and m
• modelling and describing the relationship between the units cm and m
• estimating length, using referents
• measuring and recording length, width and height.

Specific Outcome 4

Demonstrate an understanding of measuring mass (g, kg) by:

• selecting and justifying referents for the units g and kg
• modelling and describing the relationship between the units g and kg
• estimating mass, using referents
• measuring and recording mass.

Specific Outcome 5

Demonstrate an understanding of perimeter of regular and irregular shapes by:

• estimating perimeter, using referents for cm or m
• measuring and recording perimeter (cm, m)
• constructing different shapes for a given perimeter (cm, m) to demonstrate that many shapes are possible for a perimeter.

Shape & Space - 3-D Objects & 2-D Shapes (Gr. 3)
General Outcome: Describe the characteristics of 3-D objects and 2-D shapes, and analyze the relationships among them.

Specific Outcome 6

Describe 3-D objects according to the shape of the faces and the number of edges and vertices.

Specific Outcome 7

Sort regular and irregular polygons, including:

• triangles
• pentagons
• hexagons
• octagons
according to the number of sides.

Statistics & Probability - Data Analysis (Gr. 3)
General Outcome: Collect, display and analyze data to solve problems.

Specific Outcome 1

Collect first-hand data and organize it using:

• tally marks
• line plots
• charts
• lists

[ICT: C4–1.3]

Specific Outcome 2

Construct, label and interpret bar graphs to solve problems.

[ICT: C4–1.3, C7–1.3, C7–1.4]

Number (Gr. 4)
General Outcome: Develop number sense.

Specific Outcome 1

Represent and describe whole numbers to 10 000, pictorially and symbolically.

Specific Outcome 2

Compare and order numbers to 10 000.

Specific Outcome 3

Demonstrate an understanding of addition of numbers with answers to 10 000 and their corresponding subtractions (limited to 3- and 4-digit numerals) by:

• using personal strategies for adding and subtracting
• estimating sums and differences
• solving problems involving addition and subtraction.

Note: Students investigate a variety of strategies, including standard/traditional algorithms, to become proficient in at least one appropriate and efficient strategy that they understand.

Note: Through this outcome, students have the opportunity to maintain and refine previously learned addition and subtraction number facts:
Grade 3, Number SO 10 – Apply mental mathematics strategies and number properties in order to understand and recall basic addition facts and related subtraction facts to 18.

 Understand, recall and apply addition facts up to and including 9 + 9 and related subtraction facts.

Specific Outcome 4

Apply the properties of 0 and 1 for multiplication and the property of 1 for division.

Specific Outcome 5

Describe and apply mental mathematics strategies to determine basic multiplication facts to 9 × 9 and related division facts.

 Understand and apply strategies for multiplication and related division facts to 9 x 9. Recall multiplication and related division facts to 7 x 7.

Specific Outcome 6

Demonstrate an understanding of multiplication (2- or 3-digit by 1-digit) to solve problems by:

• using personal strategies for multiplication with and without concrete materials
• using arrays to represent multiplication
• connecting concrete representations to symbolic representations
• estimating products
• applying the distributive property.

Note: Students investigate a variety of strategies, including standard/traditional algorithms, to become proficient in at least one appropriate and efficient strategy that they understand.

Note: Through this outcome, students have the opportunity to maintain and refine previously learned addition and subtraction number facts:
Grade 3, Number SO 10 – Apply mental mathematics strategies and number properties in order to understand and recall basic addition facts and related subtraction facts to 18
.

 Understand, recall and apply addition facts up to and including 9 + 9 and related subtraction facts.

Specific Outcome 7

Demonstrate an understanding of division (1-digit divisor and up to 2-digit dividend) to solve problems by:

• using personal strategies for dividing with and without concrete materials
• estimating quotients
• relating division to multiplication.

Note: Students investigate a variety of strategies, including standard/traditional algorithms, to become proficient in at least one appropriate and efficient strategy that they understand.

Note: Through this outcome, students have the opportunity to maintain and refine previously learned addition and subtraction number facts:
Grade 3, Number SO 10 – Apply mental mathematics strategies and number properties in order to understand and recall basic addition facts and related subtraction facts to 18
.

 Understand, recall and apply addition facts up to and including 9 + 9 and related subtraction facts.

Specific Outcome 8

Demonstrate an understanding of fractions less than or equal to one by using concrete, pictorial and symbolic representations to:

• name and record fractions for the parts of a whole or a set
• compare and order fractions
• model and explain that for different wholes, two identical fractions may not represent the same quantity
• provide examples of where fractions are used.

Specific Outcome 9

Represent and describe decimals (tenths and hundredths), concretely, pictorially and symbolically.

Specific Outcome 10

Relate decimals to fractions and fractions to decimals (to hundredths).

Specific Outcome 11

Demonstrate an understanding of addition and subtraction of decimals (limited to hundredths) by:

• using personal strategies to determine sums and differences
• estimating sums and differences
• using mental mathematics strategies
to solve problems.

Note: Through this outcome, students have the opportunity to maintain and refine previously learned addition and subtraction number facts:
Grade 3, Number SO 10 – Apply mental mathematics strategies and number properties in order to understand and recall basic addition facts and related subtraction facts to 18
.

 Understand, recall and apply addition facts up to and including 9 + 9 and related subtraction facts.

Patterns & Relations - Patterns (Gr. 4)
General Outcome: Use patterns to describe the world and to solve problems.

Specific Outcome 1

Identify and describe patterns found in tables and charts.

[ICT: C6–2.3]

Specific Outcome 2

Translate among different representations of a pattern, such as a table, a chart or concrete materials.

Specific Outcome 3

Represent, describe and extend patterns and relationships, using charts and tables, to solve problems.

[ICT: C6–2.3]

Specific Outcome 4

Identify and explain mathematical relationships, using charts and diagrams, to solve problems.

[ICT: C6–2.3]

Patterns & Relations - Variables & Equations (Gr. 4)
General Outcome: Represent algebraic expressions in multiple ways.

Specific Outcome 5

Express a given problem as an equation in which a symbol is used to represent an unknown number.

Specific Outcome 6

Solve one-step equations involving a symbol to represent an unknown number.

Note: Through this outcome, students have the opportunity to maintain and refine previously learned addition and subtraction number facts:
Grade 3, Number SO 10 – Apply mental mathematics strategies and number properties in order to understand and recall basic addition facts and related subtraction facts to 18
.

 Understand, recall and apply addition facts up to and including 9 + 9 and related subtraction facts.

Shape & Space - Measurement (Gr. 4)
General Outcome: Use direct and indirect measurement to solve problems.

Specific Outcome 1

Read and record time, using digital and analog clocks, including 24-hour clocks.

Specific Outcome 2

Read and record calendar dates in a variety of formats.

Specific Outcome 3

Demonstrate an understanding of area of regular and irregular 2-D shapes by:

• recognizing that area is measured in square units
• selecting and justifying referents for the units cm2 or m2
• estimating area, using referents for cm2 or m2
• determining and recording area (cm2 or m2
• constructing different rectangles for a given area (cm2 or m2) in order to demonstrate that many different rectangles may have the same area.

Shape & Space - 3-D Objects & 2-D Shapes (Gr. 4)
General Outcome: Describe the characteristics of 3-D objects and 2-D shapes, and analyze the relationships among them.

Specific Outcome 4

Describe and construct right rectangular and right triangular prisms.

Shape & Space - Transformations (Gr. 4)
General Outcome: Describe and analyze position and motion of objects and shapes.

Specific Outcome 5

Demonstrate an understanding of congruency, concretely and pictorially.

Specific Outcome 6

Demonstrate an understanding of line symmetry by:

• identifying symmetrical 2-D shapes
• creating symmetrical 2-D shapes
• drawing one or more lines of symmetry in a 2-D shape.

Statistics & Probability - Data Analysis (Gr. 4)
General Outcome: Collect, display and analyze data to solve problems.

Specific Outcome 1

Demonstrate an understanding of many-to-one correspondence.

[ICT: C6–2.2, C6–2.3]

Specific Outcome 2

Construct and interpret pictographs and bar graphs involving many-to-one correspondence to draw conclusions.

Number (Gr. 5)
General Outcome: Develop number sense.

Specific Outcome 1

Represent and describe whole numbers to 1 000 000.

[ICT: C6–2.2]

Specific Outcome 2

Use estimation strategies in problem-solving contexts.

Specific Outcome 3

Apply mental mathematics strategies and number properties in order to understand and recall basic multiplication facts (multiplication tables) to 81 and related division facts.

 Understand, recall and apply multiplication and related division facts to 9 x 9.

Specific Outcome 4

Apply mental mathematics strategies for multiplication.

Specific Outcome 5

Demonstrate, with and without concrete materials, an understanding of multiplication (2-digit by 2-digit) to solve problems.

Note: Students investigate a variety of strategies, including standard/traditional algorithms, to become proficient in at least one appropriate and efficient strategy that they understand.

Note: Through this outcome, students have the opportunity to maintain and refine previously learned operations of addition and subtraction with whole numbers:
Grade 4, Number SO 3 – Demonstrate an understanding of addition of numbers with answers to 10 000 and their corresponding subtractions (limited to 3- and 4-digit numerals) by:

• using personal strategies for adding and subtracting
• estimating sums and differences
• solving problems involving addition and subtraction.

Specific Outcome 6

Demonstrate, with and without concrete materials, an understanding of division (3-digit by 1-digit), and interpret remainders to solve problems.

Note: Students investigate a variety of strategies, including standard/traditional algorithms, to become proficient in at least one appropriate and efficient strategy that they understand.

Note: Through this outcome, students have the opportunity to maintain and refine previously learned operations of addition and subtraction with whole numbers:
Grade 4, Number SO 3 – Demonstrate an understanding of addition of numbers with answers to 10 000 and their corresponding subtractions (limited to 3- and 4-digit numerals) by:

• using personal strategies for adding and subtracting
• estimating sums and differences
• solving problems involving addition and subtraction.

Specific Outcome 7

Demonstrate an understanding of fractions by using concrete, pictorial and symbolic representations to:

• create sets of equivalent fractions
• compare fractions with like and unlike denominators.

Specific Outcome 8

Describe and represent decimals (tenths, hundredths, thousandths), concretely, pictorially and symbolically.

Specific Outcome 9

Relate decimals to fractions and fractions to decimals (to thousandths).

Specific Outcome 10

Compare and order decimals (to thousandths) by using:

• benchmarks
• place value
• equivalent decimals.

Specific Outcome 11

Demonstrate an understanding of addition and subtraction of decimals (limited to thousandths).

Note: Through this outcome, students have the opportunity to maintain and refine previously learned operations of addition and subtraction with whole numbers:
Grade 4, Number SO 3 – Demonstrate an understanding of addition of numbers with answers to 10 000 and their corresponding subtractions (limited to 3- and 4-digit numerals) by:

• using personal strategies for adding and subtracting
• estimating sums and differences
• solving problems involving addition and subtraction.
Patterns & Relations - Patterns (Gr. 5)
General Outcome: Use patterns to describe the world and to solve problems.

Specific Outcome 1

Determine the pattern rule to make predictions about subsequent elements.

Patterns & Relations - Variables & Equations (Gr. 5)
General Outcome: Represent algebraic expressions in multiple ways.

Specific Outcome 2

Express a given problem as an equation in which a letter variable is used to represent an unknown number (limited to whole numbers).

Specific Outcome 3

Solve problems involving single-variable, one-step equations with whole number coefficients and whole number solutions.

Shape & Space - Measurement (Gr. 5)
General Outcome: Use direct and indirect measurement to solve problems.

Specific Outcome 1

Identify 90º angles.

Specific Outcome 2

Design and construct different rectangles, given either perimeter or area, or both (whole numbers), and make generalizations.

Specific Outcome 3

Demonstrate an understanding of measuring length (mm) by:

• selecting and justifying referents for the unit mm
• modelling and describing the relationship between mm and cm units, and between mm and m units.

Specific Outcome 4

Demonstrate an understanding of volume by:

• selecting and justifying referents for cm3 or m3 units
• estimating volume, using referents for cm3 or m3
• measuring and recording volume (cm3 or m3
• constructing right rectangular prisms for a given volume.

Specific Outcome 5

Demonstrate an understanding of capacity by:

• describing the relationship between mL and L
• selecting and justifying referents for mL or L units
• estimating capacity, using referents for mL or L
• measuring and recording capacity (mL or L).

Shape & Space - 3-D Objects & 2-D Shapes (Gr. 5)
General Outcome: Describe the characteristics of 3-D objects and 2-D shapes, and analyze the relationships among them.

Specific Outcome 6

Describe and provide examples of edges and faces of 3-D objects, and sides of 2-D shapes that are:

• parallel
• intersecting
• perpendicular
• vertical
• horizontal.

[ICT: C6–2.2, P5–2.3]

Specific Outcome 7

Identify and sort quadrilaterals, including:

• rectangles
• squares
• trapezoids
• parallelograms
• rhombuses
according to their attributes.

Shape & Space - Transformations (Gr. 5)
General Outcome: Describe and analyze position and motion of objects and shapes.

Specific Outcome 8

Identify and describe a single transformation, including a translation, rotation and reflection of 2-D shapes.

[ICT: C6–2.1]

Specific Outcome 9

Perform, concretely, a single transformation (translation, rotation or reflection) of a 2-D shape, and draw the image.

[ICT: C6–2.1]

Statistics & Probability - Data Analysis (Gr. 5)
General Outcome: Collect, display and analyze data to solve problems.

Specific Outcome 1

Differentiate between first-hand and second-hand data.

[ICT: C1–2.2, P5–2.3]

Specific Outcome 2

Construct and interpret double bar graphs to draw conclusions.

[ICT: C6–2.2, P5–2.3]

Statistics & Probability - Chance & Uncertainty (Gr. 5)
General Outcome: Use experimental or theoretical probabilities to represent and solve problems involving uncertainty.

Specific Outcome 3

Describe the likelihood of a single outcome occurring, using words such as:

• impossible
• possible
• certain.

Specific Outcome 4

Compare the likelihood of two possible outcomes occurring, using words such as:

• less likely
• equally likely
• more likely.

Number (Gr. 6)
General Outcome: Develop number sense.

Specific Outcome 1

Demonstrate an understanding of place value, including numbers that are:

• greater than one million
• less than one thousandth.

Specific Outcome 2

Solve problems involving whole numbers and decimal numbers.

[ICT: C6–2.4]

Note: Through this outcome, students have the opportunity to maintain and refine previously learned:

• multiplication and division number facts:
Grade 5, Number SO 3 – Apply mental mathematics strategies and number properties in order to understand and recall basic multiplication facts (multiplication tables) to 81 and related division facts
.  Understand, recall and apply multiplication and related division facts to 9 x 9.

• operations with whole numbers:
Grade 4, Number SO 3 – Demonstrate an understanding of addition of numbers with answers to 10 000 and their corresponding subtractions (limited to 3- and 4-digit numerals) by:
• using personal strategies for adding and subtracting
• estimating sums and differences
• solving problems involving addition and subtraction.

Grade 5, Number SO 5 – Demonstrate, with and without concrete materials, an understanding of multiplication (2-digit by 2-digit) to solve problems.

Grade 5, Number SO 6 – Demonstrate, with and without concrete materials, an understanding of division (3-digit by 1-digit), and interpret remainders to solve problems.

Specific Outcome 3

Demonstrate an understanding of factors and multiples by:

• determining multiples and factors of numbers less than 100
• identifying prime and composite numbers
• solving problems using multiples and factors.

Note: Through this outcome, students have the opportunity to maintain and refine previously learned multiplication and division number facts:
Grade 5, Number SO 3 – Apply mental mathematics strategies and number properties in order to understand and recall basic multiplication facts (multiplication tables) to 81 and related division facts.

 Understand, recall and apply multiplication and related division facts to 9 x 9.

Specific Outcome 4

Relate improper fractions to mixed numbers and mixed numbers to improper fractions.

Specific Outcome 5

Demonstrate an understanding of ratio, concretely, pictorially and symbolically.

Specific Outcome 6

Demonstrate an understanding of percent (limited to whole numbers), concretely, pictorially and symbolically.

Specific Outcome 7

Demonstrate an understanding of integers, concretely, pictorially and symbolically.

Specific Outcome 8

Demonstrate an understanding of multiplication and division of decimals (1-digit whole number multipliers and 1-digit natural number divisors).

Note: Through this outcome, students have the opportunity to maintain and refine previously learned:

• multiplication and division number facts:
Grade 5, Number SO 3 – Apply mental mathematics strategies and number properties in order to understand and recall basic multiplication facts (multiplication tables) to 81 and related division facts.

 Understand, recall and apply multiplication and related division facts to 9 x 9.

• operations with whole numbers:
Grade 4, Number SO 6 – Demonstrate an understanding of multiplication (2- or 3-digit by 1-digit) to solve problems by:
• using personal strategies for multiplication with and without concrete materials
• using arrays to represent multiplication
• connecting concrete representations to symbolic representations
• estimating products
• applying the distributive property.

Grade 5, Number SO 6 – Demonstrate, with and without concrete materials, an understanding of division (3- digit by 1-digit), and interpret remainders to solve problems.

Specific Outcome 9

Explain and apply the order of operations, excluding exponents, with and without technology (limited to whole numbers).

[ICT: C6–2.4, C6–2.7]

Note: Through this outcome, students have the opportunity to maintain and refine previously learned:

• multiplication and division number facts:
Grade 5, Number SO 3 – Apply mental mathematics strategies and number properties in order to understand and recall basic multiplication facts (multiplication tables) to 81 and related division facts.

 Understand, recall and apply multiplication and related division facts to 9 x 9.

• operations with whole numbers:
Grade 4, Number SO 3 – Demonstrate an understanding of addition of numbers with answers to 10 000 and their corresponding subtractions (limited to 3- and 4-digit numerals) by:
• using personal strategies for adding and subtracting
• estimating sums and differences
• solving problems involving addition and subtraction.

Grade 5, Number SO 5 – Demonstrate, with and without concrete materials, an understanding of multiplication (2-digit by 2-digit) to solve problems.

Grade 5, Number SO 6 – Demonstrate, with and without concrete materials, an understanding of division (3-digit by 1-digit), and interpret remainders to solve problems.

Patterns & Relations - Patterns (Gr. 6)
General Outcome: Use patterns to describe the world and to solve problems.

Specific Outcome 1

Represent and describe patterns and relationships, using graphs and tables.

[ICT: C6–2.3]

Specific Outcome 2

Demonstrate an understanding of the relationships within tables of values to solve problems.

[ICT: C6–2.3]

Patterns & Relations - Variables & Equations (Gr. 6)
General Outcome: Represent algebraic expressions in multiple ways.

Specific Outcome 3

Represent generalizations arising from number relationships, using equations with letter variables.

Specific Outcome 4

Express a given problem as an equation in which a letter variable is used to represent an unknown number.

Specific Outcome 5

Demonstrate and explain the meaning of preservation of equality, concretely and pictorially.

Shape & Space - Measurement (Gr. 6)
General Outcome: Use direct and indirect measurement to solve problems.

Specific Outcome 1

Demonstrate an understanding of angles by:

• identifying examples of angles in the environment
• classifying angles according to their measure
• estimating the measure of angles, using 45°, 90° and 180° as reference angles
• determining angle measures in degrees
• drawing and labelling angles when the measure is specified.

Specific Outcome 2

Demonstrate that the sum of interior angles is:

• 180° in a triangle
• 360° in a quadrilateral.

Specific Outcome 3

Develop and apply a formula for determining the:

• perimeter of polygons
• area of rectangles
• volume of right rectangular prisms.

Shape & Space - 3-D Objects & 2-D Shapes (Gr. 6)
General Outcome: Describe the characteristics of 3-D objects and 2-D shapes, and analyze the relationships among them.

Specific Outcome 4

Construct and compare triangles, including:

• scalene
• isosceles
• equilateral
• right
• obtuse
• acute
in different orientations.

Specific Outcome 5

Describe and compare the sides and angles of regular and irregular polygons.

Shape & Space - Transformations (Gr. 6)
General Outcome: Describe and analyze position and motion of objects and shapes.

Specific Outcome 6

Perform a combination of translations, rotations and/or reflections on a single 2-D shape, with and without technology, and draw and describe the image.

Specific Outcome 7

Perform a combination of successive transformations of 2-D shapes to create a design, and identify and describe the transformations.

Specific Outcome 8

Identify and plot points in the first quadrant of a Cartesian plane, using whole number ordered pairs.

Specific Outcome 9

Perform and describe single transformations of a 2-D shape in the first quadrant of a Cartesian plane (limited to whole number vertices).

[ICT: C6–2.1]

Statistics & Probability - Data Analysis (Gr. 6)
General Outcome: Collect, display and analyze data to solve problems.

Specific Outcome 1

Create, label and interpret line graphs to draw conclusions.

Specific Outcome 2

Select, justify and use appropriate methods of collecting data, including:

• questionnaires
• experiments
• databases
• electronic media.

[ICT: C4–2.2, C6–2.2, C7–2.1, P2–2.1, P2–2.2]

Specific Outcome 3

Graph collected data, and analyze the graph to solve problems.

[ICT: C6–2.5, C7–2.1, P2–2.1, P2–2.2]

Statistics & Probability - Chance & Uncertainty (Gr. 6)
General Outcome: Use experimental or theoretical probabilities to represent and solve problems involving uncertainty.

Specific Outcome 4

Demonstrate an understanding of probability by:

• identifying all possible outcomes of a probability experiment
• differentiating between experimental and theoretical probability
• determining the theoretical probability of outcomes in a probability experiment
• determining the experimental probability of outcomes in a probability experiment
• comparing experimental results with the theoretical probability for an experiment.

[ICT: C6–2.1, C6–2.4]

Number (Gr. 7)
General Outcome: Develop number sense.

Specific Outcome 1

Determine and explain why a number is divisible by 2, 3, 4, 5, 6, 8, 9 or 10, and why a number cannot be divided by 0.

Specific Outcome 2

Demonstrate an understanding of the addition, subtraction, multiplication and division of decimals to solve problems (for more than 1-digit divisors or 2-digit multipliers, the use of technology is expected).

[ICT: P2–3.4]

Note: Through this outcome, students have the opportunity to maintain and refine previously learned operations with whole numbers:
Grade 4, Number SO 3 – Demonstrate an understanding of addition of numbers with answers to 10 000 and their corresponding subtractions (limited to 3- and 4-digit numerals) by:

• using personal strategies for adding and subtracting
• estimating sums and differences
• solving problems involving addition and subtraction.

Grade 5, Number SO 5 – Demonstrate, with and without concrete materials, an understanding of multiplication (2-digit by 2-digit) to solve problems.

Grade 5, Number SO 6 – Demonstrate, with and without concrete materials, an understanding of division (3-digit by 1-digit), and interpret remainders to solve problems.

Specific Outcome 3

Solve problems involving percents from 1% to 100%.

[ICT: P2–3.4]

Specific Outcome 4

Demonstrate an understanding of the relationship between positive terminating decimals and positive fractions and between positive repeating decimals and positive fractions.

[ICT: P2–3.4]

Specific Outcome 5

Demonstrate an understanding of adding and subtracting positive fractions and mixed numbers, with like and unlike denominators, concretely, pictorially and symbolically (limited to positive sums and differences).

Specific Outcome 6

Demonstrate an understanding of addition and subtraction of integers, concretely, pictorially and symbolically.

Note: Through this outcome, students have the opportunity to maintain and refine previously learned operations with whole numbers:
Grade 4, Number SO 3 – Demonstrate an understanding of addition of numbers with answers to 10 000 and their corresponding subtractions (limited to 3- and 4-digit numerals) by:

• using personal strategies for adding and subtracting
• estimating sums and differences
• solving problems involving addition and subtraction.

Specific Outcome 7

Compare and order positive fractions, positive decimals (to thousandths) and whole numbers by using:

• benchmarks
• place value
• equivalent fractions and/or decimals.

Patterns & Relations - Patterns (Gr. 7)
General Outcome: Use patterns to describe the world and to solve problems.

Specific Outcome 1

Demonstrate an understanding of oral and written patterns and their equivalent linear relations.

Specific Outcome 2

Create a table of values from a linear relation, graph the table of values, and analyze the graph to draw conclusions and solve problems.

[ICT: C7–3.1]

Patterns & Relations - Variables & Equations (Gr. 7)
General Outcome: Represent algebraic expressions in multiple ways.

Specific Outcome 3

Demonstrate an understanding of preservation of equality by:

• modelling preservation of equality, concretely, pictorially and symbolically
• applying preservation of equality to solve equations.

Specific Outcome 4

Explain the difference between an expression and an equation.

Specific Outcome 5

Evaluate an expression, given the value of the variable(s).

Note: Through this outcome, students have the opportunity to maintain and refine previously learned operations with whole numbers:
Grade 4, Number SO 3 – Demonstrate an understanding of addition of numbers with answers to 10 000 and their corresponding subtractions (limited to 3- and 4-digit numerals) by:

• using personal strategies for adding and subtracting
• estimating sums and differences
• solving problems involving addition and subtraction.

Grade 5, Number SO 5 – Demonstrate, with and without concrete materials, an understanding of multiplication (2-digit by 2-digit) to solve problems.

Grade 5, Number SO 6 – Demonstrate, with and without concrete materials, an understanding of division (3-digit by 1-digit), and interpret remainders to solve problems.

Specific Outcome 6

Model and solve, concretely, pictorially and symbolically, problems that can be represented by one-step linear equations of the form x + a = b, where a and b are integers.

Note: Through this outcome, students have the opportunity to maintain and refine previously learned operations with whole numbers:
Grade 4, Number SO 3 – Demonstrate an understanding of addition of numbers with answers to 10 000 and their corresponding subtractions (limited to 3- and 4-digit numerals) by:

• using personal strategies for adding and subtracting
• estimating sums and differences
• solving problems involving addition and subtraction.

Specific Outcome 7

Model and solve, concretely, pictorially and symbolically, problems that can be represented by linear equations of the form:

• ax + b = c
• ax = b
• = b, a ≠ 0
• where a, b, and c are whole numbers.

Note: Through this outcome, students have the opportunity to maintain and refine previously learned operations with whole numbers:
Grade 4, Number SO 3 – Demonstrate an understanding of addition of numbers with answers to 10 000 and their corresponding subtractions (limited to 3- and 4-digit numerals) by:

• using personal strategies for adding and subtracting
• estimating sums and differences
• solving problems involving addition and subtraction.

Grade 5, Number SO 5 – Demonstrate, with and without concrete materials, an understanding of multiplication (2-digit by 2-digit) to solve problems.

Grade 5, Number SO 6 – Demonstrate, with and without concrete materials, an understanding of division (3-digit by 1-digit), and interpret remainders to solve problems.

Shape & Space - Measurement (Gr. 7)
General Outcome: Use direct and indirect measurement to solve problems.

Specific Outcome 1

Demonstrate an understanding of circles by:

• describing the relationships among radius, diameter and circumference
• relating circumference to pi
• determining the sum of the central angles
• constructing circles with a given radius or diameter
• solving problems involving the radii, diameters and circumferences of circles.

Specific Outcome 2

Develop and apply a formula for determining the area of:

• triangles
• parallelograms
• circles.

Shape & Space - 3-D Objects & 2-D Shapes (Gr. 7)
General Outcome: Describe the characteristics of 3-D objects and 2-D shapes, and analyze the relationships among them.

Specific Outcome 3

Perform geometric constructions, including:

• perpendicular line segments
• parallel line segments
• perpendicular bisectors
• angle bisectors.

Shape & Space - Transformations (Gr. 7)
General Outcome: Describe and analyze position and motion of objects and shapes.

Specific Outcome 4

Identify and plot points in the four quadrants of a Cartesian plane, using integral ordered pairs.

Specific Outcome 5

Perform and describe transformations (translations, rotations or reflections) of a 2-D shape in all four quadrants of a Cartesian plane (limited to integral number vertices).

[ICT: C6–3.4]

Statistics & Probability - Data Analysis (Gr. 7)
General Outcome: Collect, display and analyze data to solve problems.

Specific Outcome 1

Demonstrate an understanding of central tendency and range by:

• determining the measures of central tendency (mean, median, mode) and range
• determining the most appropriate measures of central tendency to report findings.

[ICT: P2–3.4]

Specific Outcome 2

Determine the effect on the mean, median and mode when an outlier is included in a data set.

Specific Outcome 3

Construct, label and interpret circle graphs to solve problems.

[ICT: P2–3.3]

Statistics & Probability - Chance & Uncertainty (Gr. 7)
General Outcome: Use experimental or theoretical probabilities to represent and solve problems involving uncertainty.

Specific Outcome 4

Express probabilities as ratios, fractions and percents.

[ICT: P2–3.4]

Specific Outcome 5

Identify the sample space (where the combined sample space has 36 or fewer elements) for a probability experiment involving two independent events.

Specific Outcome 6

Conduct a probability experiment to compare the theoretical probability (determined using a tree diagram, table or other graphic organizer) and experimental probability of two independent events.

[ICT: C7–3.2, P2–3.4]

Number (Gr. 8)
General Outcome: Develop number sense.

Specific Outcome 1

Demonstrate an understanding of perfect squares and square roots, concretely, pictorially and symbolically (limited to whole numbers).

Note: Through this outcome, students have the opportunity to maintain and refine previously learned operations of multiplication and division with whole numbers:
Grade 5, Number SO 5 – Demonstrate, with and without concrete materials, an understanding of multiplication (2-digit by 2-digit) to solve problems.

Grade 5, Number SO 6 – Demonstrate, with and without concrete materials, an understanding of division (3-digit by 1-digit), and interpret remainders to solve problems.

Specific Outcome 2

Determine the approximate square root of numbers that are not perfect squares (limited to whole numbers).

[ICT: P2–3.4]

Note: Through this outcome, students have the opportunity to maintain and refine previously learned operations of multiplication and division with whole numbers:
Grade 5, Number SO 5 – Demonstrate, with and without concrete materials, an understanding of multiplication (2-digit by 2-digit) to solve problems.

Grade 5, Number SO 6 – Demonstrate, with and without concrete materials, an understanding of division (3-digit by 1-digit), and interpret remainders to solve problems.

Specific Outcome 3

Demonstrate an understanding of percents greater than or equal to 0%, including greater than 100%.

Specific Outcome 4

Demonstrate an understanding of ratio and rate.

Specific Outcome 5

Solve problems that involve rates, ratios and proportional reasoning.

Specific Outcome 6

Demonstrate an understanding of multiplying and dividing positive fractions and mixed numbers, concretely, pictorially and symbolically.

Specific Outcome 7

Demonstrate an understanding of multiplication and division of integers, concretely, pictorially and symbolically.

Note: Through this outcome, students have the opportunity to maintain and refine previously learned operations of multiplication and division with whole numbers:
Grade 5, Number SO 5 – Demonstrate, with and without concrete materials, an understanding of multiplication (2-digit by 2-digit) to solve problems.

Grade 5, Number SO 6 – Demonstrate, with and without concrete materials, an understanding of division (3-digit by 1-digit), and interpret remainders to solve problems.

Patterns & Relations - Patterns (Gr. 8)
General Outcome: Use patterns to describe the world and to solve problems.

Specific Outcome 1

Graph and analyze two-variable linear relations.

[ICT: P2–3.3]

Patterns & Relations - Variables & Equations (Gr. 8)
General Outcome: Represent algebraic expressions in multiple ways.

Specific Outcome 2

Model and solve problems concretely, pictorially and symbolically, using linear equations of the form:

• ax = b
• = b, a ≠ 0
• ax + b = c
• + b = c, a ≠ 0
• a(x + b) = c
where a, b and c are integers.

Note: Through this outcome, students have the opportunity to maintain and refine previously learned operations with whole numbers:
Grade 4, Number SO 3 – Demonstrate an understanding of addition of numbers with answers to 10 000 and their corresponding subtractions (limited to 3- and 4-digit numerals) by:

• using personal strategies for adding and subtracting
• estimating sums and differences
• solving problems involving addition and subtraction.

Grade 5, Number SO 5 – Demonstrate, with and without concrete materials, an understanding of multiplication (2-digit by 2-digit) to solve problems.

Grade 5, Number SO 6 – Demonstrate, with and without concrete materials, an understanding of division (3-digit by 1-digit), and interpret remainders to solve problems.

Shape & Space - Measurement (Gr. 8)
General Outcome: Use direct and indirect measurement to solve problems.

Specific Outcome 1

Develop and apply the Pythagorean theorem to solve problems.

[ICT: P2–3.4]

Specific Outcome 2

Draw and construct nets for 3-D objects.

Specific Outcome 3

Determine the surface area of:

• right rectangular prisms
• right triangular prisms
• right cylinders
to solve problems.

Specific Outcome 4

Develop and apply formulas for determining the volume of right rectangular prisms, right triangular prisms and right cylinders.

Shape & Space - 3-D Objects & 2-D Shapes (Gr. 8)
General Outcome: Describe the characteristics of 3-D objects and 2-D shapes, and analyze the relationships among them.

Specific Outcome 5

Draw and interpret top, front and side views of 3-D objects composed of right rectangular prisms.

[ICT: C6–3.4]

Shape & Space - Transformations (Gr. 8)
General Outcome: Describe and analyze position and motion of objects and shapes.

Specific Outcome 6

Demonstrate an understanding of the congruence of polygons.

Statistics & Probability - Data Analysis (Gr. 8)
General Outcome: Collect, display and analyze data to solve problems.

Specific Outcome 1

Critique ways in which data is presented in circle graphs, line graphs, bar graphs and pictographs.

[ICT: C7–3.1, C7–3.2, F4–3.3]

Statistics & Probability - Chance & Uncertainty (Gr. 8)
General Outcome: Use experimental or theoretical probabilities to represent and solve problems involving uncertainty.

Specific Outcome 2

Solve problems involving the probability of independent events.

[ICT: P2–3.4]

Number (Gr. 9)
General Outcome: Develop number sense.

Specific Outcome 1

Demonstrate an understanding of powers with integral bases (excluding base 0) and whole number exponents by:

• representing repeated multiplication, using powers
• using patterns to show that a power with an exponent of zero is equal to one
• solving problems involving powers.

Specific Outcome 2

Demonstrate an understanding of operations on powers with integral bases (excluding base 0) and whole number exponents:

• (am)(an) = am+n
• am ÷ an = am-n, m > n
• (am)n = amn
• (ab)m = ambm
• = , b ≠ 0.

[ICT: P2–3.4]

Specific Outcome 3

Demonstrate an understanding of rational numbers by:

• comparing and ordering rational numbers
• solving problems that involve arithmetic operations on rational numbers.

[ICT: P2–3.4]

Note: Through this outcome, students have the opportunity to maintain and refine previously learned:

• operations with whole numbers:
Grade 4, Number SO 3 – Demonstrate an understanding of addition of numbers with answers to 10 000 and their corresponding subtractions (limited to 3- and 4-digit numerals) by:
• using personal strategies for adding and subtracting
• estimating sums and differences
• solving problems involving addition and subtraction.

Grade 5, Number SO 5 – Demonstrate, with and without concrete materials, an understanding of multiplication (2-digit by 2-digit) to solve problems.

Grade 5, Number SO 6 – Demonstrate, with and without concrete materials, an understanding of division (3-digit by 1-digit), and interpret remainders to solve problems.

• operations with fractions:
Grade 7, Number SO 5 – Demonstrate an understanding of adding and subtracting positive fractions and mixed numbers, with like and unlike denominators, concretely, pictorially and symbolically (limited to positive sums and differences).

Grade 8, Number SO 6 – Demonstrate an understanding of multiplying and dividing positive fractions and mixed numbers, concretely, pictorially and symbolically.

• operations with integers:
Grade 7, Number SO 6 – Demonstrate an understanding of addition and subtraction of integers, concretely, pictorially and symbolically.

Grade 8, Number SO 7 – Demonstrate an understanding of multiplication and division of integers, concretely, pictorially and symbolically.

Specific Outcome 4

Explain and apply the order of operations, including exponents, with and without technology.

[ICT: P2–3.4]

Note: Through this outcome, students have the opportunity to maintain and refine previously learned:

• operations with whole numbers:
Grade 4, Number SO 3 – Demonstrate an understanding of addition of numbers with answers to 10 000 and their corresponding subtractions (limited to 3- and 4-digit numerals) by:
• using personal strategies for adding and subtracting
• estimating sums and differences
• solving problems involving addition and subtraction.

Grade 5, Number SO 5 – Demonstrate, with and without concrete materials, an understanding of multiplication (2-digit by 2-digit) to solve problems.

Grade 5, Number SO 6 – Demonstrate, with and without concrete materials, an understanding of division (3-digit by 1-digit), and interpret remainders to solve problems.

• operations with fractions:
Grade 7, Number SO 5 – Demonstrate an understanding of adding and subtracting positive fractions and mixed numbers, with like and unlike denominators, concretely, pictorially and symbolically (limited to positive sums and differences).

Grade 8, Number SO 6 – Demonstrate an understanding of multiplying and dividing positive fractions and mixed numbers, concretely, pictorially and symbolically.

• operations with integers:
Grade 7, Number SO 6 – Demonstrate an understanding of addition and subtraction of integers, concretely, pictorially and symbolically.

Grade 8, Number SO 7 – Demonstrate an understanding of multiplication and division of integers, concretely, pictorially and symbolically.

Specific Outcome 5

Determine the square root of positive rational numbers that are perfect squares.

[ICT: P2–3.4]

Specific Outcome 6

Determine an approximate square root of positive rational numbers that are non-perfect squares.

[ICT: P2–3.4]

Patterns & Relations - Patterns (Gr. 9)
General Outcome: Use patterns to describe the world and to solve problems.

Specific Outcome 1

Generalize a pattern arising from a problem-solving context, using a linear equation, and verify by substitution.

Specific Outcome 2

Graph a linear relation, analyze the graph, and interpolate or extrapolate to solve problems.

[ICT: C7–3.1, P2–3.3]

Patterns & Relations - Variables & Equations (Gr. 9)
General Outcome: Represent algebraic expressions in multiple ways.

Specific Outcome 3

Model and solve problems, using linear equations of the form:

• ax = b
• = b, a ≠ 0
• ax + b = c
• + b = c, a ≠ 0
• ax = b + cx
• a(x + b) = c
• ax + b = cx + d
• a(bx + c) = d(ex + f)
• = b, x ≠ 0
where a, b, c, d, e and f are rational numbers.

Specific Outcome 4

Explain and illustrate strategies to solve single variable linear inequalities with rational coefficients within a problem-solving context.

Specific Outcome 5

Demonstrate an understanding of polynomials (limited to polynomials of degree less than or equal to 2).

Specific Outcome 6

Model, record and explain the operations of addition and subtraction of polynomial expressions, concretely, pictorially and symbolically (limited to polynomials of degree less than or equal to 2).

Note: Through this outcome, students have the opportunity to maintain and refine previously learned:

• addition and subtraction with whole numbers:
Grade 4, Number SO 3 – Demonstrate an understanding of addition of numbers with answers to 10 000 and their corresponding subtractions (limited to 3- and 4-digit numerals) by:
• using personal strategies for adding and subtracting
• estimating sums and differences
• solving problems involving addition and subtraction.
• addition and subtraction with integers:
Grade 7, Number SO 6 – Demonstrate an understanding of addition and subtraction of integers, concretely, pictorially and symbolically.

Specific Outcome 7

Model, record and explain the operations of multiplication and division of polynomial expressions (limited to polynomials of degree less than or equal to 2) by monomials, concretely, pictorially and symbolically.

Note: Through this outcome, students have the opportunity to maintain and refine previously learned:

• multiplication and division with whole numbers:
Grade 5, Number SO 5 – Demonstrate, with and without concrete materials, an understanding of multiplication (2-digit by 2-digit) to solve problems.

Grade 5, Number SO 6 – Demonstrate, with and without concrete materials, an understanding of division (3-digit by 1-digit), and interpret remainders to solve problems.

• multiplication and division with integers:
Grade 8, Number SO 7 – Demonstrate an understanding of multiplication and division of integers, concretely, pictorially and symbolically.
Shape & Space - Measurement (Gr. 9)
General Outcome: Use direct and indirect measurement to solve problems.

Specific Outcome 1

Solve problems and justify the solution strategy, using the following circle properties:

• the perpendicular from the centre of a circle to a chord bisects the chord
• the measure of the central angle is equal to twice the measure of the inscribed angle subtended by the same arc
• the inscribed angles subtended by the same arc are congruent
• a tangent to a circle is perpendicular to the radius at the point of tangency.

[ICT: C6–3.1, C6–3.4]

Shape & Space - 3-D Objects & 2-D Shapes (Gr. 9)
General Outcome: Describe the characteristics of 3-D objects and 2-D shapes, and analyze the relationships among them.

Specific Outcome 2

Determine the surface area of composite 3-D objects to solve problems.

Specific Outcome 3

Demonstrate an understanding of similarity of polygons.

Shape & Space - Transformations (Gr. 9)
General Outcome: Describe and analyze position and motion of objects and shapes.

Specific Outcome 4

Draw and interpret scale diagrams of 2-D shapes.

[ICT: C6–3.4]

Specific Outcome 5

Demonstrate an understanding of line and rotation symmetry.

Statistics & Probability - Data Analysis (Gr. 9)
General Outcome: Collect, display and analyze data to solve problems.

Specific Outcome 1

Describe the effect of:

• bias
• use of language
• ethics
• cost
• time and timing
• privacy
• cultural sensitivity
on the collection of data.

[ICT: F4–3.2, F4–3.3]

Specific Outcome 2

Select and defend the choice of using either a population or a sample of a population to answer a question.

Specific Outcome 3

Develop and implement a project plan for the collection, display and analysis of data by:

• formulating a question for investigation
• choosing a data collection method that includes social considerations
• selecting a population or a sample
• collecting the data
• displaying the collected data in an appropriate manner
• drawing conclusions to answer the question.

[ICT: C1–3.5, C4–3.1, C6–3.1, C6–3.2, C7–3.1, C7–3.2, P1–3.4, P2–3.1]

Statistics & Probability - Chance & Uncertainty (Gr. 9)
General Outcome: Use experimental or theoretical probabilities to represent and solve problems involving uncertainty.

Specific Outcome 4

Demonstrate an understanding of the role of probability in society.

[ICT: F4–3.3]

Appendices
Appendix 1: General & Specific Outcomes by Strand
 Appendix 1 is available at http://education.alberta.ca/teachers/program/math/educator/progstudy.aspx.

Appendix 2: Information & Communication Technology (ICT) Outcomes

The following excerpts from the Information and Communication Technology (ICT) Program of Studies provide the complete wording for outcomes that are linked to the mathematics program of studies. For the complete ICT Program of Studies, go to the Alberta Education Web site at http://education.alberta.ca/teachers/program/ict/programs.aspx.

ICT Outcomes, Division 1
 General Outcomes Specific Outcomes C4 - Students will use organizational processes and tools to manage inquiry. 1.3 organize information from more than one source C7 - Students will use electronic research techniques to construct personal knowledge and meaning. 1.1 develop questions that reflect a personal information need 1.3 draw conclusions from organized information 1.4 make predictions based on organized information P2 - Students will organize and manipulate data. 1.1 read information from a prepared database
ICT Outcomes, Division 2
 General Outcomes Specific Outcomes C1 - Students will access, use and communicate information from a variety of technologies. 2.2 organize information gathered from the Internet, or an electronic source, by selecting and recording the data in logical files or categories; and by communicating effectively, through appropriate forms, such as speeches, reports and multimedia presentations, applying information technologies that serve particular audiences and purposes C4 - Students will use organizational processes and tools to manage inquiry. 2.2 organize information, using such tools as a database, spreadsheet or electronic webbing C6 - Students will use technology to investigate and/or solve problems 2.1 select and use technology to assist in problem solving 2.2 use data gathered from a variety of electronic sources to address identified problems 2.3 use graphic organizers, such as mind mapping/webbing, flow charting and outlining, to present connections between ideas and information in a problem-solving environment 2.4 solve problems, using numerical operations and such tools as calculators and spreadsheets 2.5 solve problems requiring the sorting, organizing, classifying and extending of data, using such tools as calculators, spreadsheets, databases or hypertext technology 2.7 generate alternative solutions to problems by using technology to facilitate the process C7 - Students will use electronic research techniques to construct personal knowledge and meaning. 2.1 use a variety of technologies to organize and synthesize researched information P2 - Students will organize and manipulate data. 2.1 enter and manipulate data by using such tools as spreadsheets or databases for a specific purpose 2.2 display data electronically through graphs and charts P5 - Students will navigate and create hyperlinked resources. 2.3 navigate the Internet with appropriate software
ICT Outcomes, Division 3
 General Outcomes Specific Outcomes C1 - Students will access, use and communicate information from a variety of technologies. 3.5 analyze and synthesize information to create a product C4 - Students will use organizational processes and tools to manage inquiry. 3.1 create a plan for an inquiry that includes consideration of time management C6 - Students will use technology to investigate and/or solve problems. 3.1 articulate clearly a plan of action to use technology to solve a problem 3.2 identify the appropriate materials and tools to use in order to accomplish a plan of action 3.4 pose and test solutions to problems by using computer applications, such as computer-assisted design or simulation/modelling software C7 - Students will use electronic research techniques to construct personal knowledge and meaning. 3.1 identify patterns in organized information 3.2 make connections among related, organized data, and assemble various pieces into a unified message F4 - Students will become discerning consumers of mass media and electronic information. 3.2 understand the nature of various media and how they are consciously used to influence an audience 3.3 identify specific techniques used by the media to elicit particular responses from an audience P1 - Students will compose, revise and edit text. 3.4 use appropriate communication technology to elicit feedback from others P2 - Students will organize and manipulate data. 3.1 design, create and modify a database for a specific purpose 3.3 use a variety of technological graphing tools to draw graphs for data involving one or two variables 3.4 use a scientific calculator or a computer to solve problems involving rational numbers

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Armstrong, Thomas. 7 Kinds of Smart: Identifying and Developing Your Many Intelligences. New York, NY: Plume, 1993.

Banks, J. A. and C. A. M. Banks. Multicultural Education: Issues and Perspectives. 2nd ed. Boston, MA: Allyn and Bacon, 1993.

British Columbia Ministry of Education. The Primary Program: A Framework for Teaching. Victoria, BC: British Columbia Ministry of Education, 2000.

Caine, Renate Nummela and Geoffrey Caine. Making Connections: Teaching and the Human Brain. Alexandria, VA: Association for Supervision and Curriculum Development, 1991.

Hope, Jack A. et al. Mental Math in the Primary Grades. Palo Alto, CA: Dale Seymour Publications, 1988.

National Council of Teachers of Mathematics. Computation, Calculators, and Common Sense: A Position of the National Council of Teachers of Mathematics. May 2005. http://www.nctm.org/uploadedFiles/About_NCTM/Position_Statements/computation.pdf (Accessed May 23, 2014).

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Shaw, J. M. and M. J. P. Cliatt. "Developing Measurement Sense." In P. R. Trafton (ed.), New Directions for Elementary School Mathematics: 1989 Yearbook (Reston, VA: National Council of Teachers of Mathematics, 1989), pp. 149 - 155.

Steen, L. A., ed. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: Mathematical Sciences Education Board, National Research Council, 1990.

Western and Northern Canadian Protocol for Collaboration in Basic Education (Kindergarten to Grade 12). The Common Curriculum Framework for K - 9 Mathematics: Western and Northern Canadian Protocol. May 2006. http://www.wncp.ca/media/38765/ccfkto9.pdf (Accessed May 23, 2014).