Alberta Logonew LearnAlberta

Mathematics

Collapse All

Mathematics is a subject in which students study patterns and relationships to understand various aspects of the world. Mathematical understanding is connected to many branches of mathematics, including arithmetic, algebra, geometry, data, statistics, and probability. The procedures associated with mathematics range from counting, calculating, and measuring to analyzing, modelling, and generalizing. Communication is also fundamental to mathematics. The language of mathematics has its own system of symbolic notation and a specific vocabulary with which to communicate mathematical thinking concisely.

Mathematical skills and knowledge support the interpretation of diverse quantitative and spatial information and can be applied to solving both theoretical and practical problems. With mathematics, abstract ideas can be visualized, represented, and explained. Mathematics is a powerful tool that can be used to simplify and solve complicated real-life problems.
More Info
Collapse All
Prev
Collapse All
 
Grade 4
Grade 5
Grade 6
Next
Organizing Idea
Number: Quantity is measured with numbers that enable counting, labelling, comparing, and operating.
Guiding Question
How can place value facilitate our interpretation of number?
Guiding Question
How can the infinite nature of place value enhance our insight into number?
Guiding Question
How can the infinite nature of the number line broaden our perception of number?
Learning Outcome
Students apply place value to decimal numbers.
Learning Outcome
Students analyze patterns in place value.
Learning Outcome
Students acquire an understanding of magnitude and operations with positive and negative numbers.
Knowledge
For numbers in base-10, each place has one-tenth the value of the place to its left.

Multiplying or dividing a number by 10 corresponds to moving the decimal point one position to the right or left, respectively.

A point is used for decimal notation in English.

A comma is used for decimal notation in French.

Numbers, including decimal numbers, can be composed in various ways using place value.

A zero placed to the right of the last digit in a decimal number does not change the value of the number.

The word and is used to indicate the decimal point when reading a number.


Understanding
Decimal numbers are numbers between natural numbers.

Decimal numbers are fractions with denominators of 10, 100, etc.

The separation between wholes and parts can be represented using decimal notation.

Patterns in place value are used to read and write numbers, including wholes and parts.
Skills & Procedures
Identify the place value of each digit in a number, including tenths and hundredths.

Relate the values of adjacent places, including tenths and hundredths.

Relate place value to multiplication by 10 and division by 10.

Determine the value of each digit in a number, including tenths and hundredths.

Express numbers, including decimal numbers, using words and numerals.

Express various compositions of a number, including decimal numbers, using place value.

Compare decimal notation expressed in English and in French.

Round numbers to various places, including tenths.

Compare and order numbers, including decimal numbers.

Express a monetary value in cents as a monetary value in dollars using decimal notation.
Knowledge
A number expressed with more decimal places is more precise.

A zero in the rightmost place of a decimal number does not change the value of the number.
Understanding
Place value symmetry extends infinitely to the left and right of the ones place.

There are infinitely many decimal numbers between any two decimal numbers.

Skills & Procedures
Relate the names of place values that are the same number of places to the left and right of the ones place.

Express numbers, including decimal numbers, using words and numerals.

Relate a decimal number to its position on the number line.

Determine a decimal number between any two other decimal numbers.

Compare and order numbers, including decimal numbers.

Round numbers, including decimal numbers, to various places according to context.
Knowledge
Negative numbers are to the left of zero on the number line visualized horizontally, and below zero on the number line visualized vertically.

Positive numbers can be represented symbolically with or without a positive sign (+).

Negative numbers are represented symbolically with a negative sign (-).

Zero is neither positive nor negative.

Negative numbers communicate meaning in context, including
  • temperature
  • debt
  • elevation
Magnitude is a number of units counted or measured from zero on the number line.

Every positive number has an opposite negative number with the same magnitude.

A number and its opposite are called additive inverses.

The additive inverse of a negative number is positive.
Understanding
Symmetry of the number line extends infinitely to the left and right of zero.

Direction relative to zero is indicated symbolically with a polarity sign.

Magnitude with direction distinguishes between positive and negative numbers.
Skills & Procedures
Identify negative numbers in familiar contexts, including contexts that use vertical or horizontal models of the number line.

Express positive and negative numbers symbolically, in context.

Relate magnitude to the distance from zero on the number line.

Relate positive and negative numbers, including additive inverses, to their positions on horizontal and vertical models of the number line.

Compare and order positive and negative numbers, including fractions and decimals.
Knowledge
The set of integers includes all natural numbers, their additive inverses, and zero.

The sum of any number and its additive inverse is zero.

The sum of two positive numbers is a positive number.

The sum of two negative numbers is a negative number.

The sum of a positive number and a negative number can be interpreted as the sum of zero and another number.
Understanding
Any number can be expressed as a sum in infinitely many ways.

Skills & Procedures
Investigate addition of an integer and its additive inverse.

Express zero as sum of intergers in multiple ways.

Model the sum of two positive integers.

Model the sum of two negative integers.

Model the sum of a positive and negative integer as the sum of zero and another integer.

Add any two integers.
Knowledge
Subtracting a number is the same as adding its additive inverse.
Understanding
The difference of any two numbers can be interpreted as a sum.
Skills & Procedures
Express a difference as a sum.

Add any two numbers, including positive or negative fractions or decimals.
Knowledge
The product or quotient of two positive numbers is a positive number.

The product or quotient of two negative numbers is a positive number.

The product or quotient of a negative number and a positive number is a negative number.
Understanding
Any product can be composed of positive and negative numbers.
Skills & Procedures
Investigate situations involving the multiplication or division of positive and negative numbers.

Generalize a rule to determine the polarity sign for the product of two or more integers.

Multiply any two numbers, including positive or negative fractions or decimals.

Divide any two numbers, including positive or negative fractions or decimals.
Guiding Question
How can we extend our understanding of addition and subtraction to decimal numbers?
Guiding Question
In what ways can we articulate the processes of addition and subtraction?
Guiding Question
How can we apply the processes of addition and subtraction to problem solving?
Learning Outcome
Students add and subtract within 10 000, including decimal numbers to hundredths.
Learning Outcome
Students add and subtract within 1 000 000, including decimal numbers to thousandths, using standard algorithms.
Learning Outcome
Students solve problems using standard algorithms for addition and subtraction.
Knowledge
Standard algorithms for addition and subtraction of decimal numbers are conventional procedures based on place value.

Estimation can be used to verify a sum or difference.
Understanding
Standard algorithms are universal tools for addition and subtraction and may be used for any decimal numbers independently of their nature.
Skills & Procedures
Add and subtract numbers, including decimal numbers, using standard algorithms.

Assess the reasonableness of a sum or difference by estimating.

Solve problems using addition and subtraction, including problems involving money.
Knowledge
Standard algorithms are efficient procedures for addition and subtraction.

Understanding
Addition and subtraction of numbers with many digits is facilitated by standard algorithms.
Skills & Procedures
Add and subtract numbers, including decimal numbers, using standard algorithms.

Assess the reasonableness of a sum or difference by estimating.

Solve problems using addition and subtraction, including problems involving money.
Knowledge
Standard algorithms are reliable procedures for addition and subtraction.

Contexts for problems involving addition and subtraction can include money and metric measurement.
Understanding
Addition and subtraction of numbers in problem-solving contexts is facilitated by standard algorithms.
Skills & Procedures
Solve problems in various contexts using standard algorithms for addition and subtraction.
Guiding Question
How can we interpret multiplication and division?
Guiding Question
In what ways can we articulate the processes of multiplication and division?
Guiding Question
How can we apply the processes of multiplication and division to decimal numbers?
Learning Outcome
Students explain multiplication and division within 10 000, including with standard algorithms for multiplication and division of 3-digit by 1-digit natural numbers
Learning Outcome
Students multiply 3-digit by 2-digit natural numbers and divide 3-digit by 1-digit natural numbers using standard algorithms.
Learning Outcome
Students apply standard algorithms to multiplication and division of 3-digit natural or decimal numbers by 2-digit natural numbers.
Knowledge
A factor of a number is a divisor of that number.

A prime number has factors of only itself and one.

A composite number has other factors besides one and itself.

Zero and one are neither prime nor composite.

A number is a multiple of any of its factors.

Prime factorization represents a number as a product of prime numbers.

The order in which three or more numbers are multiplied does not affect the product (associative property).

The order in which numbers are divided affects the quotient.

Numbers can be multiplied or divided in parts (distributive property).
Understanding
A product can be composed in multiple ways.

Any natural number can be represented uniquely as a product of prime numbers, including repeated prime numbers.

Any factor of a number can be determined from its prime factorization.
Skills & Procedures
Determine the factors of a number.

Describe a number as prime or composite.

Recognize multiples of numbers within 100.

Determine the greatest common factor (greatest common divisor) of two numbers.

Compose a product in multiple ways, including with more than two factors.

Represent composite numbers as products of prime numbers.

Relate composite factors of a number to its prime factorization.

Compare the prime factorization of two natural numbers.
Knowledge
Standard algorithms are efficient procedures for multiplication and division.
Understanding
Multiplication and division of numbers with many digits is facilitated by standard algorithms.
Skills & Procedures
Explain the standard algorithms for multiplication and division of natural numbers.

Multiply and divide natural numbers using standard algorithms.

Express a quotient with or without a remainder according to context.

Assess the reasonableness of a product or quotient by estimating.

Solve problems using multiplication and division of natural numbers.
Knowledge
Standard algorithms are reliable procedures for multiplication and division of numbers, including decimal numbers.

A quotient with a remainder can be expressed as a decimal number.
Understanding
Multiplication and division of decimal numbers is facilitated by standard algorithms.
Skills & Procedures
Explain the standard algorithms for multiplication and division of decimal numbers.

Multiply and divide numbers, including decimal numbers, using standard algorithms.

Assess the reasonableness of a product or quotient by estimating.

Solve problems using multiplication and division, including problems involving money.
Knowledge
Recall of multiplication and division number facts facilitates multiplication and division strategies.

Standard algorithms facilitate multiplication and division of natural numbers that have multiple digits.
Understanding
Multiplication and division strategies can be chosen based on the nature of the numbers.

Skills & Procedures
Recall and apply multiplication number facts, with factors to 12, and related division number facts.

Multiply and divide 3-digit natural numbers by 1-digit natural numbers using personal strategies.

Examine standard algorithms for multiplication and division.

Multiply and divide 3-digit natural numbers by 1-digit natural numbers using standard algorithms.

Express a quotient with or without a remainder according to context.

Solve problems using multiplication and division.
Guiding Question
In what ways can we work flexibly with fractions?
Guiding Question
How can percentages standardize part-whole relationships?
Guiding Question
How can ratios provide new ways to relate numbers?
Learning Outcome
Students apply equivalence to the interpretation of proper and improper fractions.
Learning Outcome
Students interpret percentages.
Learning Outcome
Students interpret ratios.
Knowledge
Fractions and decimal numbers that represent the same number are associated with the same point on the number line.
Understanding
Fractions and decimal numbers can represent the same number.

Decimal numbers are fractions with denominators of 10, 100, etc.
Skills & Procedures
Relate fractions to decimal numbers, limited to tenths and hundredths.

Relate fractions and equivalent decimal numbers, limited to tenths and hundredths, to their positions on the number line.


Knowledge
Percentage is represented symbolically with %.

Decimals can be expressed as percentages by multiplying by 100.

Percentages can be expressed as decimals by dividing by 100.
Understanding
Fractions, decimals, and percentages can represent the same part-whole relationship.

One percent represents one hundredth of a whole.
Skills & Procedures
Investigate percentage in familiar situations.

Model the same part-whole relationship as a fraction, decimal, and percentage.

Express the same part-whole relationship as a fraction, decimal, and percentage symbolically.

Compare percentages within 100%.
Knowledge
A ratio can relate any two countable or measurable quantities, including
  • a part to a part
  • a part to the whole
A ratio can be expressed with a fraction or with a colon.

A proportion is an expression of equivalence between two ratios.

A percentage represents the ratio of a number to 100.

Percent of a number can be determined by mutliplying the number by the percent and dividing by 100.

Understanding
A ratio is a comparison of two quantities.

There are infinitely many equivalent ratios.

Fractions, decimals, ratios, and percentages can represent the same part-whole relationship.
Skills & Procedures
Express part-part ratios and part-whole ratios of the same whole to describe various situations.

Express a ratio as a fraction, decimal, and percentage symbolically.

Express two equivalent ratios as a proportion.

Relate percentage of a number to a proportion.

Find a percent of a number, limited to percents within 100%.
Knowledge
Equivalent fractions are associated with the same point on the number line.

Multiplication by 1 results in equivalent fractions.

Division by 1 results in equivalent fractions.

The numerator and denominator of a fraction in simplest form have no common factors.

The most efficient way to express a fraction in simplest form is using the greatest common factor of the numerator and denominator.

Understanding
There are infinitely many equivalent fractions that represent the same number.

Exactly one of infinitely many equivalent fractions is in simplest form.
Skills & Procedures
Model equivalent fractions by partitioning a whole in multiple ways.

Represent fractions equivalent to a given fraction symbolically.

Relate the position of equivalent fractions on the number line.

Relate multiplying the numerator and denominator of a fraction by the same number to multiplying by 1.

Recognize a fraction where the numerator and denominator have a common factor.

Relate dividing the numerator and denominator of a fraction by the same number to dividing by 1.

Express a fraction in simplest form.
Knowledge
Fractions greater than one are called improper fractions and can be represented by a mixed number.

Natural numbers can be expressed as improper fractions with a denominator of 1.

Decimals can be expressed as fractions with the place value of the last non-zero digit of the decimal number as the denominator.

Fractions can represent quotients.

A fraction with the same numerator and denominator represents a quotient of 1.
Understanding
Numbers greater than one can be expressed with fractions or decimal numbers.
Skills & Procedures
Count beyond 1 using fractions with the same denominator and decimal numbers.

Model improper fractions.

Express improper fractions symbolically.

Relate fractions, including improper fractions, and equivalent decimal numbers to their positions on the number line.

Convert an improper fraction to a mixed number using division.

Convert between fractions and decimal numbers.

Compare and order fractions, including improper fractions.
Guiding Question
How can we generalize the addition and subtraction of fractions?
Guiding Question
How can we extend our understanding of multiplication to fractions?
Guiding Question
How can we generalize the multiplication and division of fractions?
Learning Outcome
Students add and subtract positive fractions with like and unlike denominators.
Learning Outcome
Students interpret the multiplication of natural numbers by fractions.
Learning Outcome
Students multiply and divide positive fractions.
Knowledge
Adding and subtracting fractions is facilitated by expressing fractions with common denominators.

The product of the denominators of two fractions provides a common denominator.

The most efficient way to express two fractions with common denominators is using the least common multiple of the two denominators.

Addition and subtraction of fractions can be used to solve problems in real-life situations, such as cooking and construction.
Understanding
Any two fractions can be added or subtracted.


Skills & Procedures
Recognize two fractions where the denominator of one fraction is a multiple of the other.

Recognize two fractions where the denominators have a common factor or multiple.

Express two fractions with common denominators.

Add and subtract fractions.

Solve problems using addition and subtraction of fractions.
Knowledge
Multiplication of a natural number by a fraction is equivalent to multiplication by its numerator and division by its denominator.


Multiplication by a unit fraction is equivalent to division by its denominator.


The product of a fraction and a natural number is the fraction with
  • a numerator that is the product of the numerator of the given fraction and the natural number
  • a denominator that is the denominator of the given fraction

Understanding
Multiplication does not always result in a larger number.

Multiplication of a natural number by a fraction can be interpreted as repeated addition of the fraction.

Multiplication of a fraction by a natural number can be interpreted as taking part of a quantity.
Skills & Procedures
Investigate multiplication of a natural number by a fraction as repeated addition of the fraction.

Relate multiplication of a natural number by a fraction to repeated addition of the fraction.

Multiply a natural number by a fraction.

Model a unit fraction of a natural number.

Relate multiplication by a unit fraction to division.

Multiply a unit fraction by a natural number.

Model a fraction of a natural number.

Multiply a fraction by a natural number.

Solve problems using addition and subtraction of fractions and multiplication of a fraction and a natural number.
Knowledge
The product of two fractions is the fraction with
  • a numerator that is the product of the numerators of the given fractions
  • a denominator that is the product of the denominators of the given fractions

A reciprocal is the multiplicative inverse of a fraction.

The reciprocal of a fraction is the fraction obtained by interchanging the numerator and the denominator.

The product of a fraction and its reciprocal is 1.

The reciprocal of a natural number is a unit fraction.

The reciprocal of a reciprocal is the original fraction.

A fraction less than one has a reciprocal greater than one.

A fraction greater than one has a reciprocal less than one.

Division by a fraction is equivalent to multiplication by its reciprocal.

Division by a fraction can be computed using the formula

Understanding
Multiplication of a fraction by a fraction can be interpreted as taking part of a partial quantity.

Division of fractions can be interpreted using multiplication.
Skills & Procedures
Model a fraction of a fraction.

Multiply a fraction by a fraction.

Identify the reciprocal of a given fraction.

Investigate multiplication of a fraction by its reciprocal.

Divide a fraction by a fraction.

Solve problems using operations with fractions.
Organizing Idea
Algebra: Equations express relationships between quantities.
Guiding Question
How can equality create opportunities to reimagine number?
Guiding Question
How can expressions enhance communication of number?
Guiding Question
How can expressions support a generalized interpretation of number?
Learning Outcome
Students visualize and apply equality in multiple ways.
Learning Outcome
Students interpret numerical and algebraic expressions.
Learning Outcome
Students analyze expressions and solve algebraic equations.
Knowledge
Expressions are evaluated according to the conventional order of operations:
  • Multiplication and division are performed before addition and subtraction.
  • Multiplication and division are performed in order from left to right.
  • Addition and subtraction are performed in order from left to right.
Understanding
There are infinitely many expressions that represent the same number.
Skills & Procedures
Evaluate expressions according to the order of operations.

Create various expressions of the same number using one or more operations.

Knowledge
Expressions composed only of numbers are called numerical expressions.

Numerical expressions are evaluated according to the conventional order of operations:
  • Operations in parentheses are performed before other operations.
  • Multiplication and division are performed before addition and subtraction.
  • Multiplication and division are performed in order from left to right.
  • Addition and subtraction are performed in order from left to right.
Understanding
Numerical expressions represent a quantity of known value.

Parentheses change the order of operations in a numerical expression.
Skills & Procedures
Evaluate numerical expressions involving addition or subtraction in parentheses according to the order of operations.
Knowledge
The product of a number of identical factors can be expressed as a power (e.g., ).

A power uses a base to represent the identical factor and an exponent to indicate the number of identical factors.

Any repeated prime factor within a prime factorization can be expressed as a power (e.g., 40 can be expressed as or ).

Numerical expressions are evaluated according to the conventional order of operations:
  • Operations in parentheses are performed before other operations.
  • Powers are evaluated before other operations are performed.
  • Multiplication and division are performed before addition and subtraction.
  • Multiplication and division are performed in order from left to right.
  • Addition and subtraction are performed in order from left to right.
Understanding
Numerical expressions can include powers.
Skills & Procedures
Express repeated multiplication as a power.

Express repeated prime factors within a prime factorization as a power.

Evaluate numerical expressions involving operations in parentheses and powers according to the order of operations.
Knowledge
Equality is preserved when each side of an equation is changed in the same way (preservation of equality).
Understanding
An equation is solved by determining the value of the symbol that makes the left and right sides of an equation equal.
Skills & Procedures
Write equations to represent a situation involving one operation.

Investigate preservation of equality by adding, subtracting, multiplying, or dividing the same number on both sides of an equation without an unknown value.

Apply preservation of equality to determine an unknown value in an equation, limited to equations with one operation.

Solve problems using equations, limited to equations with one operation.

Knowledge
Expressions that include variables are called algebraic expressions.

A variable can be interpreted as a specific unknown value and is represented symbolically with a letter.

Products with variables are expressed without the multiplication sign.

Quotients with variables are expressed using fraction notation.

An algebraic term is the product of a number, called a coefficient, and a variable.

A constant term is a number.
Understanding
Algebraic expressions use variables to represent quantities of unknown value.

Algebraic expressions may be composed of one algebraic term or the sum of algebraic and constant terms.
Skills & Procedures
Relate repeated addition of a variable to the product of a number and a variable.

Express the product of a number and a variable using a coefficient.

Express the quotient of a variable and a number as a fraction.

Recognize a product with a variable, a quotient with a variable, or a number as a single term.

Recognize the sum of an algebraic term and a constant term as two distinct terms.

Write an algebraic expression involving one or two terms to describe an unknown value.

Evaluate an algebraic expression by substituting a given number for the variable.
Knowledge
Algebraic terms with exactly the same variable are like terms.

Constant terms are like terms.

Like terms can be combined through addition or subtraction.

The terms of an algebraic expression can be rearranged according to algebraic properties.

Algebraic properties include
  • commutative property of addition: a + b = b + a, for any two numbers a and b
  • commutative property of multiplication: ab = ba, for any two numbers a and b
  • associative property of addition: (a + b) + c = a + (b + c)
  • associative property of multiplication: a(bc) = b(ac)
  • distributive property: a(b + c) = ab + ac
Understanding
There are infinitely many ways to express equivalent algebraic expressions.

Algebraic properties ensure equivalence of algebraic expressions.
Skills & Procedures
Investigate like terms by modelling an algebraic expression.

Simplify algebraic expressions by combining like terms.

Express the terms of an algebraic expression in a different order in accordance with algebraic properties.
Knowledge
The process of applying inverse operations can be used to solve an equation.
Understanding
Equality is preserved by applying inverse operations to algebraic expressions on each side of an equation.
Skills & Procedures
Write equations involving one or two operations to represent a situation.

Investigate order of operations when performing inverse operations on both sides of an equation.

Apply inverse operations to solve an equation, limited to equations with one or two operations.

Solve problems using equations, limited to equations with one or two operations.

Knowledge
The equation that shows equality between the variable and a number can be interpreted as the solution.
Understanding
Algebraic expressions on each side of the equation can be simplified into equivalent expressions to facilitate equation solving.
Skills & Procedures
Simplify algebraic expressions on both sides of an equation.

Solve equations, limited to equations with one or two operations.

Compare strategies for solving equations.

Solve problems using equations, limited to equations with one or two operations.

Organizing Idea
Geometry: Shapes are defined and related by geometric attributes.
Guiding Question
In what ways can geometric properties define space?
Guiding Question
In what ways might symmetry characterize shape?
Guiding Question
How can congruence support our interpretation of symmetry?
Learning Outcome
Students interpret and explain geometric properties.
Learning Outcome
Students interpret symmetry as a geometric property.
Learning Outcome
Students relate shapes through symmetry and congruence.
Knowledge
Angle relationships, including supplementary and complementary, are geometric properties.

Two or more angles that compose 90° are complementary angles.

Two or more angles that compose 180° are supplementary angles.

Quadrilaterals include
  • squares
  • rectangles
  • parallelograms
  • trapezoids
  • rhombuses
Triangles can be classified according to side length as
  • equilateral
  • isosceles
Triangles can be classified according to angle as
  • right
  • obtuse
  • acute
Understanding
Geometric properties are measurable.

Geometric properties define a hierarchy for classifying shapes.
Skills & Procedures
Identify relationships between the sides of a polygon, including parallel, equal length, or perpendicular, by measuring.

Identify relationships between angles within a polygon, including equal, supplementary, complementary, and sum of interior angles, by measuring.

Identify relationships between the faces of three-dimensional models of prisms, including parallel or perpendicular, by measuring.

Classify triangles as equilateral, isosceles, or neither using geometric properties related to sides.

Classify triangles as right, acute, or obtuse using geometric properties related to angles.

Classify quadrilaterals in a hierarchy according to geometric properties.
Knowledge
A 2-D shape has reflection symmetry if there is a line over which the shape reflects and the two halves exactly match.

A line of symmetry can be any straight line, including a horizontal or vertical line.

A 3-D shape has reflection symmetry if there is a plane over which the shape reflects and the two halves exactly match.

A 2-D shape has rotation symmetry if it exactly overlaps itself one or more times within a rotation of 360° around its centre point.

Order of rotation symmetry describes the number of times a shape coincides with itself within a rotation of 360° around its centre point.

Central symmetry is the rotational symmetry by 180°. It may be viewed as symmetry through the centre. The straight line that connects a point with its image in the central symmetry passes through the centre.

Symmetry can be found in First Nations, Métis, and Inuit design, including
  • weavings
  • quilts
  • beading
  • architecture such as tipis or longhouses
Understanding
Symmetry is a property of shapes.

Symmetry can be created and can occur in nature.


Skills & Procedures
Recognize symmetry in nature.

Recognize symmetry in First Nations, Métis, and Inuit design.

Investigate symmetry in familiar 2-D and 3-D shapes using hands-on materials or digital applications.

Show the line of symmetry of a 2-D shape.

Describe the order of rotation symmetry of a 2-D shape.
Knowledge
Symmetrical shapes can be mapped by any combination of reflections and rotations.

The tiling of a plane with symmetrical shapes is called a tessellation.
Understanding
Symmetry is a relationship between two shapes that can be mapped exactly onto each other through reflection or rotation.
Skills & Procedures
Verify symmetry of two shapes by reflecting or rotating one shape onto another.

Describe the symmetry between two shapes as reflection symmetry or rotation symmetry.

Visualize and describe a combination of two rigid transformations that relate symmetrical shapes.

Describe the symmetry modelled in a tessellation.
Knowledge
Many shapes in the environment resemble polygons.

Rigid transformations can be used to illustrate geometric properties of a polygon.
Understanding
A shape resembling a polygon that does not share the defining geometric properties of the polygon is a close approximation.
Skills & Procedures
Show, using geometric properties, that a close approximation of a polygon is not the same as the polygon.

Verify geometric properties of polygons by translating, rotating, or reflecting using hands-on materials or digital applications.
Knowledge
A regular polygon has the same number of sides, reflection symmetries, and rotation symmetries.

A circle has infinitely many reflection and rotation symmetries.


Understanding
Symmetry is related to other geometric properties.

Skills & Procedures
Compare the number of reflection and rotation symmetries of a 2-D shape to the number of equal sides and angles.

Classify 2-D shapes according to the number of reflection or rotation symmetries.


Knowledge
Shapes related by symmetry are congruent to each other.

Congruent shapes may not be related by symmetry.
Understanding
Congruence is a relationship between two shapes of identical size and shape.

Congruence is not dependent on orientation or location of the shapes.
Skills & Procedures
Demonstrate congruence between two shapes in any orientation by superimposing using hands-on materials or digital applications.

Describe symmetrical shapes as congruent.
Organizing Idea
Coordinate Geometry: Location and movement of objects in space can be communicated using a coordinate grid.
Guiding Question
How can location enhance the ways in which we define space?
Guiding Question
In what ways can we communicate location?
Learning Outcome
Students interpret location in relation to position on a grid.
Learning Outcome
Students explain location in relation to position in the Cartesian plane.
Knowledge
Coordinate grids use coordinates to indicate the location of the point where the vertical and horizontal grid lines intersect.

Coordinates are ordered pairs of numbers in which the first number indicates the distance from the vertical axis and the second number indicates the distance from the horizontal axis.

Positional language includes
  • left
  • right
  • up
  • down
Understanding
Location can describe the position of shapes in space.

Location can be described precisely using a coordinate grid.
Skills & Procedures
Locate a point on a coordinate grid given the coordinates of the point.

Describe the location of a point on a coordinate grid using coordinates.

Describe the location of a point on a coordinate grid in relation to the location of another point using positional language.

Model a polygon on a coordinate grid using coordinates to indicate the vertices.

Describe the location of the vertices of a polygon on a coordinate grid using coordinates.
Knowledge
The Cartesian plane is named after French mathematician René Descartes.

The Cartesian plane uses coordinates,(x, y), to indicate the location of the point where the vertical line passing through (x, 0) and the horizontal line passing through (0, y) intersect.

The x-axis consists of those points whose y-coordinate is zero and the y-axis consists of those points whose x-coordinate is zero.

The x-axis and the y-axis intersect at the origin, (0, 0).

An ordered pair is represented symbolically as (x, y).

An ordered pair indicates the horizontal distance from the y-axis with the x-coordinate and the vertical distance from the x-axis with the y-coordinate.

Understanding
Location can be described using the Cartesian plane.

The Cartesian plane is the two-dimensional equivalent of the number line.


Skills & Procedures
Relate the axes of the Cartesian plane to intersecting horizontal and vertical representations of the number line.

Locate a point in the Cartesian plane given the coordinates of the point.

Describe the location of a point in the Cartesian plane using coordinates.

Model a polygon in the Cartesian plane using coordinates to indicate the vertices.

Describe the location of the vertices of a polygon in the Cartesian plane using coordinates.
Knowledge
A translation describes a combination of horizontal and vertical movements as a single movement.

A reflection describes movement across a reflection line.

A rotation describes an amount of movement around a turn centre along a circular path in either a clockwise or counter-clockwise direction.
Understanding
Location can change as a result of movement in space.

Change in location does not imply change in orientation.
Skills & Procedures
Create an image of a polygon in the Cartesian plane by translating the polygon.

Describe the horizontal and vertical components of a given translation.

Create an image of a polygon in the Cartesian plane by reflecting the polygon over the x-axis or y-axis.

Describe the line of reflection of a given reflection.

Create an image of a polygon in the Cartesian plane by rotating the polygon 90°, 180°, or 270° about one of its vertices, clockwise or counter-clockwise.

Describe the angle and direction of a given rotation.

Relate the coordinates of a polygon and its image after translation, reflection, or rotation in the Cartesian plane.
Organizing Idea
Measurement: Attributes such as length, area, volume, and angle are quantified by measurement.
Guiding Question
How can area characterize space?
Guiding Question
In what ways can we communicate area?
Guiding Question
In what ways can we relate shapes using conservation of area?
Learning Outcome
Students interpret and express area.
Learning Outcome
Students explain area using standard units.
Learning Outcome
Students analyze area of parallelograms and triangles.
Knowledge
Tiling is the process of measuring an area with many copies of a unit.

Units that tile fit together without gaps or overlaps.

The unit can be chosen based on the area to be measured.

Area can be measured with non-standard units or standard units (e.g., square centimetres).

The area of a rectangle equals the product of its perpendicular side lengths.
Understanding
Area is a measurable attribute that describes the amount of two-dimensional space contained within a region.

Area may be interpreted as the result of motion of a length.

An area remains the same when decomposed or rearranged.

Area is quantified by measurement.

Area is measured with equal-sized units that themselves have area and do not need to resemble the region being measured.

The area of a rectangle can be perceived as square-shaped units structured in a two-dimensional array.

Skills & Procedures
Model area by dragging a length using hands-on materials or digital applications.

Recognize the rearrangement of area in First Nations, Métis, or Inuit design.

Compare non-standard units that tile to non-standard units that do not tile.

Measure area with non-standard units by tiling.

Measure area with standard units by tiling with a square centimetre.

Visualize and model the area of various rectangles as two-dimensional arrays of square-shaped units.

Determine the area of a rectangle using multiplication.

Solve problems involving area of rectangles.

Knowledge
Area is expressed in the following standard units, derived from standard units of length:
  • square centimetres
  • square metres
  • square kilometres
A square centimetre (cm2) is an area equivalent to the area of a square measuring 1 centimetre by 1 centimetre.

A square metre (m2) is an area equivalent to the area of a square measuring 1 metre by 1 metre.

A square kilometre (km2) is an area equivalent to the area of a square measuring 1 kilometre by 1 kilometre.

Among all rectangles with the same area, the square has the least perimeter.
Understanding
Area can be expressed in various units according to context and desired precision.

Rectangles with the same area can have different perimeters.
Skills & Procedures
Relate a centimetre to a square centimetre.

Relate a metre to a square metre.

Relate a square centimetre to a square metre.

Express the relationship between square centimetres, square metres, and square kilometres.

Justify the choice of square centimetres, square metres, or square kilometres as appropriate units to express various areas.

Estimate an area by comparing to a benchmark of a square centimetre or square metre.

Express the area of a rectangle using standard units given the lengths of its sides.

Compare the perimeters of various rectangles with the same area.

Describe the rectangle with the least perimeter for a given area.

Solve problems involving perimeter and area of rectangles.
Knowledge
Any side of a parallelogram can be interpreted as the base.

The height of a parallelogram is the perpendicular distance from its base to its opposite side.

The area of a triangle is half of the area of a parallelogram with the same base and height.

Two triangles with the same base and height must have the same area.






Understanding
The area of a parallelogram can be generalized as the product of the perpendicular base and height.

The area of a triangle can be interpreted relative to the area of a parallelogram.



Skills & Procedures
Rearrange the area of a parallelogram to form a rectangular area using hands-on materials or digital applications.

Determine the area of a parallelogram using multiplication.

Determine the base or height of a parallelogram using division.

Model the area of a parallelogram as two congruent triangles.

Describe the relationship between the area of a triangle and the area of a parallelogram with the same base and height.

Determine the area of a triangle, including various triangles with the same base and height.

Solve problems involving area of parallelograms and triangles.








Knowledge
A common referent for a square centimetre is the area of the fingernail on the little finger.
Understanding
Area can be estimated when less accuracy is required.
Skills & Procedures
Identify referents for a square centimetre.

Estimate an area by visualizing a referent for a square centimetre.

Estimate an area by rearranging or combining partial units.
Knowledge
Area of composite shapes can be interpreted as the sum of the areas of multiple shapes, such as triangles and parallelograms.
Understanding
An area can be decomposed in infinitely many ways.
Skills & Procedures
Visualize the decomposition of composite areas in various ways.

Determine the area of composite shapes using the areas of triangles and parallelograms.
Guiding Question
How can angle broaden our interpretation of space?
Guiding Question
How can volume characterize space?
Learning Outcome
Students interpret and express angle.
Learning Outcome
Students interpret and express volume.
Knowledge
Angle defines the space in
  • corners
  • bends
  • turns or rotations
  • intersections
  • slopes
The arms of an angle can be line segments or rays.

The end point of a line segment or ray is called a vertex.


Understanding
An angle is the union of two arms with a common vertex.

An angle can be interpreted as the motion of a length rotated about a vertex.
Skills & Procedures
Recognize various angles in surroundings.

Recognize situations in which an angle can be perceived as motion.
Knowledge
Volume can be measured in non-standard units or standard units (e.g., cubic centimetres).

Volume is expressed in the following standard units, derived from standard units of length:
  • cubic centimetres
  • cubic metres
A cubic centimetre (cm3) is a volume equivalent to the volume of a cube measuring 1 centimetre by 1 centimetre by 1 centimetre.

A cubic metre (m3) is a volume equivalent to the volume of a cube measuring 1 metre by 1 metre by 1 metre.

The volume of a right rectangular prism can be interpreted as the product of the two-dimensional base area and the perpendicular height of the prism.


Understanding
Volume is a measurable attribute that describes the amount of three-dimensional space occupied by a three-dimensional object.

The volume of a prism can be interpreted as the result of perpendicular motion of an area.

Volume remains the same when decomposed or rearranged.

Volume is quantified by measurement.

Volume is measured with congruent units that themselves have volume and do not need to resemble the shape being measured.

The volume of a right rectangular prism can be perceived as cube-shaped units structured in a three-dimensional array.

Skills & Procedures
Recognize volume in familiar contexts.

Model volume of prisms by dragging or iterating an area using hands-on materials or digital applications.

Create a model of a three-dimensional object by stacking congruent non-standard units or cubic centimetres without gaps or overlaps.

Express volume in non-standard units or cubic centimetres.

Visualize and model the volume of various right rectangular prisms as three-dimensional arrays of cube-shaped units.

Determine the volume of a right rectangular prism using multiplication.

Solve problems involving volume of right rectangular prisms.
Knowledge
Superimposing is the process of placing one angle over another to compare angles.
Understanding
Two angles can be compared directly or indirectly with a third angle.
Skills & Procedures
Compare two angles directly by superimposing.

Compare two angles indirectly with a third angle by superimposing.

Estimate which of two angles is greater.
Knowledge
One degree represents of the rotation of a full circle.

Angles can be classified according to their measure
  • acute angles measure less than 90°
  • right angles measure 90°
  • obtuse angles measure between 90° and 180°
  • straight angles measure 180°
Understanding
Angle is quantified by measurement.

Angle is measured with equal-sized units that themselves are angles.

Angle measurement is based on the division of a circle.
Skills & Procedures
Measure an angle with degrees using a protractor.

Describe an angle as acute, right, obtuse, or straight.

Relate angles of 90°, 180°, 270°, and 360° to fractions of a circle.
Knowledge
A benchmark is a known angle to which another angle can be compared.

A referent is a personal or familiar representation of a known angle.

Understanding
Angle can be estimated when less accuracy is required.
Skills & Procedures
Identify referents for 45°, 90°, 180°, 270°, and 360°.

Estimate angles by comparing to benchmarks of 45°, 90°, 180°, 270°, and 360°.

Estimate angles by visualizing referents for 45°, 90°, 180°, 270°, and 360°.
Organizing Idea
Patterns: Awareness of patterns supports problem solving in various situations.
Guiding Question
How can sequence provide insight into change?
Guiding Question
How might representation of a sequence provide insight into change?
Guiding Question
How can a function enhance our interpretation of change?
Learning Outcome
Students interpret and explain arithmetic and geometric sequences.
Learning Outcome
Students relate position and terms of an arithmetic sequence.
Learning Outcome
Students acquire an understanding of functions.
Knowledge
The sequences of triangle and square numbers are examples of increasing sequences.

The Fibonacci sequence is an increasing sequence that occurs in nature.
Understanding
Sequences may increase or decrease.

Different representations can provide new perspectives of the increase or decrease of a sequence.
Skills & Procedures
Investigate increasing sequences, including the Fibonacci sequence, in multiple representations.

Create and explain increasing or decreasing sequences, including numerical sequences.

Express a numerical sequence to represent a concrete or pictorial sequence.
Knowledge
A table of values representing an arithmetic sequence lists the position in the first column or row and the corresponding term in the second column or row.

Points representing an arithmetic sequence on a coordinate grid fit on a straight line.

An algebraic expression can describe the relationship between the positions and terms of an arithmetic sequence.
Understanding
Each term of an arithmetic sequence corresponds to a natural number indicating position in the sequence.


Skills & Procedures
Represent one-to-one correspondence between positions and terms of an arithmetic sequence in a table of values and on a coordinate grid.

Describe the graph of an arithmetic sequence as a straight line.

Describe a rule, limited to one operation, that expresses correspondence between positions and terms of an arithmetic sequence.

Write an algebraic expression, limited to one operation, that represents correspondence between positions and terms of an arithmetic sequence.

Determine the missing term in an arithmetic sequence that corresponds to a given position.

Solve problems involving an arithmetic sequence.
Knowledge
A variable can be interpreted as the values of a changing quantity.

A function can involve quantities that change over time, including
  • height or weight of a person
  • height of a plant
  • temperature
  • distance travelled
A table of values lists the values of the independent variable in the first column or row and the values of the dependent variable in the second column or row to represent a function at certain points.

The values of the independent variable are represented by x-coordinates in the Cartesian plane.

The values of the dependent variable are represented by y-coordinates in the Cartesian plane.
Understanding
A function is a correspondence between two changing quantities represented by independent and dependent variables.

Each value of the independent variable in a function corresponds to exactly one value of the dependent variable.


Skills & Procedures
Identify the dependent and independent variables in a given situation, including situations involving change over time.

Describe the rule that determines the values of the dependent variable from values of the independent variable.

Create a table of values representing corresponding values of the independent and dependent variables of a function at certain points.

Represent corresponding values of the independent and dependent variables of a function as points in the Cartesian plane.

Write an algebraic expression that represents a function.

Recognize various representations of the same function.

Determine a value of the dependent variable of a function given the corresponding value of the independent variable.

Investigate strategies for determining a value of the independent variable of a function given the corresponding value of the dependent variable.

Solve problems involving a function.
Knowledge
An arithmetic sequence progresses through addition or subtraction.

A skip-counting sequence is an example of an arithmetic sequence.

A geometric sequence progresses through multiplication.

A geometric sequence begins at a number other than zero.

Understanding
An arithmetic sequence has a constant difference between consecutive terms.

A geometric sequence has a constant ratio between consecutive terms.
Skills & Procedures
Recognize arithmetic and geometric sequences.

Describe the initial term and the constant change in an arithmetic sequence.

Express the first five terms of an arithmetic sequence related to a given initial term and constant change.

Describe the initial term and the constant change in a geometric sequence.

Express the first five terms of a geometric sequence related to a given initial term and constant change.

Organizing Idea
Time: Duration is described and quantified with time.
Guiding Question
What might be the relevance of duration to daily living?
Learning Outcome
Students communicate duration with standard units of time.
Knowledge
Time of day can be expressed with fractions of a circle, including
  • quarter past the hour
  • half past the hour
  • quarter to the hour
Duration can be determined by finding the difference between a start time and an end time.

Understanding
Analog clocks can relate duration to a circle.
Skills & Procedures
Relate durations of 15 minutes, 20 minutes, 30 minutes, 40 minutes, and 45 minutes to fractions of a circle.

Express time of day using fractions.

Determine duration in minutes using a clock.

Apply addition and subtraction strategies to the calculation of duration.

Convert between hours, minutes, and seconds.

Compare the duration of events using standard units.

Solve problems involving duration.
Organizing Idea
Statistics: The science of collecting, analyzing, visualizing, and interpreting data can inform understanding and decision making.
Guiding Question
In what ways can we shape communication with our choice of representation?
Guiding Question
How might frequency bring meaning to data?
Guiding Question
How can frequency support communication?
Learning Outcome
Students apply and evaluate representation with scale.
Learning Outcome
Students analyze frequency in categorized data.
Learning Outcome
Students apply and explain relative frequency with experimental data.
Knowledge
A statistical problem-solving process includes
  • formulating statistical questions
  • collecting data
  • representing data
  • interpreting data
Understanding
Representation is part of a statistical problem-solving process.
Skills & Procedures
Engage in a statistical problem-solving process.
Knowledge
Frequency can be compared across categories to answer statistical questions.

The mode is the category with the highest frequency.
Understanding
Frequency is a count of categorized data, but it is not the data value itself.
Skills & Procedures
Examine categorized data in tables and graphs.

Determine frequency for each category of a set of data by counting individual data points.

Identify the mode in various representations of data.

Recognize data sets with no mode, one mode, or multiple modes.

Justify possible answers to a statistical question using mode.
Knowledge
Relative frequency can be used to compare the same category of data across multiple data sets.

Relative frequency can be represented in various forms.

Understanding
Relative frequency expresses the frequency of a category of data as a fraction of the total number of data values.

Skills & Procedures
Interpret frequency of categorized data as relative frequency.

Express relative frequencies as decimals, fractions, or percentages.
Knowledge
Many-to-one correspondence is the representation of many objects using one object or interval on a graph.

Graphs can include
  • pictographs
  • bar graphs
  • dot plots
Understanding
Representation can express many-to-one correspondence by defining a scale.

Different representations tell different stories about the same data.
Skills & Procedures
Select an appropriate scale to represent data.

Represent data in a graph using many-to-one correspondence.

Describe the effect of scale on representation.

Justify the choice of graph used to represent certain data.

Compare different graphs of the same data.

Interpret data represented in various graphs.
Knowledge
Closed-list response survey questions provide a list of possible responses.

Open-ended response survey questions allow any response.

Survey responses can be categorized in various ways.

Representations of frequency can include
  • bar graphs
  • dot plots
  • stem-and-leaf plots
Understanding
Frequency can be a count of categorized responses to a question.

Frequency can be used to summarize data.

Frequency can be represented in various forms.

Skills & Procedures
Discuss potential categories for open-ended response survey questions and closed-list response survey questions in relation to the same statistical question.

Formulate closed-list response survey questions to collect data to answer a statistical question.

Categorize data collected from a closed-list response survey.

Organize counts of categorized data in a frequency table.

Create various representations of data, including with technology, to interpret frequency.
Knowledge
Equally likely outcomes of an experiment have the same chance of occurring.

An event can be described as the outcome of an experiment, including
  • heads or tails from a coin toss
  • any roll of a die
  • the result of spinning a spinner
The law of large numbers states that more independent trials of an experiment result in a better estimate of the expected likelihood of an event.
Understanding
Frequency can be a count of categorized observations or trials in an experiment.

Relative frequency of outcomes can be used to estimate the likelihood of an event.

Relative frequency varies between sets of collected data.

Relative frequency provides a better estimate of the likelihood of an event with larger amounts of data.
Skills & Procedures
Identify the possible outcomes of an experiment involving equally likely outcomes.

Collect categorized data through experiments, including with coins, dice, and spinners.

Predict the likelihood of an event based on the possible outcomes of an experiment.

Determine relative frequency for categories of a sample of data.

Describe the likelihood of an outcome in an experiment using relative frequency.

Analyze relative frequency statistics from experiments with different sample sizes.