Mathematics

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.

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.

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.

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.

A zero in the rightmost place of a decimal number does not change the value of the number.

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

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.

Positive numbers can be represented symbolically with or without a

Negative numbers are represented symbolically with a

Zero is neither positive nor negative.

Negative numbers communicate meaning in context, including

- temperature
- debt
- elevation

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.

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

Magnitude with direction distinguishes between positive and negative numbers.

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.

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.

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.

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

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.

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.

Estimation can be used to verify a sum or difference.

Assess the reasonableness of a sum or difference by estimating.

Solve problems using addition and subtraction, including problems involving money.

Assess the reasonableness of a sum or difference by estimating.

Solve problems using addition and subtraction, including problems involving money.

Contexts for problems involving addition and subtraction can include money and metric measurement.

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).

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.

Describe a number as prime or composite.

Recognize multiples of numbers within 100.

Determine the greatest common factor (greatest common divisor) of

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.

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.

A quotient with a remainder can be expressed as a decimal number.

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.

Standard algorithms facilitate multiplication and division of natural numbers that have multiple digits.

Multiply and divide

Examine standard algorithms for multiplication and division.

Multiply and divide

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

Solve problems using multiplication and division.

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

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

Decimals can be expressed as percentages by multiplying by 100.

Percentages can be expressed as decimals by dividing by 100.

One percent represents one hundredth of a whole.

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%.

- a part to a part
- a part to the whole

A proportion is an expression of equivalence between two ratios.

A percentage represents the ratio of a number

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

There are infinitely many equivalent ratios.

Fractions, decimals, ratios, and percentages can represent the same part-whole relationship.

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

Multiplication

Division

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.

Exactly one of infinitely many equivalent fractions is in simplest form.

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

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

Express a fraction in simplest form.

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.

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.

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.

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.

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

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.

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.

- 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

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

Division of fractions can be interpreted using multiplication.

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.

- 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.

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

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.

Parentheses change the order of operations in a numerical expression.

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.

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.

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.

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.

Algebraic expressions may be composed of one algebraic term or the sum of algebraic and constant terms.

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.

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:
, for any two numbers*a*+*b*=*b*+*a**a*and*b* - commutative property of multiplication:
for any two numbers*ab*=*ba*,*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*

Algebraic properties ensure equivalence of algebraic expressions.

Simplify algebraic expressions by combining like terms.

Express the terms of an algebraic expression in a different order in accordance with algebraic properties.

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.

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.

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

- equilateral
- isosceles

- right
- obtuse
- acute

Geometric properties define a hierarchy for classifying shapes.

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.

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

Symmetry can be created and can occur 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.

The tiling of a plane with symmetrical shapes is called a tessellation.

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.

Rigid transformations can be used to illustrate geometric properties of a polygon.

Verify geometric properties of polygons by translating, rotating, or reflecting using hands-on materials or digital applications.

A circle has infinitely many reflection and rotation symmetries.

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

Congruent shapes may not be related by symmetry.

Congruence is not dependent on orientation or location of the shapes.

Describe symmetrical shapes as congruent.

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

Location can be described precisely using a coordinate grid.

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.

The Cartesian plane uses

The

The

An ordered pair is represented symbolically as (

An ordered pair indicates the horizontal distance from the

The Cartesian plane is the two-dimensional equivalent 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.

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.

Change in location does not imply change in orientation.

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

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.

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.

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.

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.

- square centimetres
- square metres
- square kilometres

A square

A square

Among all rectangles with the same area, the square has the least perimeter.

Rectangles with the same area can have different perimeters.

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.

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.

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

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.

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

Estimate an area by rearranging or combining partial units.

Determine the area of composite shapes using the areas of triangles and parallelograms.

- corners
- bends
- turns or rotations
- intersections
- slopes

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

An angle can be interpreted as the motion of a length rotated about a vertex.

Recognize situations in which an angle can be perceived as motion.

Volume is expressed in the following standard units, derived from standard units of length:

- cubic centimetres
- cubic metres

A cubic metre (m

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.

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.

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.

Compare two angles indirectly with a third angle by superimposing.

Estimate which of two angles is greater.

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°

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

Angle measurement is based on the division of a circle.

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

Relate angles of 90°, 180°, 270°, and 360° to fractions of a circle.

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

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°.

The Fibonacci sequence is an increasing sequence that occurs in nature.

Different representations can provide new perspectives of the increase or decrease of a sequence.

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

Express a numerical sequence to represent a concrete or pictorial sequence.

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.

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.

A function can involve quantities that change over time, including

- height or weight of a person
- height of a plant
- temperature
- distance travelled

The values of the independent variable are represented by

The values of the dependent variable are represented by

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

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.

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.

A geometric sequence has a constant ratio between consecutive terms.

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.

- formulating statistical questions
- collecting data
- representing data
- interpreting data

The mode is the category with the highest frequency.

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.

Relative frequency can be represented in various forms.

Express relative frequencies as decimals, fractions, or percentages.

Graphs can include

- pictographs
- bar graphs
- dot plots

Different representations tell different stories about the same 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.

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

Frequency can be used to summarize data.

Frequency can be represented in various forms.

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.

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

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.

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.