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.

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.

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.

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.

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 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.

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.

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.

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.

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.

- 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.

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

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.

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.

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.

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.