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Mathematics

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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.
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Organizing Idea
Number: Quantity is measured with numbers that enable counting, labelling, comparing, and operating.
Guiding Question
How can place value support our organization of number?
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?
Learning Outcome
Students interpret place value.
Learning Outcome
Students apply place value to decimal numbers.
Learning Outcome
Students analyze patterns in place value.
Knowledge
For numbers in base-10, each place has 10 times the value of the place to its right.

The digits 0 to 9 indicate the number of groups in each place in a number.

The value of each place in a number is the product of the digit and its place value.

Numbers can be composed in various ways using place value.

Numbers can be rounded in context when an exact count is not needed.

A zero in the leftmost place of a natural number does not change the value of the number.

The dollar sign, $, is placed to the left of the dollar value in English and to the right of the dollar value in French.

The cent sign, ¢, is placed to the right of the cent value in English and in French.

Understanding
Place value is the basis for the base-10 system.

Place value determines the value of a digit based on its place in a number, relative to the ones place.

Place value is used to read and write numbers.
Skills & Procedures
Identify the place value of each digit in a natural number.

Relate the values of adjacent places.

Determine the value of each digit in a natural number.

Express natural numbers using words and numerals.

Express various compositions of a natural number using place value.

Round natural numbers to various places.

Compare and order natural numbers.

Count and represent the value of a collection of nickels, dimes, and quarters as cents.

Count and represent the value of a collection of loonies, toonies, and bills as dollars.

Compare French and English symbolic representations of monetary values.

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.
Guiding Question
How can we establish processes for addition and subtraction?
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?
Learning Outcome
Students apply addition and subtraction within 1000.
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.
Knowledge
Recall of addition and subtraction number facts facilitates addition and subtraction strategies.

Estimation can be used when an exact sum or difference is not needed and to check if an answer is reasonable.

Standard algorithms for addition and subtraction are conventional procedures based on place value.
Understanding
Addition and subtraction strategies can be chosen based on the nature of the numbers.

Standard algorithms are universal tools for addition and subtraction and may be used for any natural numbers independently of their nature.
Skills & Procedures
Add and subtract natural numbers.

Estimate sums and differences.

Model regrouping by place value for addition and subtraction.

Explain the standard algorithms for addition and subtraction of natural numbers.

Add and subtract natural numbers using standard algorithms.

Solve problems using 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.
Guiding Question
How can multiplication and division provide new perspectives of number?
Guiding Question
How can we interpret multiplication and division?
Guiding Question
In what ways can we articulate the processes of multiplication and division?
Learning Outcome
Students acquire an understanding of multiplication and division within 100.
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.
Knowledge
Multiplication and division are inverse mathematical operations.

Multiplication is repeated addition.

Multiplication by two is doubling and multiplication by three is tripling.

Division is a process of sharing or grouping to find a quotient.

The order in which two quantities are multiplied does not affect the product (commutative property).

The order in which two numbers are divided affects the quotient.

Multiplication or division by 1 results in the same number (identity property).
Understanding
Quantities can be composed and decomposed through multiplication and division.
Skills & Procedures
Compose a product using equal groups of objects.

Relate multiplication to repeated addition.

Relate multiplication to skip counting.

Investigate multiplication by 0.

Model a quotient by partitioning a quantity into equal groups with or without remainders.

Visualize and model products and quotients as arrays.


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
Multiplication strategies include
  • repeated addition
  • multiplying in parts
  • compensation
Division strategies include
  • repeated subtraction
  • partitioning the dividend
The multiplication symbol is ×.

The division symbol is ÷.

The equal sign is =.

A remainder is the quantity left over after division.

Understanding
Sharing and grouping situations can be interpreted as multiplication or division.

Multiplication and division strategies can be supported by addition and subtraction.
Skills & Procedures
Investigate multiplication and division strategies.

Multiply and divide within 100.

Express multiplication and division symbolically.

Explain the meaning of the remainder in various situations.

Solve problems using multiplication and division in sharing or grouping situations.

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.
Knowledge
A multiplication table shows both multiplication and division facts.

Fact families are groups of related multiplication and division number facts.
Understanding
Multiplication number facts have related division facts.
Skills & Procedures
Examine patterns in multiplication and division, including patterns in multiplication tables and skip counting.

Recognize families of related multiplication and division number facts.

Recall multiplication number facts, with factors to 10, and related division facts.
Guiding Question
How can fractions contribute to our sense of number?
Guiding Question
In what ways can we work flexibly with fractions?
Guiding Question
How can percentages standardize part-whole relationships?
Learning Outcome
Students interpret fractions as part-whole relationships.
Learning Outcome
Students apply equivalence to the interpretation of proper and improper fractions.
Learning Outcome
Students interpret percentages.
Knowledge
Fraction notation, , relates the the numerator, a, as a number of equal parts, to the denominator, b, as the total number of equal parts in the whole.

A whole quantity can be a whole set of objects or a whole object that can be partitioned.

Each fraction is associated with a point on the number line.
Understanding
Fractions are numbers between natural numbers.

Fractions can represent part-to-whole relationships.

Skills & Procedures
Partition a whole into 12 or fewer equal parts.

Describe a whole as a fraction, limited to denominators of 12 or less.

Model fractions of a whole, limited to denominators of 12 or less.

Express fractions symbolically.

Relate a fraction less than one to its position on the number line, limited to denominators of 12 or less.

Compare fractions to benchmarks of 0, , and 1.
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
The whole can be any size and is designated by context.

Fractions can be compared by considering the number of parts or the size of parts.

Understanding
Fractions are interpreted relative to the whole.

The size of the parts and the number of partitions in the whole are inversely related.
Skills & Procedures
Recognize the whole to which a fraction refers in various situations.

Compare the same fraction of different-sized wholes.

Compare different fractions with the same denominator.

Compare different fractions with the same numerator.
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 the composition of fractions facilitate agility in operating with 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?
Learning Outcome
Students acquire an understanding of addition and subtraction of fractions with like denominators.
Learning Outcome
Students add and subtract positive fractions with like and unlike denominators.
Learning Outcome
Students interpret the multiplication of natural numbers by fractions.
Knowledge
A unit fraction is any one part of a whole divided into equal parts.

Fractions with common denominators are multiples of the same unit fraction.
Understanding
Any fraction can be interpreted as a composition of unit fractions.
Skills & Procedures
Decompose a fraction into unit fractions.

Express a fraction as repeated addition of a unit fraction.

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

Add and subtract fractions within one whole, limited to common denominators of 12 or less.

Solve problems involving fractions, limited to common denominators of 12 or less.
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.
Organizing Idea
Algebra: Equations express relationships between quantities.
Guiding Question
How can equality facilitate agility with number?
Guiding Question
How can equality create opportunities to reimagine number?
Guiding Question
How can expressions enhance communication of number?
Learning Outcome
Students interpret equality with equations.
Learning Outcome
Students visualize and apply equality in multiple ways.
Learning Outcome
Students interpret numerical and algebraic expressions.
Knowledge
The equal sign is not a signal to perform a given computation.

The left and right sides of an equation are interchangeable.
Understanding
An equation uses the equal sign to indicate equality between two expressions.

Two expressions are equal if they represent the same number.
Skills & Procedures
Write equations that represent equality between a number and an expression or between two different expressions of the same number.
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
A symbol may represent an unknown value in an equation.
Understanding
Equations can include unknown values.
Skills & Procedures
Model equations that include an unknown value.

Determine an unknown value on the left or right side of an equation, limited to equations with one operation.

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

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
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.

Organizing Idea
Geometry: Shapes are defined and related by geometric attributes.
Guiding Question
In what ways might geometric properties refine our interpretation of shape?
Guiding Question
In what ways can geometric properties define space?
Guiding Question
In what ways might symmetry characterize shape?
Learning Outcome
Students relate geometric properties to shape.
Learning Outcome
Students interpret and explain geometric properties.
Learning Outcome
Students interpret symmetry as a geometric property.
Knowledge
Geometric properties can describe relationships, including perpendicular, parallel, and equal.

Parallel lines or planes are always the same distance apart.

Perpendicular lines or planes intersect at a right angle.

Familiar representations of a right angle may include
  • the corner of a piece of paper
  • the angle between the hands on an analog clock at 3:00
  • a capital letter L
Polygons include
  • triangles
  • quadrilaterals
  • pentagons
  • hexagons
  • octagons
Regular polygons have sides of equal length and interior angles of equal measure.

Understanding
Geometric properties are relationships between geometric attributes.

Geometric properties define a class of polygon.
Skills & Procedures
Investigate geometric properties within polygons.

Describe geometric properties of regular and irregular polygons.

Sort polygons according to geometric properties and describe the sorting rule.

Classify polygons as regular or irregular using geometric properties.
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
Rigid transformations include
  • translations
  • rotations
  • reflections
Understanding
Geometric properties do not change when a polygon undergoes rigid transformation.
Skills & Procedures
Examine geometric properties of polygons by translating, rotating, or reflecting using hands-on materials or digital applications.
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.


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?
Learning Outcome
Students interpret location in relation to position on a grid.
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.
Organizing Idea
Measurement: Attributes such as length, area, volume, and angle are quantified by measurement.
Guiding Question
In what ways can we communicate length?
Guiding Question
How can area characterize space?
Guiding Question
In what ways can we communicate area?
Learning Outcome
Students explain length using standard units.
Learning Outcome
Students interpret and express area.
Learning Outcome
Students explain area using standard units.
Knowledge
The metric system, or système international d’unités (SI), is a base-10 system first adopted in France.

The basic unit of length in the metric system is the metre.

Metric units are named using prefixes that indicate the relationship to the basic unit (e.g., for length, the prefix centi- indicates there are 100 centimetres in a metre).

Metric units are abbreviated for convenience (e.g., metre is abbreviated with m and centimetre is abbreviated with cm).

Standard measuring tools show iterations of a standard unit from an origin.

The other, older, system of measurement that is also commonly used in the United States and Canada is sometimes called the imperial system and uses “Canadian units.”

“Canadian or imperial” units that are still commonly used include miles, yards, feet, inches, acres, pounds, quarts, pints, and ounces. You may encounter these in hospitals (birth announcements), housing and property (square footage/acreage), cooking and drink (pounds,ounces, quarts, pints), some roads and cars (miles, mileage, miles per hour, gallons), railways, and other contexts where integration with the United States is important.

The perimeter of a polygon is the sum of the lengths of its sides.
Understanding
Length is measured in standard units according to the metric system.

An alternative system, the imperial system, still partly in use, uses “Canadian units” (sometimes called “imperial units”). This system is important to know about because it provides core numeracy for current everyday life, understanding works of the past, and literacy concerning culture and trade with our biggest trading partner, the United States.

Length can be expressed in various units according to context and desired precision.

Length remains the same when decomposed or rearranged.
Skills & Procedures
Relate the metric system to the place value system.

Relate centimetres to metres.

Justify the choice of centimetres or metres to measure various lengths.

Measure lengths of straight lines and curves, with centimetres or metres, using a standard measuring tool.

Express length in centimetres or metres.

Convert commonly used units of measure between metric and Canadian (imperial) units within 100.

Determine perimeter of polygons.

Determine the length of an unknown side given the perimeter of a polygon.

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
A benchmark is a known length to which another length can be compared.

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

A common referent for a metre is the distance from a doorknob to the floor.
Understanding
Length can be estimated when less accuracy is required.
Skills & Procedures
Identify referents for a centimetre and a metre.

Estimate length by comparing to a benchmark of a centimetre or metre.

Estimate length by visualizing the iteration of a referent for a centimetre or metre.

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.
Guiding Question
How can angle broaden our interpretation of space?
Learning Outcome
Students interpret and express angle.
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
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 diverse representations of pattern contribute to our interpretation of change?
Guiding Question
How can sequence provide insight into change?
Guiding Question
How might representation of a sequence provide insight into change?
Learning Outcome
Students analyze pattern in numerical sequences.
Learning Outcome
Students interpret and explain arithmetic and geometric sequences.
Learning Outcome
Students relate position and terms of an arithmetic sequence.
Knowledge
Ordinal numbers can indicate position in a sequence.

Finite sequences, such as a countdown, have a definite end.

Infinite sequences, such as the natural numbers, never end.


Understanding
A sequence is a list of terms arranged in a certain order.

Sequences may be finite or infinite.
Skills & Procedures
Recognize familiar numerical sequences, including the sequence of even or odd numbers.

Describe position in a sequence using ordinal numbers.

Differentiate between finite and infinite sequences.
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
Numerical sequences can be constructed using addition, subtraction, multiplication, or division.
Understanding
A sequence can progress according to a pattern.
Skills & Procedures
Recognize skip-counting sequences in various representations, including rows or columns of a multiplication table.

Determine any missing term in a skip-counting sequence using multiplication.

Describe the change from term to term in a numerical sequence using mathematical operations.

Guess the next term in a sequence by inferring the pattern from the previous terms.

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
How can we communicate duration?
Guiding Question
What might be the relevance of duration to daily living?
Learning Outcome
Students tell time using clocks.
Learning Outcome
Students communicate duration with standard units of time.
Knowledge
Clocks relate seconds to minutes and hours according to a base-60 system.

The basic unit of time is the second.

One second is of a minute.

One minute is of an hour.

Analog and digital clocks represent time of day.

Time of day can be expressed as a duration relative to 12:00 in two 12-hour cycles.

Time of day can be expressed as a duration relative to 0:00 in one 24-hour cycle in some contexts, including French-language contexts.
Understanding
Clocks are standard measuring tools used to communicate time.

Skills & Procedures
Investigate relationships between seconds, minutes, and hours using an analog clock.

Relate minutes past the hour to minutes until the next hour.

Describe time of day as a.m. or p.m. relative to 12-hour cycles of day and night.

Tell time using analog and digital clocks.

Express time of day in relation to one 24-hour cycle according to context.
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
How can representation support communication?
Guiding Question
In what ways can we shape communication with our choice of representation?
Guiding Question
How might frequency bring meaning to data?
Learning Outcome
Students interpret and explain representation.
Learning Outcome
Students apply and evaluate representation with scale.
Learning Outcome
Students analyze frequency in categorized data.
Knowledge
Statistical questions are questions that can be answered by collecting data.
Understanding
Representation connects data to a statistical question.
Skills & Procedures
Formulate statistical questions for investigation.

Predict the answer to a statistical question.
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
Second-hand data is data collected by others.

Sources of second-hand data include
  • newspapers
  • maps
  • databases
  • websites
  • social media
  • stories
Understanding
Representation expresses data specific to a unique time and place.

Representation tells a story about data.
Skills & Procedures
Collect second-hand data using digital or non-digital tools and resources.

Represent second-hand data in a dot plot or bar graph with one-to-one correspondence.

Describe the story that a representation tells about a collection of data in relation to a statistical question.

Examine First Nations, Métis, or Inuit representations of data.

Consider possible answers to a statistical question based on the data collected.
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.