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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 quantity contribute to our sense of number?
Guiding Question
How can place value support our organization of number?
Guiding Question
How can place value facilitate our interpretation of number?
Learning Outcome
Students analyze quantity to 1000.
Learning Outcome
Students interpret place value.
Learning Outcome
Students apply place value to decimal numbers.
Knowledge
The number of objects in a set can be represented by a natural number.

The number line is a spatial interpretation of quantity.

Understanding
There are infinitely many natural numbers.

Each natural number is associated with exactly one point on the number line.


Skills & Procedures
Express quantities using words.

Represent quantities using natural numbers.

Relate a natural number to its position on the number line.
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 quantity can be skip counted in various ways according to context, including by denominations of coins and bills.
Understanding
A quantity can be interpreted as a composition of groups.
Skills & Procedures
Decompose quantities into groups of 100s, 10s, and ones.

Count within 1000, forward and backward by 1, starting at any number.

Skip count by 20, 25, or 50, starting at 0.

Determine the value of a collection of coins or bills of the same denomination by skip counting.

Knowledge
A benchmark is a known quantity to which another quantity can be compared.
Understanding
A quantity can be estimated when an exact count is not needed.
Skills & Procedures
Estimate quantities using benchmarks.

Knowledge
The less than sign is <.

The greater than sign is >.
Understanding
Each natural number is one greater than the natural number to its left on the number line, visualized horizontally.
Skills & Procedures
Compare and order natural numbers.

Describe a natural number as greater than, less than, or equal to another natural number using words or symbols.
Guiding Question
How can we interpret addition and subtraction?
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?
Learning Outcome
Students explain addition and subtraction within 100.
Learning Outcome
Students apply addition and subtraction within 1000.
Learning Outcome
Students add and subtract within 10 000, including decimal numbers to hundredths.
Knowledge
The order in which more than two numbers are added does not affect the sum (associative property).
Understanding
A sum can be composed in multiple ways.
Skills & Procedures
Compose a sum in multiple ways, including with more than two addends.


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
Familiar addition and subtraction number facts facilitate addition and subtraction strategies.
Understanding
Addition and subtraction can represent the sum or difference of countable quantities (e.g., marbles or blocks) or measurable lengths (e.g., string length or student height).

Skills & Procedures
Recall and apply addition number facts, with addends to 10, and related subtraction number facts.

Add and subtract numbers within 100.

Solve problems using addition and subtraction of countable quantities or measurable lengths.

Model transactions with money, limited to dollar values within 100 dollars or cent values within 100 cents.
Guiding Question
In what ways can composition characterize number?
Guiding Question
How can multiplication and division provide new perspectives of number?
Guiding Question
How can we interpret multiplication and division?
Learning Outcome
Students interpret even and odd quantities within 100.

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
Knowledge
An even quantity will have no remainder when partitioned into two equal groups or groups of two.

An odd quantity will have a remainder of one when partitioned into two equal groups or groups of two.
Understanding
All natural numbers are either even or odd.
Skills & Procedures
Model even and odd quantities by sharing and grouping.

Describe a quantity as even or odd.

Partition a set of objects by sharing or grouping, with or without remainders.
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
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
In what ways can parts compose a whole?
Guiding Question
How can fractions contribute to our sense of number?
Guiding Question
In what ways can we work flexibly with fractions?
Learning Outcome
Students interpret one whole using halves and quarters.
Learning Outcome
Students interpret fractions as part-whole relationships.
Learning Outcome
Students apply equivalence to the interpretation of proper and improper fractions.
Knowledge
One-half is one of two equal parts.

One-quarter is one of four equal parts.
Understanding
When a quantity is partitioned into equal groups, each group represents an equal part of the whole quantity.
Skills & Procedures
Partition an even set of objects into two equal groups and four equal groups.

Describe one of two equal groups as one-half and one of four equal groups as one-quarter.

Describe a whole set of objects as a composition of halves and as a composition of quarters.
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
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?
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.
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.
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?
Learning Outcome
Students interpret equality with equations.
Learning Outcome
Students visualize and apply equality in multiple ways.
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
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.

Organizing Idea
Geometry: Shapes are defined and related by geometric attributes.
Guiding Question
How can shape influence our perception of space?
Guiding Question
In what ways might geometric properties refine our interpretation of shape?
Guiding Question
In what ways can geometric properties define space?
Learning Outcome
Students analyze and explain geometric attributes of shape.
Learning Outcome
Students relate geometric properties to shape.
Learning Outcome
Students interpret and explain geometric properties.
Knowledge
Common geometric attributes include
  • sides
  • vertices
  • faces or surfaces
Two-dimensional shapes may have sides that are line segments.

Three-dimensional shapes may have faces that are two-dimensional shapes.
Understanding
Shapes are defined according to geometric attributes.

A shape can be visualized as a composition of other shapes.
Skills & Procedures
Sort shapes according to two geometric attributes and describe the sorting rule.

Relate the faces of three-dimensional shapes to two-dimensional shapes.

Create a picture or design with shapes from verbal instructions, visualization, or memory.

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 shape can change orientation or position through slides (translations), turns (rotations), or flips (reflections).


Understanding
Geometric attributes do not change when a shape is translated, rotated, or reflected.

First Nations, Métis, and Inuit translate, rotate, and reflect shapes in the creation of cultural art.
Skills & Procedures
Investigate translation, rotation, and reflection of two- and three-dimensional shapes.

Describe geometric attributes of two- and three-dimensional shapes in various orientations.

Recognize translation, rotation, or reflection of shapes represented in First Nations, Métis, or Inuit art inspired by the natural world.

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.
Organizing Idea
Measurement: Attributes such as length, area, volume, and angle are quantified by measurement.
Guiding Question
How can length contribute to our interpretation of space?
Guiding Question
In what ways can we communicate length?
Guiding Question
How can area characterize space?
Learning Outcome
Students communicate length using units.
Learning Outcome
Students explain length using standard units.
Learning Outcome
Students interpret and express area.
Knowledge
Tiling is the process of measuring a length with many copies of a unit without gaps or overlaps.

Iterating is the process of measuring a length by repeating one copy of a unit without gaps or overlaps.

Length can be measured more efficiently using a measuring tool that shows iterations of a unit.

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

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

Standard units enable a common language around measurement.
Understanding
Length is quantified by measurement.

Length is measured with equal-sized units that themselves have length.

The size of the unit and the number of units in the length are inversely related.

Skills & Procedures
Measure length with non-standard units by tiling, iterating, or using a self-created measuring tool.

Compare and order measurements of different lengths measured with the same non-standard units, and explain the choice of unit.

Compare measurements of the same length measured with different non-standard units.

Measure length with standard units by tiling or iterating with a centimetre.

Compare and order measurements of different lengths measured with centimetres.
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
A referent is a personal or familiar representation of a known length.

A common referent for a centimetre is the width of the tip of the little finger.
Understanding
Length can be estimated when a measuring tool is not available.
Skills & Procedures
Identify referents for a centimetre.

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

Investigate First Nations, Métis, or Inuit use of the land in estimations of length.
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 pattern characterize change?
Guiding Question
How can diverse representations of pattern contribute to our interpretation of change?
Guiding Question
How can sequence provide insight into change?
Learning Outcome
Students explain and generalize pattern.
Learning Outcome
Students analyze pattern in numerical sequences.
Learning Outcome
Students interpret and explain arithmetic and geometric sequences.
Knowledge
Change can be an increase or a decrease in the number and size of elements.

Pascal’s triangle is a triangular arrangement of numbers that illustrates multiple repeating, growing, and symmetrical patterns.

Understanding
A pattern can show increasing or decreasing change.

A pattern is more evident when the elements are represented, organized, aligned, or oriented in familiar ways.
Skills & Procedures
Describe non-repeating patterns encountered in surroundings, including in art, architecture, and nature.

Examine the representation, organization, alignment, or orientation of patterns in First Nations, Métis, or Inuit design.

Investigate pattern in Pascal’s triangle.

Create and express growing patterns using sounds, objects, pictures, or actions.

Explain the change and constancy in a given non-numerical growing pattern.

Extend a non-numerical growing pattern.

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 pattern core becomes more complex as more attributes change between elements.
Understanding
A pattern core can vary in complexity.
Skills & Procedures
Create and express a repeating pattern with a pattern core of up to four elements that change by more than one attribute.
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 duration support our interpretation of time?
Guiding Question
How can we communicate duration?
Guiding Question
What might be the relevance of duration to daily living?
Learning Outcome
Students relate duration to time.
Learning Outcome
Students tell time using clocks.
Learning Outcome
Students communicate duration with standard units of time.
Knowledge
Events can be related to calendar dates.

Comparative language for describing duration can include
  • longer
  • shorter
  • sooner
  • later
Duration can be measured in non-standard units, including events, natural cycles, or personal referents.
Understanding
Time can be communicated in various ways.

Duration is the measure of an amount of time from beginning to end.

Duration can be measured in various units according to context.
Skills & Procedures
Express significant events using calendar dates.

Describe the duration between or until significant events using comparative language.

Describe the duration of events using non-standard units.

Relate First Nations’ winter counts to duration.
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.
Knowledge
Standard units of time can include
  • years
  • months
  • weeks
  • days
  • hours
  • minutes
  • seconds
Understanding
Duration is quantified by measurement.
Skills & Procedures
Describe the relationship between days, weeks, months, and years.

Describe the duration between or until significant events using standard units of time.
Organizing Idea
Statistics: The science of collecting, analyzing, visualizing, and interpreting data can inform understanding and decision making.
Guiding Question
How can data inform representation?
Guiding Question
How can representation support communication?
Guiding Question
In what ways can we shape communication with our choice of representation?
Learning Outcome
Students relate data to representation.
Learning Outcome
Students interpret and explain representation.
Learning Outcome
Students apply and evaluate representation with scale.
Knowledge
Data can be collected by conducting a survey.

First-hand data is data collected by the person using the data.
Understanding
Data can be collected to answer questions.
Skills & Procedures
Generate questions for a specific investigation within the learning environment.

Collect first-hand data by questioning people within the learning environment.
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
Data can be recorded using tally marks, words, or counts.

Graphs can include
  • pictographs
  • bar graphs
  • dot plots
Data can be expressed through First Nations, Métis, or Inuit stories.

A graph can include features such as
  • a title
  • a legend
  • axes
  • axis labels
Understanding
Data can be represented in various ways.
Skills & Procedures
Record data in a table.

Construct graphs to represent data.

Compare the features of pictographs, dot plots, and bar graphs.
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.