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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 meaning to our daily lives?
Guiding Question
How can we communicate quantity?
Guiding Question
How can quantity contribute to our sense of number?
Learning Outcome
Children acquire an understanding of quantity to 10.
Learning Outcome
Students interpret and explain quantity to 100.
Learning Outcome
Students analyze quantity to 1000.
Knowledge
Quantity can be expressed using
  • objects
  • pictures
  • words
  • numerals
Understanding
Quantity can be the number of objects in a set.

Skills & Procedures
Recognize a number of familiar objects as a quantity.

Express a quantity in different ways.

Relate a numeral to a specific quantity.
Knowledge
The absence of quantity is represented by 0.

Canadian money includes
  • nickels
  • dimes
  • quarters
  • loonies
  • toonies
  • five-dollar bills
  • ten-dollar bills
  • twenty-dollar bills
  • fifty-dollar bills
  • hundred-dollar bills
Understanding
Quantity is expressed in words and numerals based on patterns.

Quantity in the world is represented in multiple ways, including with money.
Skills & Procedures
Express quantities using words, objects, or pictures.

Represent quantities using numerals.

Identify a quantity of 0 in familiar situations.

Express the value of each coin and bill within 100 dollars using words and numerals.

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
Quantity can be determined by counting with natural numbers.

Understanding
A quantity is always counted using the same sequence of words (counting principle: stable order).

A quantity remains the same no matter the order in which the objects are counted (counting principle: order irrelevance).

A quantity can be determined by counting each object in a set once and only once (counting principle: one-to-one correspondence).

The last number used to count represents the quantity (counting principle: cardinality).

Any quantity of like or unlike objects can be counted as a set (counting principle: abstraction).

Skills & Procedures
Count within 10, forward and backward, starting at any number, according to the counting principles.
Knowledge
Counting can begin at any number.

Counting more than one object at a time is called skip counting.
Understanding
Each number counted includes all previous numbers (counting principle: hierarchical inclusion).

A quantity can be determined by counting more than one object in a set at a time.
Skills & Procedures
Count within 100, forward by 1, starting at any number, according to the counting principles.

Count backward from 20 to 0 by 1.

Skip count to 100, forward by 5 and 10, starting at 0.

Skip count to 20, forward by 2, starting at 0.
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 small quantity can be recognized at a glance (subitized).
Understanding
Quantity can be determined without counting.

Skills & Procedures
Subitize quantities to 5.
Knowledge
Familiar arrangements of small quantities facilitate subitizing.
Understanding
A quantity can be perceived as the composition of smaller quantities.
Skills & Procedures
Recognize quantities to 10.
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
Comparative language can include
  • more
  • less
  • same
  • enough
  • too many
  • too few
Understanding
A quantity can be described relative to another quantity.

A quantity can be described in relation to a purpose or need.
Skills & Procedures
Describe quantities relative to each other using comparative language.

Describe a quantity in relation to a purpose or need using comparative language.

Solve problems in familiar situations by counting.
Knowledge
Words that describe a comparison between two quantities include
  • equal
  • not equal
  • greater than
  • less than
The equal sign is =.

The unequal sign is ≠.
Understanding
Two quantities are equal when there is the same number of objects in both sets.
Skills & Procedures
Compare quantities in two sets of objects.

Describe a quantity relative to another quantity.

Represent equal quantities symbolically.

Represent unequal quantities symbolically.
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
In what ways can we compose quantity?
Guiding Question
How can addition and subtraction provide perspectives of number?
Guiding Question
How can we interpret addition and subtraction?
Learning Outcome
Children interpret compositions of quantities within 10.
Learning Outcome
Students acquire an understanding of addition and subtraction within 20.
Learning Outcome
Students explain addition and subtraction within 100.
Knowledge
Quantity can be arranged in various ways.
Understanding
A quantity remains the same no matter how the objects are grouped or arranged (counting principle: conservation).


Skills & Procedures
Identify a quantity in various groups or arrangements.

Compose quantities within 10.
Knowledge
Addition and subtraction are opposite (inverse) mathematical operations.

Addition is a process of combining quantities to find a sum.

Subtraction is a process of finding the difference between quantities.

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

The order in which two quantities are subtracted affects the difference.

Addition of 0 to any number, or subtraction of 0 from any number, results in the same number (zero property).
Understanding
Quantities can be composed or decomposed through addition and subtraction.
Skills & Procedures
Compose quantities within 20 in various ways.


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
Strategies are meaningful steps taken to solve problems.

Addition and subtraction strategies include
  • counting on
  • counting back
  • decomposition
  • compensation
The addition sign is +.

The subtraction sign is -.

The equal sign is =.

Understanding
Addition and subtraction can show a change in quantity through joining, separating, or comparing.
Skills & Procedures
Investigate addition and subtraction strategies.

Add and subtract within 20.

Express addition and subtraction symbolically.

Solve problems using addition and subtraction in joining, separating, or comparing situations.

Model transactions with money, limited to dollar values within 20 dollars.

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.
Knowledge
Addition and subtraction number facts represent part-part-whole relationships.

In a part-part-whole relationship, the sum represents the whole and the difference represents a missing part.

Fact families are groups of related addition and subtraction number facts.
Understanding
Addition number facts have related subtraction number facts.
Skills & Procedures
Identify patterns in addition and subtraction, including patterns in addition tables.

Recognize families of related addition and subtraction number facts.

Recall addition number facts, with addends to 10, and related subtraction number facts.
Guiding Question
In what ways can we interpret the composition of number?
Guiding Question
In what ways can composition characterize number?
Learning Outcome
Students represent equal sharing and grouping of quantities within 20.

Learning Outcome
Students interpret even and odd quantities within 100.

Knowledge
Sharing involves partitioning a quantity into a certain number of groups.

Grouping involves partitioning a quantity into groups of a certain size.
Understanding
Quantity can be partitioned by sharing or grouping.
Skills & Procedures
Partition a set of objects by sharing and grouping.

Demonstrate conservation of number when sharing or grouping.
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.
Guiding Question
In what ways can parts and wholes be related?
Guiding Question
In what ways can parts compose a whole?
Learning Outcome
Students recognize one-half as a part-whole relationship.
Learning Outcome
Students interpret one whole using halves and quarters.
Knowledge
One-half can be one of two equal groups.
Understanding
In a quantity partitioned into two equal groups, each group represents one-half of the quantity.
Skills & Procedures
Identify one-half in familiar situations.

Partition an even set of objects into two equal groups.


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.
Organizing Idea
Geometry: Shapes are defined and related by geometric attributes.
Guiding Question
How can shape bring meaning to the space around us?
Guiding Question
In what ways can we characterize shape?
Guiding Question
How can shape influence our perception of space?
Learning Outcome
Children acquire an understanding of shape.
Learning Outcome
Students interpret shape in two and three dimensions.

Learning Outcome
Students analyze and explain geometric attributes of shape.
Knowledge
A shape can be represented using objects, pictures, or words.

Two-dimensional shapes include
  • squares
  • circles
  • rectangles
  • triangles
Three-dimensional shapes include
  • cubes
  • prisms
  • cylinders
  • spheres
First Nations, Métis, and Inuit name specific shapes in relation to the natural world.
Understanding
Shape is structured two-dimensional or three-dimensional space.

Skills & Procedures
Relate shapes in the natural world to various two-dimensional and three-dimensional shapes.

Identify two- and three-dimensional shapes.

Investigate three-dimensional shapes by rolling, stacking, or sliding.

Describe a shape using words such as flat, curved, straight, or round.

Sort shapes according to one attribute and describe the sorting rule.
Knowledge
Two-dimensional
shapes include
  • squares
  • circles
  • rectangles
  • triangles
Three-dimensional shapes include
  • cubes
  • prisms
  • cylinders
  • spheres
  • pyramids
  • cones
A line of symmetry indicates the division between the matching halves of a symmetrical shape.

Understanding
A shape can be modelled in various sizes and orientations.

A shape can be composed of two or more shapes.

A shape is symmetrical if it can be decomposed into matching halves.
Skills & Procedures
Identify shapes in various sizes and orientations.

Model two-dimensional shapes.

Sort shapes according to one attribute and describe the sorting rule.

Compose and decompose two- or three-dimensional shapes.

Identify shapes within two- or three-dimensional composite shapes.

Investigate symmetry of two-dimensional shapes by folding and matching.

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
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.

Organizing Idea
Measurement: Attributes such as length, area, volume, and angle are quantified by measurement.
Guiding Question
In what ways can we distinguish size?
Guiding Question
In what ways can length provide perspectives of size?
Guiding Question
How can length contribute to our interpretation of space?
Learning Outcome
Children acquire an understanding of size through direct comparison.
Learning Outcome
Students apply an understanding of size to the interpretation of length.
Learning Outcome
Students communicate length using units.
Knowledge
Measurable attributes can include
  • length
  • area
  • capacity
  • mass
Understanding
Size describes the amount of one measurable attribute of an object or a space.
Skills & Procedures
Identify measurable attributes of familiar objects to which size may refer.
Knowledge
Length may refer to the size of any one-dimensional measurable attribute of an object, including:
  • height
  • width
  • depth
  • diameter
A length does not need to be a straight line.

The length of empty space between two points is called distance.

Familiar contexts of distance include
  • distance between objects or people
  • distance between home and school
  • distance between towns or cities
Understanding
Length is a measurable attribute that describes the amount of fixed space between the end points of an object.

Length remains the same if an object is repositioned but may be named differently.
Skills & Procedures
Recognize the height, width, or depth of an object as lengths in various orientations.

Recognize the diameter of a circle as a length.

Compare and order objects according to length.

Describe distance in familiar contexts.
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
Comparative language can include
  • longer
  • taller
  • shorter
  • heavier
  • lighter
  • bigger
  • smaller
  • big enough
  • too big
  • too small
Understanding
Size may refer to only one measurable attribute at a time.

The size of two objects can be compared directly.

The size of an object can be described in relation to a purpose or need.
Skills & Procedures
Compare the length, area, mass, or capacity of two objects directly.

Order objects according to length, area, mass, or capacity.

Describe the size of an object in relation to another object, using comparative language.

Describe the size of an object in relation to a purpose or need, using comparative language.
Knowledge
Indirect comparison is useful when objects are fixed in place or difficult to move.
Understanding
The size of two objects can be compared indirectly with a third object.
Skills & Procedures
Compare the length, area, mass, or capacity of two objects directly, or indirectly using a third object.

Order objects according to length, area, mass, or capacity.

Describe the size of an object in relation to another object, using comparative language.

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.
Organizing Idea
Patterns: Awareness of patterns supports problem solving in various situations.
Guiding Question
How can we distinguish pattern?
Guiding Question
What can pattern communicate?
Guiding Question
How can pattern characterize change?
Learning Outcome
Children acquire an understanding of repeating patterns.
Learning Outcome
Students examine pattern in cycles.
Learning Outcome
Students explain and generalize pattern.
Knowledge
Patterns exist everywhere.

The elements of a pattern can include
  • sounds
  • objects
  • pictures
  • symbols
  • actions
Pattern is characterized by how the elements change or remain constant.

Repeating patterns have one or more elements that repeat.

A pattern core is a sequence of one or more elements that repeat as a unit.

Understanding
Pattern is defined by the relationship between individual elements.
Skills & Procedures
Recognize repeating patterns encountered in daily routines and play, including songs or dances.

Recognize change or constancy between elements in a repeating pattern.

Identify the pattern core, up to three elements, in a repeating pattern.

Predict the next elements in a repeating pattern.

Create a repeating pattern with a pattern core of up to three elements.
Knowledge
A cycle can express repetition of events or experiences.

Cycles include
  • seasons
  • day/night
  • life cycles
  • calendars
A pattern remains the same when elements are represented in different forms, including
  • sounds
  • objects
  • pictures
  • symbols
  • actions
Patterns can be extended by reasoning about existing elements.

Understanding
A pattern that appears to repeat may not repeat in the same way forever.

A cycle is a repeating pattern that repeats in the same way forever.

Skills & Procedures
Recognize cycles encountered in daily routines and nature.

Investigate cycles found in nature that inform First Nations, Métis, or Inuit practices.

Identify the pattern core, up to four elements, in a cycle.

Identify a missing element in a repeating pattern or cycle.

Describe change and constancy in repeating patterns and cycles.

Create different representations of the same repeating pattern or cycle, limited to a pattern core of up to four elements.

Extend a sequence of elements in various ways to create repeating patterns.


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
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.
Organizing Idea
Time: Duration is described and quantified with time.
Guiding Question
How can we make sense of time?
Guiding Question
How can time characterize change?
Guiding Question
How can duration support our interpretation of time?
Learning Outcome
Children acquire an understanding of time as a sequence of events.
Learning Outcome
Students explain time in relation to cycles.
Learning Outcome
Students relate duration to time.
Knowledge
Words to describe a sequence in time can include
  • first
  • next
  • then
  • last
  • yesterday
  • today
  • tomorrow
Ordinal numbers can indicate order in time.
Understanding
Time can be perceived as a sequence.
Skills & Procedures
Sequence events according to time using words or ordinal numbers.

Describe daily events as occurring yesterday, today, or tomorrow.
Knowledge
Time can be perceived through observable change.

First Nations, Métis, and Inuit experience time through sequences and cycles in nature, including cycles of seasons and stars.

Cycles from a calendar include days of the week and months of the year.
Understanding
Time is an experience of change.

Time can be perceived as a cycle.

Skills & Procedures
Describe cycles of time encountered in daily routines and nature.

Describe observable changes that indicate a cycle of time.

Relate cycles of seasons and stars to First Nations, Métis, or Inuit practices.

Identify cycles from a calendar.
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
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 we use data as we wonder about our world?
Guiding Question
How can data inform representation?
Learning Outcome
Students acquire an understanding of data.
Learning Outcome
Students relate data to representation.
Knowledge
Data can be collected information.
Understanding
Data can be answers to questions.
Skills & Procedures
Share wonderings about people, things, events, or experiences.

Pose questions about people, things, events, or experiences in the learning environment.

Gather data by sharing answers to questions.
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
A graph is a visual representation of data.

A graph can represent data by using objects, pictures, or numbers.
Understanding
Data can be represented in a graph.
Skills & Procedures
Collaborate to construct a concrete graph using data collected in the learning environment.

Create a pictograph from a concrete graph.
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.