Alberta Logonew LearnAlberta

Mathematics

Collapse All

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.
More Info
Collapse All
Prev
Collapse All
 
Grade 6
Next
Organizing Idea
Number: Quantity is measured with numbers that enable counting, labelling, comparing, and operating.
Guiding Question
How can the infinite nature of the number line broaden our perception of number?
Learning Outcome
Students acquire an understanding of magnitude and operations with positive and negative numbers.
Knowledge
Negative numbers are to the left of zero on the number line visualized horizontally, and below zero on the number line visualized vertically.

Positive numbers can be represented symbolically with or without a positive sign (+).

Negative numbers are represented symbolically with a negative sign (-).

Zero is neither positive nor negative.

Negative numbers communicate meaning in context, including
  • temperature
  • debt
  • elevation
Magnitude is a number of units counted or measured from zero on the number line.

Every positive number has an opposite negative number with the same magnitude.

A number and its opposite are called additive inverses.

The additive inverse of a negative number is positive.
Understanding
Symmetry of the number line extends infinitely to the left and right of zero.

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

Magnitude with direction distinguishes between positive and negative numbers.
Skills & Procedures
Identify negative numbers in familiar contexts, including contexts that use vertical or horizontal models of the number line.

Express positive and negative numbers symbolically, in context.

Relate magnitude to the distance from zero on the number line.

Relate positive and negative numbers, including additive inverses, to their positions on horizontal and vertical models of the number line.

Compare and order positive and negative numbers, including fractions and decimals.
Knowledge
The set of integers includes all natural numbers, their additive inverses, and zero.

The sum of any number and its additive inverse is zero.

The sum of two positive numbers is a positive number.

The sum of two negative numbers is a negative number.

The sum of a positive number and a negative number can be interpreted as the sum of zero and another number.
Understanding
Any number can be expressed as a sum in infinitely many ways.

Skills & Procedures
Investigate addition of an integer and its additive inverse.

Express zero as sum of intergers in multiple ways.

Model the sum of two positive integers.

Model the sum of two negative integers.

Model the sum of a positive and negative integer as the sum of zero and another integer.

Add any two integers.
Knowledge
Subtracting a number is the same as adding its additive inverse.
Understanding
The difference of any two numbers can be interpreted as a sum.
Skills & Procedures
Express a difference as a sum.

Add any two numbers, including positive or negative fractions or decimals.
Knowledge
The product or quotient of two positive numbers is a positive number.

The product or quotient of two negative numbers is a positive number.

The product or quotient of a negative number and a positive number is a negative number.
Understanding
Any product can be composed of positive and negative numbers.
Skills & Procedures
Investigate situations involving the multiplication or division of positive and negative numbers.

Generalize a rule to determine the polarity sign for the product of two or more integers.

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

Divide any two numbers, including positive or negative fractions or decimals.
Guiding Question
How can we apply the processes of addition and subtraction to problem solving?
Learning Outcome
Students solve problems using standard algorithms for addition and subtraction.
Knowledge
Standard algorithms are reliable procedures for addition and subtraction.

Contexts for problems involving addition and subtraction can include money and metric measurement.
Understanding
Addition and subtraction of numbers in problem-solving contexts is facilitated by standard algorithms.
Skills & Procedures
Solve problems in various contexts using standard algorithms for addition and subtraction.
Guiding Question
How can we apply the processes of multiplication and division to decimal numbers?
Learning Outcome
Students apply standard algorithms to multiplication and division of 3-digit natural or decimal numbers by 2-digit natural numbers.
Knowledge
Standard algorithms are reliable procedures for multiplication and division of numbers, including decimal numbers.

A quotient with a remainder can be expressed as a decimal number.
Understanding
Multiplication and division of decimal numbers is facilitated by standard algorithms.
Skills & Procedures
Explain the standard algorithms for multiplication and division of decimal numbers.

Multiply and divide numbers, including decimal numbers, using standard algorithms.

Assess the reasonableness of a product or quotient by estimating.

Solve problems using multiplication and division, including problems involving money.
Guiding Question
How can ratios provide new ways to relate numbers?
Learning Outcome
Students interpret ratios.
Knowledge
A ratio can relate any two countable or measurable quantities, including
  • a part to a part
  • a part to the whole
A ratio can be expressed with a fraction or with a colon.

A proportion is an expression of equivalence between two ratios.

A percentage represents the ratio of a number to 100.

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

Understanding
A ratio is a comparison of two quantities.

There are infinitely many equivalent ratios.

Fractions, decimals, ratios, and percentages can represent the same part-whole relationship.
Skills & Procedures
Express part-part ratios and part-whole ratios of the same whole to describe various situations.

Express a ratio as a fraction, decimal, and percentage symbolically.

Express two equivalent ratios as a proportion.

Relate percentage of a number to a proportion.

Find a percent of a number, limited to percents within 100%.
Guiding Question
How can we generalize the multiplication and division of fractions?
Learning Outcome
Students multiply and divide positive fractions.
Knowledge
The product of two fractions is the fraction with
  • a numerator that is the product of the numerators of the given fractions
  • a denominator that is the product of the denominators of the given fractions

A reciprocal is the multiplicative inverse of a fraction.

The reciprocal of a fraction is the fraction obtained by interchanging the numerator and the denominator.

The product of a fraction and its reciprocal is 1.

The reciprocal of a natural number is a unit fraction.

The reciprocal of a reciprocal is the original fraction.

A fraction less than one has a reciprocal greater than one.

A fraction greater than one has a reciprocal less than one.

Division by a fraction is equivalent to multiplication by its reciprocal.

Division by a fraction can be computed using the formula

Understanding
Multiplication of a fraction by a fraction can be interpreted as taking part of a partial quantity.

Division of fractions can be interpreted using multiplication.
Skills & Procedures
Model a fraction of a fraction.

Multiply a fraction by a fraction.

Identify the reciprocal of a given fraction.

Investigate multiplication of a fraction by its reciprocal.

Divide a fraction by a fraction.

Solve problems using operations with fractions.
Organizing Idea
Algebra: Equations express relationships between quantities.
Guiding Question
How can expressions support a generalized interpretation of number?
Learning Outcome
Students analyze expressions and solve algebraic equations.
Knowledge
The product of a number of identical factors can be expressed as a power (e.g., ).

A power uses a base to represent the identical factor and an exponent to indicate the number of identical factors.

Any repeated prime factor within a prime factorization can be expressed as a power (e.g., 40 can be expressed as or ).

Numerical expressions are evaluated according to the conventional order of operations:
  • Operations in parentheses are performed before other operations.
  • Powers are evaluated before other operations are performed.
  • Multiplication and division are performed before addition and subtraction.
  • Multiplication and division are performed in order from left to right.
  • Addition and subtraction are performed in order from left to right.
Understanding
Numerical expressions can include powers.
Skills & Procedures
Express repeated multiplication as a power.

Express repeated prime factors within a prime factorization as a power.

Evaluate numerical expressions involving operations in parentheses and powers according to the order of operations.
Knowledge
Algebraic terms with exactly the same variable are like terms.

Constant terms are like terms.

Like terms can be combined through addition or subtraction.

The terms of an algebraic expression can be rearranged according to algebraic properties.

Algebraic properties include
  • commutative property of addition: a + b = b + a, for any two numbers a and b
  • commutative property of multiplication: ab = ba, for any two numbers a and b
  • associative property of addition: (a + b) + c = a + (b + c)
  • associative property of multiplication: a(bc) = b(ac)
  • distributive property: a(b + c) = ab + ac
Understanding
There are infinitely many ways to express equivalent algebraic expressions.

Algebraic properties ensure equivalence of algebraic expressions.
Skills & Procedures
Investigate like terms by modelling an algebraic expression.

Simplify algebraic expressions by combining like terms.

Express the terms of an algebraic expression in a different order in accordance with algebraic properties.
Knowledge
The equation that shows equality between the variable and a number can be interpreted as the solution.
Understanding
Algebraic expressions on each side of the equation can be simplified into equivalent expressions to facilitate equation solving.
Skills & Procedures
Simplify algebraic expressions on both sides of an equation.

Solve equations, limited to equations with one or two operations.

Compare strategies for solving equations.

Solve problems using equations, limited to equations with one or two operations.

Organizing Idea
Geometry: Shapes are defined and related by geometric attributes.
Guiding Question
How can congruence support our interpretation of symmetry?
Learning Outcome
Students relate shapes through symmetry and congruence.
Knowledge
Symmetrical shapes can be mapped by any combination of reflections and rotations.

The tiling of a plane with symmetrical shapes is called a tessellation.
Understanding
Symmetry is a relationship between two shapes that can be mapped exactly onto each other through reflection or rotation.
Skills & Procedures
Verify symmetry of two shapes by reflecting or rotating one shape onto another.

Describe the symmetry between two shapes as reflection symmetry or rotation symmetry.

Visualize and describe a combination of two rigid transformations that relate symmetrical shapes.

Describe the symmetry modelled in a tessellation.
Knowledge
Shapes related by symmetry are congruent to each other.

Congruent shapes may not be related by symmetry.
Understanding
Congruence is a relationship between two shapes of identical size and shape.

Congruence is not dependent on orientation or location of the shapes.
Skills & Procedures
Demonstrate congruence between two shapes in any orientation by superimposing using hands-on materials or digital applications.

Describe symmetrical shapes as congruent.
Organizing Idea
Coordinate Geometry: Location and movement of objects in space can be communicated using a coordinate grid.
Guiding Question
In what ways can we communicate location?
Learning Outcome
Students explain location in relation to position in the Cartesian plane.
Knowledge
The Cartesian plane is named after French mathematician René Descartes.

The Cartesian plane uses coordinates,(x, y), to indicate the location of the point where the vertical line passing through (x, 0) and the horizontal line passing through (0, y) intersect.

The x-axis consists of those points whose y-coordinate is zero and the y-axis consists of those points whose x-coordinate is zero.

The x-axis and the y-axis intersect at the origin, (0, 0).

An ordered pair is represented symbolically as (x, y).

An ordered pair indicates the horizontal distance from the y-axis with the x-coordinate and the vertical distance from the x-axis with the y-coordinate.

Understanding
Location can be described using the Cartesian plane.

The Cartesian plane is the two-dimensional equivalent of the number line.


Skills & Procedures
Relate the axes of the Cartesian plane to intersecting horizontal and vertical representations of the number line.

Locate a point in the Cartesian plane given the coordinates of the point.

Describe the location of a point in the Cartesian plane using coordinates.

Model a polygon in the Cartesian plane using coordinates to indicate the vertices.

Describe the location of the vertices of a polygon in the Cartesian plane using coordinates.
Knowledge
A translation describes a combination of horizontal and vertical movements as a single movement.

A reflection describes movement across a reflection line.

A rotation describes an amount of movement around a turn centre along a circular path in either a clockwise or counter-clockwise direction.
Understanding
Location can change as a result of movement in space.

Change in location does not imply change in orientation.
Skills & Procedures
Create an image of a polygon in the Cartesian plane by translating the polygon.

Describe the horizontal and vertical components of a given translation.

Create an image of a polygon in the Cartesian plane by reflecting the polygon over the x-axis or y-axis.

Describe the line of reflection of a given reflection.

Create an image of a polygon in the Cartesian plane by rotating the polygon 90°, 180°, or 270° about one of its vertices, clockwise or counter-clockwise.

Describe the angle and direction of a given rotation.

Relate the coordinates of a polygon and its image after translation, reflection, or rotation in the Cartesian plane.
Organizing Idea
Measurement: Attributes such as length, area, volume, and angle are quantified by measurement.
Guiding Question
In what ways can we relate shapes using conservation of area?
Learning Outcome
Students analyze area of parallelograms and triangles.
Knowledge
Any side of a parallelogram can be interpreted as the base.

The height of a parallelogram is the perpendicular distance from its base to its opposite side.

The area of a triangle is half of the area of a parallelogram with the same base and height.

Two triangles with the same base and height must have the same area.






Understanding
The area of a parallelogram can be generalized as the product of the perpendicular base and height.

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



Skills & Procedures
Rearrange the area of a parallelogram to form a rectangular area using hands-on materials or digital applications.

Determine the area of a parallelogram using multiplication.

Determine the base or height of a parallelogram using division.

Model the area of a parallelogram as two congruent triangles.

Describe the relationship between the area of a triangle and the area of a parallelogram with the same base and height.

Determine the area of a triangle, including various triangles with the same base and height.

Solve problems involving area of parallelograms and triangles.








Knowledge
Area of composite shapes can be interpreted as the sum of the areas of multiple shapes, such as triangles and parallelograms.
Understanding
An area can be decomposed in infinitely many ways.
Skills & Procedures
Visualize the decomposition of composite areas in various ways.

Determine the area of composite shapes using the areas of triangles and parallelograms.
Guiding Question
How can volume characterize space?
Learning Outcome
Students interpret and express volume.
Knowledge
Volume can be measured in non-standard units or standard units (e.g., cubic centimetres).

Volume is expressed in the following standard units, derived from standard units of length:
  • cubic centimetres
  • cubic metres
A cubic centimetre (cm3) is a volume equivalent to the volume of a cube measuring 1 centimetre by 1 centimetre by 1 centimetre.

A cubic metre (m3) is a volume equivalent to the volume of a cube measuring 1 metre by 1 metre by 1 metre.

The volume of a right rectangular prism can be interpreted as the product of the two-dimensional base area and the perpendicular height of the prism.


Understanding
Volume is a measurable attribute that describes the amount of three-dimensional space occupied by a three-dimensional object.

The volume of a prism can be interpreted as the result of perpendicular motion of an area.

Volume remains the same when decomposed or rearranged.

Volume is quantified by measurement.

Volume is measured with congruent units that themselves have volume and do not need to resemble the shape being measured.

The volume of a right rectangular prism can be perceived as cube-shaped units structured in a three-dimensional array.

Skills & Procedures
Recognize volume in familiar contexts.

Model volume of prisms by dragging or iterating an area using hands-on materials or digital applications.

Create a model of a three-dimensional object by stacking congruent non-standard units or cubic centimetres without gaps or overlaps.

Express volume in non-standard units or cubic centimetres.

Visualize and model the volume of various right rectangular prisms as three-dimensional arrays of cube-shaped units.

Determine the volume of a right rectangular prism using multiplication.

Solve problems involving volume of right rectangular prisms.
Organizing Idea
Patterns: Awareness of patterns supports problem solving in various situations.
Guiding Question
How can a function enhance our interpretation of change?
Learning Outcome
Students acquire an understanding of functions.
Knowledge
A variable can be interpreted as the values of a changing quantity.

A function can involve quantities that change over time, including
  • height or weight of a person
  • height of a plant
  • temperature
  • distance travelled
A table of values lists the values of the independent variable in the first column or row and the values of the dependent variable in the second column or row to represent a function at certain points.

The values of the independent variable are represented by x-coordinates in the Cartesian plane.

The values of the dependent variable are represented by y-coordinates in the Cartesian plane.
Understanding
A function is a correspondence between two changing quantities represented by independent and dependent variables.

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


Skills & Procedures
Identify the dependent and independent variables in a given situation, including situations involving change over time.

Describe the rule that determines the values of the dependent variable from values of the independent variable.

Create a table of values representing corresponding values of the independent and dependent variables of a function at certain points.

Represent corresponding values of the independent and dependent variables of a function as points in the Cartesian plane.

Write an algebraic expression that represents a function.

Recognize various representations of the same function.

Determine a value of the dependent variable of a function given the corresponding value of the independent variable.

Investigate strategies for determining a value of the independent variable of a function given the corresponding value of the dependent variable.

Solve problems involving a function.
Organizing Idea
Statistics: The science of collecting, analyzing, visualizing, and interpreting data can inform understanding and decision making.
Guiding Question
How can frequency support communication?
Learning Outcome
Students apply and explain relative frequency with experimental data.
Knowledge
Relative frequency can be used to compare the same category of data across multiple data sets.

Relative frequency can be represented in various forms.

Understanding
Relative frequency expresses the frequency of a category of data as a fraction of the total number of data values.

Skills & Procedures
Interpret frequency of categorized data as relative frequency.

Express relative frequencies as decimals, fractions, or percentages.
Knowledge
Equally likely outcomes of an experiment have the same chance of occurring.

An event can be described as the outcome of an experiment, including
  • heads or tails from a coin toss
  • any roll of a die
  • the result of spinning a spinner
The law of large numbers states that more independent trials of an experiment result in a better estimate of the expected likelihood of an event.
Understanding
Frequency can be a count of categorized observations or trials in an experiment.

Relative frequency of outcomes can be used to estimate the likelihood of an event.

Relative frequency varies between sets of collected data.

Relative frequency provides a better estimate of the likelihood of an event with larger amounts of data.
Skills & Procedures
Identify the possible outcomes of an experiment involving equally likely outcomes.

Collect categorized data through experiments, including with coins, dice, and spinners.

Predict the likelihood of an event based on the possible outcomes of an experiment.

Determine relative frequency for categories of a sample of data.

Describe the likelihood of an outcome in an experiment using relative frequency.

Analyze relative frequency statistics from experiments with different sample sizes.