Quebec, Canada

The Mathematics Curriculum in Primary and Lower Secondary Grades

Mathematics is a compulsory subject in Québec throughout elementary and secondary school.^1,2,3,4 The mathematics programs comprise prescribed concepts and processes. The concepts are grouped by cycle in the three cycles of elementary school and in Secondary Cycle I, and by year for each of the three years of Secondary Cycle II. Three learning profiles or “options” are available to students in Secondary IV and V: the Cultural, Social, and Technical option, the Technical and Scientific option, and the Science option. Students who wish to pursue studies in science or certain technical training programs in college (i.e., the 12th and 13th years of study) must complete the appropriate option.

Learning mathematics enables students to:

• Use mathematical reasoning to make conjectures and to criticize, justify, or refute a proposition by drawing on an organized body of mathematical knowledge
• Communicate (i.e., interpret, produce, and convey) messages in contexts in which the subject and purpose of the message, as well as the target audience, play a significant role
• Solve situational problems by using various strategies for understanding, organizing, solving, validating, and communicating

Thus, students develop their ability to interpret reality, anticipate, generalize, and make decisions in a changing world.

Exhibits 1 and 2 present the mathematics content objectives for Elementary Cycle II (Grade 4) and Secondary Cycle I (Grade 8) in Québec.

Exhibit 1: Mathematics Content Objectives for Elementary Cycle II (Grade 4) in Québec

Content Area	Main Topics	Content Objectives
Arithmetic	Understanding and writing numbers	Natural numbers less than 100,000—Reading, writing, counting, representing, comparing, classifying, ordering numbers, writing equivalent expressions, writing numbers in expanded form, writing patterns, understanding properties (e.g., even and odd numbers, squares, prime numbers, and compound numbers), and estimating values Fractions based on a whole or a collection of objects—Reading, writing, and understanding numerator, denominator, various representations (using objects or pictures), equivalent parts, isometric parts, and comparison with 0, ½, and 1 Decimals up to two decimal places—Reading, writing, understanding various representations, ordering, writing equivalent expressions, writing numbers in expanded form, comparing, and estimating
	Meaning of operations involving numbers	Natural numbers and decimals—Choice of operation and operation sense (addition, subtraction, multiplication, division); meaning of an equality relation; meaning of an equivalence relation; relationships between the operations (addition, subtraction); properties of operations (commutative law, associative law)
	Operations involving numbers	Natural numbers—Approximating operation results, acquiring processes for mental computation, and memorizing operations (addition, subtraction); acquiring conventional processes for written computation (i.e., adding two 4-digit numbers and subtracting a 4-digit number from a 4-digit number such that the difference is greater than 0); acquiring processes for written computation (i.e., multiplying a three-digit number by a one-digit number and dividing a three-digit number by a one-digit number); and working with patterns Decimals—Written computation (i.e., addition and subtraction where the result does not go beyond the second decimal place)
Geometry	Space	Locating objects on an axis Locating objects in a plane Locating objects in a Cartesian plane
	Solids	Comparing, constructing, and identifying spheres, cones, cubes, cylinders, prisms, and pyramids Describing prisms and pyramids in terms of faces, vertices, and edges Classifying prisms and pyramids and using nets for prisms and pyramids
	Plane figures	Comparing, identifying, and describing squares, rectangles, triangles, rhombuses, trapezoids, parallelograms, and circles Describing convex and non-convex polygons Describing and classifying quadrilaterals (e.g., parallel segments, perpendicular segments, right angles, acute angles, obtuse angles, and congruent sides) Identifying and constructing parallel lines and perpendicular lines
	Frieze patterns and tessellations	Identifying congruent figures Observing and producing patterns using geometric figures Observing and producing frieze patterns and tessellations by means of reflections
Measurement	Length	Estimating and measuring  with conventional units (e.g., m, dm, cm, mm) Understanding relationships among units of measurement (e.g., m, dm, cm, mm) Calculating perimeter
	Surface area	Estimating and measuring with unconventional units
	Volume	Estimating and measuring with unconventional units
	Angles	Comparing angles (e.g., right, acute, and obtuse)
	Time	Estimating and measuring time and duration with conventional units (e.g., day, hour, minute, second, daily cycle, weekly cycle, and yearly cycle)
Probability and Statistics		Enumerating the possible outcomes of simple random experiments Interpreting and displaying data using data tables, bar graphs, pictographs, and broken-line graphs

Exhibit 2: Mathematics Content Objectives for Secondary Cycle I (Grade 8) in Québec

Content Area	Main Topics	Content Objectives
Arithmetic	Number sense with regard to decimal and fractional notation and operation sense	Reading, writing, various representations, patterns, and properties of numbers—Order of magnitude, comparison of numbers, and decomposition of numbers (e.g., additive and multiplicative) Fractional, decimal, and exponential (integral exponent) notation, percentage, and square root—Recognizing the different meanings of fractions (e.g., part of a whole, division, ratio, operator, and measurement), switching from one way of writing numbers to another, and simplifying and reducing fractions Approximation (i.e., estimating, rounding off, and truncating) Properties of divisibility—Determining the divisibility of a number (by 2, 3, 4, 5, 6, 8, 9, and 10) and using properties of divisibility (by 2, 3, 4, 5, and 10) in different situations Rules of signs for numbers written in decimal notation Equality relations: meaning, properties, and rules for transforming numerical equalities (balancing equalities) Transforming arithmetic equalities Inverse operations: addition, subtraction, multiplication, and division, and square and square root—Expressing situations using operations; mental computation with numbers written in decimal notation; written computation with numbers written in decimal notation or with positive numbers written in fractional notation Properties of operations—Commutative and associative properties, distributive property of multiplication over addition or subtraction, and factoring out the common factor; and simplifying the terms of an operation Order of operations and the use of no more than two levels of parentheses in different contexts Locating numbers on a number line or in a Cartesian plane
	Understanding proportionality	Ratio and rate—Ratios and equivalent rates; unit rate; comparison of ratios and rates; and expressing situations using ratio or rate Proportion—Equality of ratios and rates; ratios and coefficients of proportionality; recognizing and solving proportional situations by referring to their context, tables of values, or graphs Percentage—Finding a specified percentage of a number and values corresponding to 100 percent Variation—Direct and inverse
Geometry	Geometric figures and spatial sensea	Plane figures Triangles, quadrilaterals, and regular convex polygons—Segments and lines (e.g., bisector, perpendicular bisector, median, altitude, base, and height) Circles and sectors—Radius, diameter, chord, arc, and central angle Measurement—Degree (angle and arc), length, perimeter, circumference, area, lateral area, total area, choice of unit of measurement for length and area, and relationship between SI units of length and SI units of area
		Angles—Complementary and supplementary; angles formed by two intersecting lines (e.g., vertically opposite and adjacent); and angles formed by a transversal intersecting two other lines (e.g., alternate interior, alternate exterior, and corresponding)
		Solids—Right prisms, right pyramids, and right cylinders; using possible nets of solids to calculate surface area; and decomposable solids
		Congruent and similar figures—Translation, rotation, reflection, and dilatation
Algebra	Understanding algebraic expressions	Algebraic expressions—Understanding variables, coefficients, degrees, terms, and like terms; constructing and interpreting algebraic expressions; finding equivalent algebraic expressions; and performing numerical evaluation of algebraic expressions Operations on algebraic expressions—Addition, subtraction, multiplication of first-degree monomials, and division by a constant Equality, equations, and unknowns First-degree equations with one unknown expressed in the form ax+b=cx+d
	Dependence between variables	Analyzing situations using different types of representation (e.g., graphs, tables of values, words, etc.); and representing situations using graphs
Probability	Random experiments	Random experiments involving one or more steps (with or without replacement and with or without order); enumerating possible outcomes of random experiments using different types of representations (e.g., tree diagrams, networks, tables, and Venn diagrams); and sample spaces
	Events	Certain, probable, and impossible events; simple, complementary, compatible, incompatible, dependent, and independent events; calculating the probability of an event; and interpreting probabilities
	Types of probability	Theoretical probability and experimental probability
	Interpretation	Interpreting the resulting probabilities
Statistics	Statistical reports	Population and sample—Sample surveys, polls, and censuses; representative samples; sampling methods (e.g., simple random and systematic); and sources of bias Data—Gathering data, qualitative variables, discrete or continuous quantitative variables, comparing distributions, minimum and maximum, arithmetic mean, and range Tables—Characteristics, population, and frequencies Reading and drawing graphs—Bar graphs, broken-line graphs, and circle graphs