Understanding and manipulating algebraic expressions 
Student constructs knowledge with teacher guidance.
Student applies knowledge by the end of the school year.
Student reinvests knowledge. 
Elementary 
Secondary 
Cycle One 
Cycle Two 
 Algebraic expressions

6 
1 
2 
3 
4 
5 
 Describes, using his/her own words and mathematical language, numerical patterns








 Describes, using his/her own words and mathematical language, series of numbers and family of operations








 Adds new terms to a series when the first three terms or more are given








 Describes the role of components of algebraic expressions:

 unknown
Note : This concept was introduced in elementary school (although not named as such) when students were asked to find a missing term.








 variable, constant








 parameter
Note : The concept of parameter is introduced intuitively (although not named as such) in Secondary I, II and III.







CST 


TS 


S 
 coefficient, degree, term, constant term, like terms








 Constructs an algebraic expression using a register (type) of representation








 Interprets an algebraic expression in light of the context








 Recognizes or constructs equivalent algebraic expressions








 Recognizes or constructs


 equalities and equations








 inequalities








 Manipulating algebraic expressions

6 
1 
2 
3 
4 
5 

 Calculates the numeric value of an algebraic expression








 Performs the following operations on algebraic expressions, with or without objects or diagrams: addition and subtraction, multiplication and division by a constant, multiplication of firstdegree monomials








 Factors out the common factor in numerical expressions (distributive property of multiplication over addition or subtraction)








 Multiplies


 algebraic expressions of degree less than 3








 algebraic expressions







CST 


TS 


S 
 Divides


 algebraic expressions by a monomial








 a polynomial by a binomial (with or without a remainder)







CST 


TS 


S 
 a polynomial by another polynomial (with or without a remainder)







CST 


TS 


S 
 Factors polynomials by


 finding the common factor








 factoring by grouping (polynomials including decomposable seconddegree trinomials)







CST 


TS 


S 
 completing the square (factoring and switching from one type of notation to another)







CST 


TS 


S 
 using formulas for trinomials of the form ax^{2} + bx + c :
x_{1} =
and
x_{2} =







CST 


TS 


S 
 substituting seconddegree algebraic identities (perfect square trinomial and difference of two squares)







CST 


TS 


S 
 Manipulates rational expressions
Note : Rational expressions (algebraic fractions) are part of the algebraic expressions to be covered. When finding the common denominator in order to add two rational expressions, students in TS will deal only with cases in which the denominator of one fraction is the multiple of the denominator of the other fraction.







CST 


TS 


S 
 Analyzing situations using equations or inequalities

6 
1 
2 
3 
4 
5 

 Recognizes whether a situation can be translated by

 an equation








 an inequality








 Recognizes or constructs


 relations or formulas








 inequality relations and firstdegree inequalities in one variable








 Manipulates relations or formulas
(e.g. isolates an element)








 Represents a situation using

 a firstdegree equation with one unknown








 a firstdegree inequality with a variable








 Represents

 an equation using another register (type) of representation, if necessary








 an inequality using another register (type) of representation, if necessary








 Determines the missing term in an equation (relations between operations) :^{1}
a + b = □, a + □ = c, □ + b = c, a – b = □, a – □ = c, □ – b = c,
a × b = □, a × □ = c, □ × b = c, a ÷ b = □, a ÷ □ = c, □ ÷ b = c








 Transforms arithmetic equalities and equations to maintain equivalence (properties and rules for transforming equalities) and justifies the steps followed, if necessary








 Transforms inequalities to maintain equivalence (properties and rules for transforming inequalities) and justifies the steps followed, if necessary








 Uses different methods to solve firstdegree equations with one unknown of the form
ax + b = cx + d : trial and error, drawings, arithmetic methods (inverse or equivalent operations), algebraic methods (balancing equations or hidden terms)








 Solves firstdegree inequalities in one variable








 Solves the following types of equations or an inequalities in one variable

 seconddegree







CST 


TS 


S 
 exponential, logarithmic or square root, using the properties of exponents, logarithms and radicals
Note : In CST in Secondary V, students use the definitions of logarithm and change of base to solve exponential and logarithmic equations, but they are not required to solve square root equations or study the properties of radicals and logarithms. In TS, this is taught over two years, using the functional models under study.







CST 


TS 


S 
 rational







CST 


TS 


S 
 absolute value







CST 


TS 


S 
 firstdegree trigonometric involving a sine, cosine or tangent expression







CST 


TS 


S 
 trigonometric that can be expressed as a sine, cosine or tangent function







CST 


TS 


S 
 Solves a seconddegree equation in two variables







CST 


TS 


S 
 Validates a solution, with or without technological tools, by substitution








 Solves an inequality graphically and checks the feasible region of a

 firstdegree inequality in two variables







CST 


TS 


S 
 seconddegree inequality in two variables







CST 


TS 


S 
 Interprets solutions or makes decisions, if necessary, depending on the context








 Analyzing situations using systems of equations or inequalities

6 
1 
2 
3 
4 
5 

 Determines whether a situation may be translated by a system of

 equations








 inequalities








 Translates a situation algebraically or graphically using a system of

 equations








 inequalities








 Solves a system

 of firstdegree equations in two variables of the form y = ax + b by using tables of values, graphically or algebraically (by comparison), with or without technological tools








 of firstdegree equations in two variables
Note : The student chooses the method.








 composed of a firstdegree equation in two variables and a seconddegree equation in two variables
Note : In TS, these systems are solved using graphic representations, with or without the use of technological tools.







CST 


TS 


S 
 of seconddegree equations in relation to conics using changing variables, if applicable







CST 


TS 


S 
 involving various functional models (mostly graphical solutions)







CST 


TS 


S 
 Solves a system


 of firstdegree inequalities in two variables








 involving various functional models (mostly graphical solutions)







CST 


TS 


S 
 Validates the solution, with or without technological tools








 Interprets the solution or makes decisions if necessary, depending on the context








 Linear programming

6 
1 
2 
3 
4 
5 

 Analyzes a situation to be optimized
 mathematizing the situation using a system of firstdegree inequalities in two variables
 drawing a bounded or unbounded polygon of constraints to represent the situation
 determining the coordinates of the vertices of the bounded polygon (feasible region)
Note : In TS, the coordinates of points of intersection may be determined algebraically, using matrices, or approximated based on a graph.
 recognizing and defining the function to be optimized








 Optimizes a situation by taking into account different constraints and makes decisions with respect to this situation
 determining the best solution(s) for a particular situation, given a set of possibilities
 validating and interpreting the optimal solution, depending on the context
 justifying the solution(s) chosen
 changing certain conditions associated with the situation to provide a more optimal solution, if necessary







