Mathematics

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Algebra

Understanding and manipulating algebraic expressions

Understanding dependency relationships

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
  1. Algebraic expressions
6 1 2 3 4 5
  1. Describes, using his/her own words and mathematical language, numerical patterns
             
  1. Describes, using his/her own words and mathematical language, series of numbers and family of operations
           
  1. Adds new terms to a series when the first three terms or more are given
           
  1. Describes the role of components of algebraic expressions:
    1. unknown
      Note : This concept was introduced in elementary school (although not named as such) when students were asked to find a missing term.
       
    1. variable, constant
         
    1. parameter
      Note : The concept of parameter is introduced intuitively (although not named as such) in Secondary I, II and III.
      CST
TS
  S
    1. coefficient, degree, term, constant term, like terms
         
  1. Constructs an algebraic expression using a register (type) of representation
         
  1. Interprets an algebraic expression in light of the context
         
  1. Recognizes or constructs equivalent algebraic expressions
         
  1. Recognizes or constructs
 
    1. equalities and equations
       
    1. inequalities
           
  1. Manipulating algebraic expressions
6 1 2 3 4 5  
  1. Calculates the numeric value of an algebraic expression
         
  1. Performs the following operations on algebraic expressions, with or without objects or diagrams: addition and subtraction, multiplication and division by a constant, multiplication of first-degree monomials
         
  1. Factors out the common factor in numerical expressions (distributive property of multiplication over addition or subtraction)
         
  1. Multiplies
 
    1. algebraic expressions of degree less than 3
           
    1. algebraic expressions
            CST
  TS
  S
  1. Divides
 
    1. algebraic expressions by a monomial
           
    1. a polynomial by a binomial (with or without a remainder)
            CST
  TS
    S
    1. a polynomial by another polynomial (with or without a remainder)
            CST
    TS
  S
  1. Factors polynomials by
 
    1. finding the common factor
           
    1. factoring by grouping (polynomials including decomposable second-degree trinomials)
            CST
  TS
  S
    1. completing the square (factoring and switching from one type of notation to another)
            CST
  TS
  S
    1. using formulas for trinomials of the form ax2 + bx + c :
      x1 =  and x2 =
            CST
  TS
  S
    1. substituting second-degree algebraic identities (perfect square trinomial and difference of two squares)
            CST
  TS
  S
  1. 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
  1. Analyzing situations using equations or inequalities
6 1 2 3 4 5  
  1. Recognizes whether a situation can be translated by
    1. an equation
         
    1. an inequality
           
  1. Recognizes or constructs
 
    1. relations or formulas
         
    1. inequality relations and first-degree inequalities in one variable
           
  1. Manipulates relations or formulas (e.g. isolates an element)
         
  1. Represents a situation using
    1. a first-degree equation with one unknown
         
    1. a first-degree inequality with a variable
           
  1. Represents
    1. an equation using another register (type) of representation, if necessary
         
    1. an inequality using another register (type) of representation, if necessary
           
  1. Determines the missing term in an equation (relations between operations) :1
    a + b = □, a + □ = c, □ + b = c, ab = □, a – □ = c, □ – b = c,
    a × b = □, a × □ = c, □ × b = c, a ÷ b = □, a ÷ □ = c, □ ÷ b = c
           
  1. Transforms arithmetic equalities and equations to maintain equivalence (properties and rules for transforming equalities) and justifies the steps followed, if necessary
         
  1. Transforms inequalities to maintain equivalence (properties and rules for transforming inequalities) and justifies the steps followed, if necessary
           
  1. Uses different methods to solve first-degree 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)
         
  1. Solves first-degree inequalities in one variable
           
  1. Solves the following types of equations or an inequalities in one variable
    1. second-degree
      Note : In TS, this is taught over two years using the functional models under study.
            CST
TS
  S
    1. 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
    1. rational
            CST
  TS
  S
    1. absolute value
            CST
    TS
  S
    1. first-degree trigonometric involving a sine, cosine or tangent expression
            CST
  TS
    S
    1. trigonometric that can be expressed as a sine, cosine or tangent function
            CST
    TS
  S
  1. Solves a second-degree equation in two variables
    Note : In TS, this is taught over two years, using the functional models under study.
            CST
TS
  S
  1. Validates a solution, with or without technological tools, by substitution
         
  1. Solves an inequality graphically and checks the feasible region of a
    1. first-degree inequality in two variables
          CST
  TS
  S
    1. second-degree inequality in two variables
      Note : In TS, this is taught over two years using the functional models under study.
            CST
TS
    S
  1. Interprets solutions or makes decisions, if necessary, depending on the context
         
  1. Analyzing situations using systems of equations or inequalities
6 1 2 3 4 5  
  1. Determines whether a situation may be translated by a system of
    1. equations
           
    1. inequalities
           
  1. Translates a situation algebraically or graphically using a system of
    1. equations
           
    1. inequalities
           
  1. Solves a system
    1. of first-degree equations in two variables of the form ax b by using tables of values, graphically or algebraically (by comparison), with or without technological tools
           
    1. of first-degree equations in two variables
      Note : The student chooses the method.
           
    1. composed of a first-degree equation in two variables and a second-degree 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
    1. of second-degree equations in relation to conics using changing variables, if applicable
            CST
    TS
  S
    1. involving various functional models (mostly graphical solutions)
            CST
  TS
    S
  1. Solves a system
 
    1. of first-degree inequalities in two variables
           
    1. involving various functional models (mostly graphical solutions)
            CST
  TS
    S
  1. Validates the solution, with or without technological tools
         
  1. Interprets the solution or makes decisions if necessary, depending on the context
         
  1. Linear programming
6 1 2 3 4 5  
  1. Analyzes a situation to be optimized
        • mathematizing the situation using a system of first-degree 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
           
  1. 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
           
1.  Students were not shown this symbolic notation in elementary school. They did learn, however, to determine the value of a missing term, among other things, in situations that call for additive or multiplicative structures, within the limits of the elementary-level Mathematics program.

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