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Analytic Geometry

Analytic geometry provides a link between geometry and algebra. It allows students to represent geometric objects using equations and inequalities. Students therefore work on representations in a Cartesian plane.

In Secondary Cycle One, students perfect their ability to locate points in the Cartesian plane, using the types of numbers under study. They learn to represent a situation generally, using a graph.

In Secondary Cycle Two, students learn to model and analyze situations using a Cartesian reference point. They calculate distances, determine the coordinates of a point of division and study geometric loci. Depending on the option, they use coordinates to perform geometric transformations and determine results in a standard unit circle.

The following tables present the learning content associated with analytic geometry. By basing themselves on the concepts and processes targeted, students develop the three competencies of the program, which in turn enable students to better integrate the mathematical concepts and processes presented.

Analyzing situations using analytic geometry  

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. Locating
6 1 2 3 4 5
  1. Locates objects/numbers on an axis, based on the types of numbers studied
    Note : In Secondary Cycle One, students locate positive or negative numbers written in decimal or fractional notation.
       
  1. Locates points in a Cartesian plane, based on the types of numbers studied (x- and y-coordinates of a point)
       
  1. Straight lines and half-planes
6 1 2 3 4 5  
  1. Uses the concept of change to
 
    1. calculate the distance between two points
      Note : In Secondary III, students are introduced to the concept of distance between two points while studying the Pythagorean relation. In Secondary IV, the distance between two parallel lines or from a point to a line or segment is studied using concepts and processes associated with distance and equations systems.
         
    1. determine the coordinates of a point of division using a given ratio (including the coordinates of a midpoint)
      Note : In S, students can also determine the coordinates of a point of division using the product of a vector and a scalar.
          CST
  TS
  S
    1. calculate and interpret a slope
      Note : In Secondary III, students are introduced informally to the concept of slope while studying the rate of change of functions (degree 0 and 1).
         
  1. Determines the relative position of two straight lines using their respective slope (intersecting at one point, perpendicular, non-intersecting parallel or coincident)
    Note : In Secondary III, students are introduced to the concept of relative position between two lines when comparing the rate of change and graphs of functions (degree 0 and 1). The same is true for solving systems of linear equations in two variables.
         
  1. Models, with or without technological tools, a situation involving
 
    1. straight lines: graphically and algebraically
      Note : In Secondary III, students are introduced informally to the concept of lines when they study functions of degree 0 and 1. The different forms of equations of a line (standard, general and symmetric) are explored in the various options. The symmetric form of the equation of a line is not covered in CST; it is optional in TS and compulsory in S. The general form of the equation of a line is optional in CST.
         
    1. a half-plane: graphically and algebraically
           
    1. parallel lines and perpendicular lines
           
  1. Determines the equation of a line using the slope and a point or using two points
    Note : The general form of the equation of a line is optional in CST.
           
  1. Determines the equation of a line parallel or perpendicular to another
    Note : The general form of the equation of a line is optional in CST.
           
  1. Geometric transformations
6 1 2 3 4 5  
  1. Identifies, through observation, the characteristics of geometric transformations in the Cartesian plane: translations, rotations centred at the origin, reflections with respect to the x-axis and y-axis, dilatations centred at the origin, scaling (expansions and contractions)
            CST
  TS
    S
  1. Defines algebraically the rule for a geometric transformation
    Note : In TS, students may also use a matrix to define a geometric transformation.
            CST
  TS
    S
  1. Constructs, in the Cartesian plane, the image of a figure using a transformation rule
    Note :In TS, students also determine the vertices of an image using a matrix.
            CST
  TS
    S
  1. Anticipates the effect of a geometric transformation on a figure
            CST
  TS
    S
  1. Geometric loci
6 1 2 3 4 5  
  1. Describes, represents and constructs geometric loci in the Euclidian and Cartesian planes, with or without technological tools
    Note : In S, the study of geometric loci is limited to conics.
            CST
  TS
  S
  1. Analyzes and models situations involving geometric loci in the in the Euclidian and Cartesian planes
    Note : In TS, geometric loci also include plane loci, i.e. geometric loci involving lines or circles only. In S, the study of geometric loci is limited to conics.
            CST
  TS
  S
  1. Analyzes and models situations using conics
    • describing the elements of a conic: radius, axes, directrix, vertices, foci, asymptotes, regions
    • graphing a conic and its internal and external region
    • constructing the rule of a conic based on its definition
    • finding the rule (standard form) of a conic and its internal and external region
    • validating and interpreting the solution, if necessary
 
    1. parabola centred at the origin and resulting from a translation
            CST
  TS
  S
    1. circle, ellipse and hyperbola centred at the origin
            CST
  TS
  S
    1. circle, ellipse and hyperbola resulting from a translation
            CST
  TS
    S
  1. Determines the coordinates of points of intersection between
 
    1. a line and a conic
      Note : In TS, this is associated with solving systems involving the functional models under study and entails mostly graphical solutions (with or without the use of technological tools).
            CST
  TS
  S
    1. two conics (a parabola and a conic)
            CST
    TS
  S
  1. Standard unit circle
6 1 2 3 4 5  
  1. Establishes the relationship between trigonometric ratios and the standard unit circle (trigonometric ratios and lines)
            CST
  TS
  S
  1. Determines the coordinates of points associated with significant angles using metric relations in right triangles (Pythagorean relation, properties of angles: 30°, 45°, 60°)
            CST
  TS
  S
  1. Analyzes and uses periodicity and symmetry to determine coordinates of points associated with significant angles in the standard unit circle
            CST
  TS
  S
  1. Proves Pythagorean identities
            CST
  TS
  S

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