Appetizers and Lessons for Mathematics and Reason  ( Français)  
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    for education,  June 22, 2008 
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Quebec High School Mathematics Education (English Version of)

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D What to do in School & Why  

E.How to Study Mathematics

Area pages  represent an effort to follow and understand the objectives of the 1997-2005, the prior reform, and the text books required and used 1997-2005. In retrospect, the objectives and texts in question are too incoherent, too full of nonsense, for rational comprehension and for service as a base for the current reform.    A farce is a farce, is a farce


Intermediate Objectives

The global objectives of the courses are described at length in the Quebec government document found in this pdf file.   The intermediate objectives extracted below and annotated give a skill and concept checklist.  For the sake of clarity, compare and contrast with the content objectives in the current reform program.

My comments and reflections on mathematics 216 are preceded by (W:) in order to distinguish between them from the MEQ objectives as is or paraphrased. 

Intermediate Objectives:

Objectives that specify the scope of a terminal objective, intermediate objectives might also be described as "reference. objectives." They are not intended as a series of steps to be completed one after the other. Such a process would give a very fragmented picture of teaching and learning. Rather, intermediate objectives are:

  • Aspects of a theme that have been chosen for the program.

  • Clarifications to ensure that the terminal objective is clearly understood

  • Guidelines that indicate the relationship between the terminal objective and student learning

  • Prerequisites for attaining a terminal objective

1.1 Objectives

Analyze functions (situations involving them)

  1. use symbols to represent a situation involving a function, indicating a source set, a target set and a rule of correspondence.
  2. draw the Cartesian coordinate graph representing a situation involving a function, given an equivalent verbal description, table of values or rule of correspondence. prepare the table of values for a situation involving a function, given an equivalent verbal description, rule of correspondence or Cartesian coordinate graph.
  3. describe the properties of a Cartesian coordinate graph representing a function (these properties are listed below).
    - increasing or decreasing function
    - sign
    - rate of change
    - axes of symmetry, if any -
    maxima or minima, if any
    - x-intercept(s) (zeros) IF ANY
    - y-intercept
    - domain and range
  4. determine the relationships between changes in the parameters of the rule of correspondence of a function and changes in the equivalent Cartesian coordinate graph.

1.2 Objectives

transform an algebraic expression into an equivalent expression

  1. apply the theory of exponents in transforming algebraic expressions.
  2. perform operations (addition, subtraction, multiplication division and exponentiation) on algebraic expressions and on polynomials in particular.
  3. factor a given polynomial. transform rational algebraic expressions by dividing or factoring them.

1.3 Objectives

analyze linear and quadratic polynomials (polynomials of degree 0 or 1)

  1. draw the Cartesian coordinate graph (a straight line) of a real polynomial function of degree 0 or 1 (w: in plain language: graph y = ax + b where a is real, zero or not).
  2. given a linear function y = ax + b the following information: its rate of change, its x-intercept (zero), its y-intercept, its domain and range, its sign, whether it is constant, increasing or decreasing, and the member of its domain associated with a given image.
  3. graph quadratics and call the graph a parabola.
  4. for a quadratic y =ax2 + bx + c, determine its extreme (vertex of the parabola), its zeros (if any), the sum and product of the zeros, its y-intercept, its domain and range, the intervals within which it is increasing and decreasing, its sign, and the member(s) of its domain associated with a given image. use algebra to convert the general form f ( x ) = ax2 + bx + c , into the standard form f ( x ) = a ( x - h ) 2 + k  and vice versa.
  5. determine the relationships between changes in the parameters of the rule of correspondence for linear functions y = ax + b and quadratics y = a ( x - h ) 2 + k and changes in the equivalent Cartesian coordinate graph.
  6. determine the coefficients in the equation y =a x+ b of a straight line, given the slope of that line and a point on that line or given two points on that line.
  7. determine the coefficients in the equation y = ax2 + bx + c or y = a ( x - h ) 2 + k, of a quadratic given the vertex of the associated parabola and another point on that parabola or given its zeros and another point
  8. graph the sum, difference and product of constant, linear, quadratic polynomial functions, given the graph or the rule of correspondence of each of these functions.

1.4 Intermediate Objectives

Solve problems by solving linear equations in two unknowns

  1. represent a situation by a system of two first-degree equations in two variables.
  2. solve a system of two first-degree equations in two variables by graphing it.
  3. solve a system of two first-degree equations in two variables algebraically.
  4. represent a situation by a system of two equations, one being of the first degree in two variables and the other being of the second degree in two variables.
  5. use a graph to solve a system of two equations, one being of the first degree in two variables and the other being of the second degree in two variables.
  6. use algebra to solve a system of two equations, one being of the first degree in two variables and the other being of the second degree in two variables.

1.5 Objectives

Solve problems in analytic geometry

  1. determine the slope of a straight line that passes through two given points.
  2. determine the slope, x-intercept and y-intercept of a straight line from a given equation.
  3. draw a straight line in a Cartesian plane, given the slope of the line and a point on the line.
  4. determine the equation of a straight line, given any of the following combinations: its slope and a point on the line, two points on the line, the x-intercept and y-intercept, or a point on the line and the equation of a parallel or perpendicular line.
  5. transform the equation of a straight line algebraically. determine if two straight lines intersect, or if they are perpendicular, parallel and distinct, or parallel and coincident by comparing their parameters and equations.
  6. determine the distance between two points or between a point and a straight line. determine the coordinates of the point of division of a segment,
  7. given the coordinates of its endpoints and other relevant data.
  8. determine the area and the perimeter of polygons, given the coordinates of the vertices.
  9. prove propositions using analytic geometry. See Appendix 2

Appendix 2: Deductive Reasoning in Analytic Geometry

The students are assumed to have the following knowledge and skills:
  1. The formula for finding the distance between two points (based on the Pythagorean theorem)
  2. The formula for calculating the distance between a point and a straight line
  3. The formula or a method for finding the coordinates of the point of division of a segment
  4. The general form of the equation of a straight line
  5. The functional form of the equation of a straight line (slope intercept form)
  6. The symmetric form of the equation of a straight line
  7. The role of the parameters in the various forms of the equation of a straight line (general, functional and standard forms)

The following propositions are considered to be true:

  1. The x- and y-axes are orthogonal.
  2. Two straight lines that are not parallel to the y-axis are parallel if and only if their slopes are equal.
  3. Two straight lines that are not parallel to the y-axis are perpendicular if and only if their slopes are negative reciprocals.
  4. A system of axes can always be arranged so that two consecutive vertices of a given polygon are on the x-axis, one of these vertices being located at the origin.

The students can prove the following propositions using the information above.

  1. The segment joining the midpoints of two sides of a triangle is parallel to the third side and its length is one-half the length of the third side.
  2. The segment joining the midpoints of the non-parallel sides of a trapezoid is parallel to the bases and its length is one-half the sum of the lengths of the bases.
  3. The segments joining the midpoints of the opposite sides of a quadrilateral and the segment joining the midpoints of the diagonals are concurrent in a point that is the midpoint of each of these segments.
  4. A segment connecting a vertex of a parallelogram to the midpoint of one of the non-adjacent sides intersects the opposite diagonal at a point that divides both that segment and the diagonal in the ratio of 1 : 2.
  5. The midpoint of the hypotenuse of a right triangle is equidistant from the three vertices.
  6. The midpoints of the sides of any quadrilateral are the vertices of a parallelogram.
  7. The three perpendicular bisectors of the sides of a triangle are concurrent in a point that is equidistant from the three vertices.
  8. The three medians of a triangle are concurrent and trisect one another at the point of concurrency.
  9. In any triangle, if a is the length of a side opposite an acute angle, if b and c are the lengths of the other two sides and if AH is the length of the projection of side c onto side b, then the following relationship is true: a2 = b2 + c2 - 2 b(AH)

    (w) See the cosine law.
  10. In any triangle, the sum of the squares of the lengths of the medians is equal to three-quarters of the sum of the squares of the lengths of the sides.
  11. If ABCD is a parallelogram and if E is the midpoint of side AD, F is the midpoint of side AB, G is the midpoint of side BC and H is the midpoint of side CD, then the segments AH, FC, BE, and DG intersect to form another parallelogram.
  12. The sum of the squares of the distances between a given point and two opposite vertices of a rectangle is equal to the sum of the squares of the distances between this point and the other two vertices of the rectangle.

Of course, other geometric propositions can be proven.

2.1 Intermediate Objectives

Solve problems using concepts of similarity, isometry and equivalence

  1. define isometries and similarity transformations by means of geometric transformations and their composites.
  2. accurately describe the geometric transformation or the simplest composite of geometric transformations that maps one isometric or similar plane figure onto another, given two isometric or similar plane figures.
  3. characterize isometric, similar or equivalent plane figures. determine the properties (e.g. measures of angles and sides, perimeters, areas) of isometric, similar or equivalent plane figures.
  4. state the minimum conditions required for two triangles to be isometric or similar. characterize solids that are isometric, similar, equivalent or equal in total surface area.
  5. determine the properties (e.g. measures of angles and sides, perimeters, areas, volumes) of solids that are isometric, similar, equivalent, or equal in surface area.
  6. determine certain measurements of similar right solids or spheres, given other measurements of these figures, a ratio (of lengths, of surface areas or of volumes), or data that can be used to find this ratio.
  7. justify an assertion used in solving a problem. See Appendix 3.

2.2 Intermediate Objectives

Solve problems using trigonometric ratios

  1. calculate the measure of a side or an angle in a right triangle,
  2. given relevant data and using a trigonometric ratio.
  3. calculate the measure of a side or an angle in a triangle, given relevant data and using the law of sines or the law of cosines.
  4. justify an assertion used in solving a problem. See Appendix 3

Appendix 3, Annotated

Principles of Geometry Introduced in Mathematics 436

My comments are prefaced by the w colon combination (W:).  Items 1 to 9 with item 12 form a logical base for deductive geometry and trig. Appendix 2 above points to deductive geometry in the framework of analytic geometry.  That being said, in a very confusing manner, our MEQ approved textbook for English language instruction gives an awful and different development built on properties of rigid body motions and dilatations, a development that is difficult if not impossible to follow as written. Should we correct or refine that development or go with the alternative development in the site area on Euclidean Geometry, The latter provides a lean and minimal development focused on the needs of analytic geometry.  So previous objections that Euclidean geometry is too hard for students may be eased or eliminated.  I will be looking at the intermediate objectives in mathematics 116, 216 and 314 to try to see what was intended and what is feasible.  Not all is clear. Lack of time may preclude us (me or you) following some preferences.

  1. If a transversal intersects two parallel lines then: - the alternate interior angles are isometric; - the alternate exterior angles are isometric; - the corresponding angles are isometric. W: For Proof See site area on Euclidean Geometry
  2. If two corresponding (or alternate interior or alternate exterior) angles are isometric, then they are formed by two parallel lines and a transversal. W: For Proof See site area on Euclidean Geometry
  3. The corresponding parts of isometric plane figures or solids are equal in measure.
    W: With regrets, I would assume this and not prove:
  4. Plane figures or solids are isometric if and only if there is an isometry that maps one figure onto the other.  W: I understand the if part. But I have not seen a proof of the only if part. That is a gap in my education.
  5. If the corresponding sides of two triangles are isometric, then the triangles are isometric. W:  See site area on Euclidean Geometry  (SSS isometry postulate)
  6. If two sides and the contained angle of one triangle and the corresponding two sides and contained angle of another triangle are isometric, then the triangles are isometric. W:  See site area on Euclidean Geometry (SAS isometry postulate)
  7. If two angles and a side of one triangle and two angles and the corresponding side of another triangle are isometric, then the triangles are isometric. W:  See site area on Euclidean Geometry  (ASA isometry postulate)
  8. Transversals intersected by parallel lines are divided into segments of proportional lengths. W: The proof of this could be neet - depends on ....
  9. The line segment joining the midpoints of two sides of a triangle is parallel to the third side and its length is one-half the length of the third side. W: The proof of this could be neet - depends on ....
  10. Any straight line that intersects two sides of a triangle and is parallel to the third side forms a smaller triangle similar to the larger triangle. W: The proof of this could be neet - depends on ....
  11. Plane figures or solids are similar if and only if there is a similarity transformation that maps one figure onto the other. W: I understand the if part, but do not now how to prove the only if part - see the corresponding assertion above with isometry in place of similarity.
  12. If two angles of one triangle and the two angles of another triangle are isometric, then the triangles are similar.  W: AA similarity postulate
  13. If the lengths of the corresponding sides of two triangles are in proportion, then the triangles are similar. W: Another Similarity Postulate kS-kS-kS
  14. If the lengths of two sides of one triangle are proportional to the lengths of two sides of another triangle and the contained angles are isometric, then the triangles are similar. W: Another Similarity Postulate kS-A-kS
  15. In similar plane figures or solids:
    - the ratio between the measures of the corresponding angles is 1; W: That is, Corresponding angles have the same measure.
    - the ratio between the lengths of the corresponding segments is equal to the ratio k between the lengths of the corresponding sides;
    - the ratio of the areas is equal to the square k2 of the ratio between the lengths of the corresponding sides;
    - the ratio of the volumes is equal to the cube k3 of the ratio between the lengths of the corresponding sides.
  16. Plane figures or solids with a scale factor of 1 are isometric.
    W: Obvious consequence of previous:
  17. In a right triangle, the length of the side opposite a 30o angle is equal to half the length of the hypotenuse. W: A consequence of the Pythagorean Theorem. That covers the 30-60-90 right triangle. What about the isoceles 45-45 -90 right triangle?
  18. The law of sines: The lengths of the sides of any triangle are proportional to the sines of the angles opposite these sides.

    19.    a  
    sin A

    =

      b  
    sin B

    =

       c  
    sin C


    (w): the common value of the these three ratios gives the proportionality constant.
    (w): Personal Preference: T    he proof of the law of sines is an exercise in geometry that students should meet and master if they can, time permitting.

  19. The law of cosines: The square of the length of a side of any triangle is equal to the sum of the squares of the lengths of the other two sides minus twice the product of the lengths of the other two sides multiplied by the cosine of the contained angle.

a2 = b2 + c2 - 2 bc cos A
b2 = c2 + a 2 - 2 ac cos B
c2 = a2 + b2 - 2 ab cos C

w): Because some of the angles in the cosine and sine laws may be obtuse, I would prefer to introduce trig functions using the unit circle for angles from 0 to 360 degrees, and then show how similarity implies these ratios can be calculated using trig ratios in right triangles when the angles are acute, that is, from > 0 and < 90 degrees. This or the introduction of polar coordinates a simple and dismystifying geometric development of complex numbers. in a future variant of mathematics 436 or 536..

W: Heron's Formula is part of mathematics 416 but not 436 according to the above objectives. That is fine, I dislike inclusion of formulas in mathematics courses without proofs of why they workl

3.1 Intermediate Objectives

Solve problems using measures of position

  1. distinguish between measures of position, measures of central tendency and measures of dispersion.
  2. assign a quintile or a percentile rank to a data value, if necessary.
  3. determine the data value(s) that are assigned a quintile rank or a percentile rank.
  4. use measures of position to compare data.
  5. construct a box-and-whisker plot.
  6. interpret a box-and-whisker plot.
  7. find qualitative information about the dispersion of the data in a one-variable distribution, using measures of position and measures of central tendency.

3.2 Intermediate Objectives

Solve problems that involve gathering data

  1. distinguish between a sample and a population.
  2. justify the decision to prepare a census, a poll or a study to obtain information.
  3. describe the characteristics of a representative sample of a given population.
  4. choose an appropriate sampling method when gathering information.
  5. determine the possible sources of bias during the data gathering process.
  6. compare two samples from the same population.

The following words appear before the intermediates objectives.

 ... be able to assess the reliability of the sample and the relevance of the data used in making predictions about a given population. To determine if the data is relevant, one must ascertain whether or not it is representative. If the initial hypothesis is appropriate, the sample should provide an accurate picture of the population under study. The students should check the size of the sample and the data-gathering methods to ensure that a study is as unbiased and error-free as possible. The students already know several ways of describing data graphically or numerically. They must learn to follow certain principles in processing data to ensure that they draw appropriate conclusions.

They may clarify and provide a context for the intermediate objectives.

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