Mathematics 536
A Secondary V Course

This is a fifth year high school mathematics course taught in Quebec to the pre-college and pre-university students.   An abridged and sometime paraphrased version of the Quebec government document in this pdf file follows.  

 page 14 Objective 1

Use Algebra

In mathematics 536, the students will build on prior knowledge. They will begin by working with by working with inequalities and system of inequalities alone and then in the solution of linear optimization problems.  They will then meet many different types of functions, namely absolute value, step, square root, rational, exponential, logarithmic, and trigonometric. The students will learn the principal properties including asymptotes when the latter exist. The will study function composition, and operation on functions and their inverses.  Function and function operations will be used to solve problems. 

[remarks about calculator usage omitted]

The students should learn and develop emthods and procedures for solving different types of equations involving all the foregoing functions and/or inequalities involving linear and quadratic expressions, absolute values and square roots.  They will study properties of logarithms and use them to transform logarithmic expressions. They will also prove trigonometric identities. 

Objective 1.1
page 16

Solve problems using system of inequalities.

Develop the ability to represent a situation (constraints) with systems of linear inequalities in the plane,  to solve such a system for its solution set. In linear optimization problems, the solution set for a set of constraints is called the feasible set. When the constraints are linear, the latter determines a polygon-bounded convex region. The extreme values of a non-constant linear objective function occur at a vertex of the bounding polygon or along one side of the polygon.

Immediate Objectives
page 16

• represent a situation (set of constraints) by inequalities in one or two variables.

• Solve linear inequalities involving one variable.

• graph the half-plane solution set of an linear inequality in two variables.

• graph the polygon-bounded solution set of a system of linear inequality in two variables.

• Find the location(s) of the optimal value of an objective function subject to a system of linear inequalities in two variables. 

Objective 1.2
page 16

Solve problems using functions to develop a model

The objective is for students to work with different types of functions involving real variables ( absolute value, a step function such as the greatest integer less than or equal to function, truncation, rounding to so many decimal places operation, (?),  logarithmic functions, trig functions). Students should solve problems with these functions.  Students should learn to characterize or describe a function by referring to its rule of correspondence (formula?) and/or its graph  Students should study the role of parameter in the formulas for functions and the effect of these parameters, or changes in them, on the graph of the function. They will also study the sums, difference, products, quotients and composition of functions.  They will also meet inverse functions. 

Intermediate Objectives 1.2
page 19

• determine the relationship or correspondence between a change in parameter in a function and the change its graph.

• graph a function given its rule of correspondence or formulas. The function can be  absolute values, step functions, square root, rational, exponential, logarithmic, tangent or  sinusoidal (cosine and sine included).

• To describe the characteristics of a graph of a function, those in the previous item.

• Determine from a definition or rule of correspondence, the following:

  • domain and range, the inverse image of a given range

  • extrema and zeros when they exist, y-intercept, intervals where a function is increasing or descreasing, the equation of all asymptotes, if any.

  • its inverse, if the latter is a function.
    

• determine a formula or rule of correspondence for a function given sufficient information or its graph. The function may be absolute value, square root, exponential, logarithmic or sinusoidal.

• algebraically or graphically determine the sum, difference, product, quotient or composite of two real-valued functions of real variables given their graphs or rule of correspondence (formula).

• to model a situation or solve a problem with a function of a real variable.

.

Objective 1.3
page 19

Transform mathematical expressions into equivalent expressions.

Using the ability to transform expressions into equivalent expressions, students should be able to solve simple equations involving absolute value, square roots, exponents, logarithms of trigonometric expressions. Students should be able to solve simple inequalities involving absolute value, square roots and linear or quadratic expressions. Students should be shown the properties of logarithms and the close connection with exponential functions, and use those properties to transform expressions into equivalent expressions. 

Teachers should use the natural and common logarithms for  the most part. Teachers should define cotangent, secant and cosecant functions and explain the fundamental identities in trigonometry. Students will then prove simple trigonometric identities. 

Intermediate Objectives
page 21

• apply properties of logarithms in reducing or simplifying logarithmic expressions
• prove trigonometric identities
• find solution set of an equation in one real variable, an equation involving absolute values, square roots, exponents, logarithmic or trig expressions.
• find the solution set of an inequality in one real variable involving absolute values or square roots and the solution of a quadratic inequality.

Objective 1.4
page 22

description and use of conic sections

Develop the ability to solve problems involving conic sections as geometric loci and as solutions to  linear or quadratic   equations 

Ax²+Bxy+Cy²+Dx+Ex + F = 0  where B=0. 

in two variables x and y. The point is study geometric loci (straight lines, ellipses, parabolas and hyperbolas) and their equations. 

(w) The requirement that  B = 0 implies only translations found by completing the square are needed to convert the equation (*) into standard form for a  ellipse,  parabola or hyperbola when at least one of the second order coefficients A and C is non zero.  If B was not zero, a rotation would be needed as well for conversion into standard form.

Intermediate Objectives
page 23

• Determine figure corresponding to the description of a geometric locus.
• determine equation or inequality corresponding to the description of a geometric locus.
• determine equation or inequality associated with a given conic section or with the given interior or exterior of a geometric locus (w: conic section, I presume).
• describe a or each conic section and its principle elements (centre, radius, directrix, vertices, foci, minor axes or asymptotes) given its equation in standard form.

OBJECTIVE 2
page 25

analyze geometric situations

The aim is to prepare students for success in work and in college.

Mathematics 536 expands geometric knowledge to include vectors and the relationships between measurements in circles and right triangles (are we talking abut unit circle trig here?)

Students will be given problems related to the ideas and concepts they study, following by problems using their overall (cumulative?) knowledge of geometry.  The students will have to synthesize (combine) what they learnt in geometry while solving problems involving plane figures and solids. 

As in algebra, students should be require to demonstrate and prove propositions. Since students in this course will probably pursue post-secondary studies requiring mathematics,  it is important to gradually introduce them to more sophiscated mathematical procedures.  Moreover, the students will have to solve problems  with the same systematic approach required to prove a theorem. 

Objective 2.1
page 26

Solve Geometry Problems

Develop the ability to prove theorems or propositions related to circles and right triangles (See appendices).  

Develop the ability to work with vectors represented by arrows in the plane and/or ordered pairs of numbers. 

(w) Note the properties of vector addition and scalar multiplication follow from the properties of real numbers.  Representing vector via coordinate implies all the geometric properties. Starting the exposition with arrows and geometric operations on them leads to a logically incomplete exposition.

Students will study vector addition, scalar multiplication and  dot products - the scalar-product of a pair of vectors. Students will demonstrate propositions using vectors.  The aim is for students to systematically and deductively use the properties of the mathematical objects they encounter to arrive at  conclusions or to solve problems. 

2.7, Intermediate Objectives
page 27

• Prove propositions related to circles and right triangles

• Perform the following operations on vectors: addition, scalar multiplication, dot products  

• properties of vector operations

• prove propositions related to vectors

• prove propositions using vectors

• determine measurements in circles and right triangles

• justify statements (or steps) in solving a problem.

OBJECTIVE 3
page 29

analyze statistical data

In secondary IV, students learned about measures of position (percentile, quartile, quintile) and began to study dispersion (range, box and whisker plots).  In addition, students were made aware fo the problems involved in gathering data.

In this course, students will learn to draw scatter plots and determine (linear) correlation coefficients. Before this, the teacher should briefly explain the various measures of dispersion (??) focusing on the stnadard deviation and the Z-score.

Here students will learn to use data rather than produce it (or gather it). Analysis of the situation is more important than the calculation of parameters.

Objective 3.1
page 30

Solve Problems involving one or two variable statistical distributions

Students will analyze and interpret the relationship between two variable using the related distribution table (?), a scatter plot, the straight line that best represents (or fits) the scatter plot and the correlation coefficient.  

The students will also or first compare the following measures of dispersion: the range or spread, the semi-interquartile range, the mean deviation and the standard deviation.  The Z-score will be defined and used to explain the correlation (How)). To draw the regression line, the students proceed gradually; The must first make a rough sketch of the line, than make sure the line passes through the point whose coordinates represent the average of each variable and finally use (how?) modern technology 

(w) Does the use of Modern technology here imply students are being taught to use a black box with no understanding of the inner workings.  That stinks of rote learning. 

Likewise, students should learn to estimate correlation coefficients in different ways.  (A) estimate the despersion of a scatter plot by comparing it with those that have been studied (w: for which the correlation coefficient is known or given),  (B) use the best methods of approximation (ellipse or box plot);   (C) use formulas or modern technology.  The main focus should be on analysis and interpretation rather than interpretation. 

3.1 Intermediate Objectives
page 31

• compare different measure of dispersion in a given distribution.
• determine the Z-score for a data value in a distribution.
• construct scatter plots for two variable distribution
• sketch the regression line associated with a two-variable distribution
• determine the equation of the regression line associated with a two variable distribution.
• interpret the (linear) correlation coefficient for the two variables in a distribution. 

Principles of Geometry Introduced in Earlier Programs 

Between Secondary I and Secondary IV, the. students learned about the properties of two- and three-dimensional figures as were as the properties of geometric transformations. These properties are summarized below and should be included with those to be introduced in Mathematics 536. All these principles can be used to determine measurements in certain figures and to justify certain steps involved in solving problems. 

 I  PLANE FIGURES 

ANGLES 

1. Adjacent angles whose external sides are in a straight line are supplementary.
2. Vertically opposite angles are isometric
3. If a transversal intersects two parallel lines then: 
4. If two corresponding angles (alternate interior or alternate exterior) are isometric, then they are formed by two parallel lines and a transversal.

TRIANGLES

5. The sum of the measures of the interior angles of a triangle is 180".
6. In any triangle, the length of any side is less than the sum of the lengths of the other two sides.
7. In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of me lengths of the other two sides.
8. In any triangle. the longest side is opposite the largest angle.. 
9. In any isosceles triangle, the angles opposite the isometric sides are isometric..
10. In any equilateral triangle, each angle measures 60 degrees..
11.  In any right triangle, the acute angles are complementary.
12. In any isosceles right triangle, each acute angle measures 45°.
13. The axis of symmetry of an isosceles triangle contains a median, a perpendicular bisector, an angle bisector and an altitude of the triangle.
14. . The axes of symmetry of an equilateral triangle contain the medians, perpendicular bisectors, angle bisectors and altitudes of the triangle.
15. In any triangle, the length of any side is greater than the difference of the lengths of the other two sides.
16. A triangle is right-angled if the square of the length of one of its sides is equal to the sum of the squares of the lengths of the other two sides.
17.  In a right triangle, the length of the side opposite a 30D angle is equal to half the length of tbe hypotenuse.
18. The law of sines:

The lengths of the sides of any triangle are proportional to the sines of the angles opposite these sides.

  a        b        c 
----- =  -----  = -----  
sin A    sin B    sin C

19. The law of cosines:
21. 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.
23.    a² = b² + c² - 2bc cos A 
24.    b² = c² + a² - 2ac cos B 
25.    c² = a² + b² - 2ab cos C
27. The midpoint of the hypotenuse of a right triangle is equidistant from the three vertices.
28. The three perpendicular bisectors of the sides of a triangle are concurrent in a point that is equidistant from the three vertices. -
29. The three medians of a triangle are concurrent and intersect one another at the point of concurrency.

QUADRILATERALS

23. The opposite angles of a parallelogram are isometric.
24. The opposite sides of a parallelogram are isometric.
25. The diagonals of a parallelogram bisect each other.
26. The diagonals of a rectangle are isometric.
27. The diagonals of a rhombus are perpendicular to each other.
28.  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.
29.  The midpoints of the sides of any quadrilateral are the vertices of a parallelogram.

CIRCLES

30. Three non-collinear points determine one and only one circle.
31. All the perpendicular bisectors of the chords of a circle meet at the centre of the circle.
32. All the diameters of a circle are isometric.
33. In a circle. the measure of the radius is half the measure of the diameter.
34. The axes of symmetry of a circle contain its centre.
35. The ratio of the circumference of a circle to its diameter is a constant known as p.
36. In a circle, the measure of the central angle is equal to the measure of its intercepted arc.
    (w) Should that be in a unit circle?
37. In a circle, the ratio of the measures of two central angles is equal to the ratio of the measures of their intercepted arcs.
38. The area of a circle is equal to pr² where r is the radius of the circle,.
39.  In a circle, the ratio of the areas of two sectors is equal to the ratio of the measures of their central angles.

POLYGONS AND REGULAR POLYGONS

41. The diagonals from one vertex of a convex polygon form n - 2 triangIes, where n is the number of sides in that polygon.
42. In a convex polygon, the sum of the measures of the exterior angles. one at each vertex, is 360".
43. The sum of the measures of the interior angles of a polygon is 180" (n - 2) where n is the number of sides in the polygon.

II TRANSFORMATIONS OF PLANE FIGURES AND SOLIDS

ISOMETRIES AND ISOMETRIC FIGURES

43. An isometry preserves co linearity, parallelism, the order of points, distances and the measures of angles. In addition, translations and rotations preserve the orientation of the plane.
44.   Any translation will transform a straight Boo into another line parallel to it. 
45. Th e corresponding parts of isometric plane figures or solids are equal in measure. 
46. Plane figures or solids are isometric if and only if there is an isometry that maps one figure or solid onto the other.
47.   If the corresponding sides of two triangles are isometric, then the triangles are isometric.
48.   If two sides and the contained angle of one triangle and the two corresponding sides and contained angle of another triangle are. isometric, then the triangles are. isometric.
49.   If two angles and a side of one triangle and two angle5 and the corresponding side of another triangle are isometric, then the triangles are isometric. 
50. Plane. figures or solids with a scale factor of 1 are Isometric.

SIMILARITY TRANSFORMATIONS AND SIMILAR FIGURES

51.   Any similarity transformation preserves collinearity parallelism, the order of points, the orientation of the plane, the measures of angles and the ratio of the distances.
52. Any dilatation will transform a straight line into another line parallel to it. 
53. Transversals intersected by paral1el lines are divided into segments of proportional lengths.
54. 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.
55. 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. 
56. Plane figures or solids are similar if and only if there is a similarity transformation that maps one figure or solid onto the other. 
57. If two angles of one triangle and the two angles of another triangle are isometric, then the triangles are similar.
58.  If the lengths of the corresponding sides of two triangles are in proportion, then the triangles are similar.
59. If the lenghts of two sides of one triangle are proportional to the lengths of two sides of another triangle and the contained (or included) angles are isometric, then the triangles are similar. 
60. (A) In similar plane figures
    +  corresponding angles are isometric
    + the ratio K between lengths of corresponding sides is equal to the scale factor (or proportionality constant)
    + the ratio of their area is equal to the square K² of the scale factor
    B) In similar solids
    +  corresponding angles are isometric
    + the ratio K between lengths of corresponding sides is equal to the scale factor (or proportionality constant)
    + the ratio of corresponding areas is equal to the square K² of the scale factor
    + the ratio of their volumes is equal to the cube K³ of the scale factor.

SOLIDS

61. In any simple polyhedron, the sum of the number of vertices and the number of faces is eqaul to the number of edges plus 2.   V+ F= E+2
62. The lateral area of a right prism or cyclinder is equal to the product of its perimeter and height.
63. The lateral area of a regular pyramid or of a right circular cone is equal to one half the product of the perimeter of its base and slant height.
64. The area 4pr² of a sphere is equal to the product of the circumference 2pr of a great circle and it diameter d = 2r.
65. The volume of a prism or of a cylinder is equal to the product of the area of its base and its height.
66.   The volume of a regular pyramid or of a cone is equal to one-third the product of the area of its base and its height.
67. The volume of a sphere is equal to two-thirds the product of the area of its great circle and its diameter.

  IV ANALYTIC GEOMETRY 

68. The distance between two points (a,b) and (c,d) is d = sqrt( (c-a)²  + (d-b)²)

69. The distance from point P at (x,y)  to a line with equation Ax+By+C = 0 
    
        |Ax+By+C|
    d = ---------
         (A2 + B2)½
          

70. Formula for finding point P which divides segment AB:
     
71. Two straight lines that are not parallel to the y-axis are paranel if and only if their slopes are equal.
72. Two straight lines that are not parallel to the y~axis are perpendicular if and only if their slopes are negative reciprocals.

  Appendix 2 Deductions Using Vectors 

Students will be asked to prove certain properties of vectors and to prove propositions using vectors.

Using vectors prove the following propositions.

6. The diagonals of a parallelogram bisect each other. In addition, a quadrilateral whose diagonals bisect each other is a parallelogram.

7. The line segment joining the midpoints of two sides of a triangle is parallel to the third side and its length is one-half that of the third side.

Appendix III. Deductions Using Circles and Right Triangles

Quebec English Mathematics Education

A farce is a farce is a farce.

Area Intro
Copy Right Matters
Curriculum Cuts
Intermediate Objectives
MEQ Objectives

116 Textbooks
116 Objectives
116 Check List
116 Suggestions
216 Objectives
216 Check List
216 Book Review
216 Nonsense or BullShit
216 Suggestions
314 Objectives
314 Check List
314 Suggestions
416 Objectives
416 Check List
416 Suggestions
436 Objectives
436 Checklist
436 Suggestions
436 Book Reviews
436 Nonsense in
514 Objectives
514 Suggestions
514 Book Reviews
536 Objectives
536 Suggestions
536 Book Reviews

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

www.whyslopes.com   [ Back ] [ Up ] [ Next ] [Top of this Page]

Custom www Search

Road Safety Message  Do not walk on a road with your back to the traffic - rule of thumb Please report by email,  errors in mathematics or grammar or terminology to site author If a mathematics topic you need is not covered in site pages,  report that as well. Topics in most demand will be covered first in site growth.

All trademarks and copyrights on this page are owned by their respective owners.
Copyright to comments & contributions are owned by the Poster. 
The Rest © 1995 onward by site author,   Alan Selby (email form) All Rights Reserved.