Mathematics 514,
A Secondary V Course
This is a fifth year high school mathematics course taught in Quebec. An
abridged and sometime paraphrased version of the Quebec government document in
this pdf
file follows.
Relative Importance of Objectives
- 50% optimization techniques.
- 30% develop ability to analyze statistical data and data related to
probabilities.
- 20% analyze geometric situations.
Connection with Previous Programs
With continuity in learning, students can review topics they have already
studied and further develop their conceptions and representations. This
mathematics program enables students to build on the knowledge acquired in
elementary school and in the first four years of secondary school. This learning
process will be dynamic if the learning activities allow the students to use
their previously acquired knowledge and skills in new situations and help them
to become more proficient at applying what they have learned. As they acquire
new knowledge, the students will review the following skills and concepts
acquired in previous programs:
- number sense and facility with operations;
- the habit of estimating;
- proportionality;
- the concept of a variable;
- translation from one mode of representation to another;
- the relationships between variables;
- systems of linear relations;
- justifying the steps in the solution of a problem by using definitions,
properties, theorems or corollaries related to different geometric concepts;
- spatial relationships;
- the ability to organize and process statistical data;
- simulation of random events and the concept of probability
GENERAL OBJECTIVE 1
To help the students learn to apply optimization techniques
Since the beginning of secondary school, the students have learned how to
represent situations in different ways and to translate from one mode of
representation to another. Certain special graphs, called tree diagrams, were
used in Secondary II to solve counting problems.
Graphs & Optimization Theory (Combinatorial)
Terminal Objective 1.1
To solve problems using a graph
Students who have attained Terminal Objective 1.1 of this program will be
able to use graphs to represent certain situations when solving problems.
The students will have to analyze communications networks and diagrams
representing circuits, tournaments and production schedules that can be easily
modelled by means of graphs. Graphs not only describe phenomena, but also have
mathematical properties which can be used to solve problems and thereby
facilitate decision making. After solving problems using fundamental concepts
(graph, edge, vertex, path, circuit), the students will work with directed
graphs (digraphs). Problems will then be related to situations represented by
weighted graphs that may be directed or undirected. The students will complete
their study of graphs by examining tree diagrams that may or may not be
weighted.
The students will learn that the basic concepts of graph theory are simple
and effective and can be used to solve real-world problems which, at first, seem
difficult to understand.
Intermediate Objectives 1.1
- To represent a situation by a graph, a directed graph (digraph) or a
weighted graph.
- To distinguish between a path and a circuit.
- To use Euler's path or circuit, Hamilton's path or circuit, or a weighted
tree diagram to determine an optimum solution.
- To interpret a graph.
Terminal Objective 1.2
To solve problems using a system of linear inequalities from words or table
of graph
Students who have attained Terminal Objective 1.2 of this program will be
able to determine the values of the decision variables that maximize (or
minimize) a function subject to a set of constraints. These constraints
usually take the form of limitations on such things as raw materials,
production capacity, the number of employees and capital requirements.
The students will use linear programming to construct a model that will
facilitate decision making. They may have to choose from among a number of
optimum solutions and justify their choice. They can occasionally use an
algebraic approach to determine the coordinates of a vertex or the vertices of
the polygon of constraints.
Intermediate Objectives 1.2
- To represent a situation using a system of linear inequalities.
- To graph a system of linear inequalities.
- To formulate an algebraic expression that will represent the function to
be optimized.
- To determine the best solution(s) for a particular situation, given a
number of different possibilities.
- To justify the choice of values that optimize the function.
GENERAL OBJECTIVE 2
To help the students develop their ability to analyze statistical data or
data related to probabilities
In this era of rapid communication, we encounter a great deal
of qualitative and quantitative data. Indeed, raw data, graphs, rates,
percentages, probabilities, averages, predictions and tendencies have become a
part of our everyday life. They influence decisions related to health care,
the family, citizenship, employment, finance, sports and many other things. To
obtain the information we need as citizens or to work productively in today's
world, we must deal with data and be able to make intelligent decisions with
ease. It is therefore essential to make students aware of the importance of
statistics and probability in their daily lives.
With their knowledge of statistics, the students will be able to summarize
the information gathered during a study, poll or random experiment, using
different types of graphs. The students will also be able to describe a set of
data, using certain numerical summaries. In addition, they will be able to
analyze certain phenomena by evaluating the probability of a given event or
outcome, for example. .... The situations studied should be realistic, but
they should also be simple and of interest to the students.
Terminal Objective 2.1
To solve problems using the concept of correlation
In Secondary II, the students analyzed statistical data using
measures of central tendency (mean, median, mode). In Secondary IV, they
continued this analysis using measures of position. Over the last two years of
secondary school, the students have also become familiar with the concept of
dispersion of data by studying the concept of range as well as box-and-whisker
plots.
When doing a statistical study, students who have attained Terminal Objective
2.1 of this program will be able to determine if two given variables are related
and will also be able to describe that relationship, where applicable. The
students will discover that a population is characterized not only by the
distribution of variables, but also by the relationship between those variables.
A scatter plot is a useful way of representing the relationship between the
variables. It tells you whether there is a correlation between the variables and
allows you to describe some of its characteristics. The students will be asked
to describe the correlation as positive, negative or zero, high or low, or
perfect or imperfect. Calculations will be limited to estimating the correlation
coefficient by means of a graphical method.
Intermediate Objectives 2.1:
- To construct a two-variable distribution table.
- To construct a scatter plot.
- To describe the correlation between two variables in one's own words.
- To estimate the correlation coefficient.
- To interpret the correlation between two variables.
Terminal Objective 2.2
Probability and Odds
To solve problems using probabilities
Using grids, tree diagrams and networks, Secondary II
students determined the total number of possible outcomes in various
situations involving chance. As needed, they used concepts such as
complementary, mutually exclusive and non-mutually exclusive events to
determine the probability of an event. The students studied random experiments
involving one or more steps (e.g. drawing two objects in succession, with or
without replacement).
Students who have attained Terminal Objective 2.2 of this program will be
able to choose a suitable model (table, grid, tree diagram, area model,
enumeration, fundamental counting principle) and assign a probability to an
event when solving a problem. In certain cases, they will simulate the situation
in order to estimate the probability of an event. While consolidating what they
have already learned, the students will have to interpret the following ratios:
- the probability of an event (number of favourable outcomes/ total number
of possible outcomes);
- the "odds for" an event taking place (number of favourable
outcomes : number of unfavorable outcomes);
- the "odds against" an event taking place (number of unfavourable
outcomes : number of favorable outcomes).
Furthermore, in calculating the probabilities of compound events, the students
will examine conditional probability when the sample space has been restricted.
Lastly, the students are introduced to the concept of mathematical expectation,
which will simply be used to determine the fairness of a game or the possibility
of winning or losing.
... The students should be encouraged to solve problems in different ways and
to back up their conclusions.
Intermediate Objectives 2.2
- To distinguish between odds (for or against) and probability.
- To evaluate the probability that an event will occur during a random
experiment, knowing that another event has occurred during that experiment.
- To calculate the mathematical expectation of a random variable.
- To interpret the mathematical expectation of a random variable.
GENERAL OBJECTIVE 3
To have the students analyze geometric situations
From Secondary I to Secondary IV, the students built up a
system of concepts and relationships pertaining to two- and three-dimensional
figures. ... In addition, they solved problems involving the concept of
proportion or the Pythagorean theorem.
Students who have attained Terminal Objective 3.1 of this program will be
able to solve problems involving the concept of distance as well as other
geometric concepts and relationships ...
Terminal Objective 3.1
Distance Formula
To solve problems using the concept of distance
- To calculate the distance between two points.
- To determine the coordinates of the point on a line segment which divides
that segment in a given ratio.
- To compare distances.
- To justify a statement in the solution of a problem. See Appendix.
Terminal Objective 3.2
Geometric Probability
To solve problems using the concept of probability in a geometric context
From Secondary I to Secondary IV, the students established
relationships between the dimensions of different figures and between their
perimeters or their areas. In Secondary II and in Terminal Objective 2.2 of
this program, the students calculated the probability of an event during a
random experiment.
Students who have attained Terminal Objective 3.2 of this program will be
able to solve problems where an event is a set of points in a region of a
figure. The students will be able to determine probabilities by, for example,
comparing lengths or areas, using the graph of a system of inequalities or
applying concepts related to circles (e.g. central angles, measures of sectors).
In this case, the outcomes are associated with points chosen at random in one-
or two-dimensional geometric regions which represent the sample space (set of
possible outcomes).
Intermediate Objectives 3.2
- To estimate the probability of an event in a geometric context.
- To calculate the probability of an event in a geometric context.
- To justify a statement in the solution of a problem. 3.2
Appendices
Geometrically Statements and Assertions in previous courses
In the Secondary I, II, III and IV mathematics programs, the students
gradually built up a system of axioms.
MEQ: Connection to 514
Using the principles listed below, the students can deduce certain
measurements and justify certain steps involved in solving problems.
They will thus be able to structure an argument and present simple
proofs.
W: That being said by the MEQ, where proofs occur in this
course?
Secondary I Program
1. Adjacent angles whose external sides are in a straight line are
supplementary.
2. Vertically opposite angles are congruent.
3. The sum of the measures of the interior angles of a triangle is
180º.
4. In any triangle, the length of any side is less than the sum of the
lengths of the other two sides.
5. In any triangle, the length of any side is greater than the
difference of the lengths of the other two sides.
6. In any triangle, the longest side is opposite the largest angle.
7. In any isosceles triangle, the angles opposite the congruent sides
are congruent.
8. In any equilateral triangle, each angle measures 60º.
9. In any right triangle, the acute angles are complementary.
10. In any isosceles right triangle, each acute angle measures 45º.
11. The axis of symmetry of an isosceles triangle contains a median, a
perpendicular bisector, an angle bisector and an altitude of the triangle.
12. The axes of symmetry of an equilateral triangle contain the
medians, perpendicular bisectors, angle bisectors and altitudes of the
triangle.
13. The opposite angles of a parallelogram are congruent.
14. The opposite sides of a parallelogram are congruent. 15. The
diagonals of a parallelogram bisect each other. 16. The diagonals of a
rectangle are congruent. 17. The diagonals of a rhombus are perpendicular
to each other.
Secondary II Program
1. The diagonals from one vertex of a convex polygon form n - 2 triangles,
where n is the number of sides in that polygon.
2. In a convex polygon, the sum of the measures of the exterior angles,
one at each vertex, is 360º.
3. The sum of the measures of the interior angles of a polygon is 180º
(n - 2), where n is the number of sides in the polygon.
4. Three non-collinear points determine one and only one circle.
5. All the perpendicular bisectors of the chords of a circle meet at
the centre of that circle.
6. All the diameters of a circle are congruent. 7. In a circle, the
measure of the radius is half the measure of the diameter.
8. The axes of symmetry of a circle contain its centre.
9. The ratio of the circumference of a circle to its diameter is a
constant known as .
10. In a circle, the measure of the central angle is equal to the
measure of its intercepted arc.
11. In a circle, the ratio of the measures of two central angles is
equal to the ratio of the measures of their intercepted arcs.
Secondary III Program
1. In a right triangle, the square of the length of the hypotenuse is
equal to the sum of the squares of the lengths of the other two sides.
2. 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.
3. In any convex polyhedron, the sum of the number of vertices and the
number of faces is equal to the number of edges plus two.
4. Any translation and any dilatation will transform a straight line
into another line parallel to it.
5. Isometries or dilatations have one or more of the following
properties :
- they preserve collinearity;
- they preserve parallelism;
- they preserve the order of points;
- they preserve the orientation of the plane;
- they preserve distances and measures of angles.
Appendix
Secondary IV Program
- If two corresponding (or alternate interior or alternate exterior)
angles are congruent, then they are formed by two parallel lines and a
transversal.
- If a transversal intersects two parallel lines then:
- the alternate interior angles are congruent;
- the alternate exterior angles are congruent;
- the corresponding angles are congruent.
- The angles and sides of isometric figures 1 are equal in measure.
- Figures are isometric if and only if there is an isometry or a
composite of isometries that makes one figure coincide with the other.
- Two triangles whose corresponding sides are congruent must be
congruent.
- If two sides and the contained angle of one triangle are congruent
to two sides and the contained angle of another triangle, then the
triangles must be congruent.
- If two angles and the contained side of one triangle are congruent
to two angles and the contained side of another triangle, then the
triangles must be congruent.
- Transversals intersected by parallel lines are divided into segments
of proportional lengths.
- Any straight line that intersects two sides of a triangle and is
parallel to a third side forms a smaller triangle similar to the
larger triangle.
- 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.
- Similar figures have congruent corresponding angles and proportional
corresponding sides.
- Two figures are similar if and only if there is a dilatation or a
composite of transformations that preserves the order of points, the
measures of the corresponding angles and the ratio of the
corresponding sides.
- If two angles of one triangle are congruent to two angles of another
triangle, then the triangles must be similar.
- If the lengths of the corresponding sides of two triangles are in
proportion, then the triangles must be similar.
- 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.
- In similar plane figures or solids:
- the ratio between the measures of the corresponding angles is 1;
- the ratio between the lengths of the corresponding segments is equal
to the ratio between the lengths of the corresponding sides;
- the ratio of the areas is equal to the square of the ratio between
the lengths of the corresponding sides;
- the ratio of the volumes is equal to the cube of the ratio between
the lengths of the corresponding sides.
- Plane figures or solids with a scale factor of 1 are
isometric.
- In a right triangle, the length of the side opposite a 30o
angle is equal to half the length of the hypotenuse.
- Hero's or Heron's Formula
The area of a triang whose sides measure a, b and c is
S = sqrt( p(p-a)(p-b)(p-c) )
where p = (½)(a+b+c) is the semi- or half-perimeter of the
triangle.
- The law of sines: The lengths of the sides of any triangle
are proportional to the sines of the angles opposite these
sides.
a
sin A |
= |
b
sin B |
= |
c
sin C |
(w): the common value of the these three ratios gives the
proportionality constant.
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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
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