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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
| |
Mathematics 436
A Secondary IV Course
See too: Secondary
IV - Functions to Trig & Statistics - support for maths 436
This is a fourth year high school mathematics course taught in Quebec. An
abridged and sometime paraphrased version of the Quebec government document in
this pdf
file follows.
Three Objectives %
- algebra. 55%
- geometry. 35%
- statistics, develop critical attitude towards a statistical study. 10%
OBJECTIVE 1, algebra
In Secondary II, the students learned that algebra was a
powerful and useful language or communication tool. They were introduced to
different modes of representation (e.g. numerical expressions, images or
drawings, tables of values, graphs or diagrams, algebraic expressions,
equations, formulas) which highlighted certain aspects of problems they had to
solve.
In Secondary III, the students used algebra to derive
general rules from a number of specific situations. Conversely, they applied
general rules to individual cases. The students discovered the type of
dependence characterizing the relationship between certain variables,
especially the dependence represented by the graph of a straight line. They
also continued learning about algebraic manipulations.
In Mathematics 436, the students build on this knowledge in a more formal
way. They will analyze different ways of representing functions. More
specifically, they will determine the properties of a function given its
Cartesian coordinate graph. In addition, given the rule of correspondence of a
function, the students will analyze the relationships between changes in the
parameters of that rule and changes in the equivalent Cartesian coordinate
graph. To save time and help the students better understand this material, it
would be worthwhile to use a graphing calculator or a computer when exploring
these topics in class.
Like any other language, algebra has its own rules and syntax, and it is
important that the students observe them. The students will continue
developing their ability to perform algebraic operations, exploring those that
involve the laws of exponents, radicals, operations on algebraic expressions,
factoring and systems of first- or second-degree equations in two variables. The
students will thus have several effective problem-solving tools. They will then
analyze quadratcis - polynomial functions of a degree less than three, using
both the rules of correspondence and the graphs of these functions. Here again,
a graphing calculator or a computer would be very useful and effective.
Analytic geometry illustrates the relationship between algebra and geometry.
The students will begin by studying straight lines in the Cartesian plane and
their equations in various forms. In the process, they will define the concepts
of distance and slope and then use them to prove geometric propositions.
(w) Note the text says: the students will define. It does
not say the student will meet or be introduced to distance and slope. What
were the authors smoking?
The formal approach to mathematics in this course should result in the use
of set notation and logical symbols and connectives, which will enable the
students to express mathematical ideas more precisely and concisely.
See Québec, ministère de l'Éducation, Information Document, Graphs, Notation
and Symbols Used in Secondary Mathematics, Code 16-3306A, May 1982. (Québec:
ministère de l'Éducation, 1982).
(w) Most students (2002-5) enter 436 with a poor command of
fractions.
The teacher should therefore present and explain this notation and symbolism as
the need arises and encourage the students to use it often. With practice, the
students will find it easy to understand and apply.
(w) Why not just let the student define the notation as well, like they did
defined slope and the concept of distance between points.
Objective 1.1, Analyze functions
In Secondary II, the students had to use different modes of representation to
describe and represent a situation. They learned to translate a situation into a
first-degree equation. In studying ratios and proportions, they explored
situations involving direct variation. In Secondary III, they studied situations
in which the variables are directly or inversely proportional or in which one of
the variables is proportional to the square of the other. In particular, they
analyzed situations in which the relationship between the variables is linear
(i.e. situations involving direct or partial variation). They were able to do
this with a minimum of complex symbolism.
Develop the ability to use different modes of representation to analyze
situations involving functions in a more formal way. After defining a function
as the relationship between one independent real number variable and one
dependent real-number variable, the students will learn to write this
symbolically as y = f(x). They will learn to state the properties of a function
given its Cartesian coordinate graph. They will also learn to associate certain
changes in the parameters of the rule of correspondence of a function with
changes in the equivalent Cartesian coordinate graph. The students can explore a
variety of situations involving different types of functions (e.g. polynomial,
inverse variation, rational, square root, step and exponential functions).
However, the students will not have to distinguish between these types of
functions or classify the situations from which they are derived. This part of
the course provides an introduction to different types of functions, focusing on
the concept of a function and its various modes of representation. The
terminology used should be formally defined. In addition, it is important that
the students examine many different situations.
The use of a wide variety of situations giving rise to discussions and
questions that involve analyzing functions is encouraged.. The students will
develop their powers of observation and their ability to analyze and synthesize
a situation. They will learn to interpret graphs and understand the
relationships between symbolic, graphic and numerical representations of the
same situation. It may be useful (even necessary) to employ different learning
aids and methods (i.e. "pencil-and-paper" exercises, graphing
calculators and computers).
Intermediate Objectives 1.1
- use symbols to represent a situation involving a function, indicating a
source set, a target set and a rule of correspondence.
- 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.
- 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
- determine the relationships between changes in the parameters of the rule
of correspondence of a function and changes in the equivalent Cartesian
coordinate graph.
Objective 1.2, Equivalent Expressions
transform an algebraic expression into an equivalent expression
In Secondary I, the students developed their understanding
of the four operations on rational numbers as well as their ability to perform
these operations. In Secondary II, they began to perform operations on certain
expressions containing a variable. In Secondary III, they were able to perform
operations on expressions containing exponents and to apply certain laws
pertaining to positive integral exponents. They also performed operations on
polynomials (added and subtracted polynomials, multiplied a monomial by a
polynomial and a binomial by a binomial, and divided a polynomial by a
monomial).
Students who have attained Terminal Objective 1.2 will be able to do the
calculations involved in converting algebraic expressions into equivalent
expressions. When transforming algebraic expressions, they should be able to
apply the definitions and properties of rational exponents. The theory of
radicals should not be emphasized. Exercises should simply involve operations on
square roots as well as the rationalization of numerators and denominators (the
numerators and denominators should consist of no more than binomials). The
students should be able to perform standard operations on algebraic expressions
such as simple rational expressions. Nevertheless, operations on polynomials
will be emphasized. Divisions will be limited to those that involve finding the
quotient of a polynomial and a binomial. Conversely, students should be able to
factor polynomials by removing the common factor, by grouping, by finding a
difference of squares or a second-degree trinomial with integral coefficients or
by completing a square.
A geometric approach (e.g. algebra tiles) may be useful for
introducing factoring and will make these techniques more concrete.
(w) algebra tiles complicate matters - students
Lastly, bear in mind that the students must be able to apply
these techniques regularly in order to master them. It is therefore important
to give the students the opportunity to use these methods as often as
possible.
Activities in which the students gradually develop their
knowledge of the structure of algebra, their understanding of the laws of
algebra and their ability to apply its techniques are encouraged. While
ensuring that the students understand these methods, the teacher should also
help them develop certain habits. The students will have many opportunities to
use the skills they have already acquired or developed.
Intermediate Objectives 1.2
- apply the theory of exponents in transforming algebraic expressions.
- perform operations (addition, subtraction, multiplication division and
exponentiation) on algebraic expressions and on polynomials in
particular.
- factor a given polynomial. transform rational algebraic expressions by
dividing or factoring them.
Objective 1.3, Linear & Quadratic Functions
In Secondary II, the students began to study algebra and
used various modes of representation to describe and represent a situation. In
Secondary III, they analyzed various situations involving relationships
between two variables. In the earlier Objective 1.1, the students analyzed
several situations involving functions (more specifically, they formally
defined the concept of a function, the properties of functions given the
equivalent Cartesian coordinate graphs, and the terminology used). They also
studied the relationships between changes in the parameters of the rule of
correspondence of a function and changes in the equivalent Cartesian
coordinate graph.
Develop the ability to analyze real polynomial functions (w: linear or
quadratic, I presume), focusing on the relationships between the various forms
of the rule of correspondence of a function and the equivalent Cartesian
coordinate graph. The students will learn to use a graphing calculator or a
computer to determine the coordinates of important points on a graph with the
required degree of precision. They will also study formulas for finding the
zero(s) of polynomial functions (the roots of the corresponding equations) and
their extremes if any. Students should be able to transform the rule of
correspondence of a function, draw the Cartesian coordinate graph of a function,
give its main characteristics and determine the rule of correspondence of a
function if its Cartesian coordinate graph is given or described. This type
of in-depth analysis will be limited to polynomial functions of degree 0, 1 or
2.
Furthermore, the sum, difference and product of two
polynomial functions will be examined. The students could create their own
functions and then graph them. They can then observe the properties of the
resulting Cartesian coordinate graphs, comparing and describing them. The
student should explore and observe many of these functions, using the
appropriate terminology to describe the resulting graphs without trying to
formalize what they have learned. A graphing calculator or a computer would be
very useful.
Activities that bring about discussions and questions and in
which the students will develop their powers of observation and their ability
to analyze and synthesize situations are encouraged. The students will learn
to classify a polynomial function by its rule of correspondence and Cartesian
coordinate graph and to understand the relationships between these two modes
of representation. Graphing calculators and computers are efficient teaching
tools for this topic because they make it possible to analyze a greater number
of functions, thereby facilitating the task of synthesizing this material. In
addition, students will be able to use their knowledge and skills to solve a
variety of problems based on mathematical, real, realistic and/or imaginary
situations.
Intermediate Objectives 1.3
- draw the Cartesian coordinate graph (a straight line) of a real polynomial
function of degree 0 or 1,
- 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.
- graph quadratics and call the graph a parabola.
- 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 , nto the standard form f ( x ) = a ( x -
h ) 2 + k , a … o and vice versa.
- 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.
- 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.
- 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
- 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.
Terminal Objective 1.4, Systems of Equations
solve and apply system of equations in two variables, both linear, or
one linear and one quadratic.
In Secondary II, the students acquired certain skills that
enabled them to represent a situation by a first-degree equation and then
solve it. In Secondary III, they continued developing their ability to perform
operations on algebraic expressions. In Objectives 1.1 and 1.3 of this
course, they used algebra to analyze functions. In Objective 1.2 of this
course, they became familiar with new algebraic techniques
w: the transformation of expressions into equivalent
expressions
and developed their ability to use them.
Develop the ability to represent a problem by a system of equations and solve
them algebraically or by graphing. A graphing calculator or a computer can be
used to solve systems of equations graphically when the equations must first be
transformed into rules of correspondence. The students will study several
algebraic methods of solving a system of two first-degree equations in two
variables, but they will be free to use the methods they prefer.
The substitution method should be emphasized for linear systems in two
variables and for quadratic systems consisting of linear equation dx+ey= f and
quadratic y = ax2 + bx + c (or the equivalent). In the
Cartesian plane, the solution of these system will correspond to the
intersection of the two graphs representing these two equations (i.e. an empty
set, a singleton or a pair). In all cases, the students should be able to solve
problems arising from mathematical, real, realistic and/or imaginary situations.
(w): Students may solve linear and quadratic systems
algebraically or graphically. Both skills should be acquired.
A wide variety of situations in which students must analyze the relationships
between the data in the problem, formulate a system of equations, solve it
graphically or algebraically and interpret the results is encouraged.
Intermediate Objectives 1.4
- represent a situation by a system of two first-degree equations in two
variables.
- solve a system of two first-degree equations in two variables by graphing
it.
- solve a system of two first-degree equations in two variables
algebraically.
- 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.
- 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.
- 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.
Objective 1.5, Analytic Geometry Intro
solve problems in analytic geometry
Since Secondary II, the students have been acquiring
algebraic knowledge and skills (i.e. first-degree equations in one variable,
operations on polynomials, the relationship between variables in a situation,
functions, transformations of algebraic expressions, and systems of
equations). In Objectives 1.1 to 1.4 of this course, they learned
to use advanced algebraic methods. The students have also been studying
geometry since Secondary I. See Appendix 1, page 41.
Develop the ability to use their algebraic and geometric knowledge and skills
to solve problems in the Cartesian plane in a more formal way.
Begin by using analytic geometry to study straight lines. This will involve
drawing straight lines, examining the role of parameters, and finding different
forms of equations, the distance between two points (absolute value must be
defined), the distance between a point and a straight line, the point of
division of a segment as well as the area and perimeter of polygons. The
students will have to show and justify all the steps in their solutions. Then go
on to prove geometric propositions formally (with the teacher at first, then in
teams and eventually by themselves). Appendix 2 (page 43) gives a list of simple
propositions that are useful and easy to deal with.
When introducing the students to the task of presenting a proof, it is
important to guide them through the process by helping them acquire skills that
will be developed later. Proofs should not be emphasized in summative
evaluation.
(w) In other words, there are no consequences if a 436 course skips proofs.
However, the students should be able to apply these propositions in specific
cases.
Activities that will help the students gradually expand
their knowledge of analytic geometry, while developing their ability to
justify and prove propositions are encouraged. The students should focus on
understanding proofs rather than on the mechanics of presenting them. They
will be able to use their knowledge and skills to solve a variety of problems.
They will also discover the effectiveness of methods that show the
relationship between algebra and geometry.
Intermediate Objectives 1.5
- determine the slope of a straight line that passes through two given
points.
- determine the slope, x-intercept and y-intercept of a straight line from a
given equation.
- draw a straight line in a Cartesian plane, given the slope of the line and
a point on the line.
- 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.
- 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.
- 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,
- given the coordinates of its endpoints and other relevant data.
- determine the area and the perimeter of polygons, given the coordinates of
the vertices.
- prove propositions using analytic geometry. See Appendix 2, page 43.
OBJECTIVE 2, Geometry
analyze geometric situations
One of the major reasons for teaching geometry "is to build the kind of
strong geometric intuition that has been shown to be an important factor for
success on the job and in college.
The students gradually develop their geometric thinking
skills. They first learn to recognize shapes and then analyze the different
properties of these shapes before establishing relationships between the
properties and making simple deductions. Through numerous activities involving
active exploration and observation, the students create a system of
relationships pertaining to triangles, quadrilaterals, circles, regular
polygons, isometric transformations, dilatations and solids. See Appendix 1,
page 41.
In Mathematics 436, this system will be expanded to include the concepts of
isometry, similarity, and equivalence as well as trigonometric ratios.
When formally defining the concepts of "isometry" and
"similarity transformation" and studying them in greater detail,
the students will apply their knowledge of geometric transformations. In
the process, they will discover the minimum conditions required for
triangles to be isometric or similar (students can discover this through
experimentation and observation) as well as the properties of isometric or
similar plane figures. See Appendix 3, page 43.
Here the students can once again apply their knowledge
of ratios and proportions. Equivalent plane figures will be defined as
"figures with the same area" and their main properties will be
studied.
W: The call here to develop minimal conditions for triangles to be
isometric means an extra step in geometry in which the SSS, SAS and ASA
isometry conditions are not assumed, but derived from the properties of
transformations. The extra step, over-rated, is a complication and
not a simplification undertaken in a time where proofs in geometry have
been criticized as being too demanding. Further more the
derivation in the MEQ approved textbook package depends vaguely on
properties of transformations that the student or reader of the approved
texts is suppose to discover. In accordance with the modern discovery
methods for instruction but in opposition to the classical and
modern axiomatic development of mathematics, the approved text
mention properties, using them in proofs lie axioms, all without formally
stating them. That is paradoxical. |
In Mathematics 436, the students must continue honing the spatial sense they
began developing in Secondary III. This skill, which makes it possible to create
and manipulate images of objects, will be used to study isometric and similar
right solids. In the process, the geometric transformations studied up to now
will be very informally examined in three-dimensional space.
Equivalent solids will be defined as "solids with the same
volume." The students will then establish relationships between
the total surface area and the volume of different solids. See Appendix 4,
page 47.
Trig (Important)
Other activities will help the students discover new techniques for
solving problems related to triangles (i.e. sine, cosine and tangent ratios in
right triangles; the law of sines and the law of cosines in any triangle).
Naturally, all these definitions, properties and laws will be used to solve
geometric problems involving two- or three dimensional figures. As a rule, the
students should justify any statement or problem-solving procedure. The geometry
in this course should be logical and well thought-out and should prepare the
students for the formal proofs they will encounter later on.
"Students need to have many informal experiences that
involve reasoning and arguing to support conjectures before they are likely to
understand the need for, or the value of, a formal proof." (from 8.
Arthur F. Coxford, et al., "Geometry from Multiple Perspectives," in
Curriculum and Evaluation Standards for School Mathematics, Addenda Series
Grades 9 to 12 (Reston, Va.: National Council of Teachers of Mathematics,
1991), 51.) As a result, students should be given "more chances to
investigate and think about geometric questions on their own or in small
groups." (from 8. Arthur F. Coxford, et al., "Geometry from Multiple
Perspectives," in Curriculum and Evaluation Standards for School
Mathematics, Addenda Series Grades 9 to 12 (Reston, Va.: National Council of
Teachers of Mathematics, 1991), 51.)
Modern technology can be very useful at this point, since some computer
programs can be used to explore geometric problems, thereby enabling students to
formulate conjectures, discuss them and test them without outside help.
Objective 2.1, Transformation Geometry
Solve problems using the concepts of isometry, similarity and equivalence
Since Secondary I, the students have been constructing plane figures using
isometric transformations, dilatations and their composites. They have also
explored the properties of these transformations. They have made these
constructions using geometry instruments as well as a rule for transforming the
coordinates of points in a Cartesian plane (reflection about the axes or the
bisectors of the quadrants, rotations centred at the origin and for which the
rotation angle is a multiple of 90o, and dilatations
centred at the origin). The students have studied the concept of an inverse
transformation and identified the one transformation equivalent to a composite
of transformations. In addition, they have developed their spatial sense and
increased their knowledge of solids (they have created, represented, classified,
constructed and analyzed them and split them into sections). They have also
determined the measurements of these solids.
Students who have attained Terminal Objective 2.1 of this program will have
increased their knowledge of isometric, similar and equivalent two- or
three-dimensional figures so that they can solve different problems. First, the
students will describe isometries or similarity transformations and then try to
identify the isometry or similarity transformation (or the composite of two
geometric transformations) that maps one isometric or similar plane figure onto
another. They will then state the main properties of isometric, similar or
equivalent plane figures and use them to solve problems. Lastly, they will apply
these concepts to solids that are isometric, similar, equivalent or equal in
total surface area. For both solids and polygons, the students will solve
problems by organizing their solutions, justifying the steps in their reasoning
and referring to definitions, theorems and properties they have already studied.
The students can gradually learn how to present formal proofs by endeavoring to
provide structured solutions involving sound, systematic argumentation.
W: The subject of transformation geometry, that is
First, the students will describe isometries or similarity
transformations and then try to identify the isometry or similarity
transformation (or the composite of two geometric transformations) that maps
one isometric or similar plane figure onto another. They will then state the
main properties of isometric, similar or equivalent plane figures and use them
to solve problems.
is a complication. For most students and teachers in Quebec, the above
development of transformation geometry is out of context. The inclusion of the
whole thread of transformation geometry in the Quebec high school mathematics
program is unsound given the current lack of development of fraction and algebra
skills and sense of most student who enter secondary IV mathematics in
mathematics 416 or 426.
... students will have to identify properties or theorems, prove them and
then use them to solve problems are encouraged. The students will learn to
distinguish a conjecture from a certainty and a hypothesis from a conclusion. By
always making an effort to explain their logic when analyzing a geometric
situation or solving a problem, they will gradually learn to use the formal
reasoning required to present proofs.
Intermediate Objectives 2.1
- define (construct?) isometrics and similarity transformations by means
of geometric transformations and their composites.
- 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.
- characterize isometric, similar or equivalent plane figures. and how to
determine the properties (e.g. measures of angles and sides, perimeters,
areas) of isometric, similar or equivalent plane figures.
- 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.
- 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.
- 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.
- justify an assertion used in solving a problem. See Appendix 3, page 45.
Objective 2.2, Trig
Solve problems using trigonometric ratios
In the first cycle of secondary school, the students learned the concepts
of ratio and proportion. In studying the material covered in Terminal
Objective 2.1 of this course, they learned about similar figures and developed
their ability to reason more systematically.
Students who have attained Terminal Objective 2.2 of this program will be
able to use trigonometric ratios, the law of sines and the law of cosines to
solve problems that involve determining measurements in a triangle. Exercises
that simply involve finding the measure of a side or an angle in a triangle will
not suffice. Instead, the students should be required to use these measurements
to solve problems.
Activities in which the students can discover that trigonometric ratios are
derived from ratios of similitude in similar right triangles are encouraged.
The students will work out the law of sines and the law of cosines and use
these tools to solve a variety of problems that involve determining the
measurements in triangles (e.g. finding measurements that cannot be determined
directly, locating a position, land surveying). Here too, the students will
gradually learn to use formal reasoning by always making an effort to justify
each step in their solutions.
Intermediate Objectives 2.2
- calculate the measure of a side or an angle in a right triangle,
- given relevant data and using a trigonometric ratio.
- 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.
- justify an assertion used in solving a problem. See Appendix 3, page 45.
OBJECTIVE 3. Statistics
develop a more critical attitude towards a statistical study
To be informed and productive, a person must have a certain
ability to handle data and make intelligent decisions based on quantitative
and qualitative arguments. In addition to knowing how to find the right
numerical answer, he or she must be able to analyze situations critically. The
students will learn how to ask pertinent questions and present an analysis
while developing their critical sense.
In the first cycle of secondary school, the students organized and presented
data in tables and graphs. They also saw that they could use certain descriptive
measures (mean, median, mode, range) to synthesize data and thus provide
information on various phenomena. Statistical analysis calls for other data
besides measures of central tendency. In Secondary IV, the students will
therefore begin examining measures of position and will be introduced to the
concept of dispersion.
The students should also be encouraged to analyze the way in which data is
gathered from various sources and to assess the strengths and weaknesses of the
data-gathering process. To do this, the students will have to learn certain
methods of analyzing data.
With this approach, the students will learn to use data rather than produce
it. They should be given the opportunity to investigate and discuss such things
as public opinion polls, media ratings and census data.
Terminal Objective 3.1
Solve problems using measures of position
In the first cycle of secondary school, the students learned
to use certain tools (measures of central tendency and range) to analyze
information and presented data in the form of tables or graphs (bar,
broken-line and circle graphs as well as histograms).
Develop the ability to solve problems using the graphic and numerical tools
they have to analyze information. They are to use these tools to study the
variability of a distribution. The students will use measures of position to
determine the rank of a data value in relation to other values in a
distribution, or to identify the possible variations among the data values in
the distribution. In continuing to explore methods of analyzing data, the
students will increase their knowledge of mathematical models by studying the
box-and-whisker plot. This graph not only highlights certain characteristics of
a distribution, but also gives the students an idea about the dispersion of the
data.
... students will learn to interpret graphs and understand
the connections between graphic and numerical representations of the same
situation.
Intermediate Objectives 3.1
- distinguish between measures of position, measures of central tendency and
measures of dispersion.
- assign a quintile or a percentile rank to a data value, if
necessary.
- determine the data value(s) that are assigned a quintile rank or a
percentile rank.
- use measures of position to compare data.
(w: examples please.)
- construct a box-and-whisker plot.
- interpret a box-and-whisker plot.
(w: serious examples please.)
- find qualitative information about the dispersion of the data in a
one-variable distribution, using measures of position and measures of
central tendency.
page 36, Objective 3.2
Solve problems that involve gathering data
In the first cycle of secondary school, the students organized data in the
form of tables or graphs, usually working with given information. They continued
to study phenomena involving chance and also used certain measures to describe
this data (mean, median, mode, range).
Students who have attained Terminal Objective 3.2 of this program will 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.
Activities in which the students will develop a critical attitude towards
survey data are consistent with the global objectives, General Objective 3 and
the guiding principles. The students should become aware that a survey consists
of several elements, all capable of affecting the accuracy of the results.
During their discussions and investigations, the students should watch for
biases in the selection of data, for errors in measurement and for distortions
in graphic or numerical representations of data, both in the media and in their
own work.
page 37, 3.2 Intermediate Objectives
- distinguish between a sample and a population.
- justify the decision to prepare a census, a poll or a study to
obtain information.
- describe the characteristics of a representative sample of a given
population.
- choose an appropriate sampling method when gathering information.
- determine the possible sources of bias during the data gathering
process.
- compare two samples from the same population.
page 41-42, Appendix 1
Principles of Geometry Studied in the First Cycle of Secondary School
Expanding their knowledge gradually, the students have studied properties of
two- and three-dimensional figures as well as the properties of geometric
transformations. These properties are summarized below and should be included
with those to be introduced in Mathematics 436 . Since the students started
using the term "isometric transformation" in Secondary III, statements
related to the concept of congruence (2, 7, 13, 14, 16 and 23) have been
changed. In these cases, the concept of congruence has been replaced by the
concept of isometry, which is a broader term. The principles given below can be
used to determine measurements in certain figures and to justify certain steps
involved in solving problems.
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Note: When the segments, sides or angles involved are equal in measure,
they can be described as congruent. |
- Adjacent angles whose external sides are in a straight line are
supplementary.
- Vertically opposite angles are isometric.
- The sum of the measures of the interior angles of a triangle is 180o.
- In any triangle, the length of any side is less than the sum of the
lengths of the other two sides.
- In any triangle, the length of any side is greater than the difference of
the lengths of the other two sides.
- In any triangle, the longest side is opposite the largest angle.
- In any isosceles triangle, the angles opposite the isometric sides
are isometric.
- In any equilateral triangle, each of the angles measures 60o.
- In any right triangle, the acute angles are complementary.
- In any isosceles right triangle, each of the acute angles measures
45o.
- The axis of symmetry of an isosceles triangle contains a median, a
perpendicular bisector, an angle bisector and an altitude of the
triangle.
- The axes of symmetry of an equilateral triangle contain the medians,
perpendicular bisectors, angle bisectors and altitudes of the
triangle.
- The opposite angles of a parallelogram are isometric.
- The opposite sides of a parallelogram are isometric.
- The diagonals of a parallelogram bisect each other.
- The diagonals of a rectangle are isometric.
- The diagonals of a rhombus are perpendicular to each other.
- The diagonals from one vertex of a convex polygon form n - 2 triangles,
where n is the number of sides in that polygon.
- In a convex polygon, the sum of the measures of the exterior angles, one
at each vertex is 360o.
- 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.
- Three non-collinear points determine one and only one circle.
- All the perpendicular bisectors of the chords of a circle meet at the
centre of the circle.
- All the diameters of a circle are isometric.
- In a circle, the measure of the radius is half the measure of the
diameter.
- The axes of symmetry of a circle contain its centre.
- The ratio of the circumference of a circle to its diameter is a
constant known as p.
- In a circle, the measure of the central angle is equal to the measure of
its intercepted arc.
- In a circle, the ratio of the measures of two central angles is equal to
the ratio of the measures of their intercepted arcs.
- The area of a circle is equal to pr2.
- In a circle, the ratio of the areas of two sectors is equal to the ratio
of the measures of their central angles.
- 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 sides.
- 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.
- 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.
- An isometry preserves collinearity, parallelism, the order of
points, distances and measures of angles. In addition, translations and
rotations preserve the orientation of the plane.
- Any dilatation preserves collinearity, parallelism, the order of points,
the orientation of the plane, the measures of angles and the ratio of the
distances.
- Any translation and any dilatation will transform a straight line into
another line parallel to it.
page 43, Appendix 2
Deductive Reasoning in Analytic Geometry
The students are assumed to have the following knowledge and skills:
- The formula for finding the distance between two points (based on the
Pythagorean theorem)
- The formula for calculating the distance between a point and a straight
line
- The formula or a method for finding the coordinates of the point of
division of a segment
- The general form of the equation of a straight line
- The functional form of the equation of a straight line (slope intercept
form)
- The symmetric form of the equation of a straight line
- 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:
- The x- and y-axes are orthogonal.
- Two straight lines that are not parallel to the y-axis are parallel if and
only if their slopes are equal.
- Two straight lines that are not parallel to the y-axis are perpendicular
if and only if their slopes are negative reciprocals.
- 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 on
the previous page.
- 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.
- 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.
- 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.
- 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.
- The midpoint of the hypotenuse of a right triangle is equidistant from the
three vertices.
- The midpoints of the sides of any quadrilateral are the vertices of a
parallelogram.
- The three perpendicular bisectors of the sides of a triangle are
concurrent in a point that is equidistant from the three vertices.
- The three medians of a triangle are concurrent and trisect one another at
the point of concurrency.
- 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.
- 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.
- 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.
- 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.
Appendix 3,
Principles of Geometry Introduced in Mathematics 436
- 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.
- If two corresponding (or alternate interior or alternate exterior) angles
are isometric, then they are formed by two parallel lines and a
transversal.
- The corresponding parts of isometric plane figures or solids are equal in
measure.
- Plane figures or solids are isometric if and only if there is an isometry
that maps one figure onto the other.
- If the corresponding sides of two triangles are isometric, then the
triangles are isometric.
- 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.
- 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.
- Transversals intersected by parallel lines are divided into segments of
proportional lengths.
- 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.
- 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.
- Plane figures or solids are similar if and only if there is a similarity
transformation that maps one figure onto the other.
- If two angles of one triangle and the two angles of another triangle are
isometric, then the triangles are similar.
- If the lengths of the corresponding sides of two triangles are in
proportion, then the triangles are 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.
- 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.
- 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.
a 2 = b 2 + c 2 - 2 bccosA
b 2 = c2 + a 2 - 2 accosB
c 2 = a 2 + b 2 - 2 abcosC
page 47, Appendix 4
Properties of Two- and Three-Dimensional Figures
- Regular polygons have the smallest perimeter of all equivalent polygons
with n sides.
- Of two equivalent convex polygons, the polygon with the most sides will
have the smaller perimeter. (As a limiting case, an equivalent circle will
have the smaller perimeter.)
- Cubes have the largest volume of all rectangular prisms with the same
total surface area.
- Spheres have the largest volume of all solids with the same total surface
area.
- Cubes have the smallest total surface area of all rectangular prisms with
the same volume.
- Spheres have the smallest total surface area of all solids with the same
volume.
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