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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

Objectives of Mathematics 416

This is a high school mathematics course taught in Quebec. The Quebec government document in this pdf file presents the course objectives for delivery and content  in a very hard to follow manner.. An abridged and often paraphrased version of the objectives follow  If you would like to focus on the content objectives, see the intermediate objectives collected here in a single webpage or embedded below.   

Connections With Previous Studies

• 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 types of dependence characterizing the relationship between variables;
• 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

page 15,  OBJECTIVE 1

apply algebra

Students must develop an intuitive understanding of the way variables affect one another. .. Drawing on what they learned in previous courses, students can determine if a phenomenon is best represented by a continuous or discontinuous line or curve.

Students should use functions to create models of familiar situations. The students should be able to identify and compare families of functions as well as understand, analyze and use systems of functions, including the functions studied in Secondary III.

By using technology, students do not have to master algebraic manipulations, ...

w:  The latter may imply algebraic manipulations are optional in this course. Ouch. 

Relations and Their Representations

page 16,  Objective 1.1

analyze variations using different modes of representation In 

Secondary III, students illustrated the type of dependence characterizing the relationship between variables. They studied situations in which the variables are directly or inversely proportional, as well as other situations in which one of the variables is proportional to the square of the other. 

Develop the ability to use different modes of representation to analyze a situation and thereby differentiate between families of functions. Through exploration activities, the students learn to identify the different families of functions intuitively. 

The students will also study situations where the relationship between variables is represented by exponential or step functions. The students are not expected to be able to write an equation for a given situation or name the type of relation involved. The following table indicates the different types of translations from one mode of representation to another; the shaded boxes indicate the translations covered in this objective. 

Note that the students have used this approach to study different types of relations since Secondary II. The Roman numerals indicate the level(s) at which these translations were studied. 

page 17, immediate objectives

• determine the dependent variable and the independent variable in a given situation.
• make a table of values for a given situation. 
• determine the most appropriate scale for the graph of a given situation. 
• draw a graph representing a particular situation, given a table of values. 
• compare different situations expressed by means of the same mode of representation.

Objective 1.2

Solve problems dealing with systems of linear relations

 In Secondary III, the students solved problems involving direct or partial variation.

Students who have attained Terminal Objective 1.2 of this program will be able to use different modes of representation to solve problems involving systems of linear relations. The concept of a function studied up to now can be applied to more complex situations in which several functions are considered simultaneously. At this point, however, the students will simply study situations that can be represented by straight lines.

Skills

• To represent a situation by a system of linear relations.
• To describe a real-life situation expressed as a system of linear relations.
• To make a table of values for a system of linear relations.
• To determine the most appropriate scale for the graph of a system of linear relations.
• To draw a graph representing a system of linear relations.
• To justify the interpretation of a system of linear relations by using one or more modes of representation.
• To determine specific values of a system of linear relations with the degree of precision required for that situation

page 21,OBJECTIVE 2

To enable students to analyze geometric situations

The study of geometry provides an ideal opportunity to introduce the students to the deductive method and help them understand it so they can use it to solve problems. From the beginning of secondary school, the students progress through a hierarchy of levels in developing their geometric thinking skills. They first learned to recognize shapes and then analyzed the properties of these shapes before making deductions by establishing relationships between these properties. They must now discover that the reasoning used to solve a problem is similar to the sound, structured argumentation needed to present a proof. Formal reasoning as such should be emphasized so that students can learn to present more organized proofs. Require the students to work out relatively simple proofs.

In the course of their studies, the students have established a system of geometric relationships pertaining to angles, triangles, quadrilaterals, circles, polygons and solids. After studying transformations and their characteristics in the first cycle of secondary school, the students are now prepared to use the concepts of isometry and similarity to solve problems in Secondary IV.

After applying the concept of similarity to different figures, the students will discover that certain trigonometric ratios are derived from ratios of similitude involving similar right triangles. With the help of these tools and proportional reasoning, the students can determine measures and solve problems pertaining to measurements that cannot be found directly.

page 22, Objective 2.1

 master the concept of similarity

Students who have attained Terminal Objective 2.1 will be able to solve problems involving the concepts of similarity and isometry by structuring their solutions and, if necessary, justifying the steps in their reasoning by referring to relevant definitions, theorems or corollaries. A close connection should be established with Terminal Objective 2.1 of the Secondary III mathematics program to ensure that students understand that the concept of similarity is directly derived from the characteristics of geometric transformations. By defining the concept of similarity in this way, we can apply it to any two- or three-dimensional figure. As a result, cases involving similar or isometric triangles, which were examined as theorems in Euclidean geometry, become properties of similarity transformations in transformational geometry.

Given similar or isometric figures, the students will discover that there is always at least one similarity transformation or isometry which maps one figure onto another. The proofs assigned to the students should be within their capability. For both solids and similar polygons, the students will be asked to deduce certain measures or ratios required to solve problems.

Activities in which the students must organize the work involved in solving problems are consistent with the global objectives

page 23, Immediate Objectives:

• To distinguish similar or isometric figures from those that are not.
• To describe a similarity transformation or an isometry involving two polygons.
• To support an assertion used in presenting a proof involving the concepts of similarity or isometry.
• To deduce certain measures in similar figures from an appropriate geometric principle.
• To justify an assertion used to solve a problem involving the concept of similarity

 Objective 2.2

 solve problems using trigonometric ratios

Develop the ability to use trigonometric ratios to find the measures needed to solve a variety of problems. Problems should not be limited to determining the measure of a side or an angle in a right triangle or in another type of triangle. They should also involve using such information to deduce other data needed to solve a problem. To help students establish connections between mathematical concepts they already know, it is important to show them that trigonometric ratios are derived from ratios of corresponding sides in similar right triangles. By using a calculator, the students can concentrate on geometric reasoning rather than on calculations. Geometric properties of right triangles with an acute angle of 30°, 45° or 60° can be used to deduce certain measures and easily establish the trigonometric ratios for these angles. Using certain trigonometric principles, students can determine distances, lengths and heights that would be more difficult to measure directly.

Activities in which the students learn to use different modes of representation of a problem, estimate results and evaluate ratios mentally are consistent with the global objectives, General Objective 2 and the guiding principles. Through these activities, the students will discover that if they are given two measurements in a right triangle, they have enough information to find a third measurement in that triangle. By solving a variety of problems related to different fields of activity in the real world, the students can establish many connections between the different mathematical skills and concepts they have learned.

page 25, Immediate Objectives

• To deduce the measures of a right triangle using trigonometric ratios.
• To deduce the measures of triangles from various geometric principles.
• To justify an assertion used to solve a problem

Statistics - Gathering Data - Representing Data

page 27, OBJECTIVE 3

To help students develop the ability to analyze statistical data

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. In Secondary IV, the students will begin examining measures of position and will be prepared to study the concept of dispersion. It would also be advisable to examine the source of the data to see how it was obtained. The students will learn to assess the strengths and weaknesses of the data-gathering process and then acquire some tools for analyzing situations.

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.

page 28, Objective 3.1

To 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 summarize data (mean, median, mode and range).

Students who have attained Terminal Objective 3.1 will be able to assess the reliability of a sample and the relevance of the data used when solving problems involving predictions about a population. If the initial hypothesis is valid, 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 summarizing data graphically or numerically. They must learn to follow certain principles in processing data to ensure that they draw appropriate conclusions. When presenting their results or conclusions, the students can use everyday language to support their arguments.

Activities in which the students learn to develop a critical attitude towards survey data are consistent with the global objectives, General Objective 3 and the guiding principles. They should become aware that a survey consists of several elements, all of which can affect 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 29, Immediate 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 data.
• determine the possible sources of bias when gathering data.
• compare two samples from the same population

Percentiles, Box & Whisker Plots

page 30, Objective 3.2

solve problems using measures of position

Students who have attained Terminal Objective 3.2 will be able to solve problems using the graphic or 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 various 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.

Activities in which the students can present information about a set of data are consistent with the global objectives, General Objective 3 and the guiding principles. Technology should be used to facilitate the analysis and interpretation of the situation. Emphasis should be placed on analyzing and presenting the situation. In this way, the students will learn to interpret graphs and understand the connections between graphic and numerical representations of the same situation.  

page 31, Immediate Objectives:

• distinguish between measures of central tendency, measures of position and measures of dispersion.
• assign a quintile, a quartile or a percentile rank to a data value in a distribution.
• determine the data value(s) that are assigned a given rank.
• use measures of position to compare data.
• construct a box-and-whisker plot.
• interpret a box-and-whisker plot.
• find qualitative information about the dispersion of the data in a one-variable distribution, using measures of position and measures of central tendency.

Appendices - Geometric Principles

Principles Related to Themes Introduced in Mathematics 416 

Through their activities in geometry, the students increase their understanding of concepts and perfect several skills. Using definitions, properties, theorems and corollaries related to similarity or certain relations dealing with measurements in a triangle, they can deduce measures and justify an assertion used to present a proof or solve a problem.

1. The term figure designates a plane figure or a solid below. 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.

In the Secondary I and Secondary II programs, the students began to build up a system of axioms. In order to deduce certain measurements and justify certain steps involved in solving problems, the students must apply the following principles as well as those studied in Secondary III

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