his folder has a tree like structure. The child, same level and parent level webpages for this webpage follow..
More Links:
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 216
A Secondary II Course
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 paraphrased version of the objectives follow
My comments begin with W.
Relative Importance of Objectives
page 47,
- use algebra to solve problems. 25%
- proportional-reasoning ability. 25%
- apply knowledge of geometric figures. 35%
- phenomena involving chance or probability. 15%
If you would like to focus on the content details, see the intermediate objectives or find them embedded below. Compare and contrast the content objectives with the course content indicated by the current reform. There is an overlap.
In speaking of earlier studies, what students are expected to know, I have inserted the word should. When students lack prerequisites for a topic, essential skills and concepts should be developed and verified to provide a solid base for further studies at the start of studies or in the course of studies.
W NotesThe site lesson plans for Secondary I mathematics- fractions & allied concepts (decimals, percentages) and Secondary II - Algebra (arithmetic versus algebraic methods, backward use of formulas and proportionality equations) - provide support for maths 116 and 216.
W: Again for practical ideas on how to develop secondary II mathematics, hopefully in accordance with the old if not new MEQ objectives and the current reform (whatever that may be), see site lesson plans for secondary II and secondary I. The former plans extend the latter. Site lessons plans for secondary I and II are designed to give a solid base for secondary IV mathematics.
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Objective 1.
page 16
use algebra to solve problems
MEQ: Mathematical models have always been an excellent tool for representing reality. Developing these models involves observing a situation and selecting the characteristics of that situation which can be expressed mathematically. It then becomes easier to establish relationships, to formulate and test hypotheses, and to generalize and present the results.
MEQ: Numerical expressions, pictures or drawings, tables of values, graphs or diagrams, algebraic expressions, equations and formulas are some of the different modes of representation .. The goal is to help the students discover the advantages of using several modes of representation ..
MEQ: The students will first learn how to use the different modes of representation and to switch from one to the other Then the students gradually learn how to use algebra by working with literal expressions ...
page 17, Objective 1.1Translate one representation of a situation into another
In Secondary I, the students acquired the (arithmetic) skills prerequisite to the study of algebra. ... In statistics, they also used tables and graphs to represent a situation. Develop and verify the ability to use different modes of representation already familiar to them in order to interpret data logically. Students must also be able to translate one mode of representation into another. Whether the students give a description of a situation (first shaded column) conveyed through words, a drawing, a table of values or a graph, or represent a situation (first shaded row), they will have to give a comprehensive picture of that situation.
Intermediate Objectives
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page 18, Objective 2.1
Solve problems that can be expressed as a Linear equation
MEQ: Develop and verify the ability to use a linear equation to solve a problem. If the students are to appreciate the power and usefulness of algebra, it is important to assign problems that can be solved more efficiently with algebra than with arithmetic.
MEQ: Students should be given real-life problems whose solution involves generating and manipulating first-degree equations. The situations may be expressed using more than one unknown, but the students must be able to convert them into equations containing one unknown that are of the form ax + b = cx + d.
W: See the site area Solving Linear Equations with Stick Diagrams for ideas on how to develop
page 19, Intermediate Objectives
- Translate a verbal problem into an equation.
- Translate an equation into a verbal problem.
- Add and subtract expressions containing one variable and constants.
- Multiply and divide by a constant expressions containing one variable and constants.
- Solve a first-degree equation containing one unknown.
W: Emphasis items 1 and 5 above. Items 3 and 4 are almost implicit in 5. Item 2 is important to re-enforce item 1. I would emphasize item 2 only with students have difficulty with item 1.
Objective 2:
page 21
develop proportional-reasoning ability
MEQ: Proportionality is a basic mathematical concept and many aspects of our world operate according to proportional rules.
MEQ: In Secondary I, the students developed number and operation sense. Among other things, they were expected to be able to compare numbers and to arrange them in a particular order. In Secondary II, the students continue studying these topics by learning how to compare numbers or quantities so as to establish ratios or rates. The objective here is not to start a semantic debate about the terms "ratio" and "rate," but rather to introduce the students to situations in which they will compare different types of elements in order to establish a rate and to situations in which they will compare the same types of elements in order to establish a ratio.
W: Establishing rates and ratios could be subsumed into the calculation of proportionality constants. After a while, the students may find the calculation of proportionality constants K and the forward and backward use of proportionality equations Y = K X repetitive. The solution pattern is as follows. Find K from data (X,Y) satisfying the equation. Then use the equation to find a Y given a X or vice versa, find X given a Y.
MEQ: .. students develop the ability to reason proportionally.
W: To reason proportionally might means here to recognize when an equation Y = KX holds for some quantities or numbers X and Y. Then proceed with the the solution pattern mentioned above: Find K from data (X,Y) satisfying the equation. Then use the equation to find a Y given a X or vice versa, find X given a Y.
MEQ: The word "proportion" often denotes a relation of equality between two ratios (four quantities, a, b, c, d, are in proportion if a:b = c:d). Hence, given three terms of a proportion, the fourth can be found.
W: Keep it simple. Suppose in the equality a:b = c:d, one of the letters c or d denotes an unknown. Then K = a/b is known and a = K b. Now K = c/d. So we have c = K d. In other words, in a "proportion" a:b = c:d and in the equivalent fractions a/b = c/d, the numerators or first terms a and c are proportional to the denominators or second terms b and d. Here again we meet the proportionality relations Y = KX and its use. Analysis of the equality a:b = c:d, is an old-fashioned and awkward view of proportionality. Much ado about nothing.
The cross multiplication method for solving a/b = c/d follows from putting both fractions over the common denominator bd. Here the reasoning is more important than the concept.Once students understand how to use a proportionality relation Y+ KX and how to solve a/b = c/d for one of the numbers a, b, c or d given the other three, there is no mathematical purpose in any further discussion. There might be a minor purpose in providing students some vocabulary. That could be done in a review period before a final examination.
MEQ: If the students are to learn to reason proportionally, it is important to assign not only simple missing value problems (given three quantities, find the fourth), but also problems involving proportional situations.
MEQ: Concrete activities, questions, discussions, examples and counterexamples should be used ..
MEQ: Premature emphasis on algorithmic learning may prevent the students from assimilating and correctly applying concepts. While students should learn to solve problems efficiently, it is important to introduce these concepts by focusing first and foremost on comprehension rather than exclusively on efficiency.
W: The MEQ is saying do not teach by rote. Good stuff. Comments above point to the alternative.
MEQ: The Secondary II program covers many topics that involve applying the concept of ratio (e.g. percentages, similarity transformations, the circumference of a circle, converting from one unit of linear measure or surface area to another, certain applications of the concept of probability). After learning how to evaluate ratios qualitatively (W: What does that mean?), the students must then solve problems involving proportional situations or percentages
W: The discussion of rates and proportionality constants may involve numerators and denomintors. See the discussion of unit in calculations in the Fractions, Ratios, Rates, Proportions & Units site area of www.whyslopes.com
W: Mathematics 216 has transformation but not functions as part of the curriculum. The avoidance of the function strikes me as odd or inconsistent.
Objective 2.1
page 22,
Solve Problems using ratios and rates
MEQ: In Secondary I, the students learned to abstract the concept of a number written in the form a/b This type of number was expressed as a part of a whole or as a quotient. Ratios as such were not studied.
Develop and verify the ability to make qualitative evaluation of the data in problems involving ratios or rates
W: That is, how changes in the data affect results.
MEQ: The students must establish, read, interpret and compare ratios or rates and realize that a ratio or a rate represents a relation. They will be working with situations in which they must analyze how changes to the numerator or the denominator will alter a relation:
MEQ Example! I have mixed in a certain quantity of pigment with white paint. If I add more pigment, will the resulting colour be darker, lighter, or unchanged?
MEQ: Students may also be assigned problems that involve describing the changes made to the numerator or the denominator, given the qualitative direction of change in the value of the relation:
MEQ Example: To increase his speed, what change(s) must an athlete make in terms of distance travelled or the time consumed?
MEQ: In these problems, it is important that the students give qualitative rather than quantitative answers even if simple calculations may be required to support their line of reasoning.
MEQ: By analyzing ratios and rates, the students learn to take in a variety of information and to interpret it critically. In these activities, the focus should be on reasoning rather than on calculations
Activities in which the students make multiple numerical and non-numerical comparisons and examine the qualitative aspect of data are encouraged.
Intermediate Objectives
W: Here is a ratio is being identified with and written as a fraction
- Translate a situation into a ratio or a rate.
- To interpret a ratio or a rate.
W: What is the physical significance, what proportionality constant does it represent or give?
- To compare ratios or rates.
W: Which interest rate or speed or rate is greater in the sense of magnitude. Avoid mention and comparison of ratios and rates that may be negative.
- To interpret, for a given situation, the effect of a change in one of the
quantities that form a ratio or a rate.
W: Here is the qualitative aspect. For ratios or rates involving positive or unsigned quantities, increasing the numerator or decreasing the denominator increase a fraction or rate while decreasing the numerator or increasing the denominator descreases a fraction or rate.
- To indicate the change(s) made to the quantities that form a ratio or a
rate, given the qualitative direction of change in the value of that ratio
or rate.
W: See the previous comment.
W: The discussion of rates and proportionality constants may involve numerators and denomintors. See the discussion of unit in calculations in the Fractions, Ratios, Rates, Proportions & Units site area of www.whyslopes.com
Objective 2.2
page 24:
Solve problems involving proportions and percentages
MEQ: In Secondary I, the students saw that a percentage was one way of writing a rational number. They carried out transformations involving percentages and expressed percentages as fractions, most notably in statistics when drawing circle graphs. Percentages viewed as ratios were not studied. In the Secondary I program, the students also developed an understanding of the four operations and the ability to perform these operations on rational numbers.
MEQ: Develop and verify the ability of students to give quantitative solutions to problems involving proportions or percentages. To achieve this objective, students follow a two-step process. Step one focuses on proportions and proportional situations. The students analyze situations to determine what characterizes a proportional situation. They then use different procedures (unit-rate method, factor of change, additive procedure) to solve problems. Note that a procedure like the cross-product algorithm will prove useful in certain situations.
W: The cross-product algorithm here may the the same as the cross- multiplication rule for equivalent fractions a/b = c/d that yields ad = bc.
Recall the rule follows from by multiplying both sides by bd or by using bd as a common denominator.
MEQ: In step two, various situations are used to help the students consolidate their understanding of percentages, which can be viewed as a specific type of ratio. As a result, students can now use proportions to solve many problems involving percentages.
W: Translation required for step two.
Activities in which students learn how to evaluate situations more critically and that involve analysis, discussion and reasoning are encouraged. These activities should give students the opportunity to use new problem-solving methods and should help the students understand the basic concepts essential to the development of proportional-reasoning skills.
W: Powers that be, please provide examples.
Intermediate Objectives
- To distinguish situations that involve proportions from situations that do not.
- To establish a proportion.
- To establish a series of proportions.
- To apply the properties of equal ratios.
- To express the ratio between two numbers as a percentage.
- To calculate a given percentage of a number.
- To determine the number corresponding to one hundred percent, given a number and the percentage value it represents.
Objective 3
page 27
apply knowledge of geometric figures
MEQ: As outlined in the Secondary I program, the students progress through a hierarchy of levels in developing their geometric thinking skills. The students 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 various active exploration and observation activities, the students establish a system of relationships that enables them to see geometry as a tool for creating, understanding and representing.
MEQ: In the Secondary II program, this system will be expanded to include circles and regular polygons. Students will learn to analyze circles and regular polygons by discovering their properties and will learn to establish certain mathematical relationships. They will continue to study geometric transformations by studying similarity transformations, which can then become a geometric tool for reinforcing their understanding of proportional situations. The students will learn how to construct a reduced or an enlarged image of a figure accurately.
W. The foregoing study of similarity transformation comes after the study of polygons and circles, and proportionality relations. Contrary to that, our MEQ approved textbook puts the study of similarity first. The remedy is obvious.
MEQ: In exploring geometric transformations more systematically, they will perform operations on the coordinates of a figure drawn in a Cartesian plane, thereby enabling them to derive and apply algebraic rules.
W: Our MEQ textbook package for secondary II begins with an introduction to coordinates in the plane. That it introduces postive and negative dilatations of the plane without mentioning the coordinate description (x,y) -- > (kx,ky) where k is a positive or negative scale factor or proportionality constant. Thus operations on coordinates are absent in contradiction to the MEQ request.
W: Further the textbook game of calculating the scale k from various data would be best put at the end of course after students have learnt to use proportionality relations Y = K X forwards and backwards. The textbook placement of dilatation before lessons on solving linear equations and the discussion of proportionality appears out of sequence.
page 51, Appendix
In studying geometry, the students analyze regular polygons and circles. Through learning activities, they increase their grasp of a number of concepts and improve their skills. They should also become familiar with the definitions and some of the properties (see below) of the figures they are studying. They will use these definitions and properties to calculate measurements and justify any assertions used in solving problems involving regular polygons and circles. Objective 3.3
Objective 3.4
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Objective 3.1
page 28
Solve problems that involve enlarging or reducing a figure
Through activities that involve grouping together figures of the same shape, students were informally introduced to the concept of similarity in the early years of elementary school.
W: Formality may begin with a definition.
Develop and verify the ability of students to use their knowledge of enlarged or reduced figures to solve problems.
They must accurately construct the image of a figure, given instructions calling for a similarity transformation. Through exploration and observation, the students learn about the ratio of similitude and the centre of similitude. Construction activities will enable them to explore the properties of similarity transformations and to develop their understanding of similar figures.
Activities in which students analyze a construction, observe the properties of a similarity transformation, use appropriate symbols and apply the concepts of "ratio" and "proportion" are encouraged.
MEQ: For all objectives pertaining to geometry, "to construct" means to draw a figure, using a ruler, compass, set square or protractor.
W: The construction activities depart from the earlier objective of introducing students to operations on figure with the aid coordinates. The MEQ approved textbook in accordance with the MEQ directions above takes students through a coordinate-free exposition or path.
Intermediate Objectives
page 29
- Construct the image of a figure under a similarity transformation. The
ratio of similitude may be positive or negative.
W: According to my Collins Dictionary of Mathematics, a similtude is a transformation (x,y) ==> (kx,ky) where k = the ratio of simultude = the scale factor is positive. Negative values give an extension of the concept.
- Determine the ratio of similitude, given a figure and its image.
W: Given if two points are joined to their by straight lines, and the straight lines are not collinear, the centre of similtude or fixed point is the point of intersection.
W: If the image of point has distance R' from the fixed point, and its image has distance R, then both points and the fixed lie on straight line and R' = |k| R. The latter represents a proportionality relation between the distance of image points to the fixed point and the distance of the original or preimage points. The sign of k is positive if the image and preimage lie on the same side of the fixed point on the line through all three points. And, the sign is negative otherwise. The absolute value or magnitude of the scale factor, that is |k| can be determine a single pair of values for R' and R. Then it can be applied to find the location of image when the preimage is given, and vice-versa.
W: If the MEQ had taken a coordinate view of this transformation, that is use the property (x,y) ==> (X, Y) = (kx,ky) then the value and sign of k would follow from one or both of the equations X = kx and Y = ky.
- Distinguish figures that are similar from those that are not, given a set
of figures.
W: By defining a 1 to 1 correspondence between the vertices of two n-gons (triangles, quadrilaterals), a bijection or one to one and onto mapping, we obtain a correspondence between sides and angles. The concept of a correspondence needs to be explained before any definition of similarity is given. Then two n-gons are similar when and only when (or if you like, if and only if) there exists a correspondence between their vertices such that corresponding angles are equal and corresponding sides are proportional. So here again for the length of sides, if R is the length of side in one and R' is the the length of the other, there is a proportionality constant K such that R' = KR. I prefer to talk about proportionality constant in place of 'proportions" that involve ratios. The discussion of the latter distract students from proportional thinking with proportionality constants.
Objective 3.2
page 30
Solve problems involving isometric or similar figures in a Cartesian plane
MEQ: In the elementary school program, the students should have learnt how to use a Cartesian plane to describe the position of an image resulting from the geometric transformation of a figure (first quadrant only).
W: That follows Descartes use of numbers, unsigned or positive for coordinates. The use of negative numbers for coordinates presumably came later.
In Secondary I and in this program (Objective 3.1), students explored
isometric and similarity transformations by making numerous observations about a
single construction. By using geometry instruments to construct different
figures, students improved their accuracy and precision.
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W: Students may arrive from elementary
school with the inability to properly use a ruler to measure. Check
that students can measure a known length between two points with the aid
of ruler where placing the end of the ruler on one of the point leads to
error - a too small value for the length. See diagram below. |
MEQ: Develop and verify the ability to transform figures in a Cartesian plane using the relationships they have established between algebra and geometric transformations. To begin with, students will transform a figure by performing a given operation on its coordinates and applying a transformation rule. Then, given a completed transformation, they will discover the relationship between the points of the original figure and the points of its image.
MEQ: Activities in which students develop their powers of observation and their ability to locate figures on a surface are encouraged.
3.2 Intermediate Objectives
- Determine the position of a point in a Cartesian plane.
W: Presumably this means, locate a point given its coordinates, and for point or dot in the Cartesian plane, provide it coordinates as an ordered pair.
- Express the relationship between a point and its image by means of
variables. (W: coordinates). The relationship (W: rule or
function) may represent
- a translation,
W: (x,y) ==> (X,Y) = (x+a, y+b) = t(a,b)(x,y)
- a similarity transformation (the centre must be at the origin),
W: (x,y) ==> (X,Y) = (kx, ky)
- a rotation (the rotation angle must be a multiple of 90° and the
centre must be at the origin)
(x,y) ==> (X,Y) = (-y, x)
- a reflection (with respect to the axes or the bisectors of the
quadrants).
W: Reflection about x-axis (x,y) ==> (X,Y) = (x,-y) or
Reflection about y-axis (x,y) ==> (X,Y) = (-x,y)
Page 18 of the MEQ document for secondary III, includes the following statement:Functional notation should not be used, because it is important that students be able to observe and explore situations without being distracted by overly complex symbolism.
The notation in the Secondary II, MEQ approved textbooks for translations, rotations and reflections contradicts this rule for Secondary III.W: Second thoughts: the appearance of function notation for transformations of plane in secondary II could be encouraged and developed alongside function notation y = f(x) for real valued-functions of a single variable. The latter does not have to wait for secondary IV mathematics 436. Function notation such as f(x) = 3x+5 encourages the view that a letter x may stand as place holder in a computation rule or formula for a number or quantity.
- a translation,
- Identify a transformation by providing the rule that describes it, given a figure and its image.
- Construct the image of a given figure by performing a given operation on its coordinates, an operation given by a transformation rule.
Objective 3.3
page 32
Solve problems involving polygons
MEQ: In the Secondary I program, the students learned to construct certain polygons (triangles and quadrilaterals), given specific data. In addition, they analyzed these polygons by discovering the properties of their angles, sides, altitudes and diagonals.
Develop and verify the ability to identify the characteristics and properties of a given polygon that make it possible to derive the information required to solve a problem. Students will learn how to construct 5-, 6-, 8- and 10-sided polygons. By analyzing these different polygons, they will be able to describe certain important lines (diagonals and apothems), observe axes of symmetry and deduce certain properties from these features. The students will also be able to establish the relationship between the dimensions of a figure and its perimeter and area.
W: calculate perimeters and areas, and/or give formulas for the same quantities.
The teacher should ensure that the students know how to find the square root of a number.
W: Does this mean exactly given the prime decomposition of the radicand, for instance sqrt(196) = sqrt(72 22) = 7*2 = 14 and sqrt( 75) = 5 *sqrt(3). Does it mean with a calculator? Does it mean teach a decimal algorithm for obtaining the square root.
The students should to support their reasoning with relevant definitions and properties. See Appendix.
Activities in which students assimilate the terminology pertaining to polygons and learn how to construct figures as well as establish relationships that make it possible to solve problems are encouraged.
In learning the formulas for calculating perimeters and areas, the students continue to learn how algebra can be used to generalize situations.
W: I would say describe calculations that might be done.
Intermediate Objectives
page 33
- Construct a 5-, 6-, 8- or 10-sided regular polygon, given sufficient
data.
- Construct the axes of symmetry of a regular polygon.
- Express the relationship (W: in words or with a formulas) between the
perimeter of a regular polygon and the measure of its side, using variables.
(W: using letters to denote physical quantities).
W: use the formulas directly and indirectly in problem solving
- Express the relationship between the area of a regular polygon and some of
its dimensions, using variables.
W: use the formulas directly and indirectly in problem solving
- Calculate the perimeter and area of a regular polygon, given sufficient
data.
- Determine the square root of a number.
- Calculate the measure of one of the dimensions of a triangle, a
trapezium or a regular polygon, given its area and sufficient data.
W: use the formulas directly and indirectly in problem solving
- To justify (See appendix) an assertion used in solving a problem
involving regular polygons.
Appendix Objectives for Objective 3.3 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 360°.
3. 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.
Objective 3.4
page 34
solve problems involving circles
Through various exploration activities in the elementary school, the students learned met the relationship between the diameter of a circle and its circumference.
W: circumference p = p x diamater = 2p x radius
Develop and verify the ability to analyze a circle to discover its main elements, as well as establish the relationship among some of these elements, the circumference of the circle and its area.
W area A = p r2 = (¼) p d2
The students should be able to construct a circle and identify the
characteristics and properties (See Appendix)* of a circle that make it possible
to solve problems.
W: There is no mention of the radius r the MEQ? So its introduction above is an extra.
Appendix: Characteristics and Properties of Circles 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 p.
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.
Construction, exploration, observation and discussion activities in which students can derive properties that can be used to support their reasoning are encouraged. ... Various activities give them the opportunity to establish relationships between geometric concepts and the concept of proportionality. *
3.4 Intermediate Objectives
- To construct a circle, given sufficient data.
W: the centre and radius directly or indirectlty
- To express the relationship between the circumference of a circle and its
radius
W: p = p d = 2p r (Use backward and forward)
- To calculate the circumference of a circle, given sufficient
data.
- To express the relationship between the area of a circle and its
radius.
W: Use the formulas A = p r2 = (¼) p d2 backwards and forwards.
- To calculate the area of a circle, given sufficient data.
W: Use the formulas A = p r2 = (¼) p d2 directly
- To calculate the radius of a circle, given sufficient data.
W: Use A = p r2 = (¼) p d2 or p = p d = 2p r to find r.
- To justify an assertion used in solving a problem involving
circles. See appendix.
Appendix: Characteristics and Properties of Circles 4. Three non-collinear points determine one and only one circle.
W: The centre will be at the intersection of the perpendicular bisectors of the line segments between the point. That point can be found by construction now. The point is equidistant from each of the three points. Therefore a circle centred at it passsed through the three points.
The coordinate-based study of straight lines in secondary IV provides an algebraic approach which implies the
5. All the perpendicular bisectors of the chords of a circle meet at the centre of that circle.
W: The centre is equidistant from the endpoints of each chord. Therefore it lies on a or the perpendicular bisectors of each chord.
6. All the diameters of a circle are congruent.
W: d = 2r as a circle consists of all points at distant r from the centre and two collinear radii form each. diameter.
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.
W: In this courses, that is an assumption
9. The ratio of the circumference of a circle to its diameter is a constant known as p.
W: p = p d = 2p r (Use backward and forward)
10. In a circle, the measure q of the central angle is equal to the measure s of its intercepted arc.
W: Concretely, the arclength s = k q for some proportionality constant k. That can be implied by examples. Since a full circle has perimeter 2p r, we have 2p r = k *360 degrees. So when
q = n degrees we havek = p r
180 degreesand s p r q
180 degrees= p r n
180
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.
W: See 10.
Remark: The MEQ approved textbook goes beyond the MEQ curriculum in covering the proportionality of the area of a sector in a circle to its central area.
W: Concretely, the A = k q
for some proportionality constant k. That can be implied by
examples. Since a full circle has area p
r2, we have p r2 =
k *360 degrees. So with
q = n degrees we have
| k | = | p r2 360 degrees |
and | A | p r2
q 360 degrees |
= | p r2 n 360 |
Objective 4
page 37
Mathematical interpretation of phenomena involving chance
W: Here a chance to apply and re-enforce fraction skills and sense. Use it.
We are sometimes required to analyze situations and make decisions whose validity depends on our ability to summarize, compare results and make projections. In addition, we often encounter situations involving probabilities (e.g. games of chance, lotteries, betting, card games, weather forecasts). By studying realistic and very simple situations, students can become aware of the myths associated with chance, with its occurrences, and with the sometimes faulty interpretations of random events.
W: Here in Quebec, casinos and lotteries provide the government revenue and jobs for many people. But it also makes many poorer and may lead to gambling addiction and suicides. Critical discussion underlying ethical issues might serve the cross-curricular component of the forthcoming reforms
A variety of dynamic learning activities can be used to study probability. A concrete and very visual approach involving the use of experiments, concrete situations, games, graphs, and diagrams . In studying probability, the students learn about certain concepts related to phenomena involving chance by repeating an experiment. Often, several simulations are required before students can deal with phenomena that are not equi-probable, appreciate the significance of certain assertions or detect possible ways of tampering with the rules of a game, with betting schemes or with survey results.
An experimental approach should be used to develop the students' probabilistic thinking skills. The students should be motivated to verify their predictions. It is important to get the students to ask themselves questions during simulation activities so that they can discover the relationships between the facts they have deemed relevant. The activities should enable students to discuss ideas, modify them and develop models on their own.
W: Try the discovery approach if you can. But at the end of day, if it does not work. Get to the point, and present the theory directly and clearly, stopping along the way to check comprehension and provide feedback.
Objective 4.1
page 38
Calculate probabilities of the outcomes*
In elementary school, the students should have first explored probability by estimating and verifying outcomes in cases where they knew the probability value intuitively. In Secondary I, activities involving descriptive statistics helped the students learn to think critically, logically and analytically.
Develop and verify the ability to develop counting models that will enable them to describe, understand and even predict outcomes. Students should learn to think about the various aspects of a situation involving chance and to use tables, tree diagrams and networks to depict reality so that they will be able to identify all the possibilities. With this information, they can then determine the probability of an outcome. Nevertheless, the students should realize that the fundamental counting principle is very useful when a random experiment involving several steps has too many possible outcomes to be represented by a tree diagram.
Activities in which students develop the ability to formulate a mathematical interpretation of a phenomenon involving chance are encouraged. To simplify matters, "outcome" in this case refers to a simple event.
4.1 Intermediate Objectives
page 38
- distinguish experiments that are random from those that are not.
- enumerate some of the outcomes of a random experiment.
- list all the possible outcomes of a random experiment.
- assign a probability value to one of the outcomes of a random experiment.
Objective 4.2
page 39
Solve problems that involve calculating the probability of certain events during a random experiment
The prerequisites for this material were not acquired in previous programs, but rather by covering the topics related to the previous Objective 4.1
Students who have attained the current objective this program have gone beyond the manipulation stage and are able to calculate the probability of an event. In conducting random experiments, the students will develop their ability to analyze given hypotheses critically and be required to formulate predictions. By examining certain situations that focus on qualitative rather than quantitative considerations, they will acquire a greater understanding of fairness.
In a random experiment involving several steps, the students must determine whether or not the situation involves repetition. If necessary, they can calculate the probability of an event by determining whether or not events are complementary or mutually exclusive.
Activities in which students must simulate reality are consistent encouraged.. These simulations call for interaction between the students and the teacher. Problems presented numerically, graphically, symbolically and verbally should be assigned as often as possible so that the students can use the various ways of representing, interpreting and solving a problem-situation.
W: The set of possible outcomes is called not an outcome space, but a sample space. I would prefer a change of nomenclature in probability theory. Here events are proper or improper subsets of the outcome or sample space.
4.2 Intermediate Objectives
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- identify complementary events.
w: two events are complementary if they include no incomes in common, and their union provides all possible outcomes in the sample space.
- identify mutually exclusive events.
w: two events are complementary if they include no incomes in common,
- calculate the probability of an event.
w: when all outcomes are equally likely, and the outcome space is finite, the the number of outcomes in the event divided by the total number of outcomes gives the probability of the event.
W: The calculation of probabilities by counting, enumerating and listing
outcomes provides another opportunity to develop and verify fraction sense and
skills. While there is a call for technology in mathematics, exact and
efficient skills with fractions are still required. Directly and
indirectly, fraction sense and skills need to be maintained. Anything less
leads to difficulty in further mathematics and all quantitative disciplines. The
MEQ curriculum for mathematics 116 and 216 I am pleased to say indicates
fraction sense and skills are prerequisites to algebra. They are
also prerequisite for true success in physical science 436, mathematics 436 and
536. Students who enter CEGEP calculus without fraction skills and sense will
suffer. I will go further and state the following. Mathematics instruction
at the high school level in which fraction skills are not developed is a waste
of time for students. A spade is a spade.
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whyslopes - notes on algebra For triangles and quadrilaterals, we may use of letters as shorthand the area and perimeters, side lengths and where defined, heights. With them, student may see a shorthand formula or algebraic description of how some numbers and quantities, here lengths and heights, can be used to compute other numbers and quantities, here areas and perimeters.
The use of letters as abbreviations for lengths and areas in polygons and circles provides an easier introduction to algebraic ways of writing and reasoning than the context-free phrase. Let x, q and r be numbers. The novice may react in an offended manner to this phrase and say give m the numbers. Yet less offense will be taken, if we say Let x, q and r be the lengths of three line segments or Let s be the number of units in the area of that circle. The geometric or physical significance of the letters turns them into placeholder or pronouns for numbers and quantities easily visualized. Again, it is easier for students to accept the height of a rectangle and to say it is h units, than it is for them to say let h be a number. With the use of letters to denote quantities or numbers, expression involving those letters become meaningful. They describe calculations that could be done. By using letters to denote lengths or non-negative numbers, the commutative law for multiplication represents the notion that two different ways to compute the area of a rectangle should provide the same result, the distributive law and the foil method represent two different ways to calculate the areas of a rectangle as a whole or as the union of subrectangles. The commutative law for addition represents the ideas that the order in which two line segments are placed or measured does not affect the overall length. The distributive law can also be associated with the notion that a change of units (change of currency) should not affect a sum. Geometric significance here provides a scaffolding for the introduction of algebra with positive or non-negative quantities. By algebra in the first instance, we mean the role of shorthand notation in denoting numbers and quantities, and beyond that in describing calculations and the equality of calculations. The simplest context for introducing algebra appears before or apart from the use of negative numbers and lengths and areas are non-negative. The site area Solving Linear Equations with Stick Diagrams may be used to introduce and re-enforce the skills and concepts in class. |
For practical ideas on how to develop secondary II mathematics, hopefully in
accordance with the old if not new MEQ objectives and the current reform
(whatever that may be), see site lesson plans for secondary
II and secondary I. The
former plans extend the latter.