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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
The collection of intermediate objectives below provide a skills and concept checklist for the pre-reform 216 course. For the sake of clarity, compare and contrast with the content objectives in the current reform program.
The immediate objectives here come from the Quebec government course objectives online in a pdf file
My comments and reflections on mathematics 216 are begin with W:
| Intermediate Objectives:
Objectives that specify the scope of a terminal objective, intermediate objectives might also be described as "reference. objectives." They are not intended as a series of steps to be completed one after the other. Such a process would give a very fragmented picture of teaching and learning. Rather, intermediate objectives are:
|
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%
Intermediate Objective 1.1
Translate one representation ( w: description)
of a situation into another
- Express the relationships among the data in a problem, using his or her own words or a drawing.
- Give a comprehensive description of a situation represented by a table of values. (w: does this mean interpret or apply a table of values?)
- To give a comprehensive description of a situation represented by a graph. (w: does this mean interpret or apply a table of values?)
- To represent a situation, using a table of values.
- To represent a situation comprehensively, using a graph.
W: My preference is to embed these objectives in the rest of the course instead of devoting a single chunk of a textbook or course time to it.
Intermediate Objectives 2.1
Solve problems that can be expressed as a Linear equation
- Multiply and divide by a constant, expressions containing one variable and constants.
- Solve a first-degree equation containing one unknown.
- Translate a verbal problem (w: word problem) into an equation.
- Translate an equation into a verbal problem.
- Add and subtract expressions containing one variable and constants.
W: See the site area Solving Linear Equations with Stick Diagrams, the solution of linear equations ax+b = cx+ d and the discussion of systems of equations in essentially one unknown. In the latter discussion, students will meet the more complication systems in essentially one unknown where the distributive law is required where mastery of objective 1 is an embedded requirement - a natural part of the solution process.
Objective 2:
develop proportional-reasoning ability
Intermediate Objective 2.1
Solve Problems using ratios and rates
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
Intermediate Objective 2.2
Solve problems involving proportions and percentages
- 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.
Intermediate Objective 3.1
Solve problems that involve enlarging or reducing a figure
MEQ: For all objectives pertaining to geometry, "to construct" means to draw a figure, using a ruler, compass, set square or protractor.
- 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
Solve problems involving isometric or similar figures in a Cartesian plane
|
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. |
Intermediate Objectives 3.2
- 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)
- 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.
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.
Intermediate Objective 3.3
Solve problems involving polygons
- 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: 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
solve problems involving circles
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.
Objective 4
Mathematical interpretation of phenomena involving chance
Intermediate Objectives 4.1
Calculate probabilities of the outcomes
- 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.
Intermediate Objective 4.2
Solve problems that involve calculating the probability of certain events during a random experiment
4.2 Intermediate Objectives
page 39
- 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 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.
| 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. |
FROM MEQ:
Students who have attained the current objective (w: 4) 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.