Mathematics 216
Unauthorized ideas
|
The second year course consists of the following
topics
- Algebra: Modes of representation, concept of a
variable, solution of problems of first degree with one unknown,
algebraic manipulations
- Proportional Reasoning: ratios and rates,
solution of problems involving proportions and percents.
- Probability: Random Experiments,
probability of a outcome, probability of an event.
- Geometry: Transformations (reflection,
translations, rotation, dilatations); construction of
circles and regular polygons and circles; calculation of perimeters,
areas and angles in regular polygons and in or for sectors of circles.
|
| The above description comes from a page describing
mathematics 116, 216 and 314. I am not sure of its origin. |
A guiding focus for high school and college
mathematics could be preparation for calculus.
Preparation for calculus prepares for all arts, trades and disciplines involving
mathematics.
The following site areas include ideas useful for
mathematics 116, 216 and 314.
Logic
& Algebra Solving
Linear Equations with Stick Diagrams, Fractions,
Ratios, Rates, Proportions & Units Euclidean
Geometry, Number
Theory.
Students in the first year of high school may come with a weak
to non-existence command of the times table (addition table too) and with a weak
to non-existence fraction sense and abilities. The most important service
of first year mathematics in high school is to consolidate fraction sense and
skills. See Solving
Linear Equations with Stick Diagrams if your students have a weak
command of fractions or if you want to develop algebraic thinking skills.
The ability to follow a multi-step process in a repeatable
and reproducible manner, modulo some accidents, is a sign that the students
master further multi-step operations in and outside of arithmetic. That is the
skill or intelligence we seek. Start emphasizing in it in arithmetic.
Calculators betray students by allowing them to skip a first example of a
multi-step process in which accuracy is demanded at each and every
step. The last topic, statistics, should be exploited as much as
possible to develop and reinforce fraction skills and sense.
See the more recent lesson plans Secondary
I - fractions & allied concepts (decimals, percentages) and Secondary
II - Algebra (arithmetic versus algebraic methods, backward use of
formulas and proportionality equations) before reading the further material
below.
Details or Rambling Suggestions
The concepts of variable is best explained in the Logic
& Algebra site area.
The site area Logic
& Algebra includes a discussion of three skills for algebra and of
what is a variable. It further compares and contrasts numerical solution of
problems with algebraic solutions in an delicate manner that may extend both
arithmetic and algebra skills. In mathematics, the description of
numbers and quantities as known, unknown, variable, constant, etc has been
confused with the use of letters or symbols to denote them. This site area
offers a greater clarity in the matter.
Solving
Linear Equations with Stick Diagrams may correspond to the subtopic
solution of problems of first degree with one unknown.
The site area Solving
Linear Equations with Stick Diagrams may be covered in full in
mathematics 116, 216 and 314 to introduce algebra and to build and
consolidate fraction sense and skills. The coverage of essentially one-unknown
problems points to an easier way to solve essentially one unknown word
problems. The inclusion here of triangular systems provides optional
exercises.
Algebraic manipulations can be represented by simplification of products and
fractions involving units instead of or besides variables. Such manipulations
serve the calculation of proportionality constants (if they be in mathematics
216).
The site area Fractions,
Ratios, Rates, Proportions & Units includes ideas for inclusion
in mathematics 116, 216 and 314. The area explains operation on
fractions in terms of line segments (more sticks). It further points out the
difference between multiple and binary ratios. The latter can be identified
with a fraction. Likewise binary, but not multiple proportion can
be identified with a fraction or a rate. The discussion of units in
computation, how to do calculations with them, with some or without all needs
to extended.
Extra: The site area Logic
& Algebra includes a math-free coverage in logic of implication
rules, chains of reason, longer chains of reason and islands and divisions of
knowledge.
- seeing the difference between one and two way implications (a.k.a
conditional and biconditional statements) may improve communication skills -
promote precision reading and writing,
- seeing how implication rules can be used in chains of reason and
longer chains of reason one at a time and one after another to arrive at a
conclusion may be compare to a multi-step process in arithmetic for arriving
at a results, an error in one step makes all the rest wrong.
- Seeing or declaring how rule-based knowledge may divide into separate
islands may imply that many islands can be mastered separately.
The Quebec government calls for cross-curricula themes could be served here
by having English and mathematics emphasize precision reading and writing.
In probability, the calculation of probabilities of events and
outcomes in mathematics 216 may serve the consolidation of fraction skills and
sense, efficient operations on fractions without a calculator. Anything
less is ill-advised.
In Geometry, dilatations are introduced apart from coordinates, and the
calculation of scale coefficient in it represents an exercise in
proportionality. This dilatation subtopic provides a first introduction to
similarity of triangles and polygons. Special needs students could skip this
subtopic as it not necessary for further learning except for a brief
re-appearance in secondary IV classes where again its presence is not necessary
for further learning. In contrast to dilatations, secondary II geometry provides
coordinate-based notation for reflection, translations, rotation. The
Quebec text in particular provides a four-variable, function-like notation to
represent translations without any prior explanation of function notation.
Implicit there is a sudden leap in the algebraic maturity demanded of
students. Dilatations could be dropped by the Quebec government for the
sake of a leaner and more effective curriculum.
In Geometry, the coverage of circles involves proportional reasoning
in the calculation of areas of sectors and their perimeters (minus the two
bounding radii). Here the area and arc-lengths in question are proportional to a
central angle. Deriving the formula from the existence of proportionality
constants and exact formulas for areas and circumference of circles points to
the algebraic reasoning skills student need.
In Geometry, some proportional reasoning may be used in the
description of regular polygons.
In Geometry, the use of letters a to z to denote lengths,
areas and volume provides students a more accessible view of equations
and calculations instead of the more abstract approach of using letters
to denote any number. The natural reaction to the latter is what number.
It is easier for students to understand x as a letter denoting a length
in a diagram than to say let x be number.
|
More Notes
- Fractions Continued: In previous years, students should have
learnt to add, multiply and subtract fractions exactly in an efficient
fashion. Efficient ways to Add and Multiply; how to divide fractions
by replacing division by multiplying by a reciprocals. Identification of
whole numbers or integers N with the fraction N/1, How division by a
fraction a/b is equivalent to multiplying by its reciprocal. How
1/(a/b) = b/a.
Site Reference: Fractions,
Ratios, Rates, Proportions & Units & Solving
Linear Equations with Stick Diagrams
- More Algebra: What is a variable, Solution of single
unknown and essentially one unknown problems, algebraic description of the
properties of rational numbers (associative, commutative and distributive
laws).
Site Reference: Solving
Linear Equations with Stick Diagrams & Three
Skills for Algebra
- The Algebra of Units. Distributive Law A- Units +
B Units = (A+B) Units for additions & subtractions involving like units
or quantities. How to multiply Quantities by each other and by pure
numbers. Use exponents to Express the Result. Fractions with units and
quantities - how to multiply, divide and simplify them. How to express
a fraction involving units with integer exponents and how to rewrite a
fraction so that only positive exponents appear in the numerator and
denominators. (Here is an alternative to practice with exponents using
monomials x y z and their ratios, an alternative immediately useful in
the representation and calculation of proportionality constants.)
Imperfect Site Reference: Site Area: Fractions,
Ratios, Rates, Proportions & Units
- Proportional Reasoning: How proportions, rates and ratios
involving pairs of numbers have fraction like properties. How triple,
quadruple or multiple proportions and ratios differ from fractions.
Calculation of proportionality constants and expressing them as fractions
involving units. Show how to find proportionality constants and use the
associated formulas backwards and forwards with specific numerical examples
(arithmetic solutions) and general algebraic examples. Illustrate with
constant Rates and speeds examples. For the example of joint
proportionality, introduce the example that work W done is proportional to
the time T or duration of work, the work group size, assuming each is
equally productive. That leads to the formula W = KNT and an inverse
proportionality relation between N and T when the work to be done is given.
Show how to do calculations with and without units by expressing
proportionality relationship in terms of numbers and/or in terms of amounts
and quantities.
Imperfect Site Reference: Fractions,
Ratios, Rates, Proportions & Units & Three
Skills for Algebra. In the latter, the first chapter on compound
interest compares and contrast arithmetic and algebraic methods for the
forward and backward use of formulas. That provides a model for your
in-class development of algebraic skills.
Note: An amount or quantity M is given by a number N
times a unit. So all equations involving the quantity M can be expressed in
terms of the number N in a way that the unit of measurement for the quantity
N disappears. The disappearance or cancellation of units is aide by
using a single scientific system of measurement for lengths, mass, time and
money, etc. So all equations or systems of equations can be expressed with
unit or without. That being said, in the discussion of the compound
interest formula, interest rates may be given as number while monetary
amount may use a unit of currency. So we have a hybrid system in which
the description of some quantities involve units while units have been
eliminated or dropped from the description of other quantities M by picking
a unit and using the corresponding pure number N where-ever a reference to M
would have occurred.
- Geometry and Algebra: Formulas for perimeters, areas and
angles in circles and regular polygons can be used to emphasize
proportionality and develop further skills in the forward and backward use
of formulas. For instance length of an arc and the area of a sector are both
proportional to the central angle - an observation that can be made by
discussion in class and translated into equations. The derivation of
formulas where possible further introduces algebra or mathematical reasoning
to students. That being said, in ordinary classes, one may present or
develop formulas and test students on their mastery of the formula and not
mastery of the derivation.
- Geometry: The use of ordered pairs [a,b] of rational or real
numbers to locate points in a plane relative to a pair of axes, one vertical
and one horizontal, provides motivation or a context the use of signs as
prefixes to unsigned numbers - only the first quadrant would have
coordinates if negative numbers were not used. That leads to a
numerical description of translations.
Site Reference: Analytic
Geometry & Complex
Numbers
An Alternate to well-trodden paths. Most schools do
not introduce polar coordinates (r,q) to locate
points in the plane. but doing so would permit the numerical description of
rotations about the origin (0,0) of a plane and of reflections about lines
through the plane. Moreover, if one shows students how to add points in the
plane using rectangular coordinates and how multiply points in the plane
using polar coordinates, they will become familiar with operations related
to translation and rotations, and beyond that see the law of signs for real
numbers as a consequence or special case of the rule for multiplying points
in the plane. The development of algebra and geometry could be further
extended by saying each point [a,b] in the plane represents a complex number
a+ib and beyond give the field properties of complex numbers. Proofs
are optional - best left to references. Each angle q
determines a point cis(q) on the unit circle, a
point to be identified with cos(q) + i sin(q)
after or besides the much later unit circle view and extension of
right-triangle defined trig functions. The foregoing view of addition and
multiplication in the plane would immediately answer many questions.
- Probability: Use counting principles and tree diagram to emphasize
and illustrate exact calculations with whole numbers and fractions.
Probability is not essential for further learning in mathematics at the high
school level, not a vital part of preparation for calculus, but the
opportunities it provides for verification and re-enforcement of fraction
sense and skills should not be missed.
- Dilatations and Proportionality in Geometry Here in Quebec, the
curriculum emphasizes dilatations of the plane. In the first instance a
dilatation is a radial expansion or contraction from a fixed point determine
by a scale factor K > 0. The limiting case K = 1 gives the identity
map - all points are left unmoved. The further cases where K < 0
radially reflect points through the fixed point, so the image of a point P
lies on a line through the fixed point on the half-line that does not
contain the pre-image point P. Student may find it fun to dilate the
image of their initials using several positive values of the scale
factor. From drawing examples, students may see the image
of three collinear point are collinear with no change of order, the
image of a line segment is a line segment, so the image of a polygon is a
polygon, and that angles are preserved. Student may further see that the
image of a polygon which divides the plane into parts is similar
polygon and so provides evidence for the later discussion of
similarity for triangles. The scale factor K provides a proportionality
constant between the image and pre-images of line segments. So students can
go back and forth between finding the scale factor from given data or
measurement and then using it. Examples can be chosen to emphasize exact
arithmetic. But calculators may also be used to speed up drawings. the
image of a house under dilatation (expansion, isometry or contraction with K
positive or negative) shows the effect of dilatations on orientation -
Orientation is preserved for positive scale factors and inverted for
negative scale factors. Dilatations can be used to explain how
overhead projectors work, perspective drawings with grid
diagrams, how lens worked by inverting images, and how maps can be viewed as
perspective drawings. The question of how areas are affected by dilatations
(multiplied by the square of the K factor) may also given or hinted at in
examples.
That being said, the rich or poor discussion of dilatations is not necessary
for future success in mathematics in Quebec, except for brief mention in
fourth year mathematics. The time current spent on dilatations in
Quebec would be better spent ensuring that students have perfected their
abilities with fraction - what they mean and how to do exact arithmetic with
them. The treatment of dilatations, a nice topic by itself, is
mostly a distraction from the mastery of core skills and concepts.
What theoretical or practical purpose does this topic serve? Dropping
the topic from the curriculum in secondary II and IV would allow students
and teachers to focus on more important matters.
Site Reference: Analytic
Geometry - See coordinate view of dilatations.
Students in the first year of high school may come with a weak to
non-existence command of the times table (addition table too) and with a weak to
non-existence fraction sense and abilities. The most important service of
first year mathematics in high school is to consolidate fraction sense and
skills. That is a prerequisite to algebra, geometry, trig and calculus. High
school mathematics literally becomes a waste of time if fraction sense and
skills are not consolidated and maintained in all years.
| |
Quebec English Mathematics Education
A farce is a farce is a farce.
Area Intro
Copy Right Matters
Curriculum Cuts
Intermediate Objectives
MEQ Objectives
116 Textbooks
116 Objectives
116 Check List
116 Suggestions
216 Objectives
216 Check List
216 Book Review
216 Nonsense or BullShit
216 Suggestions
314 Objectives
314 Check List
314 Suggestions
416 Objectives
416 Check List
416 Suggestions
436 Objectives
436 Checklist
436 Suggestions
436 Book Reviews
436 Nonsense in
514 Objectives
514 Suggestions
514 Book Reviews
536 Objectives
536 Suggestions
536 Book Reviews
Area pages represent an effort to follow and understand the objectives of the
1997-2005, the prior reform, and the
text books required and used 1997-2005. In retrospect, the objectives and texts
in question
are too incoherent, too full of nonsense, for rational comprehension and for
service as a base for the current reform. A farce is a farce,
is a farce
|