Secondary II Mathematics
year of algebra and proportionality
Electronic calculators can be used to aid exact
calculations with whole numbers and fractions without lessening skills
that would be required if no electronic calculators were
allowed.
The second year of high school mathematic may be called the year of
algebra. Students should learn how to use directly and indirectly, or
forwards and backwards the formulas that appear for perimeters and areas
of common shapes (squares, circles, triangles, trapezoids and
parallelograms) and formulas that appear in the discussion of
proportionality. Teachers should tell students the following:
Every formula high school mathematics will be used forwards and
backwards. For the backward use problems they are numerical and
algebraic solutions to see and master. Using the two phrases direct
and indirect use and/or forward and backwards use vocalizes
a hitherto silence theme which runs through the algebra in high school
and college mathematics.
Students in the first year or years 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. 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
in first and second year, high school mathematics.
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,
probability, may be be exploited to develop and reinforce
fraction skills and sense.
Again, in place of a complete thought-based development of mathematics,
each secondary course in mathematics should aim to show student how to
use rules and patterns, one at a time and in combination, one after
another another to arrive at numerical results or further rules and
patterns in a repeatable and reproducible manner. The ability to
combine rules and patterns to arrive at or justify further ones should
be presented in class even if not required of students to illustrate to
the thought-based development and connection of skills and concepts
where some rules and patterns are assumed (learnt by rote if need-be)
and others derived.
Aims in Brief
The second year course consists of the following
topics
-
Algebra: concept of a variable, solution of
linear equations in one unknown and solution of linear systems of
equations in essentially one unknown, plus algebraic
manipulations - including the difference between arithmetic and
algebraic (or symbolic) solutions of problem. Repeatedly inform
students each formula met will be used forwards and backwards, that is
directly and indirectly. See below.
-
Proportional Reasoning: ratios and rates,
solution of problems involving proportions and percents.
-
Probability: Random Experiments,
probability of a outcome, probability of an event. Here is an
opportunity to reinforce fraction skills and to show how decision or
outcome trees can count possibilities and/or yield
probabilities.
-
Synthetic Geometry: construction and
duplication of circles and regular polygons and circles; calculation of
perimeters, areas and angles in regular polygons and in or for sectors
of circles. Show when Side-Side-Side, Angle-Side-Angle and
Side-Angle-Side methods fail or do not work as expected. The latter
provides motivation for the parallel line postulate. See site section
on Euclidean
Geometry
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Transformation Geometry: Transformations
(reflection, translations, rotation, dilatations. This can done without
or with coordinates. If done with coordinates, consider the
introduction of both rectangular and polar coordinates. Then (radical
innovation for high school if not students in technical trades or adult
education) show students how to add and multiply points or arrows in
the plane without necessarily justifying the algebraically described,
arithmetic properties of Complex
Numbers.
The above description comes from a booklet describing
mathematics 116, 216 and 314 in Quebec. I am not sure of its origin. Quebec
teachers should see the comments below in item 10 on geometry in Quebec
English Instruction.
Lessons and Lesson Plans
Preparation for calculus prepares for all arts,
trades and disciplines involving mathematics. A guiding
focus for high school and college mathematics could be
preparation for calculus.
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.
Algebra Lesson Plans, the
first three steps, written earlier, compliment this page. The remaining
steps are for later years. If some of your students have not seen the
secondary I material
in Solving
Linear Equations with and then without Stick Diagrams to the
level of solving systems in essentially one unknown which require mastery
of the distributive law, you should include that material in this year of
algebra. Students in your class who have seen the material could be
assigned to cover the higher level material (that geared to later years) in
Solving Linear
Equations with and then without Stick Diagrams
Step 1. Algebra and Fraction Skills
The site page Fractions by Rote
may lead to efficient operational command of fraction skills, a
command sufficient for the second year of mathematics.
Comprehension can come or be emphasized later.
The first assignments could review arithmetic skills
with whole numbers and fractions. Students need to meet the message
that fraction sense and skills are important. Giving assignments and
correcting them in and out of class is recommended. If students object
to a review of fraction skills in class, give them the assignments. On
the return of the marked assignments, students will be interested in
what they did right or wrong, or inefficiently. Fraction lessons can
also woven into the return of marked assignments or the in-class
correction of the questions.
Step 2. Words before and Besides Symbols
Because arithmetic and algebraic expressions are better
seen and read silently in a glance than read aloud symbol by symbol,
mathematics has taken a non-verbal nature. A partial remedy comes from
what I called the first skill for algebra, namely our ability to talk
about and describe numbers without doing and without describing
arithmetic.
Show students how to talk about numbers and quantities and cover the
question of what is a variable, constant or parameter in class or in
assigned readings. The next reference provides a very good model for
this, a significant innovation that clarifies the use of words in
introducing algebra.
Reference: Chapter
9, Talking about numbers and quantities in Volume 2. Three Skills
for Algebra and the
Words Before Symbols (What is a variable) postscript what is a
variable in the online. Algebra Lesson
Plans, steps II, gives a longer account of words before
symbols.
3. Algebra and Formula Evaluation, the forward or
direct use of formulas
Show students how to develop (where feasible) and how to evaluate
formulas for perimeters and areas of triangles, rectangles, squares,
trapezoids and circles. Emphasize the word and algebraic (letter
and symbol) shorthand description of these calculations. Tell
students when they use these formulas, they should write out the formula,
substitute the value of numbers and quantities into the formulas and then
evaluate. Here you should show students how to carry units of measurement
through the calculations, and how to convert one unit of measurement of
length, mass and time into another unit of measurement. Then give
students rectangular area calculation problems with dimensions given in
different units (say centimeters and meters) to point out the need
to convert units before substitution into formulas or while
carrying them through calculations. In corporate into exercises or
examples illustrations of how areas given by a whole number of square
centimeters may be given by a mixed number of square meters. Formulas for
perimeters, areas and distance (time of journey times average speed)
demonstrate the first service of algebra to other subjects, the shorthand
description of calculations that may be done.
Reference: Volume
2, Three Skills for Algebra, Chapter 10:
4. Algebra: the indirect, inverse or backward use of formulas
Reference: Algebra Lesson Plans,
steps I to III
So far, students have seen how to use a formula directly to obtain a
perimeter, area and even a volume. From such examples, students
expect formulas to be used directly. Yet, all formulas given in high
school mathematics and science can and will be used directly and
indirectly. So in your explanation of formulas identify
the forward or direct use, and identify the backward or indirect use.
Students know how to compute the area of a rectangle
from given values for length and width, its dimensions. That
represents the forward use of a formula. The backward use of the
formula gives the area (the value of formula) and gives one of the
dimensions, the length or the width, of the rectangle. Finding the
missing dimension becomes the problem. That problem has arithmetic
solutions (one arithmetic solution for each time it is met). That
problem also has an algebraic solution - the formula that says that the
area divided by the given dimension yields the value of the missing
dimension. Details: How
to use the Rectangular Area formula backwards - algebraic viewpoint
only - add(?) a few numerical examples before or besides this
treatment in class.
In site Volume 2, Three Skills for Algebra, Chapter 14
covers Algebra versus
Arithmetic in using the Compound Interest Formula and Chapter
15 (first section) goes from numerical to algebraic solution for x
of linear equations ax+ b = c. The coverage in chapter
14 may be too advanced for most secondary II students (material is
secondary III or above) but it shows you the teacher or tutor what is
meant by numerical and algebraic solution methods. For calculations
simpler than the compound growth or interest formula, Your task is to
show students arithmetic solutions first and then point out how the
algebraic solution method solves many similar backward use problems all
at once, a power of algebraic shorthand method of reasoning with letters
and symbols. The theme of arithmetic versus algebraic solution methods
should continue through out rest of secondary II and above in order to
develop your students algebraic thinking and reasoning skills. A
few to several examples follow.
Two examples A and B follow. They can be presented to show students
arithmetic and then algebraic solutions for the problems involving the
indirect or backward use of formulas for perimeters and areas.
A. Forward and Backward use of formulas for perimeter and area of a
square.
A square with side of length x has area A = x2 and perimeter p
= 4x. Given the value of x, students can calculate area A and
perimeter p. Ask students to memorize the squares of whole numbers from 2
to 15.
From the value of perimeter p, students should obtain x from the backward
use of the formula p = 4x. Then they should obtain area A from the value
of x. Describing the foregoing backward calculation step by step in
shorthand algebraic notation would lead to a formula for A in terms of p,
and would illustrate the power of algebra to solve many backward
problems at once.
From the value of the area A = x2 , students would need the
concept of a square root to find the value of the side length x and
from that the value of perimeter p. Describing the foregoing backward
calculation step by step in shorthand algebraic notation would lead to a
formula for A in terms of x, and would illustrate the power of
algebra to solve many backward problems at once. Here is an opportunity (
to show students how to use the prime decomposition of a number to get an
exact representation of its square roots, and (ii) how to use a
calculator to obtain square roots exactly or approximately.
B. Forward and Backward use of formulas for perimeter and area of a
circle
The formulas for the perimeter and area of a circle can be used
similarly. Both perimeter and area are proportional to the radius and the
radius squared, respectively. But the number p
appears.
A circle with radius of length R has area A =
pR2 and perimeter
p = 2pR. Given
the value of R, students can calculate area A and perimeter p
directly.
From the value of perimeter p , students should obtain R
from the backward use of the formula p = 2pR. Then they should obtain area A from
the value of R. Describing the foregoing backward calculation
step by step in shorthand algebraic notation would lead to a formula
for A in terms of p, and would illustrate the power of algebra to
solve many backward problems at once.
Finally, from the value of the area A =
pR2 , students
would need the concept of a square root to find the value of the
radius R and from that the value of perimeter p. Describing the
foregoing backward calculation step by step in shorthand algebraic
notation would lead to a formula for A in terms of R and would again
illustrate the power of algebra to solve many backward problems at
once.
The direct and indirect use of Formulas for the area of triangles,
rectangles and trapezoids can also be met in class examples or exercises.
Reference: Algebra Lesson Plans,
steps I to III.
5. Proportional Reasoning - the algebraic perspective
Definition (1). A single quantity Y is proportional to a
second quantity X when and only when there is a non-zero
constant K such that Y = K X.
Here the direct use of Y = KX is to calculate the value
of Y from those of K and X. But the typically two step problem gives
the values (X1,
Y1) first, from which the value of the
proportionality K can be computed via a backward use of the formula.
And after K is known, the formula Y = K X can be used directly or
indirectly to compute Y or X respectively. The foregoing represents a
two step recipe for finding and then using the proportionality constant
K. The discussion of rates of changes can be included in this subject
along with development of algebraic computation skills with
units. See the site section Fractions, Ratios, Rates,
Proportions & Units.
Students may have met proportional reasoning unknowingly in the
following nine examples or situations. The proportionality can be
suggested by numerical examples or questions, and the graphing of one
quantity by another. Pick and choose the examples you like for
presentation in class, and then give the rest or further ones in
exercises.
- Average speed S for a journey is given by distance D traveled divided
by time T taken for the journey. Whence the distance traveled is
the product of speed and time.. That is D = ST. Here S is the
proportionality constant.
In the forward use of the formula D = ST, the values of
S and T are given and the value of D is computed. In the backward use,
the value of D and one of S and T are given. A typical two step
problem may say an object travels at a constant average speed over a
time interval of length T2 and ask how far the object has
traveled if the time T1 to travel an given distance
D1 is known. The first step of the solution computes
the proportionality constant K =S from the given values of (D, T) =
(D1,T1). The second step uses the formula D
= ST directly using T2 and the computed value of
S.
- The length S of arc of a circle of radius R subtended by a central
angle is proportional to the number of degrees N in the subtended
angle. The foregoing relation S = KN can be suggested via drawing
small angles and then considering multiples of them. The proportionality
constant K can be found from the fact that semi-perimeter (number of
degrees N = 180) is pR
where R is the radius of the circle. So
pR = K 180
Whence
and hence
is proportional to the product RN and hence jointly proportional to
both quantities N and R. Mastery of the latter formula means being
able to describe the suggestive geometric proportionality involved in
its derivation, and being able to use the formula
directly and indirectly, that is backwards and forwards. See Volume 2, Chapter
20. and express the calculation in chapter 20 in terms of degrees
only (not radians)
Definition A single quantity Z is jointly proportional
to two quantities X and Y when and only when there is a
non-zero constant K such that Z = K XY.
- The area A of a sector of a circle of radius R is proportional
to the number of degrees N in central angle.
The foregoing relation A = KN can be suggested via drawing
small angles and then considering multiples of them. The
proportionality constant K can be found from the fact that area
of a full circle, the case where the number of degrees N =
360 is pR2 . So
pR2 = K 360
Whence
and hence
|
S
|
= (
|
pR2
360
|
) N
|
= (
|
p
360
|
)N R2
|
is proportional to the product N R2 and hence jointly
proportional to the number of degrees N in the central angle and the
square R2 of the radius R. Mastery of the latter
formula means being able to describe the suggestive geometric
proportionality involved in its derivation, and being able to use the
formulas
directly and indirectly, that is backwards and forwards.
-
Division of Fractions Example: The question of how many times T
a line segment of length X unit lengths can be divided in line segments
of fixed length D unit lengths can be viewed from a proportionality
perspective. Geometric drawings suggest that T = KX.
To find K observe T = 1 when X = D. So the proportionality
equation K in T = K X satisfies 1 = K D. Hence K =
1/D. So T = (1/D)X.
In the case D = A/B, the relation 1 = K(A/B) implies K =
B/A and hence
T = (B/A)X = X (B/A).
The foregoing argument supports the rule that division by a fraction D
= (A/B) has the same effect as multiplying by its reciprocal B/A.
-
From Direct to Inverse Proportionality: The work W done in
many situations is jointly proportional to the number of workers N and
the interval of time T worked. That can be suggested by a few
well-posed questions. So
W = KNT.
That being said, this joint proportionality relationship can be used
backwards to find the value of K from values of N, T and W. Then with
the latter value of K it can be used directly or indirectly to find any
one of the quantitiies W, N and T when the other two are given or
implied by the circumstances at hand.
Now the algebraic view of the backward use of equation W = KNT.
implies the time T required to accomplish work W with N workers is
So the quantity T is proportional to W and inversely proportional to
N, jointly
Now the algebraic view of the backward use of equation W =
KNT. implies the time N of workers required to accomplish work W
in a time interval of length T s
So the number N required is proportional to W and inversely
proportional to time interval T worked.
The foregoing shows students who have mastered the algebraic
viewpoint of solving equations from earlier topics how inverse
proportionality relations may follow from direct proportionality
relations.
-
When are two simple or compound fractions equal? The proportionality
connection: The question of when a fraction C/D, compound or
not, has the same value as another fraction A/B, that is the
question of when
has a simple answer. Put
Then
and so the numerator
C = KD
is proportional to the denominator D and the proportionality constant
K = A/B.
-
Proportionality and change of units. Show students that
the number of centimeters in a length is proportional to the number of
meters, and vice versa with a proportionality constant k. Show students
that the number of square centimeters in a length is proportional to
the number of square meters, and vice versa with a proportionality
constant K2. The foregoing could lead to the discussion of
the relationship between lengths and areas in scale drawing, that is
plans and maps, and the actual lengths or areas. A further
generalization in exercises, if not in class, see next item,
might connect material use, volumes, areas and lengths in scale models,
larger or smaller, to unit or full scale models.
-
Proportionality and Map or Model Features. In maps and plans,
and 3D models, the scale of 1 to K implies lengths and distance
in the plan, map or model is one K-th (1/K) of the actual lengths or
distance, or that the latter are K times the former. In consequence,
the area of actual real regions or surface is =
K2 the area of the corresponding map, plan or 3D model
region In consequence, the volume of actual real solids are
K2 the volumes of the corresponding map, plan or 3D model
region or representation.
-
Binary and Multiple Ratios and Rates. A
discussion of binary ratios a:b and multiple ratios a:b:c appears
in the site section Fractions, Ratios, Rates,
Proportions & Units. The notation a:b and
a:b:c is archaic but still in common use. While I am quite content to
use ratio as an alternative term for fraction - all fractions are
ratios, but some ratios (those of parts to parts) are not
fractions. Something more needs to be said here. I would
emphasize the difference between the ratio of part to whole
(identifiable with a fraction) and the ratio of complementary or
overlapping parts of a whole (not identifiable with a simple fraction).
To make the distinction between ratios and fractions even
clearer, I would discuss, time permitting, multiple ratios and
multiple proportions. However, the discussion of ratios is, as
indicated, an archaic topic in mathematics courses, one that remains
due to later requirements and common conventions in society. To add to
the confusion, or lack of distinction between fractions and ratios, the
ratios of a pair of numbers, whole or not, may be called a fraction, a
habit I still keep. The site author needs further schooling in
this matter.
Reference: Fractions, Ratios, Rates,
Proportions & Units
6. Arithmetic Properties, Algebraically Described
In modern and post-modern mathematics
curricula, axioms (assumed patterns or properties) of real numbers were
given to provide a thought-based foundation for algebra. But
comprehension assumes students and teachers understand the algebraic
shorthand description of the patterns or properties. That is one
assumption too many for most students. We need to introduce the
algebraic codification or description of properties of fractions and
real numbers gradually.
Starting with fractions, we may
describe how arithmetic with fractions, that is the numerical operations
of addition, subtraction, multiplication, comparison, raising and
lowering terms, the rules for these operations, are described
algebraically with the aid of formulas or equations. For many numerical
examples, the correspondence between numbers in those examples and the
letters used in the formulas or equation need to be made explicit, so
that students see how to connect the algebraic description of a
calculation or the equality of two calculations with numbers. In
using each formula or equation, identify for many examples, the numerical
value or role of each letter in the formula or equation for the rule
describing the underlying calculation.
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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 the
calculation of numbers and quantities, named or not, and the
equality of calculations - when one calculation can replace another
because both give the same result.. The simplest context for
introducing algebra appears before or apart from the use of
negative numbers as lengths and areas are non-negative.
The
Logic & Algebra site area in discussing how a box
volume formula V = hA and V = h (WL) can be transformed
into each other illustrates and may introduce the notion of
equivalent expressions. The law applied here is A = WL is a
geometric law rather than an algebraic law (like the distributive
law). None, the idea that an expression represents a number
or quantity and that there may be more than one ways to compute
the number or quantity is key to the notion of
equivalence.
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 me 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, or Let y denote the
number or amount of money in this container. The geometric or
physical or monetary 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 or h is the
number of unit in its length, than it is for them to say let h
be a number. The introduction to algebra will
come more easily if letters are introduced as abbreviations or
shorthand for number or quantities, or their longer descriptions,
and algebra is done in the first instance with letters that have a
more concrete meaning than the phrases let x be integer or
suppose a, b and c are real numbers. The abstract
meaning of these phrases leaves student asking for and insisting
being given the numbers. They see not the need to describe
calculations in general. Letters with meaning are more
understandable even though they may denote an unknown or
unspecified quantity.
For more details, casual or rigorous, see the site
page on complex numbers and the
site sections on number theory and
analytic
geometry.
The question of Rigor
a compromise or two.
By making assumptions that different ways to count the number of
elements in a set produce the same result, and by making
assumptions that the area of a rectangle is equal to the sum of
areas of any covering by sub-rectangles with disjoint interiors,
and that the area of a rectangle or 3D box is given by the product
of its dimensions, we may geometrical suggest and imply arithmetic
properties of non-negative numbers. The foregoing provides a
chain of reason that is easily understood and repeated. Next we
assume that the algebraically described patterns and methods thus
implied (see Algebra Lesson
Plans, steps I to III, step III, and geometric implication for
algebra) also hold for both positive and negative numbers, that is
we may assume the field axioms for real numbers and more for ease
of exposition and comprehension. The foregoing leads to
suggestive lines of reason sufficient to give students an
operational understanding and mastery quickly in a manner that is
deductive in part but not in full. Operational mastery means
results are verifiable in the sense that they repeatable and
reproducible. Here-in lies a compromise between teaching by rote
with know-how, but no know-why offered, and and teaching with
a full development of skills and concepts in rigorous and
hence slow manner .
Compromises appear elsewhere in mathematics course design. If you
want more rigor in high school mathematics before calculus should
look at the exposition of right-triangle based trig and area
arguments in calculus for the limit as x --> 0 of sin(x)/x. They
should also look at formulas taught by rote that appear from time
to time, too frequently, in secondary mathematics be fore calculus.
Rigour itself may be left perhaps to advance courses in calculus
and beyond. That is food for thought.
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7. Probability
The calculation of probabilities provides an opportunity to reinforce the
calculator-free fraction skills of students. In modeling or
representation of random experiments, the calculation of
probability of a outcome a probability of an event can and should use
fractions and percentages alone and in products. The mastery of
exact and efficient arithmetic with fractions is prerequisite to algebra.
Aside: Event or decision generator trees can be
used to enumerate outcomes (possibilities) and calculate probabilities.
Similar trees (let us call them factor trees) can be use to calculate
all pairs of factors of a whole number from prime factorization. The
latter may be useful in factoring quadratics x2+bx+c in x
with integral coefficients b and c in later mathematics courses.
8. Transformation Geometry
Transformations (reflection, translations, rotation,
dilatations); construction of circles and regular polygons
and circles can all be described using rectangular and polar coordinate
systems or without coordinate system. More to come. ...
The introduction of complex numbers could
provide a setting for the underlying operations without an emphasis on
transformation geometry.
9. An Alternative Base for Senior High School Mathematics
Derivation of properties of real (and complex) numbers in site
section Number Theory
and this Complex Numbers pages departs
from earlier modern mathematics curricula, 1955 onward, which
assumed and then used the properties as axioms (patterns to follow). Pure
modern mathematics with its context free development and codification of
numbers and coordinate systems apart from the connection of the latter to
the physical space we habit is I suspect, a codification of the
empirically and thus inductively established skills and concepts.
Derivation of properties of real (and complex) numbers in site
section Number Theory
and this Complex Numbers pages provides
an inductive development of arithmetic using a mix of enumerative and
geometric assumptions. There-in lies an alternative to the modern
mathematics curricula of the 1955-80, and post-modern successors.
The modern mathematics curricula also departed from the
pure mathematics in drawing right triangles to introduce trig
functions. In other words, the modern mathematics curricula were
inconsistent with the pure mathematics they supposed echoed and also
inconsistent with the continuation and extension of the common
knowledge of decimal arithmetic and geometry. So course designers
today, site author included, are free to consider
alternatives.
While advanced university students may see modern
mathematics in a derived in a context-free manner, all students need an
inductive introduction to mathematics and its algebraic,
deductive, pattern-recognizing and -employing ways of reason for
the sake of quantitative disciplines outside of pure mathematics and
for the sake of acquiring the algebraic-deductive maturity for
understanding, if wanted, axiomatic codification of mathematics based
on set theory or an alternatives that may yet supplant it or
not.
The introduction to mathematics does not have to be context-free. It has
to be accessible, as as much as possible, empirically sound and
practical. The introduction of mathematics from counting to calculus
may aim to provide the algebraic-deductive maturity and the
context needed for the optional study of the or an axiomatic codification
of mathematics while supporting and extending, not constraining,
the common knowledge of decimals and geometry with and without
coordinates. The introduction of Complex
Numbers by showing how to add and multiply points in the plane with
the aid of rectangular and polar coordinates, and the statement of the
field axioms of complex numbers, with or with geometric justifications,
provides a shortcut for the development of senior high school
mathematics. The shortcut can be used where rote learning is emphasized
or where details of a geometric thought-based development of the axioms
(assumed patterns) is optional.
10. Quebec Mathematics
216 Geometry in English Schools
and its black hole for learners and teachers.
The MEQ objectives in mixing course objectives with
delivery instruction make course content unclear. So the Quebec English
progam for secondary II in practice is implied by the Minister of
Education approved textbook package (two books and a teacher's guide)
for secondary II instruction plus examples of past final
examinations.
Second year high school Math 216 in Quebec
introduces rectangular coordinates and then immediately drops the use
of coordinates to describe dilatations and (?) other geometric
transformations in a coordinate free manner. Frst reading of the
secondary II to IV coverage of this topic in Quebec high school texts,
those available in English, leaves me with a lack of understanding of
what is intended, that is with a doctorate in Mathematics.
There-in lies a black hole for mathematics education in Quebec, one for
further study and if possible removal. Watch this space for some
enlightment or black hole removal.
The Quebec English program for secondary II begins with the use of
rectangular coordinates in the plane to locate points and to identify
four quadrants.
The Quebec English program for secondary II then launches into a
coordinate-free discussion or introduction of dilatations. Each
dilatation has a fixed point, its centre, and a scale factor k which may
be positive or negative, and which may be greater than, equal to or less
than one in magnitude. Dilatation exercises can be used to imply that
collinear points go into collinear points, that lines and lines segments
are also lines and line segments respectively, that the distance between
a pair of image points are magnitude of k times the distance between
corresponding pre-image points; and that angles (at the intersection of
line segments) are preserved.
All the foregoing provides a framework for the definition of similarity
for triangles and more generally polygons in terms of corresponding
angles being equal and corresponding the lengths of corresponding sides
being proportional with the proportionality constant being given by the
magnitude of k. The foregoing implies that a dilatation is a similarity
transformation. However dilatations followed or preceded by rigid body
motion (translation, rotation or reflection) also yields a similarity
transformation. Maps and plans drawn to scale provide examples of
similarity transformations (mappings, correspondence rules) with positive
scale factors or ratios. Overhead projectors, telescopes and
microscopes may provide examples of dilatations or similarity mappings.
While there is a introduction of dilatations in the plane comes
immediately after the introduction of coordinates for the plane, the
Quebec English program does not introduce algebraic function notation
(x,y) ---> (kx, ky) to describe dilatations. Yet at the same
time, Quebec English program introduces function notation for
translations, reflections and rotations. There-in lies an inconsistency,
an out-of-sequence development of notation and concepts.
Teachers B ( More on Function Notation)
Single-Variable Function notation y = f(x) appears in secondary
IV, but before that (i) parameter dependent function notation for
translations T(a,b)(x,y) = (a+x, b+y) for reflections
across x-axis, y-axis and (?) the line y = x; and for rotations
through a few multiples of 45 degrees in appears in secondary II.
These functions have ordered pairs for values in place of real
numbers. At the secondary II and III level, The Quebec
mathematics program as seen in English instruction follows two
separates paths in its development of real-valued functions of a
real-numbers and point-valued transformations of points in the plane.
The separate threads here need to be united. Unification might
follow from the introduction of function notation for real and
ordered-pair value functions of one, two and more numbers in
secondary II, and numerical substitution exercises with this notation.
Function composition is not required. This introduction would also set
the stage for the introduction of parameteris in secondary IV
mathematics, if not before.
All the foregoing would be simpler if the scale or similarity factor k
was restricted to positive values. However Quebec program allows for
negative values in which the image of point.
Teachers C: Dilatations are determined by the
location of their centres or fixed points and their scale factors, a
proportionality constant. Secondary II mathematics may be characterized
by the forward and backward (direct and indirect) use first of
formulas and use second of proportionality
relations. The calculation of scale factors for dilatations from
information about distances between image and pre-image, and their
placement on the same side or opposite sides of the dilatation centre
should come in sequence after the second item as a re-enforcement of
it. There-in lies a correction or refinement for the current Quebec
English mathematics program for secondary II. Likewise, the
calculation of scale, similarity or proportionality factor in
similarity should come after the second item an another reinforcement
of it.
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