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HIP,
HIP, HIP, Hooray
YOU are better than YOU think. Show yourself how:
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Logic
chapters 1 to 5 re- appear not in sequence, as is or longer,
in Volume 1A, Pattern Based
Reason, Bon Appetite.
Logic
Mastery
Amazing, Amusing, Amorous, Delicious, Delightful, Edifying,
Strengthening Elixir.
It eases work & learning difficulties Makes the hard easier. Opens eyes.
Leads to greater precision.
in reading and
writing
Logic
mastery makes the hard, easier. Logic
mastery leads to better, stronger and richer comprehension. Logic
mastery improves reading and writing. Logic
mastery ease learning difficulties. Logic
mastery gives a headstart. In sum, logic
mastery will develops critical thinking, improve reading and writing,
and give a firmer base for work and studies at many levels. Good luck.
After logic,
(a) continue reading Three
Skills for Algebra, chapters 8 to 14 and do so alongside site area on solving
liinear2007 Equations ; or (b) see this calculus
starter lesson and Volume 3, Why
Slopes & More Math, chapters 2 to 6;
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Caution: Site advice is approximately
correct, for some circumstances, not all. That leaves room for thought |
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What may be learnt and when depends on how skills
and concepts are developed. Making the hard easier and clearer will allow
earlier & richer development of skills and concepts.
Try the Twiddla
Whiteboard. In principle, it allows
to people to draw and chat together online on a copy of this webpage or a clean
sheet. The chat may be via text or audio. Visit www.twiddla.com
to set up whiteboards to work with the webpage of your choice.
For online automated help in senior high school maths & calculus,
visit quickmath.com For Automatic
Calculus and Algebra Help with derivatives, integrals, graphs, linear equations,
matrix algebra, visit calc101.com
With overlap, each site quickmath
& calc101offers a different range of
services, some free, some not, all based on webmathematica. Good luck.
| |
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
- 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. |
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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