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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
liinear 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.
| |
Intermediate Objectives
The global objectives of the courses are described at length in
the Quebec government document found nn this pdf
file. The intermediate objectives extracted below and annotated
give the details necessary to identify most course content.
The correspondence between these intermediate
objectives, current practices and the MEQ approved textbook for English and
French language instruction is not clear - more thought is required. Has
course content drifted away from these objectives under the influence of the
MEQ approved textbooks and evolution of final examinations?
Watch for links, overlap or duplication between the objectives involving
similarity and isometry.
Some question or vagueness remain on what should be assumed and
what should be developed or derived.
1.1 Objectives
Analyze functions (situations involving them)
- use symbols to represent a situation involving a function, indicating a
source set, a target set and a rule of correspondence.
- draw the Cartesian coordinate graph representing a situation involving a
function, given an equivalent verbal description, table of values or rule of
correspondence. prepare the table of values for a situation involving a
function, given an equivalent verbal description, rule of correspondence or
Cartesian coordinate graph.
- describe the properties of a Cartesian coordinate graph representing a
function (these properties are listed below).
- increasing or decreasing function
- sign
- rate of change
- axes of symmetry, if any -
maxima or minima, if any
- x-intercept(s) (zeros) IF ANY
- y-intercept
- domain and range
- determine the relationships between changes in the parameters of the rule
of correspondence of a function and changes in the equivalent Cartesian
coordinate graph.
1.2 Objectives
transform an algebraic expression into an equivalent expression
- apply the theory of exponents in transforming algebraic expressions.
- perform operations (addition, subtraction, multiplication division and
exponentiation) on algebraic expressions and on polynomials in particular.
- factor a given polynomial. transform rational algebraic expressions by
dividing or factoring them.
1.3 Objectives
analyze linear and quadratic polynomials (polynomials of degree 0 or 1)
- draw the Cartesian coordinate graph (a straight line) of a real polynomial
function of degree 0 or 1 (w: in plain language: graph y = ax + b where a is
real, zero or not).
- given a linear function y = ax + b the following information: its rate of
change, its x-intercept (zero), its y-intercept, its domain and range, its
sign, whether it is constant, increasing or decreasing, and the member of
its domain associated with a given image.
- graph quadratics and call the graph a parabola.
- for a quadratic y =ax2 + bx + c, determine its extreme (vertex
of the parabola), its zeros (if any), the sum and product of the zeros, its
y-intercept, its domain and range, the intervals within which it is
increasing and decreasing, its sign, and the member(s) of its domain
associated with a given image. use algebra to convert the general form f ( x
) = ax2 + bx + c , into the standard form f ( x ) = a ( x - h ) 2
+ k and vice versa.
- determine the relationships between changes in the parameters of the rule
of correspondence for linear functions y = ax + b and quadratics y = a ( x -
h ) 2 + k and changes in the equivalent Cartesian coordinate
graph.
- determine the coefficients in the equation y =a x+ b of a straight line,
given the slope of that line and a point on that line or given two points on
that line.
- determine the coefficients in the equation y = ax2 + bx + c or
y = a ( x - h ) 2 + k, of a quadratic given the vertex of the
associated parabola and another point on that parabola or given its zeros
and another point
- graph the sum, difference and product of constant, linear, quadratic
polynomial functions, given the graph or the rule of correspondence of each
of these functions.
1.4 Intermediate Objectives
Solve problems by solving linear equations in two unknowns
- represent a situation by a system of two first-degree equations in two
variables.
- solve a system of two first-degree equations in two variables by graphing
it.
- solve a system of two first-degree equations in two variables
algebraically.
- represent a situation by a system of two equations, one being of the first
degree in two variables and the other being of the second degree in two
variables.
- use a graph to solve a system of two equations, one being of the first
degree in two variables and the other being of the second degree in two
variables.
- use algebra to solve a system of two equations, one being of the first
degree in two variables and the other being of the second degree in two
variables.
1.5 Objectives
Solve problems in analytic geometry
- determine the slope of a straight line that passes through two given
points.
- determine the slope, x-intercept and y-intercept of a straight line from a
given equation.
- draw a straight line in a Cartesian plane, given the slope of the line and
a point on the line.
- determine the equation of a straight line, given any of the following
combinations: its slope and a point on the line, two points on the line, the
x-intercept and y-intercept, or a point on the line and the equation of a
parallel or perpendicular line.
- transform the equation of a straight line algebraically. determine if two
straight lines intersect, or if they are perpendicular, parallel and
distinct, or parallel and coincident by comparing their parameters and
equations.
- determine the distance between two points or between a point and a
straight line. determine the coordinates of the point of division of a
segment,
- given the coordinates of its endpoints and other relevant data.
- determine the area and the perimeter of polygons, given the coordinates of
the vertices.
- prove propositions using analytic geometry. See Appendix 2
Appendix 2: Deductive Reasoning in Analytic Geometry
The students are assumed to have the following knowledge and skills:
- The formula for finding the distance between two points (based on
the Pythagorean theorem)
- The formula for calculating the distance between a point and a
straight line
- The formula or a method for finding the coordinates of the point of
division of a segment
- The general form of the equation of a straight line
- The functional form of the equation of a straight line (slope
intercept form)
- The symmetric form of the equation of a straight line
- The role of the parameters in the various forms of the equation of a
straight line (general, functional and standard forms)
The following propositions are considered to be true:
- The x- and y-axes are orthogonal.
- Two straight lines that are not parallel to the y-axis are parallel
if and only if their slopes are equal.
- Two straight lines that are not parallel to the y-axis are
perpendicular if and only if their slopes are negative reciprocals.
- A system of axes can always be arranged so that two consecutive
vertices of a given polygon are on the x-axis, one of these vertices
being located at the origin.
The students can prove the following propositions using the information
above.
- The segment joining the midpoints of two sides of a triangle is
parallel to the third side and its length is one-half the length of
the third side.
- The segment joining the midpoints of the non-parallel sides of a
trapezoid is parallel to the bases and its length is one-half the sum
of the lengths of the bases.
- The segments joining the midpoints of the opposite sides of a
quadrilateral and the segment joining the midpoints of the diagonals
are concurrent in a point that is the midpoint of each of these
segments.
- A segment connecting a vertex of a parallelogram to the midpoint of
one of the non-adjacent sides intersects the opposite diagonal at a
point that divides both that segment and the diagonal in the ratio of
1 : 2.
- The midpoint of the hypotenuse of a right triangle is equidistant
from the three vertices.
- The midpoints of the sides of any quadrilateral are the vertices of
a parallelogram.
- The three perpendicular bisectors of the sides of a triangle are
concurrent in a point that is equidistant from the three vertices.
- The three medians of a triangle are concurrent and trisect one
another at the point of concurrency.
- In any triangle, if a is the length of a side opposite an acute
angle, if b and c are the lengths of the other two sides and if AH is
the length of the projection of side c onto side b, then the following
relationship is true: a2 = b2 + c2 -
2 b(AH)
(w) See the cosine law.
- In any triangle, the sum of the squares of the lengths of the
medians is equal to three-quarters of the sum of the squares of the
lengths of the sides.
- If ABCD is a parallelogram and if E is the midpoint of side AD, F is
the midpoint of side AB, G is the midpoint of side BC and H is the
midpoint of side CD, then the segments AH, FC, BE, and DG intersect to
form another parallelogram.
- The sum of the squares of the distances between a given point and
two opposite vertices of a rectangle is equal to the sum of the
squares of the distances between this point and the other two vertices
of the rectangle.
Of course, other geometric propositions can be proven. |
2.1 Intermediate Objectives
Solve problems using concepts of similarity, isometry and equivalence
- define isometries and similarity transformations by means of geometric
transformations and their composites.
- accurately describe the geometric transformation or the simplest composite
of geometric transformations that maps one isometric or similar plane figure
onto another, given two isometric or similar plane figures.
- characterize isometric, similar or equivalent plane figures. determine the
properties (e.g. measures of angles and sides, perimeters, areas) of
isometric, similar or equivalent plane figures.
- state the minimum conditions required for two triangles to be isometric or
similar. characterize solids that are isometric, similar, equivalent or
equal in total surface area.
- determine the properties (e.g. measures of angles and sides, perimeters,
areas, volumes) of solids that are isometric, similar, equivalent, or equal
in surface area.
- determine certain measurements of similar right solids or spheres, given
other measurements of these figures, a ratio (of lengths, of surface areas
or of volumes), or data that can be used to find this ratio.
- justify an assertion used in solving a problem. See Appendix 3.
2.2 Intermediate Objectives
Solve problems using trigonometric ratios
- calculate the measure of a side or an angle in a right triangle,
- given relevant data and using a trigonometric ratio.
- calculate the measure of a side or an angle in a triangle, given relevant
data and using the law of sines or the law of cosines.
- justify an assertion used in solving a problem. See Appendix 3
Appendix 3, Annotated
Principles of Geometry Introduced in Mathematics 436
My comments are prefaced by the w colon combination
(W:). Items 1 to 9 with item 12 form a logical base for deductive
geometry and trig. Appendix 2 above points to deductive geometry in the
framework of analytic geometry. That being said, in a very confusing
manner, our MEQ approved textbook for English language instruction gives
an awful and different development built on properties of rigid body
motions and dilatations, a development that is difficult if not impossible
to follow as written. Should we correct or refine that development or go
with the alternative development in the site area on Euclidean
Geometry, The latter provides a lean and minimal development
focused on the needs of analytic geometry. So previous objections
that Euclidean geometry is too hard for students may be eased or
eliminated. I will be looking at the intermediate objectives in
mathematics 116, 216 and 314 to try to see what was intended and what is
feasible. Not all is clear. Lack of time may preclude us (me or you)
following some preferences.
- If a transversal intersects two parallel lines then: - the alternate
interior angles are isometric; - the alternate exterior angles are
isometric; - the corresponding angles are isometric. W:
For Proof See site area on Euclidean
Geometry
- If two corresponding (or alternate interior or alternate exterior)
angles are isometric, then they are formed by two parallel lines and a
transversal. W: For Proof See site area on Euclidean
Geometry
- The corresponding parts of isometric plane figures or solids are
equal in measure.
W: With regrets, I would assume this and not prove:
- Plane figures or solids are isometric if and only if there is an
isometry that maps one figure onto the other. W: I understand
the if part, but do not now how to prove the only if part.
- If the corresponding sides of two triangles are isometric, then the
triangles are isometric. W: See site area on Euclidean
Geometry (SSS isometry postulate)
- If two sides and the contained angle of one triangle and the
corresponding two sides and contained angle of another triangle are
isometric, then the triangles are isometric. W:
See site area on Euclidean
Geometry (SAS isometry postulate)
- If two angles and a side of one triangle and two angles and the
corresponding side of another triangle are isometric, then the
triangles are isometric. W: See site area on Euclidean
Geometry (ASA isometry postulate)
- Transversals intersected by parallel lines are divided into segments
of proportional lengths. W: The proof of this could
be neet - depends on ....
- The line segment joining the midpoints of two sides of a triangle is
parallel to the third side and its length is one-half the length of
the third side. W: The proof of this could be neet -
depends on ....
- Any straight line that intersects two sides of a triangle and is
parallel to the third side forms a smaller triangle similar to the
larger triangle. W: The proof of this could be neet
- depends on ....
- Plane figures or solids are similar if and only if there is a
similarity transformation that maps one figure onto the other. W:
I understand the if part, but do not now how to prove the only if part
- see the corresponding assertion above with isometry in place of
similarity.
- If two angles of one triangle and the two angles of another triangle
are isometric, then the triangles are similar. W:
AA similarity postulate
- If the lengths of the corresponding sides of two triangles are in
proportion, then the triangles are similar. W:
Another Similarity Postulate kS-kS-kS
- If the lengths of two sides of one triangle are proportional to the
lengths of two sides of another triangle and the contained angles are
isometric, then the triangles are similar. W:
Another Similarity Postulate kS-A-kS
- In similar plane figures or solids:
- the ratio between the measures of the corresponding angles is 1; W:
That is, Corresponding angles have the same measure.
- the ratio between the lengths of the corresponding segments is equal
to the ratio k between the lengths of the corresponding sides;
- the ratio of the areas is equal to the square k2 of the
ratio between the lengths of the corresponding sides;
- the ratio of the volumes is equal to the cube k3 of the
ratio between the lengths of the corresponding sides.
- Plane figures or solids with a scale factor of 1 are isometric.
W: Obvious consequence of previous:
- In a right triangle, the length of the side opposite a 30o
angle is equal to half the length of the hypotenuse. W:
A consequence of the Pythagorean Theorem. That covers the 30-60-90
right triangle. What about the isoceles 45-45 -90 right triangle?
- The law of sines:
The lengths of the sides of any triangle are
proportional to the sines of the angles opposite these sides.
|
a
sin A |
= |
b
sin B |
= |
c
sin C |
(w): the common value of the these three ratios gives
the proportionality constant.
(w): Personal Preference: The proof of the law of sines is an
exercise in geometry that students should meet and master if they can,
time permitting.
- The law of cosines:
The square of the length of a side of any
triangle is equal to the sum of the squares of the lengths of the other
two sides minus twice the product of the lengths of the other two sides
multiplied by the cosine of the contained angle.
a2 = b2 + c2 - 2 bc cos A
b2 = c2 + a 2 - 2 ac cos B
c2 = a2 + b2 - 2 ab cos C
w): Because some of the angles in the cosine and sine laws may be
obtuse, I would prefer to introduce trig functions using the unit circle
for angles from 0 to 360 degrees, and then show how similarity implies
these ratios can be calculated using trig ratios in right triangles when
the angles are acute, that is, from > 0 and < 90 degrees. This
or the introduction of polar coordinates a
simple and dismystifying geometric development of complex numbers.
in a future variant of mathematics 436 or 536..
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3.1 Intermediate Objectives
Solve problems using measures of position
- distinguish between measures of position, measures of central tendency and
measures of dispersion.
- assign a quintile or a percentile rank to a data value, if necessary.
- determine the data value(s) that are assigned a quintile rank or a
percentile rank.
- use measures of position to compare data.
- construct a box-and-whisker plot.
- interpret a box-and-whisker plot.
- find qualitative information about the dispersion of the data in a
one-variable distribution, using measures of position and measures of
central tendency.
3.2 Intermediate Objectives
Solve problems that involve gathering data
- distinguish between a sample and a population.
- justify the decision to prepare a census, a poll or a study to obtain
information.
- describe the characteristics of a representative sample of a given
population.
- choose an appropriate sampling method when gathering information.
- determine the possible sources of bias during the data gathering process.
- compare two samples from the same population.
The following words appear before the intermediates objectives.
... be able to assess the reliability of the sample and the relevance of
the data used in making predictions about a given population. To determine if
the data is relevant, one must ascertain whether or not it is representative.
If the initial hypothesis is appropriate, the sample should provide an
accurate picture of the population under study. The students should check the
size of the sample and the data-gathering methods to ensure that a study is as
unbiased and error-free as possible. The students already know several ways of
describing data graphically or numerically. They must learn to follow certain
principles in processing data to ensure that they draw appropriate
conclusions.
They may clarify and provide a context for the intermediate objectives.
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www.whyslopes.com
Lesson & Lesson Plans for
Sec IV (Maths 436)
a reference for learning and teaching functions, polynomials, solving linear
systems,
powers + exponents + bases + radicals (roots) , quadratic formulas, equations
of straight lines
1A. Master Logic
1B. Problems Solving Method
2A Solve Linear Equations i
2B.Solve Linear Equation II
2C Use Equal Sign Properly
2D. Perfect Arithmetic Skills
3 Words & Symbols
3 Goals to Set for Students
4 Use Equations Backwardly
5. Master Functions & Relations
6. Exponents & Radicals I
6 Exponents & Radicals II
7. Straight Lines
8. Polynomials (x,/,+/-)
9. Quadratics
10 Prove it
13 Similarity Scale Factors
12 Trig & Triangles
14 Statistics
MEQ Intermediate Objectives
Remarks for Teachers
Sit down and study - no one else can do that for you.
Advice and Directions
What to do in School & Why
How
to Study Maths & Why
Preparing
for Science
Good News: If you can learn to follow a multi-step
methods in any subject precisely, you should be able to do so in other
subjects, as well. Hint: Start with arithmetic
Words Before Symbols:
What is a Variable?
Level: Secondary II to VI, or Grades 7 to 12)
Introduction
Variation between Examples
Variation of Letters
A letter denotes a variable
Cases of Double Variation
Three Notions of a Variable
Constants, Parameters
& Variables
Talking about numbers
Dependent
or Independent
Variable, a Matter of Choice
Complex number starter lesson
Arithmetic Videos
Fractions
Primes
Greatest Common Divisors
Least Common Multiples
Square Root Simplification
Arithmetic Videos
Decimal Addition Methods
Decimal
Subtraction Methods
Decimal
Multiplication Methods
Decimal Division Methods
Fraction
Starter Lesson
(simplify, multiply, divide &
then add or subtract)
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