Decimal Lesson - Reference
Counting & Addition (8
lessons)
Comparison
to Subtraction (9 lessons)
Multiplication
( 11 lessons)
Long Division
(12 lessons)
Decimals and Primes (8 lessons)
-Primes
& Composites
-Primes
Factorization
-Greatest Common Divisors & Multiples.
-Prime Factorization Aids (Learn how to find factors
quickly)
-Prime Factorization Examples
-Counting
& Generating. Factors.
-Divisibility
Rules and Remainders for Division by 2, 3, 5, 9 and 11.
Integers (12 lessons)
Intro to Signed Numbers
Fractions
(< 20 lessons) Essential
Skills
& Concepts
Ratios & Fractions (3 lessons):
Similarities & Differences
Units
in calculations
Fractions
with Units
Solving Linear Equations
- in one unknown.
Intro
with stick diagrams?
the
normal way
& with good nttn.
(the nttn that reappears in Gaussian Elimination. |
-in more unknowns:
simultaneous equations essentially one unknown.
the let algebra do the work view of word
problems.
- still in more unknowns:
Gaussian Elimination via substitution,
by equality
or comparison, by
operations on equations.
Words before symbols:
See if U like the lengthy chapters
8 to 12 in Volume 2, Three
Skills for Algebra
What
is a Variable. The answer here is a simple prequel to the modern mathematics
viewpoint.
|
First, every rule & pattern U meet in math, logic &
science will be used forwards and backwards. Get a head start with this theme by
reading Chapter
14 in Three
Skills for Algebra. Second, in the study
of Proportionality
Relations
(3 dense lessons here) finding the proportionality constant
gives an initial backward use of the proportionality
formula. |
| Talking about words before symbols and the
forward and backward use of formulas gives words to make
algebra simpler & clearer. |
|
If you can not
read or write precisely, you will have difficulty in following
instructions. One wordy remedy is given by chapters
2 to 5 in Three
Skills for Algebra. Where
does Logic or a geometric model for reason Appear in Mathematics?
The answer lies in Euclidean-Geometry
In North America, Euclidean Geometry
disappeared from high school mathematics as it was too
hard. The light treatment here is a possible remedy.
The
Pythagorean Theorem. Chapter
17 from in Three
Skills for Algebra uses algebra and geometry to
show why the Pythagorean equation for right
triangles holds. Its forward and backward use is
common exercise.. At a more theoretical
level, the Pythagorean theorem leads the discovery that not all
lengths can be fractional multiples of a unit length. That
geometrically implies a need for and even existence of irrational numbers.
Analytic Geometry:
Common Practices with Maps
and Plans drawn to scale give coordinate-dependent base
for senior high school development of similarity, trig,
vectors and straight lines.
Complex Numbers: This lesson on Complex
Numbers draws on Euclidean and Analytic geometry.
Sbortcuts simplifiy trig identities, the cosine law;
and trig formulas for 2D dot- and
cross-products.
|
F. Logarithms, Exponentials,
Roots & Powers
|
Logarithms, exponentials, rational and real powers for secondary
students. This
complete Operational Viewpoint. (Sufficient for the
precalculus forward and backward use of compound growth and decay
formulas in biology, physics, chemistry, personal finance,
and calculus. To learn more, if you study calculus, see chapter 19 of Volume 3,
Why Slopes and More.Math
|
In Volume 2, Three Skills for Algebra, chapters
- Geometric
Sums Etc,
- Notation
For Sums,
- Personal
Money Maths and
- Some
Finite Mathematics
identify methods useful in money computations,
methods needed for calculus. Your teachers or other writer may
present the same ideas with greater clarity and detail - A site to
do.
|
H.
Polynomial & Quadratics
|
Analytic
Geometry:
- Slopes
and Lines
- Take 1. Take 2 appears in site section Maps
and Plans. Two views are better than one. I
may combine them later.
-In my school days, slopes appeared year after year. This Why
Slopes calculus preview on graphs of functions y = f(x)
explains why. Enjoy.
Quadratics and
Polynomials:
- Operations
on Polynomials: Meet a light and
ultraquick geometric introduction to multiplication,
addition and subtraction of polynomials. Then see how the
foregoing combine to permit long division of
polynomials.
Compare
Fractions
with Units. Enrichment: A Plus: The Geometric
introduction here gives or is almost identical to a
justification for column methods in decimal
arithmetic.
Geometric Derivation of the Quadratic
Formula: The account here gives a
starter lesson for the more algebraically harder
geometric-free derivation. If you study physics, chemistry
or trigonometry, you will need to know about quadratics,
their factorization and the quadratic formula.
Technical Value:
The study of polynomials high school mathematics has
technical value as part of the senior high school
mathematics preparation for calculus. This simple
account of Why
Factor Polynomials (Chapters 2 to 6 in Volume 3 .Why.Slopes.&.More.Math.)
will give a context for the study of polynomials, their factorization, and sign analysis of functions, all in
a way that should improve your algebraic thinking and
reasoning skills.
|
Vectors
in the Plane (2 simple lessons)
- Navigation
with vectors or arrows
- Sum
of Motions
- more lessons to be added later.
Operations on movement or vectors along the line and in the
plane have value in mathematics in defining and implying the
properties of real and complex numbers before the assumption
of those properties as axioms. Vectors and their
properties appear in physics, its mathematical description
and formulation.
- Functions
- Forwards & Backwards. Here is a full technical
reference (24 lessons) for use in a calculus or precalculus
course as needed. In it, the set viewpoint of functions expression of
modern pure mathematics.
comes from the set-based codification and
In the mathematics
education reforms of the 1960s in North America, primary and
secondary school mathematics were expressed in terms of
sets. That expression has now retreated from primary and
secondary school texts. But it still lingers on, and can be
very useful, a source of clarity and precision, in the
situations where it should be retained: Counting with the
aid of sets and functions; the description of functions; the
high school account of probability theory; and in the
discussion or illustration of ideas in logic.
|
J.
Pre-Calculus Skill Check |
-
Arithmetic
Skill Check. In the calculus courses I taught 1983-89,
too many students had weak skills in arithmetic. I would
give and carefully correct these exercises to tell students
what they needed to review and master.
- All the skills and concepts in
Chapters 1 to 24 or Volume 2, Three
Skills for Algebra: Look for those you do not understand and
fill the gaps. Do so quickly while balancing this advice
with your other duties. Good luck.
|
