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Have your gifted students read logic
chapters 1 to 5 in online volume Three
Skills for Algebra for greater skills & confidence
in work
and study.
tell students to read notes and textbooks like a lawyer, so that no nuance, no
subtlety and no clause escapes their attention. |
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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
Tell students that 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. In Volume 2, Three
Skills for Algebra, a 4th skill for algebra appears in Chapter 14. It
provides a unifying theme for high school mathematics - equations and formulas
may be used forwards and backwards, directly and indirectly, numerically in arithmetic
solutions & with literals in algebraic solutions.
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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.
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Chapter 11: Decimal Notation and Decimal Fractions
The positional decimal representation of whole numbers is easily extended to
decimal fractions via the introduction of a decimal point and the introduction
of positions for tenths, one hundredths and one thousandths, and so on. A
decimal fraction by definition is a multiple of one tenth, one hundredth, and so
on. Here we can even speak of improper and proper decimal fractions. The
fraction [(p)/(10k)] is improper if p > 10k.
Finite decimal expansions (decimal fractions, proper or not) have a role in
the approximate recording of measurements. The uncertainty of the last digit
should be less than or equal half a unit if no error bound is specified.
Decimal Methods for Division
The Euclidean algorithm for division is represented in decimal computations.
The algorithm is discussed more generally in higher mathematics. Given a whole
number m, one can ask what is the largest multiple kn of another whole number n,
the divisor, which equals or does not exceed m. The Euclidean division algorithm
provides an answer. The computed number k is called the quotient and the
difference r=n-kn is called the remainder. The first examples can be
constructed so that the remainder is zero. And when the remainder is nonzero,
the division is not exact. The computation can be checked by computing the
product kn and the difference r=n-km is less than the divisor m
A modification of the Euclidean division algorithm yields the long division
process of decimal arithmetic for ratios of whole numbers and then for ratios of
other real numbers.
(1) To compute the decimal expansion of m/n where both m and n are whole
numbers, the decimal division algorithm can be continued until a zero
remainder is obtained or sufficient accuracy is obtained or until patience is
exhausted.
(2) To compute the decimal expansion of m/n where n is represented by
a decimal expansion with p places after the decimal point, an equivalent
fraction M/N with N a whole number can be obtained by setting M = 10pm
and N = 10pn This corresponds to shifting
the decimal point in the numerator and denominator.
(3) In general, the decimal expansion of m/n can be computed to any desired
precision by approximating m and n by decimal expansions of sufficient
precision.
Note the foregoing represents an exercise in rule and pattern-based reason.
Calculators can be used in place of teaching the division algorithm. But
teaching the division algorithm has one advantage. It can be recalled in the
discussion of polynomial division algorithm that the decimal expansion of a
whole number can be viewed as a special polynomial in powers of 10.
Decimal notation just provides a shorthand way of writing the polynomial
expansion.
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www.whyslopes.com
Volume 1B, Mathematics Curriculum Notes,
Foreword + Chapters 1 to 10 + 12
Book Entrance
Inductive Principles
1 Introduction
2 For & Against Math
3 Algebraic Thought
4 Why Slopes & SQRT of -1
5 Books & Articles to Read
6 Unruly Origins of Algebra
6. Axiomatic Civilization
7 Geometry, 2 Ways
8 Modern Instruction
9 The Two Ends
10 The Transition
10 Explaining Logic
10 Explaining Algebra
10 Why Sets in Math.
12 Four Phases
Links
Chapter 11: Primary School Mathematics
11 Primary Math
11 Cue Cards
11 Counting
11 Decimals - Addition
11 Decimals -Times
11 Decimals & Subtraction
11 Fractions and Division
11 Notational Conflict
11 Reciprocals Etc
11 Decimals - Ratios
11 Size Comparison
11 Numbers, +ve or -ve
11 Rename < Sign
11 Complex Numbers
Will provide an alternative to Chapter 11 later, most likely in the Parent's
Area: Help Your Child or Teen
Learn
Most students in high school are not heading for calculus,
but most topics in high school mathematics are present due to calculus.
Preparation for calculus demands their coverage at full strength.
See too, this site 55+, Math
Education Essays. Site areas and pages provide pieces of the a Mathematics
Education, Jigsaw Puzzle, in formation.
-Inductive
principles for course design & delivery require a clear
description of where and how skills and concepts may rest on earlier ones, so
that difficulties may be explained and remedied by looking for what was
missed or forgotten in earlier studies.
Mathematics is a demanding subject. All errors in notation
and comprehension need to be identified and corrected. In
reading, spelling and writing, students have to learn all the letters in the
alphabet, not just some. and memorize spelling. Anything less implies
difficulty.
Likewise in mathematics, students have to master key skills
and concepts, one at a time and one after another. Anything less implies
difficulty.
Modern mathematics curricula introduced an inconsistency
into course design and delivery. They did not sanction the use of decimals nor
the use of diagrams in skill and concept development but decimal arithmetic
and diagrams are needed for student comprehension and for an operational
mastery of quantitative skills. That implies the need for an mixed-math
curricula based on a systematic development of operational skills, sufficient
for applications and sufficient to provide a base & context
for the very optional study of pure mathematis.
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