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YOU are better than YOU think. Show yourself
how:
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Read logic
chapters 1 to 5 in online volume Three
Skills for Algebra for greater skills & confidence
in work
and study.
Learn to read notes and textbooks like a lawyer, so that no nuance, no
subtlety and no clause escapes your 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
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.
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Chapter 11: Counting
In words, if not on paper, students may learn to count from 1 to a high single
digit, then 1 to two digits, and then 1 to beyond 100. Here they may learn the
pattern of counting aloud, before or besides making marks or symbols on paper.
The idea that counting can go on forever, not stop, appears. There is no largest
(whole) number. The questions of how far one can count or what is the largest
number that can be counted may lead to this realization.
Writing the numbers on paper introduces decimal notation, and the concept
that the value of digit in the decimal representation of a number is determined
by its position. Thus the student becomes familiar with the unit, tens, hundreds
and thousands positions in the decimal coding and expression of whole numbers.
Whole numbers can be used to count the number of objects in a group, for
instance the number of feet (or meters) between two points along a measuring
tape or the number of marbles in a bag. Numbers written on paper and the
rules for arithmetic represent the first symbolic manipulations that appear in
mathematics.
Multiplication and addition are present in the decimal system. In particular,
to envision the number 34, we can envision three groups of 10 squares counted
together with another 4 squares. The image of a rectangle covered by 4 rows,
each consisting of ten squares, leads to the notion of area – exactly how many
squares are needed to cover a region. The area of this rectangle is forty
squares. Units can be introduced: square inches, square centimeters, etc. (How
to compute the area of a rectangle is being introduced or hinted at in the
latter example.)
To physically represent the notion of multiplication, cue cards or pictures
of rectangles and squares with varying heights and widths, as indicated by the
number of squares in a horizontal row, or vertical column can be employed. These
images can illustrate or define the 10, 12 and 16 times table, etc. They can be
used to observe that 4 times 6 gives the same result as 6 times 4 – the order
of multiplication is not important. Further examples of a similar or different
kind will be generated by a teacher. The object of each is to introduce a new
idea or to widen and reinforce a previous one.
Before multiplication can be described further, addition needs to be
discussed. Addition can be initially viewed physically as the combination of two
or more objects, or groups of objects together. This is a physical definition
that is easily understood by students before the addition of decimal numbers has
been explained or even mentioned. The addition of various numbers of objects can
be illustrated with small groups of marbles, dots, squares, etc. By this method,
the addition of pairs and even triplets of the numbers 1 to 9 can be introduced
and illustrated. For instance, in a repeatable and reproducible fashion, a child
may see two plus three is five simply by combining a group of two marbles with a
group of three. For a child and possibly some adults, such considerations
inductively show why 1+1=2 or 2+2=4. No deep philosophy is required. The
meaning, justification and interpretation here is a consequence of the
adjectival role of numbers in counting how many and the conservation of objects,
say marbles.
The grouping concept is further useful in developing or explaining the
distributive law of multiplication over addition. For example, five bags of 4
marbles plus three bags of 4 marbles gives five plus three, that is, eight bags
of 4 marbles. The physically observed conservation of marbles now suggest the
distributive law
This physically seen or induced distributive law whether it is recognized or
not, provides an informal basis for the thought-based development discussion of
numbers and their decimal representations in elementary mathematics. It may be
implicit in the explanation of multiplication.
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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
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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