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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: Fractions and Divisions
Fractions appear via the concept say of division, the division say of a pie,
an amount or a group of objects into separate but equal portions. This leads to
the concept of fractions: enjoying a fifth of a pie, consuming two quarters of
pie, or being sick on three quarters of a pie. And when there are several pies,
one can introduce improper fractions, for instance eating 7 quarters of more
than two pies, the latter having being physically divided into quarters. Here
fractions can be added or grouped together in a physical sense. Here the
mathematical operation of division is linked to the semantic and physical
concept or root.
In discussing pies, we observe eating [6/18]th or [4/12]ths of a pie is
almost the same as eating a third, only the number of divisions may be
different. Like wise, eating 15 out of 20 portions of a pie is the same as each
3 quarters of the pie. Pie based examples like this lead to the idea of
physically equivalent fractions and the simplest form of a fraction - the
form that requires that the least number of divisions in a pie.
Examples can be generated using marbles, distances, and measures of mass,
area and so on. Further, some improper fractions, like [10/2] can be identified
with whole numbers. That is, 10 halves of a pie (more precisely of several pies)
is equivalent to 5 pies in total. The significance of [11/2] can also be
explained. These observations when the numerator and denominator are small, can
be made with the aid of simple examples.
A fraction is said to be simplest form when the numerator and denominator
have no common divisors. The fraction [60/100] can be reduced to [6/10] and then
to [3/5]. The calculation of the greatest common divisor (g.c.d) of two whole
numbers finds an application and thus its motivation in this reduction: the
simplification of fractions. The calculation of the g.c.d requires mastery of
division with decimals and the two further notions of (i) prime numbers and (ii)
prime number decomposition.
The addition of fractions can also be introduced by taking pieces of a pie.
The question of how to take the fraction [1/3]and also the fraction [2/5] of a
pie can be resolved by cutting a pie (or pies) into 15ths. Here students may see
that 5 times 3 is fifteen. The question of how to take the fraction [3/4] along
with the fraction [1/6] of a pie can also be resolved by cutting the pie into 6
times 4 = 24 equal portions or into 12 portions. The latter number is the least
common multiple of the denominators. The addition of fractions together provides
motivation for the discussion and computation of the lowest least common
multiple (l.c.m) and the computation of the latter via prime number
decomposition.
Products of Fractions
Taking [2/5] of a whole pie, an amount or a group of objects, involves a
division of the pie, amount or group into 5 equal portions and then taking two
portions. Now if the original amount is given by [2/3] of a pie, we need to
divide the [2/3] into five equal pieces. Here
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2
3 |
= |
2 ×5
3 ×5 |
= |
10
15 |
= 10 fifteenths |
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Now [1/5] of ten is two; and [2/5] of ten is 4. Thus [2/5] of 10 fifteenths in 4
fifteenths or [4/15]. This provides a physical introduction to multiplication of
fractions.
Examples like this may suggest the rule.
or in more words, the numerator and denominator of a product of two fractions is
given by the product of the numerators over the product of the denominators.
Digression. When the factors [(a)/(b)] and
[(c)/(d)] are both in reduced form, a shortcut for the
simplification of the product comes from observing the product also equals [(a)/(d)]
·[(c)/(b)], where the fractions may be further reduced or
simplified using the l.c.m of a and d, and the l.c.m of c
and b.
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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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