Employ an online or offline tutor at your own risk
from
AU:
tutorfinder.com.au
CDN :
findatutor.ca
CDN: .i-tutor.ca
CDN: Montreal
Tutors
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findatutor.co.nz
UK:
tutorhunt.com
UK: tutors4me.co.uk
USA:
wiziq.com
USA: ziizoo.com
or employ the site author - View
his WiZiQ profile
- Calculus students are very welcome.
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YOU are better than YOU think. Show
yourself how:
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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
Do not leave here without it - Logic
mastery will develops critical thinking, improve reading and
writing, and give a firmer base for work and studies at many levels.
Good luck.
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Caution: Site advice
is approximately correct, for some circumstances, not all.
Site How-TOs are
logically developed, but not tried and tested. That leaves
room for thought and refinement.. |
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After logic,
(a) continue reading Three
Skills for Algebra, chapters 8 to 14 and do so alongside
site area on solving
linear2007 Equations ; or (b) see this calculus
starter lesson and Volume 3, Why
Slopes & More Math, chapters 2 to 6;
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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Explore collaborative whiteboards
from groupboard,
twiddla or
scriblink.
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Chapter 11: Positive and Negative Numbers
the coordinate perspective
Thermometers with temperatures above or below a reference point
labelled zero provide an example of a numbered line where the numbers have
positive and negative signs in front and a physical significance. A positive
temperature indicates so many steps or units above the zero mark while a
negative sign indicates some many steps below the zero mark. The addition of a
positive number now corresponds to be and may be defined as an upward movement
of so many steps. The addition of a negative number and the subtraction of a
positive number corresponds to a downward movement. These additions and
subtractions can be done at any point on the scale.
The subtraction of a negative number in the first instance is undefined. But
one can define negation for a number as the reversal of direction, and regard
subtraction of a number as the addition of it negation. Students can be shown
that this applies to the subtraction of a positive number before the subtraction
of negative numbers is considered.
Two negations or reversals further result in the original number. Here the
negative -a of a number a will be the
number or point obtained by subtracting the number a from 0. The
foregoing provides a physical concept of addition and subtraction.
Note that multiplication of a number a by a whole number n, can
be viewed as the result of the addition of a to itself, a whole number n
times. Multiplication by nonnegative proper and improper fractions, and then
positive decimals can also be physically interpreted. Next every negative number
b is the negation of a positive number a. In consequence,
multiplication by a negative number b = -a
can be defined as the multiplication by a followed by a negation
(reversal of direction).
Coordinates along a horizontal line (the real numbers) can be represented by
signed decimal numbers
The dots ... represent digits not written. The sign indicates a location to the
left or right of the zero mark (origin) of the line while the unsigned expansion
gives the magnitude of the displacement. A signed number +akak-1¼a1a0.a-1a-2¼
is just the same as the number, akak-1¼a1a0.a-1a-2¼
with the sign removed. The latter provides the distance of the number to the
origin - a quantity that students can visualize and possibly measure. Signs can
be linked with coordinate displacement in one direction or the other from a
reference point or origin; in connection with the computation of assets and
debts etc.
More on Subtraction.
Subtraction of n from m yields a number k with the
property that n+k = m. When m and n are
given, subtraction of n from m answers the question, what number k
when added to n yields m? Examples for subtraction when m
> n can be revisited in this context. Such examples imply that k
equal m - n when n + k = m.
The calculation k = -(n-m)
can be given when n > m as a means for computing k = m-n.
Then again n + k = m. Here n-m
is computed using previously taught methods for the subtraction of decimals or
fractions. (When m and n are fractions, subtraction answers the
question: what fraction k when added to n gives m? The
fraction can be signed.)
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www.whyslopes.com
Volume 1B, Mathematics Curriculum Notes,
Foreword + Chapters 1 to 10 + 12
Area Map
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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