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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Fractions Etc
The mastery of mathematics from algebra to
calculus demands fraction sense and skills fully and efficiently. Providing
that is the first objective of this site area. See how much you can understand
below alone or with help.
Area pages provides a reference for teaching students about
fractions, ratios, rates, proportions, proportionality constants and arithmetic
with units.
The starter
lesson (fraction summary page) points to fraction know-how. Mastery
of simplification, cross-cancellation in multiplication (an exercise in
simplification), division of fractions (another exercise in efficient
multiplication and simplification), and then addition and subtraction with
least common denominators and more simplification. Simplification may employ
rules for recognizing multiples of 2, 3, 5 and 10, and exploit or emphasize 10
or 12 times table. Simplification and more simplification (lowering terms) is
the theme. However, raising terms appears in the addition and subtraction of
fractions with unlike denominators as an aid to these operations and via the
choice of least common denominators, to simplification. (Teachers:
The introduction to simplification here uses real player videos - providing a
text or html form of this introduction is a site to do. The following steps
are in html format.)
Except for the starter lesson emphasizing know-how,
the site pages on Fractions aim to develop know-why. The development of know-how
provide a context for the know-why (perhaps).
| The Arithmetic
Videos (Realplayer format) may be viewed apart from or besides
fraction lessons 1 to 12. Exercises on Mostly
Fractions will test fraction know-how. The site area Solving
Linear Equations may help students visualize fractions while you
meet a geometric approach to algebra. I hope you can follow and enjoy
the underlying ideas. |
Teachers: Put fraction know-how first to
emphasizes development of efficient figuring skills, to build students skills
and confidence by showing how they can obtain results in a repeatable and
reproducible and therefore verifiable ways. Once the skills and confidence are
here, start weaving explanations of why or how fraction operations are
justified. Then see the more general discussion of fractions in the Number
Theory area.
Before or besides the simple use of
formulas in primary school and of the mastery via numerical examples
of methods for addition, subtraction, multiplication and division of
fractions, the algebraic description of
the latter operations on fractions (rules for them) may give a taste of
the later shorthand role of letters and symbols in describing
associative, commutative and distributive properties of arithmetic with real
and complex numbers.
Instead of besides working with division of
monomials in variables and addition and subtraction of like monomials in
variables, give examples of division and addition of monomials in units
of mass, length, time and even money. Such examples are or could be useful in
senior high school or college level chemistry and science where calculations
with units appear. Besides that, the aforementioned units may have (slightly)
more significance to students than letters (at least for students who are ill
ease with the use of letters as placeholders for numbers).
Learn More
See the following chapters in site book Three Skills for Algebra:
8
The Three Skills For Algebras
9 The
First Skill
10 Two
More Skills
11 Why
Shorthand
14
Compound Interest
15
Linear Equations
16
Painless Proofs
17
Pythagoras
This site area on linear equations, Chapters 14
Compound Interest and 15
Linear Equations all provide routes to introduce and extend algebraic way of
writing and reasoning. Which one to follow first is a matter of taste. Nibbling
at all in parallel is an option until their digestion is complete.
Chapter 14
Compound Interest could be rewritten in terms of compound growth and decay
of populations and radioactive material, and and/or connected to exponential
growth and decay without any mention of compound interest.
Technical Notes
-
While ratios a:b involving a pair of
numbers can be identified with a fraction a/b and even a
proportionality constant k = a/b = a fraction equivalent to a/b,
ratios a:b:c may appear in the discusion of propotions, but they
cannot be identified with fractions of the form A/B. So there is
a difference between fractions and the concept of what a ratio or
proportion when more than two numbers or quantities appear.
-
A quantity Q = a number N
say times a unit of measure U. The discussion of how to add,
subtract, multiply, and divide units by themselves and by
(real) numbers and how to change or rescale units leads to
products and quotients of units, and thus to unit-based
representation of rates and proportionality constants as a real
multiples of a product or quotient of units. Calculations with units
provide monomial like exercises with units instead of variable in the
numerators and denominators of fractions in which simplification
relies on the addition and subtraction of exponents.
Calculations with units provide a greater context for the algebra of
monomials. They also provide the algebraic ways to represent
rates and proportionality constants as quantities and to extend
algebra beyond the realm of real numbers to the realm of calculation
with units or quantities. This use of algebra emphasizes the power of
algebra with quantities and put aside the artificial requirement
and extra work to express proportionality problems only in terms
of numbers via unit elimination, a complication for students who are
just learning algebra.
Hint: If a whole number < 121 is not a
divisible by the primes 2, 3, 5 and 7, it is prime. Here you need
to remember decimal notation (representation) based rules for
recognizing multiples of 2, 3 and 5. Here you need to remember all
multiples of 7 < 121, namely 14, 21, 28, 35, 42, 49, 56, 63,
70, 77, 84, 91, 98, 105, 112 and 119, or the
shorter list 49, 91 and 119 of multiples < 121
not divisible by 2, 3 and 5. In general, a number < N2
is prime if that number is not a multiple of each and every prime <
N. Why can be can explained in algebraic discussion of number theory.
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Teachers
The new site page Teaching
Algebra describes a program to follow.
The ideas here could be woven in early high school or late
primary school class for students ages 10 to 15 say. Past mathematics
curriculums called for an efficient mastery and comprehension of on paper
methods for arithmetic with whole numbers and fractions to serve as a basis for
algebra. Recent practice in classroom leaves many students without a
fraction sense or comprehension and without the ability to do and understand
arithmetic with fractions. The lack of fraction sense after the first year
of high school implies failure or a waste of time and energy in further
mathematics courses. I kid you not.
Note(1) : The discussion of two term
ratios a:b (read a to b) and multiple-term ratios a:b:c (read a to b to c)
historically (?) may have come before the discussion and physical
interpretation of fractions a/b. Fractions themselves can be identified
with twp-term ratios (and may be called ratios) but a three or more term
ratio cannot be identified with a single fraction. There-in
lies a difference. Some ratios are not fractions.
Note (2): The discussion of ratios here
is link to proportionality - In equivalent fractions, the
numerators are proportional to the denominators with proportionality constant
equal to the common value of the fractions. In equivalent two-term
ratios, the the first term is proportional to the second term with
proportionality constant equal to common value of the associated equivalent
fractions.
Note (3): In the evaluation of formulas
for perimeters and areas, etc, students may see letters replaced by numerical
values. Seeing such substitutions could be part of the development of the
algebraic way of writing and reasoning. The shorthand description of how
to add, multiply and divide fractions provide further opportunities to
describe or summarize computations that could be done, and a further chance
for students to see letters as place holders for numbers, place holders than
may be identified with or replaced by numbers in actual computations.
If you are teacher or tutor, I hope you will see how to generate
more examples and illustrations. Those here give the main ideas but
more examples would help. The examples below are based on division of lengths or
rectangles due to the convenience or inconveniences of html in web page
production.
Teachers: This a reference.
Student may first obtain an efficient mastery of fractions and decimals
operations by rote to aid their learning of algebra in a thought-based
fashion. See the rest of this site. Then they may revisit explanations like
the following to obtain a thought-based understanding of fractions as
well. But as a teacher, you should have this thought-based
understanding, so that you drop hints of it while most students are learning
by rote. In arithmetic, with or without rote learning, students need to learn
to figure well in a repeatable and reproducible manner.
More Notes - for experts
Pure mathematics talks about and applies the
properties numbers and avoids the use of units. Units can be factored out of
calculations with quantities by expressing all in terms of pure numbers. For a
distance or length D can be expressed a d meters. So all calculations
involving the distance D can be expressed in terms of the pure number
d. That process applied to all quantities results in equations or
calculations with pure numbers in which units have been factored
away. That elimination of units requires the student to choose a system of
measurement and convert all data to its units before starting the solution of a
problem. But less work is a required if units are carried through
calculations in a careful and precise manner in which units conversion are done
only when needed and in which the development of algebraic skills with
units. The treatment of units in this site area shows how to multiply and
divide units to define calculations with products and quotients of units (unit
monomials and their fractions involving them), and beyond shows how to add like
products and like quotients. Two products or quotients are said to be
alike when the exponents of all units in them coincide.
Ratios of a pair of numbers can be identified
with fractions. From that identification follows the equivalent properties
or equality of ratios involving different terms and the equality of fractions
involving different denominators. Yet ratios involving more than two
numbers cannot be identified with fractions, as fractions only involve a pair of
numbers. But a ratio of N real numbers (say N > 3) can be
identified with an ordered N-tuplet (triplet if N = 3) of numbers. Two N-tuplets
are equivalent when and only when one is a positive real multiple k of the
other. Likewise, two N term ratios are equivalent when and only when
there exists a single positive number k, such each term in the first is a
positive real multiple k of the corresponding term in the other. That
brings the discussion of multiple terms ratios in the realm of coordinates
(projective geometry) and away from fractions. Here the equivalent N-tuplets
and equivalent N-term ratios altogether form a ray through the origin in N
dimensions. More generally, you could allow k to be negative or positive
in the foregoing.
Ratios of a pair of quantities can be identified
with rates and and extended concept of fractions in which numerator or
denominators involve units. Proportion involving N quantities (say N >
3) may be identified with N-tuplets of quantities. Two N-tuplets are
equivalent when and only when one is a positive multiple k of the other where k
is product or quotient of units and/or real numbers.. Likewise, two
N term proportion are equivalent when and only when there exists a single
quantity k, given by product or quotient of units and/or real numbers, such each
term in the first is a positive real multiple k of the corresponding term in the
other. Here the quantity k is called a proportionality constant,
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www.whyslopes.com
Fractions, Ratios, Units, Rates
& Proportionality
Fraction
Starter Lesson
(simplify, multiply, divide & then add or subtract)
An Alternative Starter
Lesson
(take your pick, or try both)
Area Map & Intro
Fraction Starter Lesson A
Fraction Starter Lesson B
1 What is a Fraction
2 Multiplication I
3 Multiplication II
4 Multiplication III
5 Equivalent Fractions
6. Mixed Numbers
7 Comparison
8 Addition I
9 Addition II
10 Addition III
11 Multiplication IV
12 Division
13 Two Term Ratios
14 Implied Ratios
15 Multiple Ratios
16 Units in Arithmetic
16 Longer Explanation
16 Change Units
16 Products of Quantities
16. Fractions with Units
16. Division+Reciprocals
17 Proportionality
17 Examples
18 Rates & Slopes EGs
18 Constant Rate
18 Varying Rate
18 Velocity Calc., EGs
18 Changing Units
18 Slopes and Units
18 Slopes, No Units
19 RealPlayer Videos
Links
Arithmetic Videos - Real Player Format
Decimal Addition
Methods
Decimal
Subtraction Methods
Decimal
Multiplication Methods
Decimal Division
Methods
Fractions
Primes
Greatest Common
Divisors
Least Common Multiples
Square Root
Simplification
Area Content Summary
- Fraction Starter Lesson
- Real Player Videos on Operations with Primes and
Fractions
- Continuous Ruler & Line Segment
model for fractions and operations on fractions - Number Theory Area
points to the general model.
- Distinction between Ratios and Fractions, a nuance:
While binary ratios a:b may be identified with a fraction, triple
ratios a:b:c and further multiple ratios cannot.
- Saying how to add and subtract like monomials in
units and their powers, and saying how multiply and divide like and
unlike monomials leads to fraction like expressions involving units
and a framework for discussion rates - ratios of quantities - a
framework for handling proportionality constants, and framework for
carrying units through calculation in quantitative disciplines
Hint: See site area on solving linear equations to strengthen
fraction sense and algebra skills together. Good luck.
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