Mathematics Concept & Skill Development Lecture Series:
Webvideo consolidation of site
lessons and lesson ideas in preparation. Price to be determined.
Bright Students: Top universities
want you. While many have
high fees: many will lower them, many will provide funds, many
have more scholarships than students. Postage is cheap. Apply
and ask how much help is available.
Caution: some programs are rewarding. Others lead
nowhere. After acceptance, it may be easy or not
For students of reason in society, science and technology:
Pattern Based Reason describes
origins, benefits and limits of rule- and pattern-based thought and
actions. Not all is certain. We may strive for objectivity, but not
reach it. Postscripts offer
a story-telling view of learning: [
A ] [
B ] [
C ] [
D ] to suggest how we share theories and practices.
These online chapters may amuse while leading to greater precision and comprehension in reading and
writing at home, in school, at work and in mathematics.
1 versus 2-way implication rules - A different starting point - Writing or introducting
the 1-way implication rule IF B THEN A as A IF B may emphasize
the difference between it or the latter, and the 2-way implication A IF and ONLY IF B.
Deductive Chains of Reason - See which implications can and cannot be used together
to arrive at more implications or conclusions,
Mathematical Induction - a light romantic view that becomes serious.
Responsibility Arguments - his, hers or no one's
Islands and Divisions of Knowledge - a model for many arts and
disciplines including mathematics course design. Site Theme: Different entry
points may be easier or harder for knowledge mastery.
Deciml Place Value - funny ways to read multidigit decimals forwards and
backwards in groups of 3 or 6, US-CDN, UK-German and Metric SI style.
Decimals for Tutors - lean how to explain or justify operations.
Long division of polynomials is easier for student who master long
division with decimals.
Primes Factors - Efficient fraction skills and later studies of
polynomials depend on this.
Fractions + Ratios - See how raising terms to obtain equivalent fractions leads to methods for
addition, comparison, subtraction, multiplication and division of
Arithmetic with units - Skills of value in daily life and in the
further study of rates, proportionality constants and computations in
science & technology.
a Variable? - this entertaining oral & geometric view
may be before and besides more formal definitions - is the view mathematically
Formula Evaluation - Seeing and showing how to do and
record steps or intermediate results of multistep methods allows the
steps or results to be seen and checked as done or later; and will
improve both marks and skill. The format here
allows the domino effects of care and the domino effects of mistakes
to be seen. It also emphasizes a proper use of the equal sign.
Linear Eqns with & then without fractional operations on line segments - meet an visual introduction and learn how to
present do and record steps in a way that demonstrate skill; learn
how to check answers, set the stage for solving word problems by
by learning how to solve systems of equations in essentially one
unknown, set the stage for solving triangular and general systems of
Function notation for Computation Rules - another way of looking
at formulas. Does a computation rule, and any rule equivalent to it, define a function?
Axioms [some] as equivalent Computation Rule view - another way for understanding
and explaining axioms.
Formulas Backwards - Most rules, formulas and relations may be used forwards and backwards.
Talking about it should lead everyone
to expect a backward use alone or plural, after mastery of forward use. Proportionality
relations may be use backward first to find a proportionality constant before being
used forwards and backwards to solve a problem.
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www.whyslopes.com >> Arithmetic and Number Theory Skills >> 6 Fractions and Ratios
Lessons below may revise or consolidate fraction skills and concepts with
operations on lengths. In the process, algebraic descriptions of
operations are indicated not as requirement for fraction mastery but as
enrichment option. Methods for comparison, addition, subtraction,
multiplication and division are all developed or justfied below
by raising terms to transform a general case into simpler cases.
Algebraic view of those operations, click here.
Step-by-Step Skill Development
all methods are likely to be well-known in the mathematical sense,
the methods themselves may be new to many students and teachers. These raising
term methods may overwhelm some students but help others. Instructors and
tutors should emphasize
fraction sense and skill mastery by rote when and where explanations of why methods work do
not help. Partial or full comprehension may be left to older or gifted students, and to
students who need to understand the development or origins of methods before using
What is a Fraction:
Meaning of a Fraction - A whole number counts how many ones, A
fraction counts how many parts of equal value. Algebraic
Fraction Multiplication I. What is a Unit Fraction of a Unit
Fraction? What is a half of a third? What is a unit fraction of a
unit fraction? A unit fraction has one in the numerator. Algebraic
Fraction Multiplication II. Unit Fraction of a Simple
Fraction: what is a half of two thirds? What is a quarters of
Fraction Multiplication III. : What is a Fraction of a
Fraction: what is seven quarters of three tenths?
Fractions: What is the difference between two quarters and
one half? What is the difference between six eights and three
quarters? What is the difference between an apple and four quarters
of the apple? The thought that there is no difference, that different
fractions may describe the same amount, quantity, leads to the idea
of equivalent fractions. Examples of fraction simplifying and
Algebraic Development. The first, easy case treats the case where
the denominator of one fraction is a divisor of the numerator of the
second fraction The general case follows by raising terms in the
second factor to apply the easy case.
Numbers and Equivalent Fractions: We may describe a distance as 3
half meters or we may describe the same length as 1½ meters.
Physically, there is no difference in the distance, only its
descriptions. The descriptions 3/2 meters and 1½ meters both have the
same value physically. So we declare the numerals 3/2 and 1½ to have
the same value, or to be equivalent. Likewise 3 fifties
(half-hundreds) is the same a 1½ hundreds. There is another instance
where the fraction 3/2 and the mixed numeral 1½ may be identified as
adjectives for the same count or measure.
Comparison of Fraction
and comparison of Mixed Numbers: Algebraic Description
included of the first included.
Fraction Addition I: Easy
case of like denominators - the easy case Algebraic
Fraction Addition II: General
Case of unlike denominators. the general case follows from
raising terms (as little as possible) to use the easy case.
Examples to show "raising terms" similarities
between comparison, addition and subtraction of fractions.
Fraction Addition III: Methods for adding
and subtracting Efficiently - Questions and Problems: (a) How is
the list method used to obtain a least common denominator = the least
common multiple of a pair of denominators? (b) How can the prime
number decomposition (also known as factorization) be used to
calculate the LCM and GCD of a pair of whole numbers? (c) How can
Euclid's algorithm be used forwards and backwards to calculate the
GCD (the normal result) and then the LCM of a pair of whole numbers?
(d) Employ the M-method to find the sum of eighteen 21st and nine
14-ths? Say which of the latter is more than the other, and find how
Fraction Multiplication IV: Efficient Ways to Multiply
Fractions: Learn how to calculate a few products of fractions
with and with the cancellation methods described below for
"efficient" multiplication, or more precisely efficient or easier
simplification after (cross) cancellation of common factors.
Fraction Division, Compound Fractions and Reciprocals:
Introduction to Fraction Division: Explanations, twice-over, are
given in the next lesson.
Fraction Division Methods Explained. See the previous page for
an introduction of the fraction division methods or formulas
- Two Step
Development Option - Explanation in two smaller steps, first with
the easy like denominators case and second with unlike denominators
(Raising terms in dividend and divisor fractions turns the second
case into the easy case).
How to do Arithmetic with
Rational Numbers, that is signed fractions, signed whole numbers
and signed mixed numbers. Here a model for introducing arithmetic
with real numbers.
A. Ratios And Fractions
Similarities and Differences
Fractions may be identified with two term ratios, and vice-versa
as well, sometimes. Two fraction are equal or equivalent when and only
when the corresponding two term ratios are equal or equivalent. But
fractions can be added, subtracted, multiplied and divided while the
same operations are not defined for two- and multiple term ratios.
While we may call a fraction, a ratio or a rational number, ratios are
different. Triple term ratios exist, but triple term
fractions do no exist. Three and more -term ratios cannot be
identified with fractions.
The following lesson cover the properties of two term and
multiple term ratios.
Fractions As (two term)
Ratios and Fractions Versus Ratios: Fractions are often called
ratios, and vice-versa. But the vice-versa only holds for two term
ratios. This lesson identifies fractions with two-term ratios and
contrasts the properties of fractions and two-term ratios. (Ratios
cannot be added, subtracted or compared, but like fractions, the
terms in ratios can be raised or lowered).
Implied or Derived Ratios - New Fractions and Ratios from Old: If
two fractions are equivalent then their reciprocals are also
equivalent. Likewise if a pair of two-term ratios are equivalent,
interchanging the first and second terms of each ratio in the pair
leads to a pair of equivalent ratios. Beyond that, more equivalent
ratios can also be generated from a pair of ratios. Food for
thought: How may equivalent fractions or ratios may be formed
from the relations ad = bc?
Ratios: Multiple Term Ratios - Three Term Ratios to be
precise. We read the triple ratio a : b :c as a to b to c. We
a : b: c :: A: B:
to say two triple ratios a : b: c and
A: B: C are equal or equivalent when and only
there are other ways to say when two triple ratios are
equal or equivalent.
Note: Triple ratios or triple proportionalities
occur between the sides of similar triangles. More generally,
multiple ratios or proportionalities occur between the sides of
The discussion of ratios or multiple ratios is best
understood besides a discussion of proportionality.
Inner Versus Outer Terms - small point: In the
discussion of equality of ratios a : b = A: B written in
that order, the inner terms are small b and big A while the outer terms
are small a and big B. In contrast, if we rewrite the equality as
A: B = a : b, we find the inner and outer terms are
interchanged. However, the equality requires the product of the inner
and outer terms be equal, that is aB = Ab. That equality is not
affected by rewriting a : b = A: B as A: B = a
: b, and the resulting swap of inner and outer terms
www.whyslopes.com >> Arithmetic and Number Theory Skills >> 6 Fractions and Ratios
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Road Safety Messages
for All: When walking on a road, when is it safer to be on
the side allowing one to see oncoming traffic?
1996 - Magellan, the McKinley
Mathphobics, this site may ease your fears of the subject, perhaps even
help you enjoy it. The tone of the little lessons and "appetizers" on
math and logic is unintimidating, sometimes funny and very clear. There
are a number of different angles offered, and you do not need to follow
any linear lesson plan. Just pick and peck. The site also offers some
reflections on teaching, so that teachers can not only use the site as
part of their lesson, but also learn from it.
2000 - Waterboro Public Library, home schooling section:
CRITICAL THINKING AND LOGIC ... Articles and sections on topics such as
how (and why) to learn mathematics in school; pattern-based reason;
finding a number; solving linear equations; painless theorem proving;
algebra and beyond; and complex numbers, trigonometry, and vectors. Also
section on helping your child learn ... . Lots more!
2001 - Math Forum News Letter 14,
... new sections on Complex Numbers and the Distributive Law
for Complex Numbers offer a short way to reach and explain:
trigonometry, the Pythagorean theorem,trig formulas for dot- and
cross-products, the cosine law,a converse to the Pythagorean Theorem
2002 - NSDL Scout Report for Mathematics, Engineering, Technology
-- Volume 1, Number 8
Math resources for both students and teachers are given on this site,
spanning the general topics of arithmetic, logic, algebra, calculus,
complex numbers, and Euclidean geometry. Lessons and how-tos with clear
descriptions of many important concepts provide a good foundation for
high school and college level mathematics. There are sample problems that
can help students prepare for exams, or teachers can make their own
assignments based on the problems. Everything presented on the site is
not only educational, but interesting as well. There is certainly plenty
of material; however, it is somewhat poorly organized. This does not take
away from the quality of the information, though.
2005 - The
NSDL Scout Report for Mathematics Engineering and Technology -- Volume 4,
... section Solving Linear Equations ... offers lesson ideas for
teaching linear equations in high school or college. The approach uses
stick diagrams to solve linear equations because they "provide a concrete
or visual context for many of the rules or patterns for solving
equations, a context that may develop equation solving skills and
confidence." The idea is to build up student confidence in problem
solving before presenting any formal algebraic statement of the rule and
patterns for solving equations. ...
Maps + Plans Use - Measurement use maps, plans and diagrams drawn
Euclidean Geometry - See how chains of reason appears in and
besides geometric constructions.
Coordinates - Use them not only for locating points in the plane
Complex Numbers - Learn how rectangular and polar coordinates may
be used for adding, multiplying and reflecting points in the plane,
in a manner known since the 1840s for representing and demystifying
"imaginary" numbers, and in a manner that provides a quicker,
mathematically correct, path for defining "circular" trigonometric
functions for all angles, not just acute ones, and easily obtaining
their properties. Students of vectors in the plane may appreciate the
complex number development of trig-formulas for dot- and
Lines-Slopes [I] - Take I & take II respectively assumes no
knowledge and some knowledge of the tangent function in
What is Similarity - another view of using maps, plans and
diagrams drawn to scale in the plane and space. May buildings in
space are similar by design.
Why study slopes - this fall 1983 calculus appetizer shone in many
classes at the start of calculus. It could also be given after the intro of slopes
to introduce function maxima and minima at the ends of closed intervals.
Why factor polynomials - this 1995-96 lesson introduces calculus
skills and concepts. It may also may be given to introduce further function maxima
and minima both inside and at the ends of closed intervals.
Check Arith. Skills - too many calculus and precalculus
students do not have strong arithmetic and computation skills. The
exercises here check them while numerically hinting at
equivalent computation rules.
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