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
to switch.
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
fractions.
Arithmetic with units - Skills of value in daily life and in the
further study of rates, proportionality constants and computations in
science & technology.
What is
a Variable? - this entertaining oral & geometric view
may be before and besides more formal definitions - is the view mathematically
correct?
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.
Solve
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
equations algebraically.
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.
Using
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 >> 2 Arithmetic with Decimals >> B Decimal Comparing and Subtracting Methods
Notes, Lessons 1 to 8
are described next. Conversions here are also called borrows.
-
Lessons 1 & 2: Direct comparison and subtraction of whole
numbers where no conversions (borrows) are needed. The more
than symbol > and the less than symbol < are
introduced to compare numbers. Comparisons and subtractions can
be done directly one the subtracted number has fewer ones, tens and
hundreds etc than the decimal from which it is subtracted. In this case
no knowledge of lexicographic ordering of decimals is required.
Exercises are included.
Teachers and Tutors:
In this section, the symbol
> is called the more than symbol here, as that is consistent with
calling the symbol <. the less than symbol. More the name more
than for the symbol > avoids or minimizes a later confusion in the
comparison of signed numbers, a confusion that results from a
divergence between the common meaning of greater than as a comparison
of size and the technical definition of more than (or greater
than).
-
Lessons 3 to 5: These lesson explains when and how conversions
(borrows) can be done to extend decimal notation through the
introduction of a J symbol (value 10), so that comparison and
subtraction can be done directly after a conversion. Exercises are
included.
Teachers and Tutors: While students may
still learn or be taught how to compare decimals lexicographically, the
conversion approach given here justifies the latter and provides a
common path for comparison and subtraction, one that makes "borrows"
easy to understand and explain.
The conversion results in an add 2, subtract one, arithmetic
problem. Exercises are included. Both examples and
exercises involve decimal counts (whole numbers) and decimal fractions
(mixed numbers with fractional parts provided by tenths and/or
hundredths). Borrows or conversion appear on both sides and across
decimal points in examples, if not in the exercises. Exercises
are included. (Students may want to see Lesson 5 first - It covers a
method to make subtractions easier.)
Teachers and Tutors: Here is the first
instance of a conversion in arithmetic to allow an operation to be
done. Further conversions will be met in the addition, subtraction,
comparison and division of fractions. There-in lies a
unifying thread or theme for showing and saying how arithmetic will be
done with decimals and fractions. Operations are based on
conversions. Instead of writing Convert to
Compare/Subtract, lesson labels could have used the phrase
Compare/Subtract Indirectly. .
-
Lesson 6: The how-to Subtract Efficiently, Option.
Calculating 9 + 3 - 5 as 12 -5 is less efficient in the eye of
this beholder than calculating 9- 5 +3 = 4 + 3 - 3. The latter
avoids subtraction of a single digit number from a 2 digit number.
Exercises are included.
Teachers and Tutors: When
numbers are converted to make subtraction more direct, expressions of
the form 9 + A - B and J + A - B appear where A < B are
digits 0 to 9, and J has the value 10. Evaluation of the two
expressions 9 + A - B and J + A - B may be obtained more quickly
from the left to right calculation of 9 - B + A and J
- B + A, respectively.
-
Lesson 7: Subtraction of Decimal Fractions involving tenths and
hundreths with Conversion before and after decimal point.
-
Lesson 8: Working with Mixed Units: Lesson 7 gives several
examples of column methods for subtraction of mixed unit of measures
for money, length, time, working hours, and angles. Exercises
are included.
-
Lesson 9: Here is a review or repeat of ideas in lessons 1 to 4
and 6. View this video to set expectations for arithmetic subtraction
skills.
Teachers Tutors: The older site area Number Theory Area explores
three methods for subtraction. In the latter page, conversion or
borrows when needed, the first method, is illustrated with
examples. I suspect that the convert to compare and subtract method
here might help in the explanation of (1) how electronic chips store
numbers and perform remainder arithmetic while catching under- and
over-flows; and (2) the use of the Abacus.
www.whyslopes.com >> Arithmetic and Number Theory Skills >> 2 Arithmetic with Decimals >> B Decimal Comparing and Subtracting Methods
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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?
Site Reviews
1996 - Magellan, the McKinley
Internet Directory:
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,
Number 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
to scale.
Euclidean Geometry - See how chains of reason appears in and
besides geometric constructions.
Coordinates - Use them not only for locating points in the plane
or space.
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
cross-products.
Lines-Slopes [I] - Take I & take II respectively assumes no
knowledge and some knowledge of the tangent function in
trigonometry.
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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