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 >> D Decimal Long Division Methods
Notes
Long Division (LD) Examples and Theory
Teaching and Tutoring Tip: (1) The long division method and
checks for it requires at full strength, multiplication, subtraction
and addition methods for decimals. So long division represents a check
of addition, subtraction and multiplication skills as well.
Difficulties with the latter operations need to caught and remedied
before preferably or while students do long division calculations. (2)
Preparation for calculus requires students to add, multiply, subtract
and divide polynomials. In that preparation, long division of
polynomials resembles long division of decimals . So mastery of the
latter will help with mastery of the former.
-
Lesson 1. Where Does Division Appear? The division question of
asking how many times does one quantity, number or unit go into
another appears in counting, grouping and measurement.
-
For Distances and Length Measurement: How many times does
a shorter length (a unit length perhaps) go into (go into) a
longer length. The longer length is view a whole number multiple
of the shorter length with a remainder.
-
For Dot Counting and Grouping: How many same size groups
of say three dots can be formed from sixteen dots provide a first
example.
-
For Volume Measurement: How many times does a small unit
of volume go into a larger volume.
-
Lessons 2 to 4 - Worked Examples of Long Division with 1
and 2 digit divisors.
(2) 6847 = 2282 × 3 + 1 (Dividend: 6847, Divisor: 3, Quotient: 2282,
Remainder: 1)
(3) 7834 = 1119 × 7 + 1 (Dividend: 7834, Divisor: 7, Quotient: 1119,
Remainder: 1)
(4) 89463 = 6881 × 13 + 10 (Dividend: 7834, Divisor: 7, Quotient:
1119, Remainder: 1)
These examples illustrate the long division method and format met in
the site author's school days. These examples introduce the helpful
practice of listing the first 10 multiples of the divisor. Single
digit multiples are needed in the method. These examples also show
how to check (test) that a computed or given quotient and remainder
are correct.
Quotable quotes: Even if we don't know why long division
works, we can check its results by testing whether or not
Dividend = Quotient × Divisor + Remainder.
Remember: If a check fails, there is an error between the start of
the long division itself and the end of the check. And if the check
fails, you need double check the check and the long division.
For single digit divisors, there is a short division
format that is more compact (uses less space) than the long division
format. Learn it if you wish.
-
Lesson 5: Examples in which LD (long division) is done with
& without extra zeroes to further indicate how, if not why, LD
works.
-
Lesson 6: Why the long division methods works. Here is a
closer inspection illustrated with the long division (with extra
zeroes) of 7275 by 2 to 7275 = 3637 x 2 + 1. This first explanation
and lesson 8 are for people who like to understand long division
fully as well as learning to do it and check results.
-
Lesson 7. Long Division With a Mistake - How to handle
mistakes in mathematics solutions
Long Division Method here suggests
5626 = 308 × 15 + 6
Observe how the error is spotted and ignore the extra zeroes that
appear in the solution. Their presence will be explained below.
Tip 1: In class, when a check spots an error, the error itself
will between the start of your solution and the end of your check.
The error could be in your check.
Tip 2: In a test, do not erase your solution if you a spot an
error or think you have made one. Do not erase your solution until
you have written and checked a replacement. Better yet, cross out the
solution instead of erasing it when you have the replacement. And if
on a test, you do not have time to provide a replacement, give the
solution and check, and write Oops! do not have time to
correct in large print or boldly besides the check. The student
who submits wrong a clearly written solution with a clearly written
check, right or wrong, gets more respect and show a greater command
of subject matter than the students who does not present work
clearly. The advice do not erase your solution may help as well with
instructors who are looking for reasons to reward your work. Good
luck.
Tip 3: In homework, if you have time to correct your work, do
so. If not follow the advice in tip 2, and change your error
declaration to : Oops! Did not make time to correct.
-
Lesson 8. Correcting the Error. Doing long division
properly (with extra zeroes) to get
5626 = 375 × 15 + 1
and then checking the results. It works.
-
Lesson 9. Yet another insight into why long division method
works. This lesson inspects the numbers and subtractions that
appear in long division of the dividend 5626 by a divisor 15 to show
why 5626 = 375 × 15 + 1
-
Lessons 10 and 11: More examples of long division.
-
Lesson 12. The long division method in expanded form gives yet
another explanation of why the LD method works.
Not Done Here or Yet - Later is anticipated: The treatment of
long division for decimal fractions (mixed numbers plus a proper
fraction equivalent to a fraction whose denominator is a power of ten)
belongs after the coverage of fractions and in that coverage, a
discussion of decimal fractions.
www.whyslopes.com >> Arithmetic and Number Theory Skills >> 2 Arithmetic with Decimals >> D Decimal Long Division 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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