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Home < Arithmetic and Number Theory Skills < 2 Arithmetic with Decimals << D Decimal Long Division Methods
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D Decimal Long Division Methods
1 Divsion Physical Examples
2 Division with Single Digit Divisors
3 Division Single Digit Divisor Example
4 Division with 2 Digit Divsors
5 Long Division - Include Zeroes or not
6 Why Decimal Long Division Methods Works - Take I
7 Long Divison Mistake Catching
8 Correcting the Mistake
9 Why Long Division Works - Take II
10 Division by Five Long and Short Ways
11 Another Single Digit Divisor Example
12 Why Long Division Works - Take III
Long Division forwards and backwards Example 1
Long Division forwards and backwards Example 2
Long Division forwards and backwards Example 3
Division with Counts and Lengths
Long Division Backwards
Long Division Backwards more
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.
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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.
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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.
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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.
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Lesson 5: Examples in which LD (long division) is done with
& without extra zeroes to further indicate how, if not why, LD
works.
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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.
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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.
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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.
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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
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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.
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Secondary
Mathematics for Ages 11+, A Practical Approach for home-tutoring or -schooling, or for schools & colleges
with local curriculum control. Study how to include site content - its skill development how-TOs and innovations
into present or future lesson plans - some reading required.
Road
Safety Messages and Questions: When and why should you face
traffic when walking along a road or cycle path? Is it a good
idea to hang limbs outside of cars etc? What gives more
protection in a crash: a car, motorbike or bicycle?
See too, the BBC-Belgium story Texting and
Driving - texting & the impossible test - the article links to a gruesome utube video on the subject
The Logic of Injustice:
How Texas sent
an innocent man to his death - The wrong Carlos. Some judgments are irreversible. Procescution: Where and when prosectors play to win rather than for
justice, guilt beyond a reasonable doubt goes unrespected due to prosecutors who putting winning
first, those innocence before the law may be convicted. Some procescutors offices in continuing to accuse after a pardon
due to reasonable doubt or innocent being shown, may sucessfully oppose compensaton for false convictions
by asserting a pardon individual is still under suspicion. Then the pardoned individual or the latter's estate
is not compensation for years or decade
of improper or false imprisonment, or for execution. Site chapters on Logic
and some in Pattern
Based Reason may slowly lead to greater precision in reading, applying and
writing laws.
May 2012, Composition Starting:
Pre-School and Primary Mathematics - Quantitative Skills, An
Intellectual View, Feedback Welcome:
The 8 Most Popular Site Inlinks
Parent Center: Help your child or teen
learn:
Parent-friendly
Work Booklets for ages 3+ to 13 Use these or others to check
or build skills. Other booklets are available but these booklets
allow parents unsure of themselves in mathematics to help their
children. The selection acquired in Canada is published in the
USA. So it has a US orientation. In retrospect, the selection
shows parents what to check with the booklets or by other ways,
the choice is theirs. But in retrospect, the selection does not
cover integral and fractions liquid weights and measures - ask
the publishers to correct that! For ages 9 to 12 say, parents may
compensate by showing boys and girls how to use weights or mass,
and further measures in food preparation. Beyond that children
may be shown how to measure and calculate angles, lengths and
areas [proportional amounts too] directly or by using maps and
plans drawns to scale. Learning how to gather and measure all the
ingredients, pots and pans for a dish or a meal, along with
cleaning up sets the stage for like activities or experiments in
science courses, and in developing organizational skills,
gives boys and girls a head start. Good luck. At the other
extreme, more comprehensive than light, if your motto is
McCainian: drill, drill, drill then Toronto
mathematician and actor John Mighton's jump math organization has jump math
workbooks for at least grades 3 to 8 for at-home and in-school
use - training sessions for teachers available. Jump math has
been expanding to cover older students. Jump Math Samples: plus
Fractions for
Grades 3-4 & Grades 5-6 [Read] Free Resources grades 1 to 8
[unread - likely to be good]. and
Mathematics
Skills For Ages 3 to 14 - technical!
Skills with take
home value - A few ideas
Basic skills include
time-date-calendar Matters; money matters; map, plan and
scale diagram matters;counting, measuring and figuring;
decision making with logic and likelyhood; being careful and
being aware of the domino effect of mistakes; reading and
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Is your child able to add, subtract and multiply amounts
of money, work with fractions, work with clocks and calendars,
work with maps and plans, and measure length, weight-mass and
volume? Schools may promote your son or daughter without
providing basic skills in reading, writing and
arithmetic.
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Calculus Starter Lessons
Calculus Lessons Elsewhere:
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How to Ace Calculus: Street Wise Guide - Mostly
Text.
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Flash
Video for Calculus Phobics
They cover basic topics in ways likely to complement your
notes, your textbooks and site material. When Goldilocks
trespassed in the house of the three bears, she found three bowls
of porridge, two not to her liking, and one just right. Different
bears have different tastes. As invited guest here and elsewhere,
if one or more explanations is not to liking, try another. It may
be better or just right.
Unsolicited Advice
Learning to do and high marks if it comes to easy is often
deceptive - light rather than deep. For that reason, students
with learning difficulties determined not to let it get in their
way may go deeper and farther than those with none. High marks,
if the come easy, may be deceptive - provide a too light and not
a deep mastery. That could have been your problem in secondary
school, one that leads to comprehension shock or difficulties in
calculus and more generally in the first year of college. Bon
Appetite.
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