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for college students, gifted teens, home-tutoring and K1-12 schooling, with chapters
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Teachers & Tutors: This December 2011, 5-phase framework
offers a context for mathematics & logic education. Phases 1 to
3 may focus on skills with actual or potential local value for
adult & daily life. College-oriented phases 5 & 4 focus on
calculus & preparation for it. Phases 1 to 4 may also serve
trades & professions not dependent on calculus. Reform: look
before you leap - plan all in detail first.
Site Review: Math resources ... span ... arithmetic, logic,
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and how-tos .... provide a good foundation for high school and
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Home < Arithmetic and Number Theory Skills < 2 Arithmetic with Decimals << B Decimal Comparing and Subtracting Methods
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B Decimal Comparing and Subtracting Methods
1 Comparison and Subtraction - Easy Direct Cases
2 Subtraction Easy Case Examples
3 Harder Cases - Convert to Compare and Subtract
4 Subtraction with Conversions Borrows and Letter J
5 A Tip for Efficent Subtraction
6 Subtraction with Conversion Example with Exercises
7 Subtraction for Decimal Fractions with Exercises
8 Subtraction with Units of Measure
9 22 Minute Subtraction Review Video
-Subtraction Another Video Lesson
-Subtraction with J Conversions Examples
Appendix 1 Decimals Comparison Method Take II
Appendix 2 Three Decimal Subtraction 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.
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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
writing with precision.
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.
Arithmetic
and Number Theory Skills
Algebra
Starter Lessons
Geometry
- maps plans trigonometry vectors
More
Algebra
70
Calculus Starter Lessons
Calculus Lessons Elsewhere:
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How to Ace Calculus: Street Wise Guide - Mostly
Text.
-
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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