14. Adding and Subtracting with like Denominators
Instead of counting how much or how is present in terms of whole units we
may count in terms of 1-numerator fractions: half, third, quarter, fifth,
sixth, ...., one hundredths, and son on:
Addition: For a quarter of a divisible object, 3
times a quarter + 2 times a quarter = (3+2) times a
quarter = 5 quarters
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Line Segment Example: In the following diagram we
show a green line segment divided into four
quarters. Then we show two more line segments, one
three quarters of the first and the next two quarters.
What is the length of the two line segments side by side?
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Line Segment
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Three Quarters of it
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3 quarters
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Two Quarters of it
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2 quarters
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Three Quarters & Two Quarters
Side by Side
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=>
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5 quarters
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In equation form: 3 quarters + 2 quarters = 5
quarters, or
at least when we are talking about or describing fractions of a
line segment and their addition.
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Example 1. Counting thirds
2
3
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+
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4
3
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=
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2+4
3
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=
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6
3
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=
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2
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The foregoing says 2 thirds plus 4 thirds (of a unit of measure)
gives six thirds (of the unit of measure) and the latter is
equivalent to 2 (of the unit of measure).
Example 2. Counting Tenths
18
10
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+
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15
10
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=
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18+15
10
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=
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33
10
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=
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3
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3
10
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since 33 = 3×10+3. The foregoing says 18 tenths plus 15 tenths (of
a unit of measure) gives 33 tenths (of the unit of measure) and the
latter can be regrouped in to 3 units (as ten tenths equal 1) plus 3
tenths.
Example 3. Subtracting 12ths.
11
12
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-
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5
12
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=
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11-5
12
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=
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6
12
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=
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6×1
6×2
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=
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1
2
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since 33 = 3×12+3. The foregoing says 11 twelfths minus 5 twelfths
is 6 twelfths (of a unit of measure). The latter gives a half. The
following diagram attempts to illustrate the subtraction and the fact
that 6 twelfths is a half.
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11
12
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6
12
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5
12
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6
12
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6
12
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Exercise- food for thought: Express the sum of 21 hundredths
and 39 hundredths in terms of hundredths, tenths and fifths.
Example 4: 7 times a third - 5 times
a third = (7-5) times a third = 2 thirds
Geometric Exercise: Take the length of a
line segment (measured or not) to be a unit length. Illustrate
the foregoing by drawing 7 thirds and subtracting 5 thirds.
show by counting thirds, that the result is 2 thirds
long.
>
Algebraic View
for reading as part of algebra skill development - optional
reading for now
Addition and Subtraction with like denominators - Algebraic Shorthand
Pattern
The algebraic pattern for addition of fractions with like
denominators i
A
N
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+
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B
N
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=
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A+B
N
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For addition of fractions with the same
denominator,
add numerators and keep the denominator
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The general pattern or rule for subtraction in terms of shorthand
letters is as follows:
A
N
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-
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B
N
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=
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A-B
N
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For subtract of fractions with the same
denominator,
subtract numerators and keep the denominator
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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
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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
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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
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ingredients, pots and pans for a dish or a meal, along with
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McCainian: drill, drill, drill then Toronto
mathematician and actor John Mighton's jump math organization has jump math
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Mathematics
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Skills with take
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time-date-calendar Matters; money matters; map, plan and
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Is your child able to add, subtract and multiply amounts
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Arithmetic
and Number Theory Skills
Algebra
Starter Lessons
Geometry
- maps plans trigonometry vectors
More
Algebra
70
Calculus Starter Lessons
Calculus Lessons Elsewhere:
-
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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