Basic Arithmetic with Decimals & Fractions
This is a new site area - posted online November 14. Most lessons are based on flash videos -
access requires JavaScript and a flash viewer. Many chains
of reason are present
Most videos are two to six minutes long, usually one per lesson.
If you plan to watch a video twice, leave its browser window open, before watching the video again. That
avoid reloading and save bandwidth
Each chapter or lesson group give a very detailed and enriched view of its
topic.
A Decimal Arithmetic Explained
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Arithmetic is required
at full strength in consumer mathematics to handle calculations with masses,
weights, measures and money; to acquire and show the skill, ability and
patience to follow instructions steps by step; and to expose students to
the consequences of mistakes, so they realize the need to be
careful. Mastery of arithmetic methods and conventions at full
strength is also required fully and completely by college mathematics.
The latter employs arithmetic, geometry algebra at full strength, and
the full strength command of algebra requires the full strength command of
arithmetic. Where and when course design tries to be kind by omitting skills
and concepts, courses designers should have a critical path knowledge of the
requirements of consumer mathematics and in the form of calculus, college
mathematics.
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Subject: Counting, addition, comparison, subtraction, multiplication, long division
Four chapters with 40 lessons and more than 40 flash videos lessons follow.
- (8 lessons) counting and
adding whole numbers Counting and Addition of Decimals
- (9 lessons) comparing
and subtracting whole numbers and decimal fractions Comparing and
Subtracting with Decimals
- (11 lessons) multiplication
of whole numbers and decimal fractions - Multiplication Methods and Theory for Decimals
- (12 lessons) long division for
whole numbers - Long Division Methods and Theory for Decimals - Whole
numbers only,
Decimal notation and methods for counting, comparison and arithmetic are met
in primary school and should be reviewed and mastered in full by students 12 to
14 years of age in school.
B. Integer Arithmetic Explained
Lessons on Integers and three
appendices include
exercises to consolidate and extend understanding - 15 plus flash videos.
Total viewing time for the webvideos will be about one hours - an hour per
chapter.
The lessons provide a
hands-on, thought based development. Lessons assign three geometric roles to integers
to develop and explain rules for integer arithmetic.
- Role I: Integer are first introduced as coordinates for points on a
line, where adjacent points are a unit distance apart.
- Role II. Integers then serve as multipliers in the definition of integer multiples
of a unit movement, integer multiples that can be added and multiplied by
whole numbers and then integers.
- Role III. Integers themselves may describe movements, how many steps to
the left or right, along a straight line, and so can be identified with
movement, integer multiples of a unit movement, now called a step. That
third role or identification leads allows integers to be added and
multiplied.
Exercises are included in most lessons.
Three appendices cover an
associative law, division quotient and remainder options for integers, and how
remainder arithmetic explains the alternating sum of digits test for
divisibility of decimals by 11.
Extension: This three role,
geometric development and explanation of integers starting with unsigned whole
numbers provides an example to follow for the development and
explanation of (i) rational numbers starting with unsigned fractions; and (ii)
real numbers starting from unsigned (positive) real numbers or their decimal
representation. The site exposition of complex numbers continues the geometric
development of number theory.
C. Prime Numbers
Decimals and Prime Number Decomposition
New webvideo lessons to be posted will replace or complement the current site
coverage of the following.
- Primes
& Composites
Composite numbers less than 101
Prime Numbers less than 101
- Primes
Factorization
Unique Prime Factorization Theorem (Cryptic Statement)
- Primes
& Composites
Calculation of Greatest Common Divisors and Least Common Multiples from
Prime Factorizations
- Prime
Factorization Aids & Prime
Factorization Examples (two pages) If a whole number N < 121 is
not divisible by each prime 2, 3, 5 and 7 < 121 = 112 then N
is prime. Alternatively, if N < 121 is composite then it will be a single
or repeat multiple at least one of the primes 2, 3, 5 and 7. Whence
recognizing whether or not a whole number N < 121 is divisible by 2, 3, 5
or 7 gives a quick test for primality and a quick method for prime
factorization. See examples.
- Counting
Whole No. Factors. How to count and generate all possible
factors of a whole number from its prime factorization. Introduces the
concept of proper and improper factors - For the latter, is there a more
standard terminology?
- Divisibility
Rules and Remainders for Division by 2, 3, 5, 9 and 11. There are many
rules for recognizing when whole numbers are multiples of 2, 3, 4, 5, 6, 7,
8, 9,10 and 11. Those rules are consequences of modulo or remainder
arithmetic
D. Fractions
New webvideo lessons to be posted will replace or complement the current site
coverage of the following.
1 What is a Fraction
2 Multiplication I
3 Multiplication II
4 Multiplication III
5 Equivalent Fractions
6. Mixed Numbers
7 Comparison
8 Addition I
9 Addition II
10 Addition III
11 Multiplication IV
12 Fraction Division & Reciprocals
13 Compound Fractions
14 Operations with Units I
15 Operations with Units II
16 Operations with Units III
17 Operations Unit IV
18. Operations with Units V
19. Operations with Units VI.
20. Operations on Units VII
Pages 1 to 13 review or consolidate fraction skills and
concepts via operations on lengths. In the process, algebraic descriptions of
operations are indicated not as requirement for fraction mastery but as
enrichment option.
Pages 14 to 20 extend arithmetic with fractions to include
arithmetic with units of measure that may appear in daily life or science.
Rates of change and proportionality constants involving quantities without
unlike measures may be expressed as "fractions" in which numerators
and denominators are multiples of units of measure to first or higher powers.
Operations with units, their products and quotients, is needed for calculations
with numbers, amounts and measures in daily life and in senior high school
mathematics. These operations are very similar to operations with polynomials
and so may serve as preparation or a prequel to the latter.
Again this is a new site area - posted online November
14. It complements and replace lessons in the Arithmetic
Reference Page. The latter page points to a 100 or so site
resource in arithmetic with no links at the present time to material in
this site area.
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Basic Arithmetic with
Decimals & Fractions
Counting & Addition
Compare & Subtract
Multiplicaton
Long Division
Integers - Intro to Signed No.s
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