RealPlayer Videos: Primes, Fractons, LCD, GCD -- Fractions, Ratios, Proportions

Arithmetic Videos - Real Player Format

Four Groups of Videos follow.

For quicker results, Start with fraction videos first and cover the others as needed.

Primes, How to Recognize Them. Extras include statement and justification of rules for division by 2, 3, 5, 9 and 11, and the calculation of remainders for division by 2, 3, 5, 9 and 11.

Fractions, Operations With. Addition, Multiplication and Reduction (Simplification) using primes, LCM, GCD. Euclid's Algorithm for computing the GCD of a pair of whole numbers provides a method for simplifying fractions, quickly without using prime decomposition of numerators and denominators.

Greatest Common Divisors, Calculation using Primes or Euclid Algorithm.

Least Common Multiples, Calculation using Primes or Greatest Common Divisor

Pen and pencil arithmetic skills is a must for algebra and a plus for the use of arithmetic in daily life.

Primes may be used in simplifying expressions involving fractions and square roots. See the calculation of GCDs and LCMs below.

[Play Video] 5 minutes - A Times Table (10 x 10) and how a number is not prime (composite) if it is in the interior of the table, that is if it is a product of smaller natural numbers. Some where in here is a Definition for Primes. A Natural number is composite if it is not prime.
[Play Video] 9½ minutes - Digit- Based Rules for recognizing divisibility by the divisors 2, 3, 5, 9, 10 and 11 or calculating the remainders on division by these divisors. These rules follow from 10 = 0 mod 2 or 5, and 10 = 1 mod 3 or 9, and 10 = -1 mod 11. Exercise: (1) Use 100 = 2 mod 49 to develop a digit-based rule for division by 49 or 7. (2) Give digit-based rules for division by 2, 3,5, 7, 11 and 13 that apply to the hexadecimal representation of whole numbers.
Square Root Rule: A number N is prime if it is not divisible by all primes p whose square p² is less than or equal to N. On the other hand if a number N is not prime, it will be divisible by a prime p with p² less than N+1. With a calculator, the best bet is check where all primes p < sqrt(N) starting with the smallest. Here if N = Mq where all primes < p are not divisor of the prime N then all primes < p will not be divisors of M. With the aid of a calculators and rules for divisibility by 2,3, 5, and 11, you can quickly get the prime decomposition of a whole number N.
[Play Video] 10 minutes - Recognizing Primes in the interval to 100 by eliminating all numbers that are multiples of primes < 11 = the first prime with square 11² = 121 > 100. (The Sieve of Erasothenes)

If a first number N is a product of two factors, the square of the larger factor will be greater than or equal to the first number, and the square of the smaller will be less than or equal the first number N. So if the first number N can be factored, there will be a divisor, the smallest factor in a product with square < the first number N. That in turn implies there will be a prime < the smallest factor which divides N and whose square is < N. From the study of logic (the contrapositive of an implication rule), if all primes with square < N do not divide N, N cannot be written as a product of factors - natural numbers smaller than N.
[Play Video] 2½ minutes - Prime Factorizations (also called decomposition) for numbers in the interval 2 to 15.
[Play Video] 3 minutes - Prime Factorizations for numbers in the interval 16 to 30.
[Play Video] 4½ minutes - Prime Factorizations for numbers in the interval 31 to 49.
[Play Video] 4 minutes - Prime Factorizations for numbers in the interval 50 to 66. Note: 51 = 3 x 17 is not prime as stated in video. Oops.
[Play Video] 5½ minutes - Prime Factorizations for numbers in the interval 67 to 82.
Note: 76 = 2 x 38 = 2 x 2 x 19. Video shows 17 instead of 19. Oops
[Play Video] 5½ minutes - Prime Factorizations for numbers in the interval 83 to 100.
Note: 90 = 6 x 15 = 2 x 3 x 3 x 5 = 2 3²5 Video write 4 x 15 instead of 6 x 15. Oops

Start here if you wish and refer to methods for obtaining Prime Factorization, GCDs, LCDs as needed.

[Play Video] 3-4 minutes. Equivalent fractions - Lowering and raising terms (the values of numerators and denominators) to obtain equivalent fractions. Simplification involves lowering terms - cancelling common factors or divisors on top and bottom. Addition & subtraction of fractions may involve raising terms to obtain a common denominators. See below.
[Play Video] 2-3 minutes A few examples of Simplifying Fractions - lowering terms by canceling common factors until there are no more common factors, so that the numerator and denominator are relatively prime, that is there prime decompositions have no primes in common.
[Play Video] 2-3 minutes. Multiplying Fractions with cancellation of common factors done first (recommended) or not, with more simplification to be done later.
[Play Video] 5 minutes. How to add fractions using common denominators. Here the common dominators is the lowest or least common denominator (LCD) and its given by the least common multiple (LCM) of the denominators in the fractions added together. Here the listing multiples method is used to compute the LCM. The alternative of not using the LCD for the fractions is explored to show what happens when the LCD is not used.
[Play Video] 3 minutes Another example of how to add fractions with and without the least common denominators with an explanation that not using the LCD (least common denominator) leads to ratios that can be simplified. So use of LCDs is promoted.
[Play Video] 3 minutes - Comparison of Fractions Size or Magnitude, and more examples of the use of common denominators in addition and subtraction.
[Play Video] 3 minutes - Another example of the listing multiples method to find the LCM and thus the LCD for the sum of two fractions.
[Play Video] 4 minutes - Factorization method to obtain a common denominator, here the LCM and thus the LCD for the sum of two fractions. See if you can recognize the GCD of the denominators here. It is not mentioned here. In this example, the LCD is given by a product that does not have to be evaluated explicity due to cancellation of common terms after addition of fractions.
[Play Video] 2 minutes - Fraction Simplification using Prime Decomposition (factorization) to identify common factors for cancellations.
[Play Video] 5 minutes - Product Simplification using Prime Decomposition by Canceling Common Primes, thus avoiding some denominator and numerator multiplication. An alternative common factors as they appear, more opportunistic, is given and is to be recommended.
[Play Video] 5 minutes - How to use Prime Factorization or Decomposition for LCM and LCD for a pair of denominators, an example.

The simplification, multiplication and addition of Fractions may depend on recognition and cancellation of common factors, prime or not. See how GCDs and LCMs (or LCDs) may be used in the addition and multiplication of fractions.

See how to compute greatest common divisors with and without the use of prime factorizations.

[Play Video] 7 minutes. Finding All Divisors of a Natural number from its Prime Factorization/Decomposition
[Play Video] 6 minutes. Computing GCD for pairs of Natural Numbers from their Prime Factorizations /Decompositions)
[Play Video] 2½ minutes Computing GCD from Prime Factorizations /Decompositions, another example.
[Play Video] 3 minutes. Computing GCDs using Primes, yet another example.
[Play Video] 6½ minutes. Euclid Algorithm computes GCDs not using Prime Factorization.
[Play Video] 3 minutes. Another Euclid Algorithm GCD example with result confirmed using Prime Decomposition.
[Play Video] 1½ minutes. Two numbers are relatively prime when and only when they have GCD =1 when and only when the numbers have no prime divisors in common. Euclid algorithm leads to a quick identification of relatively prime whole numbers in the numerators and denominators of fractions by themselves or products.
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[Play Video] 4 minutes. Two Ways to Find the GCD of a pair of numbers. Both lead to the same result.

Euclid's algorithm provides a means to compute the GCD without mentioning prime factorization. The latter is best for computations with large numbers - numbers for which the prime factorization is not immediately obvious. Euclid algorithm can be implemented on calculator.

For a pair of denominators, the greatest common dominator is given by their least common multiple.

[Play Video] 2¼ minutes. Common Multiples and Least Common Multiples for a par of natural numbers, finding by listing mutliples of first and second number - the list method.
[Play Video]2¼ minutes. Least Common Multiple for a pair of Natural numbers from Prime factorizations of each, and then by list method.
[Play Video]1 minute. Least Common Multiple for a pair of Natural numbers by computing the GCD divisor with the aid of Prime Factorization of each.
[Play Video] 4 minutes. Least Common Multiple for a pair of Natural numbers by computing the GCD divisor with the aid of Euclid's Algorithm, 1st Example.
[Play Video] 3 minutes. Least Common Multiple for a pair of Natural numbers by computing the GCD divisor with the aid of Euclid's Algorithm, 2nd Example. Note use of calculator.

A. More Arithmetic a must for algebra etc

D. Logic In Mathematics

G. Algebra with Take Home Value

I. Vectors & Functions

Decimal Lesson - Reference
Counting & Addition   (8 lessons)
Comparison to Subtraction (9 lessons)
Multiplication ( 11 lessons)
Long Division (12 lessons)
Decimals and Primes (8 lessons)
-Primes & Composites
-Primes Factorization
-Greatest Common Divisors & Multiples.
-Prime Factorization Aids (Learn how to find factors quickly)
-Prime Factorization Examples
-Counting & Generating. Factors.
-Divisibility Rules and Remainders for Division by 2, 3, 5, 9 and 11.
Integers (12 lessons) Intro to Signed Numbers
Fractions (< 20 lessons)  Essential Skills & Concepts
Ratios & Fractions (3 lessons): Similarities & Differences
Units in calculations Fractions with Units

B. Basic Algebra

Solving Linear Equations
- in one unknown. Intro with stick diagrams?
the normal way & with good nttn. (the nttn that reappears in Gaussian Elimination. |
-in more unknowns: simultaneous equations essentially one unknown. the let algebra do the work view of word problems.
- still in more unknowns: Gaussian Elimination via substitution, by equality or comparison, by operations on equations.

C. More Algebra

Words before symbols: See if U like the lengthy chapters 8 to 12 in Volume 2, Three Skills for Algebra
What is a Variable. The answer here is a simple prequel to the modern mathematics viewpoint.

First, every rule & pattern U meet in math, logic & science will be used forwards and backwards. Get a head start with this theme by reading Chapter 14 in Three Skills for Algebra. Second, in the study of Proportionality Relations (3 dense lessons here) finding the proportionality constant gives an initial backward use of the proportionality formula.

Talking about words before symbols and the forward and backward use of formulas gives words to make algebra simpler & clearer.

If you can not read or write precisely, you will have difficulty in following instructions. One wordy remedy is given by chapters 2 to 5 in Three Skills for Algebra. Where does Logic or a geometric model for reason Appear in Mathematics? The answer lies in Euclidean-Geometry In North America, Euclidean Geometry disappeared from high school mathematics as it was too hard. The light treatment here is a possible remedy.

E. More Geometry

The Pythagorean Theorem. Chapter 17 from in Three Skills for Algebra uses algebra and geometry   to show why the Pythagorean equation for right triangles holds. Its forward and backward use is common exercise.. At a more theoretical level, the Pythagorean theorem leads the discovery that not all lengths can be fractional multiples of a unit length. That geometrically implies a need for and even existence of irrational numbers.
Analytic Geometry: Common Practices with Maps and Plans drawn to scale give coordinate-dependent base for senior high school development of similarity, trig, vectors and straight lines.
Complex Numbers: This lesson on Complex Numbers draws on Euclidean and Analytic geometry. Sbortcuts simplifiy trig identities, the cosine law; and   trig formulas for 2D dot- and cross-products.

F. Logarithms, Exponentials,
Roots & Powers

Logarithms, exponentials, rational and real powers for secondary students. This complete Operational Viewpoint. (Sufficient for the precalculus forward and backward use of compound growth and decay formulas in biology, physics, chemistry, personal finance, and calculus. To learn more, if you study calculus, see chapter 19 of Volume 3, Why Slopes and More.Math

In Volume 2, Three Skills for Algebra, chapters

identify methods useful in money computations, methods needed for calculus. Your teachers or other writer may present the same ideas with greater clarity and detail - A site to do.

H. Polynomial & Quadratics

Analytic Geometry: - Slopes and Lines - Take 1.   Take 2 appears in site section Maps and Plans.   Two views are better than one. I may combine them later. -In my school days, slopes appeared year after year.   This Why Slopes calculus preview on graphs of functions y = f(x) explains why. Enjoy.
Quadratics and Polynomials: - Operations on Polynomials: Meet a light and ultraquick geometric introduction to multiplication, addition and subtraction of polynomials. Then see how the foregoing combine to permit long division of polynomials. Compare Fractions with Units. Enrichment: A Plus: The Geometric introduction here gives or is almost identical to a justification for column methods in decimal arithmetic.
Geometric Derivation of the Quadratic Formula:   The account here gives a starter lesson for the more algebraically harder geometric-free derivation. If you study physics, chemistry or trigonometry, you will need to know about quadratics, their factorization and the quadratic formula.
Technical Value: The study of polynomials high school mathematics has technical value as part of the senior high school mathematics preparation for calculus. This simple account of Why Factor Polynomials   (Chapters 2 to 6 in Volume 3 .Why.Slopes.&.More.Math.) will give a context for the study of polynomials, their factorization, and sign analysis of functions, all in a way that should improve your algebraic thinking and reasoning skills.

Vectors in the Plane (2 simple lessons)
- Navigation with vectors or arrows
- Sum of Motions
- more lessons to be added later.
Operations on movement or vectors along the line and in the plane have value in mathematics in defining and implying the properties of real and complex numbers before the assumption of those properties as axioms. Vectors and their properties appear in physics, its mathematical description and formulation.
- Functions - Forwards & Backwards. Here is a full technical reference (24 lessons) for use in a calculus or precalculus course as needed. In it, the set viewpoint of functions expression of modern pure mathematics. comes from the set-based codification and
In the mathematics education reforms of the 1960s in North America, primary and secondary school mathematics were expressed in terms of sets. That expression has now retreated from primary and secondary school texts. But it still lingers on, and can be very useful, a source of clarity and precision, in the situations where it should be retained: Counting with the aid of sets and functions; the description of functions; the high school account of probability theory; and in the discussion or illustration of ideas in logic.

J. Pre-Calculus Skill Check

- Arithmetic Skill Check. In the calculus courses I taught 1983-89, too many students had weak skills in arithmetic. I would give and carefully correct these exercises to tell students what they needed to review and master.
- All the skills and concepts in Chapters 1 to 24 or Volume 2, Three Skills for Algebra: Look for those you do not understand and fill the gaps. Do so quickly while balancing this advice with your other duties. Good luck.

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Arithmetic Videos

Recognizing Primes

Operations with Fractions

Greatest Common Divisors

Least Common Multiples