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Ages 15+: Why study slopes Polynomials Quadratics Why factor polynomials Logarithms Functions
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What is a variable 5. Fraction Operations by Raising Terms Solving Linear Equations: Take I Take II

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Home < Arithmetic and Number Theory Skills << 3 Prime Factorization Skills

3 Prime Factorization Skills

     1 video how Products are bigger than factors
     2 Prime and Composites less than 16
     3 video Primes and Composites from 9 times table
     4 video Prime Factorization Introduction
     5 Prime Factorization and a Square Rule
     6 Sieve-of-Eratosthenes and Square Rule
     7 Calculator Usage Notes and Cautions
     8 video Prime Factorization upto 19
     9 video Prime Factorization upto 19 squared
     10 video Prime Factorization upto 23 squared
     11 Efficient Square Rule Use
     12 LCD GCD and LCM using Primes
     13 video Factors of 24 using primes
     14 video Factors of 24 Take II
     15 video Factors of 20 using Prime Factorization
     16 video Factors of 980 using primes
     17 Identify and Count Factors using Primes
     18 video Count Factors given Prime Factorization
     19 video Prime Factorization Unique
     20 Uniqueness of Prime Factorization

More Notes

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Quick Prime Identification and Composite Number Factorization Method

If a whole number N less than 169 = 132 is not divisible by 2, 3, 5, 7 nor 11 [the primes smaller than 13] then the number is prime. Otherwise, it - the number N - is a product of 2, 3, 5, 7 or 11 with another factor N' less than 169.

Knowledge of times tables, divisibility rules or a calculator - one displaying results to two plus decimals, may be used to recognize multiples of 2, 3, 5, 7 and 11. Two decimals are sufficient because the fractions one half to one eleventh are all more than 0.01 = one hundredth. See Efficient Square Rule Use to quickly learn more.

In learning and applying algebra exactly, one usually needs to compute with fractions or ratios of whole numbers less than 169 or so. Learning how to efficiently use the above method for recognizing primes and obtaining prime factorization of of whole numbers less than 169 is sufficient for most ends and purposes purposes in high school and college mathematics and science courses. The above method is based on the square method below.

If a whole number N less than the square of a given prime is not divisible by all the the primes smaller than given one then the number is prime. Otherwise, it - the number N - is a product of one smaller prime and another factor N' less than the square of the given prime.

Recognition of prime factors and the prime decomposition of whole numbers speeds calculations of LCM, GCD and LCD in exact arithmetic with whole numbers and fractions. They also lead to cosmetic simplification of square roots. The recognition of common factors in numerators and denominators of fractions alone or in products helps with the reduction of fractions and via cancellation leads to efficient methods for multiplying fractions.

Description of Folder Lessons

  1. [video] how Products are bigger than factors. This observation simplifies the identification of primes and composite numbers.

  2. Prime or Composite less than 16.. Instead of saying a whole number is prime when and only when it is not a product of whole numbers larger than one, we say a whole number is prime when and only when it is not the product of two or more smaller whole numbers larger than one. The addition of the word smaller, redundant and technically not required due to lesson 1, none the less makes prime number identification easier to learn and teach. This webpage employs the definition of primes and the 12 times table to recognize that the whole numbers 2, 3, 5, 7, 11 and 13 as primes.

  3. [video] Primes and Composites from 9 times table. Saying a whole number is prime when and only when it is not the product of two or more smaller whole numbers larger than one allow small primes in the 9 times table to be quickly identified. Here is a short form video version or variant of the previous lesson.

  4. [video] Prime Factorization Introduction. This video lesson introduces and illustrates prime factorization for a few whole numbers. In the last example, the equivalence between tree notation and the use of equal signs in the development of prime factorization is emphasized.

  5. Prime Factorization and a Square Rule. This lesson provides a more detailed discussion of prime factorization and introduces a well-known, but I suspect hitherto nameless rules, for identifying primes and obtaining prime factorization. The name square rule is coined. This rule or method provides students with a quick manual method for obtaining the prime factorization of whole numbers. This quick method can be employed with the aid of the divisibility rules for the small primes 2, 3, 5 and 11, when or if mastered, or with the aid of calculators that display sufficiently many digits after the decimal point. See the calculator usage notes and cautions below.

  6. Sieve-of-Eratosthenes upto 100. Discussion of this Sieve (or filter) shows how striking out of multiples of 2, 3, 5 and 7 is sufficient by the square rule to identify all primes in a list or table of whole numbers 1 to 100.

  7. Calculator Usage Notes and Cautions. The square rule for identifying primes and obtaining prime factorizations, two side of the same coin, can be employed with the aid of calculators that display sufficiently many digits after the decimal point. Here are notes on how with some cautions.

  8. [video] Prime Factorization upto 19. Given the identification of all primes less than 100, this video is redundant. However, it is retained to illustrate the use of the square rule. An alternative approach would be to use the 18 times table alone or a subtable of a larger times table.

  9. [video] Prime Factorization upto 19 squared. The square of 19 is 361. provides examples of prime factorizaton with the square rule for a few numbers less 361.

  10. [video] Prime Factorization upto 23 squared. The square of the prime 23 is 5629. provides examples of prime factorizaton with the square rule for a few numbers less 529.

  11. Efficient Square Rule Use. The last part of the lesson Prime Factorization and a Square Rule provides a few hints or directions for the efficient use of the square rule. This lesson on efficient use provides examples.

  12. LCD GCD and LCM calculations using Primes. Finding least ommon ddominators, least ommon multiples and greatest common divisors are of sets of whole numbers, two or more, represent number theory practices of service in simplifying, adding, comparing, subtracting, multiplying and dividing fractions, and of service in cosmetic conventions for the exact representation of square, cube and higher roots of whole numbers. This webpage gives some examples of LCD GCD and LCM calculations using Primes, or more precisely prime factorizations.

    Lessons 13 to 18 present and illustrate methods to count and find all whole number factors of a given whole number using tables, products and trees. In the study of quadratics, factoring by inspection requires the identification of all integer factors of a given integer. The methods here or variant of them turn that identification part of this factoring a quadratic by inspection into a systematic art.

  13. [video] Factors of 24 using primes

  14. [video] Factors of 24 Take II

  15. [video] Factors of 20 using Prime Factorization

  16. [video] Factors of 980 using primes

  17. Identify and Count Factors using Primes

  18. [video] Count Factors given Prime Factorization

  19. [video] Prime Factorization Unique. This video lesson introduces the notion that the prime factorization of a whole number will be independent of the route that gives the factorization. Knowing about the uniqueness and how it implies the latter is nuance that may included for completeness, or not.

  20. Uniqueness of Prime Factorization. This lesson describes the uniqueness in general - offers reason for it.

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

20 Times Table - the most popular site page - popular pages - unexpected.
Fractions & Ratios - with lesson on raising terms to introduce & justify times, division & comparison as well addition & subtraction
Parent Center - See below
Volume 1, Elements of Reason - Intro to all site books.
What is a Variable - best for ages 13+
Written work formats for Arithmetic and Algebra - a skill method and standard!
Complex Numbers Visually - best for ages 13+
Natural Logs, Exponentials, Powers, Roots

Division of Labour: This site offers advice and directions with pointers to resources elsewhere, if known, when they help or lessen the need to write more.

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

1 Working With Sets
2 Formula Forward Use - Evaluation
3 Solving Linear Equations - Skip first step with students able to solve 1 eqn in 1 unknown.
4 Computation Rules and Function Notation
5 Real Numbers
6 More Less Greater Than Inequalities and Comparison
7 Axioms Logic and Equivalent Equations
8 Unifying Theme For Algebra
9 Proportionality Backwards and Forwards
10 Examples of Algebraic Reasoning
A Origins of Counting and Figuring Methods
B Real Numbers Extrinsic Development


Site coverage of formuala evaluation format, of computation rules and axioms, and of the forward and backward use of formulas and proportionality relations lessens the amount of natural talent needed to understand and explain algebra.

Geometry - maps plans trigonometry vectors

1 Maps Plans Measurement
2 Euclidean Geometry - Constructions + extras
3 Cartesian and Polar Coordinates
4 Lines and Slopes Take 1
5 What is Similarity
6 Trigonometry first steps
7 Complex Numbers
8 Unit-Circle Trigonometry
9 Lines and Slopes Take 2 with tangent function
10 Intersecting Straight Lines and Transversals
11 Parallel Straight Lines and Transversals
12 Function Translating and Rescaling
13 Vectors
14 Degrees to Radians and Radians to Degrees
15 Arc or Inverse Trigonometric Function

Pre-Teen and young teen mastery of skills and practices which should be common with map-plans-diagrams drawn to scale, contour interpretation included, has actual or potential take-home value for daily- and adult-life in solving routine problems. Elevating some practices to principles, axioms or postualates, provides a base for analytic and Euclidean geometry, an analytic view of similarity, and an efficient mastery of trigonometry and complex numbers. Right triangle trigonometry provide an analytic alternative to solving geometric problems by drawing diagrams to scale.

More Algebra

Natural-Logarithms Exponentials Powers Roots
Five Polynomial Operations
Quadratics Geometrically
Functions
5 Factored Polynomial Sign Analysis Examples
Rewriting algebraic substitution as function substitutions

The first topic leads to a full high school level theory for the forward and backward mastery of growth and decay models and for definition, range and domains of radicals, roots and powers. The next two topics make quadratics and polynomials easier to learn and teach. Site coverage of functions turns vertical and horizontal line rules into computation methods for evaluating functions.

70 Calculus Starter Lessons

Calculus Lessons Elsewhere:

  1. How to Ace Calculus: Street Wise Guide - Mostly Text.

  2. 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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Home < Arithmetic and Number Theory Skills << 3 Prime Factorization Skills

Logic-Reason for all
Careful Thinking
Chains of Reason
Mathematical Induction
Responsibility
Bodies-of-Knowledge

Arithmetic - Ages 10+
1. Deciml Place Value - fun
2. Decimals for Tutors
3. Prime Factors - quickly
4. Fractions + Ratios
5. Arith with units - science

Geometry
1 Maps + Plans Use
2 Euclidean Geometry
3 Rct +Polr Coordinates
4 Lines-Slopes [I]
5. What is Similarity 		Algebra Starters - the base
1. Better Work Format
2. Solve Linear Eqns
3. Computation Rules
4. Axioms, Item 3 Viewpnt
5. Formulas Backwards 		More Algebra
Logarithms-ax & m/nth roots
Five Polynomial Operations
Quadratics Geometrically
Functions || Vectors too
Arith. Skill Check+Answers 		Calculus Prep/Preview
What is a Variable
Why study slopes
Why factor polynomials
Complex Numbers
Limits + Continuity

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