Two message to help you build skills and confidence

• Do you know about the domino effect of mistakes and errors in arithmetic? If not, ask parents, teachers or fellow students to explain what that might be. If yes, if you know about it, then you know about the need to be careful in each arithmetic step you take. The domino effect of errors appears both in and outside of arithmetic. In it, an error in one step likely makes all following steps and any result wrong. Taking care to avoid the domino effects is a must for building skills and confidence in all skill based subjects at home, at work and in school. Good luck.
  

• Understanding exactly or precisely what is said or written will avoid confusion when you are following instructions at home, at school and at work. Learning to write and speak exactly or precisely will avoid confusion of others when you explain or give instructions as well. When you are old enough (or now), read the site math free chapter on two logic puzzles. The chapter may help you read, write and speak with precision. That in turn may help you avoid the domino effects in many subjects.
  

Good Luck.

Arithmetic Steps

Some good practices for skill development in arithmetic appear in site arithmetic and number theory steps. Where these good practices are not best, say so.

1. Definition of Primes, Simplified : A simpler definition of prime numbers which takes advantage of the 12 times tables to identify small primes upto 13. In particular, a whole number is said to be prime if it is not the product of two smaller whole numbers. With the word smaller in the definition, the whole numbers 11 and 13 are prime because it is not given by a product inside the 10 and 11 times table. [This was the small example given above]

2. Quick Prime Factorization of Small Whole Numbers: Emphasis of a square or square root rule to provide QUICK prime factorization skills for whole numbers less than 169 = 13². In particular, a whole number less 169 is prime if and only is it is not a multiple of the primes 2, 3, 5, 7 and 11 less than 13. Simple divisibility rules and calculators (an overkill) here may be used to recognize multiples of 2, 3, 5 and 11. Quick prime factorization of whole numbers is a key to exact and efficient fraction practices employed in mathematics from algebra to calculus. There is no escape.

3. Fraction Operations Explained: A thought-based development of addition, comparison, subtraction, multiplication and division operations starting with simpler cases where operations are easily explained, and continuing on to general cases where all operations are justified by raising terms. In higher mathematics, if not elementary mathematics, comprehension of why methods work is highly valued, it is part of the spirt of mathematics mastery. Understanding how and why operations are justified should move you away from learning by rote. Reference: fraction operations by raising terms

4. Arithmetic and Fractions With Units: Figuring with denominate numbers, that is multiples of units of measure for physical quantities and units of value for monetary quantities. This practical value for calculations involving speed, rates in general and associated proportionality constants in daily life and also in practical and applied arts and sciences. (In algebra taught by rote, you may see similar figuring with multiples and powers of variables in products and quotients. The path here has more meaning and is very practical)

5. Oral Dimension of Arithmetic: Verbal description and extension of common practices for finding counts, totals and products by forming and adding or multiplying subcounts, subtotals and subproducts. Here calculation practices are introduced and described orally instead of symbollically, the latter being harder for many to grasp. For many, how to calculate averages and how to calculate perimeters of polygon are best described with words, the use of letter or symbols being to complicated to understand in the first instance. Mastery of common practices for counting, totaling and multiplying do not have to wait for their algebraic description. Instead, the verbal forms can be given. [These arithmetic notes expand on part of the big example given above.]

6. Place Value Revisited: An exposition of place value in decimals with places before an after the decimal point in groups of three may amuse and inform. In it, students in North America may learn how to read aloud and write on paper the decimal

   6,571,045,375,905,333,034,412.450,033,870

   as 6 sextrillions, 571 quintrillons, 45 quadrillionths, 375 trillionths, 905 billionths, 333 millions, 34 thousands, 421 ones, 450 thousandths, 33 millionths and 870 billionths. In contrast, students elsewhere may use the following "SI" (system international) method how to read aloud and write on paper the decimal form of 6 zettaunits, 571 exaunits, 45 petaunits, 375 teraunits, 905 giga-units, 333 megaunit, 34 kilounits 421 ones, 450 milliunits, 33 microunits and 870 nanounits.

7. Addition, comparison, subtraction, multiplication and division of decimals: The site development may covered more lightly than presented. The development of place value methods for all but long division is thought-based. Why methods work are both indicated. Long division method is given without justification, but with a method to check results. In all methods, students will meet the domino effects of care and mistakes. Avoiding the latter provides an end, value and tool for skill mastery, an echo of the old fashion idea that figuring well is a sign of practical intelligence.

8. Signed Numbers: The site description of arithmetic with integers and arithmetic with signed numbers is not bad. The site objective so far has not been to cover everything in mathematics, but to develop and express ideas on how mathematics should be learnt or taught. That being done, a clearer account of arithmetic with signed numbers is due.

9. More Steps To Elaborate - not in site material: Talk about scientific notation, arithmetic with, and arithmetic with mixed decimals - that is, decimals with multiple places before and after the decimal point. Relate foregoing to fraction skills and practices. Explain the comparison, addition and subtraction of scientific in terms of of finding a common factor or denominator.

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