YOU are better than YOU think. Show yourself how:
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Logic
Mastery Do not leave here without it - Logic mastery will develops critical thinking, improve reading and writing, and give a firmer base for work and studies at many levels. Good luck.
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-/[]\- After logic, (a) continue reading Three Skills for Algebra, chapters 8 to 14 and do so alongside site area on solving linear2007 Equations ; or (b) see this calculus starter lesson and Volume 3, Why Slopes & More Math, chapters 2 to 6; For online automated help in senior high school maths & calculus, visit quickmath.com For Automatic Calculus and Algebra Help with derivatives, integrals, graphs, linear equations, matrix algebra, visit calc101.com With overlap, each site quickmath & calc101offers a different range of services, some free, some not, all based on webmathematica. Good luck. |
Fractions Etc
The mastery of mathematics from algebra to calculus demands fraction sense and skills fully and efficiently. Providing that is the first objective of this site area. See how much you can understand below alone or with help.
Area pages provides a reference for teaching students about fractions, ratios, rates, proportions, proportionality constants and arithmetic with units.
The starter lesson (fraction summary page) points to fraction know-how. Mastery of simplification, cross-cancellation in multiplication (an exercise in simplification), division of fractions (another exercise in efficient multiplication and simplification), and then addition and subtraction with least common denominators and more simplification. Simplification may employ rules for recognizing multiples of 2, 3, 5 and 10, and exploit or emphasize 10 or 12 times table. Simplification and more simplification (lowering terms) is the theme. However, raising terms appears in the addition and subtraction of fractions with unlike denominators as an aid to these operations and via the choice of least common denominators, to simplification. (Teachers: The introduction to simplification here uses real player videos - providing a text or html form of this introduction is a site to do. The following steps are in html format.)
Except for the starter lesson emphasizing know-how, the site pages on Fractions aim to develop know-why. The development of know-how provide a context for the know-why (perhaps).
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Explore the pages one at a time and one after another. Pages 13 to 15 cover ratios, simple or multiple. Pages 16 to 18 introduce units in calculations and provide a setting for the discussion and definition of rates and proportionality constants. The coverage of units considers addition and subtraction of like monomials and multiplication and division of monomials, like or unlike, with or without numbers. Here a monomial refers a real multiple of a unit or a product of units. |
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| The Arithmetic Videos (Realplayer format) may be viewed apart from or besides fraction lessons 1 to 12. Exercises on Mostly Fractions will test fraction know-how. The site area Solving Linear Equations may help students visualize fractions while you meet a geometric approach to algebra. I hope you can follow and enjoy the underlying ideas. |
Teachers: Put fraction know-how first to emphasizes development of efficient figuring skills, to build students skills and confidence by showing how they can obtain results in a repeatable and reproducible and therefore verifiable ways. Once the skills and confidence are here, start weaving explanations of why or how fraction operations are justified. Then see the more general discussion of fractions in the Number Theory area.
Before or besides the simple use of formulas in primary school and of the mastery via numerical examples of methods for addition, subtraction, multiplication and division of fractions, the algebraic description of the latter operations on fractions (rules for them) may give a taste of the later shorthand role of letters and symbols in describing associative, commutative and distributive properties of arithmetic with real and complex numbers.
Instead of besides working with division of monomials in variables and addition and subtraction of like monomials in variables, give examples of division and addition of monomials in units of mass, length, time and even money. Such examples are or could be useful in senior high school or college level chemistry and science where calculations with units appear. Besides that, the aforementioned units may have (slightly) more significance to students than letters (at least for students who are ill ease with the use of letters as placeholders for numbers).
Learn More
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See the Purple math links for lessons on arithmetic, algebra and geometry.
See the following chapters in site book Three Skills for Algebra:
8 The Three Skills For Algebras
9 The First Skill
10 Two More Skills
11 Why Shorthand
14 Compound Interest
15 Linear Equations
16 Painless Proofs
17 Pythagoras
This site area on linear equations, Chapters 14 Compound Interest and 15 Linear Equations all provide routes to introduce and extend algebraic way of writing and reasoning. Which one to follow first is a matter of taste. Nibbling at all in parallel is an option until their digestion is complete.
Chapter 14 Compound Interest could be rewritten in terms of compound growth and decay of populations and radioactive material, and and/or connected to exponential growth and decay without any mention of compound interest.
Technical Notes
Hint: If a whole number < 121 is not a divisible by the primes 2, 3, 5 and 7, it is prime. Here you need to remember decimal notation (representation) based rules for recognizing multiples of 2, 3 and 5. Here you need to remember all multiples of 7 < 121, namely 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98, 105, 112 and 119, or the shorter list 49, 91 and 119 of multiples < 121 not divisible by 2, 3 and 5. In general, a number < N2 is prime if that number is not a multiple of each and every prime < N. Why can be can explained in algebraic discussion of number theory. |
Teachers
The new site page Teaching Algebra describes a program to follow.
The ideas here could be woven in early high school or late primary school class for students ages 10 to 15 say. Past mathematics curriculums called for an efficient mastery and comprehension of on paper methods for arithmetic with whole numbers and fractions to serve as a basis for algebra. Recent practice in classroom leaves many students without a fraction sense or comprehension and without the ability to do and understand arithmetic with fractions. The lack of fraction sense after the first year of high school implies failure or a waste of time and energy in further mathematics courses. I kid you not.
Note(1) : The discussion of two term ratios a:b (read a to b) and multiple-term ratios a:b:c (read a to b to c) historically (?) may have come before the discussion and physical interpretation of fractions a/b. Fractions themselves can be identified with twp-term ratios (and may be called ratios) but a three or more term ratio cannot be identified with a single fraction. There-in lies a difference. Some ratios are not fractions.
Note (2): The discussion of ratios here is link to proportionality - In equivalent fractions, the numerators are proportional to the denominators with proportionality constant equal to the common value of the fractions. In equivalent two-term ratios, the the first term is proportional to the second term with proportionality constant equal to common value of the associated equivalent fractions.
Note (3): In the evaluation of formulas for perimeters and areas, etc, students may see letters replaced by numerical values. Seeing such substitutions could be part of the development of the algebraic way of writing and reasoning. The shorthand description of how to add, multiply and divide fractions provide further opportunities to describe or summarize computations that could be done, and a further chance for students to see letters as place holders for numbers, place holders than may be identified with or replaced by numbers in actual computations.
If you are teacher or tutor, I hope you will see how to generate more examples and illustrations. Those here give the main ideas but more examples would help. The examples below are based on division of lengths or rectangles due to the convenience or inconveniences of html in web page production.
Teachers: This a reference. Student may first obtain an efficient mastery of fractions and decimals operations by rote to aid their learning of algebra in a thought-based fashion. See the rest of this site. Then they may revisit explanations like the following to obtain a thought-based understanding of fractions as well. But as a teacher, you should have this thought-based understanding, so that you drop hints of it while most students are learning by rote. In arithmetic, with or without rote learning, students need to learn to figure well in a repeatable and reproducible manner.
More Notes - for experts
Pure mathematics talks about and applies the properties numbers and avoids the use of units. Units can be factored out of calculations with quantities by expressing all in terms of pure numbers. For a distance or length D can be expressed a d meters. So all calculations involving the distance D can be expressed in terms of the pure number d. That process applied to all quantities results in equations or calculations with pure numbers in which units have been factored away. That elimination of units requires the student to choose a system of measurement and convert all data to its units before starting the solution of a problem. But less work is a required if units are carried through calculations in a careful and precise manner in which units conversion are done only when needed and in which the development of algebraic skills with units. The treatment of units in this site area shows how to multiply and divide units to define calculations with products and quotients of units (unit monomials and their fractions involving them), and beyond shows how to add like products and like quotients. Two products or quotients are said to be alike when the exponents of all units in them coincide.
Ratios of a pair of numbers can be identified with fractions. From that identification follows the equivalent properties or equality of ratios involving different terms and the equality of fractions involving different denominators. Yet ratios involving more than two numbers cannot be identified with fractions, as fractions only involve a pair of numbers. But a ratio of N real numbers (say N > 3) can be identified with an ordered N-tuplet (triplet if N = 3) of numbers. Two N-tuplets are equivalent when and only when one is a positive real multiple k of the other. Likewise, two N term ratios are equivalent when and only when there exists a single positive number k, such each term in the first is a positive real multiple k of the corresponding term in the other. That brings the discussion of multiple terms ratios in the realm of coordinates (projective geometry) and away from fractions. Here the equivalent N-tuplets and equivalent N-term ratios altogether form a ray through the origin in N dimensions. More generally, you could allow k to be negative or positive in the foregoing.
Ratios of a pair of quantities can be identified with rates and and extended concept of fractions in which numerator or denominators involve units. Proportion involving N quantities (say N > 3) may be identified with N-tuplets of quantities. Two N-tuplets are equivalent when and only when one is a positive multiple k of the other where k is product or quotient of units and/or real numbers.. Likewise, two N term proportion are equivalent when and only when there exists a single quantity k, given by product or quotient of units and/or real numbers, such each term in the first is a positive real multiple k of the corresponding term in the other. Here the quantity k is called a proportionality constant,
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www.whyslopes.com
Fractions, Ratios, Units, Rates
& Proportionality
Fraction
Starter Lesson
(simplify, multiply, divide & then add or subtract)
An Alternative Starter
Lesson
(take your pick, or try both)
Arithmetic Videos - Real Player Format
Decimal Addition
Methods
Decimal
Subtraction Methods
Decimal
Multiplication Methods
Decimal Division
Methods
Fractions
Primes
Greatest Common
Divisors
Least Common Multiples
Square Root
Simplification
Area Content Summary
- Fraction Starter Lesson
- Real Player Videos on Operations with Primes and Fractions
- Continuous Ruler & Line Segment
model for fractions and operations on fractions - Number Theory Area points to the general model. - Distinction between Ratios and Fractions, a nuance: While binary ratios a:b may be identified with a fraction, triple ratios a:b:c and further multiple ratios cannot.
- Saying how to add and subtract like monomials in units and their powers, and saying how multiply and divide like and unlike monomials leads to fraction like expressions involving units and a framework for discussion rates - ratios of quantities - a framework for handling proportionality constants, and framework for carrying units through calculation in quantitative disciplines
Hint: See site area on solving linear equations to strengthen fraction sense and algebra skills together. Good luck.