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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. |
Chapter 11: Fractions and Divisions
Fractions appear via the concept say of division, the division say of a pie, an amount or a group of objects into separate but equal portions. This leads to the concept of fractions: enjoying a fifth of a pie, consuming two quarters of pie, or being sick on three quarters of a pie. And when there are several pies, one can introduce improper fractions, for instance eating 7 quarters of more than two pies, the latter having being physically divided into quarters. Here fractions can be added or grouped together in a physical sense. Here the mathematical operation of division is linked to the semantic and physical concept or root.
In discussing pies, we observe eating [6/18]th or [4/12]ths of a pie is almost the same as eating a third, only the number of divisions may be different. Like wise, eating 15 out of 20 portions of a pie is the same as each 3 quarters of the pie. Pie based examples like this lead to the idea of physically equivalent fractions and the simplest form of a fraction - the form that requires that the least number of divisions in a pie.
Examples can be generated using marbles, distances, and measures of mass, area and so on. Further, some improper fractions, like [10/2] can be identified with whole numbers. That is, 10 halves of a pie (more precisely of several pies) is equivalent to 5 pies in total. The significance of [11/2] can also be explained. These observations when the numerator and denominator are small, can be made with the aid of simple examples.
A fraction is said to be simplest form when the numerator and denominator have no common divisors. The fraction [60/100] can be reduced to [6/10] and then to [3/5]. The calculation of the greatest common divisor (g.c.d) of two whole numbers finds an application and thus its motivation in this reduction: the simplification of fractions. The calculation of the g.c.d requires mastery of division with decimals and the two further notions of (i) prime numbers and (ii) prime number decomposition.
The addition of fractions can also be introduced by taking pieces of a pie. The question of how to take the fraction [1/3]and also the fraction [2/5] of a pie can be resolved by cutting a pie (or pies) into 15ths. Here students may see that 5 times 3 is fifteen. The question of how to take the fraction [3/4] along with the fraction [1/6] of a pie can also be resolved by cutting the pie into 6 times 4 = 24 equal portions or into 12 portions. The latter number is the least common multiple of the denominators. The addition of fractions together provides motivation for the discussion and computation of the lowest least common multiple (l.c.m) and the computation of the latter via prime number decomposition.
Products of Fractions
Taking [2/5] of a whole pie, an amount or a group of objects, involves a
division of the pie, amount or group into 5 equal portions and then taking two
portions. Now if the original amount is given by [2/3] of a pie, we need to
divide the [2/3] into five equal pieces. Here
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Examples like this may suggest the rule.
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Digression. When the factors [(a)/(b)] and [(c)/(d)] are both in reduced form, a shortcut for the simplification of the product comes from observing the product also equals [(a)/(d)] ·[(c)/(b)], where the fractions may be further reduced or simplified using the l.c.m of a and d, and the l.c.m of c and b.
www.whyslopes.com
Volume 1B, Mathematics Curriculum Notes,Foreword + Chapters 1 to 10 + 12
Book Entrance Inductive Principles 1 Introduction 2 For & Against Math 3 Algebraic Thought 4 Why Slopes & SQRT of -1 5 Books & Articles to Read 6 Unruly Origins of Algebra 6. Axiomatic Civilization 7 Geometry, 2 Ways 8 Modern Instruction 9 The Two Ends 10 The Transition 10 Explaining Logic 10 Explaining Algebra 10 Why Sets in Math. 12 Four Phases Links
Chapter 11: Primary School Mathematics
11 Primary Math 11 Cue Cards 11 Counting 11 Decimals - Addition 11 Decimals -Times 11 Decimals & Subtraction 11 Fractions and Division 11 Notational Conflict 11 Reciprocals Etc 11 Decimals - Ratios 11 Size Comparison 11 Numbers, +ve or -ve 11 Rename < Sign 11 Complex Numbers
Will provide an alternative to Chapter 11 later, most likely in the Parent's Area: Help Your Child or Teen LearnMost students in high school are not heading for calculus, but most topics in high school mathematics are present due to calculus. Preparation for calculus demands their coverage at full strength.
See too, this site 55+, Math Education Essays. Site areas and pages provide pieces of the a Mathematics Education, Jigsaw Puzzle, in formation.
-Inductive principles for course design & delivery require a clear description of where and how skills and concepts may rest on earlier ones, so that difficulties may be explained and remedied by looking for what was missed or forgotten in earlier studies.
Mathematics is a demanding subject. All errors in notation and comprehension need to be identified and corrected. In reading, spelling and writing, students have to learn all the letters in the alphabet, not just some. and memorize spelling. Anything less implies difficulty.
Likewise in mathematics, students have to master key skills and concepts, one at a time and one after another. Anything less implies difficulty.
Modern mathematics curricula introduced an inconsistency into course design and delivery. They did not sanction the use of decimals nor the use of diagrams in skill and concept development but decimal arithmetic and diagrams are needed for student comprehension and for an operational mastery of quantitative skills. That implies the need for an mixed-math curricula based on a systematic development of operational skills, sufficient for applications and sufficient to provide a base & context for the very optional study of pure mathematis.