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-/[]\- Logic chapters 1 to 5 re- appear not in sequence, as is or longer, in Volume 1A, Pattern Based Reason, Bon Appetite. Logic
Mastery Tell students that Logic mastery makes the hard, easier. Logic mastery leads to better, stronger and richer comprehension. Logic mastery improves reading and writing. Logic mastery ease learning difficulties. Logic mastery gives a headstart. In sum, logic mastery will develops critical thinking, improve reading and writing, and give a firmer base for work and studies at many levels. Good luck. After logic, (a) continue reading Three Skills for Algebra, chapters 8 to 14 and do so alongside site area on solving liinear Equations ; or (b) see this calculus starter lesson and Volume 3, Why Slopes & More Math, chapters 2 to 6. In Volume 2, Three Skills for Algebra, a 4th skill for algebra appears in Chapter 14. It provides a unifying theme for high school mathematics - equations and formulas may be used forwards and backwards, directly and indirectly, numerically in arithmetic solutions & with literals in algebraic solutions.
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-/[]\- What may be learnt and when depends on how skills and concepts are developed. Making the hard easier and clearer will allow earlier & richer development of skills and concepts. Try the Twiddla
Whiteboard. In principle, it allows
to people to draw and chat together online on a copy of this webpage or a clean
sheet. The chat may be via text or audio. Visit www.twiddla.com
to set up whiteboards to work with the webpage of your choice. |
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Multiplication by Reciprocals. The number of times 4 pies goes into 11 pies can be obtained by dividing the 11 into groups of four and then counting them. The answer is two and a three quarter times, or the improper fraction [11/1]×[1/4] = [11/4] simplified.
The number of times 3 goes into 7 pies can be obtained by dividing the 7 into groups of three and then counting them. The answer is two and a third times, or the improper fraction [7/3] simplified.
The number of times 1.5 or [3/2] pies goes into 5 pies can be visualized by dividing the 5 each into halves, and then dividing the resulting halves into groups of three, and then counting the groups. The answer is then three groups with one half leftover or the answer is three and a third times. The third leftover is one third of a group of three halves. The latter gives the same result as [5 ×2/3] = [5/1] ×[2/3] = [10/3] = 3[1/3] as before. So division of 5 by [3/2] gives the same result as multiplication by the reciprocal [2/3].
The foregoing [4] inductively suggests that division by a fraction gives the same result as multiplication with the fraction’s reciprocal.
Now an answer to the question, what fraction when multiplied by [5/2] yields the result [2/3] is obtained as follows. Multiply the desired result [2/3] by the reciprocal [2/5] of [5/2], and simplify. The answer to the question is [2/3]×[2/5] = [4/15]. This may provide some motivation for the further or later notion that division and multiplication (thanks to the associative law) are inverse operations.
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.
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