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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. |
Decimals and SubtractionGiven a bag of 12 marbles, 5 can be taken out. This leaves a remainder of seven. This subtraction can be done physically. We note that adding those taken to those leftover yields the original number 12. This gives the first physical ideas of subtraction. Besides, we note that it is not possible to take more than 12 marbles from a bag. This physical subtraction has its limitations. Now with decimal numbers, rules for the subtraction of a smaller number, the subtrahend, from a large number can be given and practiced. The results again are repeatable, reproducible and thus verifiable. Beyond this the result of the subtraction can be added to the subtrahend, and the sum should be the original number. This situation provides a corroboration of the marble example above. The operation of addition and subtraction can be used to undo each other: subtracting a group of marbles from a set, and then returning the group to the set leaves the original count unchanged; adding a group of marbles to another set and subtracting the same also leaves the original count unchanged. With whole numbers, the comparison of numbers, what is smaller or greater, depends on the decimal representation. A whole number whose decimal representation contains more digits than another is larger than the other. And when two, decimal representations have the same number of digits, we compare the leading digits (lexicographically) to identify the greater or smaller. The justification comes from the positional basis of decimal notation. Leading digits are assumed to be nonzero. The subtraction of a larger number from a smaller number appears impossible. More precisely, it cannot be understood via the bag of marbles analogy in a simple fashion. Other physical interpretations of subtraction consistent with the marble interpretation are possible. For instance, the partial payment of a debt of ten marbles with a partial payment of seven leaves a debt or residue of three to be paid. As with some merchants of the middle ages, signs can be used to record credits and/or debts. With these, it is possible to visualize subtraction of any whole number from another larger one. This is the first context for subtraction.42
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