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Volume 1B |
| Vol. 1B. Math
Curriculum Notes, describes inductive principles for |
Lesson Planning Available (some one U know will need it sooner or later) |
This explanation of What is a Variable differs from the common view (letters in algebra are variables) and is compatible with the pure math view (variable in maths are placeholders for element of a set).
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Chapter 11: Positive and Negative Numbers
the coordinate perspective
Thermometers with temperatures above or below a reference point labelled zero provide an example of a numbered line where the numbers have positive and negative signs in front and a physical significance. A positive temperature indicates so many steps or units above the zero mark while a negative sign indicates some many steps below the zero mark. The addition of a positive number now corresponds to be and may be defined as an upward movement of so many steps. The addition of a negative number and the subtraction of a positive number corresponds to a downward movement. These additions and subtractions can be done at any point on the scale.
The subtraction of a negative number in the first instance is undefined. But one can define negation for a number as the reversal of direction, and regard subtraction of a number as the addition of it negation. Students can be shown that this applies to the subtraction of a positive number before the subtraction of negative numbers is considered.
Two negations or reversals further result in the original number. Here the negative -a of a number a will be the number or point obtained by subtracting the number a from 0. The foregoing provides a physical concept of addition and subtraction.
Note that multiplication of a number a by a whole number n, can be viewed as the result of the addition of a to itself, a whole number n times. Multiplication by nonnegative proper and improper fractions, and then positive decimals can also be physically interpreted. Next every negative number b is the negation of a positive number a. In consequence, multiplication by a negative number b = -a can be defined as the multiplication by a followed by a negation (reversal of direction).
Coordinates along a horizontal line (the real numbers) can be represented by signed decimal numbers
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More on Subtraction.
Subtraction of n from m yields a number k with the property that n+k = m. When m and n are given, subtraction of n from m answers the question, what number k when added to n yields m? Examples for subtraction when m > n can be revisited in this context. Such examples imply that k equal m - n when n + k = m.
The calculation k = -(n-m) can be given when n > m as a means for computing k = m-n. Then again n + k = m. Here n-m is computed using previously taught methods for the subtraction of decimals or fractions. (When m and n are fractions, subtraction answers the question: what fraction k when added to n gives m? The fraction can be signed.)
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