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YOU are better than YOU think. Show yourself how:
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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 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;
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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.
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Generalized Distributive LawVisual Aids and Column Multiplication MethodsThe association of products of whole numbers with counting subrectanglar divisions of a larger rectangle leads to visual aids for developing and remembering the generalized distributive law for whole numbers. We will see the same visual aids with refinement for fractions, proper or not, and nonnegative real numbers.
First Example (Alternative to Foil Method)How do we express a product NM = (a+b)(c+d) as a expression of the terms a, b, c and d? Solution: The number NM gives the number of subrectangles in the blue rectangle below.
The BLUE rectangle can be divided into 4 intermediate size subrectangles
Each intermediate rectangle labelled I to VI contains a number of the NM subrectangles we are counting. Each of the NM subrectangles we are counting belongs to one and only one of the intermediate size rectangles. See below.
Column Multiplication MethodWe have use the product rule for counting subrectangles to find the number of subrectangles of the total MN in each intermediate subrectangle. The intermediate size rectangles lead to four groups of subrectangles with counts ac, ad, bc and bd we can be add to obtain the total number MN.
So MN= (a+b)(c+d) = ab+ad + bc + bd. We may introduce a column multiplication method to obtain the product
Here ab+ad + bc + bd and (a+b)(c+d) give two different ways to compute a single number, the number of subrectangles MN. The equality of two different ways to compute a single number gives many formulas in mathematics. Second ExampleHow do we express a product NM = (a+b+c)(e+f) as a expression of the terms a to f giving each factor. Solution: The number NM gives the number of subrectangles in the blue rectangle below.
The BLUE rectangle can be divided into 6 intermediate size subrectangles
Each intermediate rectangle labelled I to VI contains a number of the NM subrectangles we are counting. Each of the NM subrectangles we are counting belongs to one and only one of the intermediate size rectangles. See below.
We have use the product rule for counting subrectangles to find the number of subrectangles of the total MN in each intermediate subrectangle. The intermediate size rectangles lead to six groups of subrectangles with counts ea, eb, ec, fa, fb and fe we can be added in any order to obtain the total number MN.
So MN= (a+b+c)(e+f) = ea + eb + ec+ fa+ fb +fc We may introduce a column multiplication method to obtain the above product
Column Methods for MultiplicationWe may replace the rectangles above by multiplication tables in which the terms in the factors provide the initial entries in rows and columns.
Further table entries are obtained via products. The foregoing can be tabulated as a column method for multiplication: a + b + c Third Example - Product of PolynomialsUse rectangles to expand the product P = (50+ 6b +4b2)(8+ 6 b + 4b2+ 10b3) where b is a whole number. Solution: Form intermediate size rectangles
and compute the number of subrectangles in each. The latter operation first gives
and then this
The foregoing could have been done in one step. Now instead of add the intermediate rows by rows, we will add them along the diagonals in the powers of b are identical. If we have made no mistakes, the foregoing gives the result
Whether or not the powers of b increase or decrease in the result is a cosmetic convention - some prefer one way, others the other, and some either. Our conclusion follows. The product That is computation of the left hand side for a given value of b gives the same result as computation of the right. Multiplication Table Approach. The above geometric approach suggests a table method:
which holds for real number b as well as whole numbers since the generalized distributive law holds for real numbers as well as whole numbers (why to come later). Here the rows and columns of the table come from the terms in the factors of the product to be computed, here (50+ 6b +4b2)(8+ 6 b + 4b2+ 10b3) while the entries inside list or tabulate the products of pairs of terms, one from each factor. Column Method for Multiplication of PolynomialsHere we modify the table approach and dedicate a column to each power of b as follows
In this modified table approach, we compute all possible products as before, but align the products in each row according to their power of b. That makes addition and collecting like powers of b (with the aid of the distributive law) simpler. We add an extra row for the sum.
So the product P = 300 + 348b + 268b2 + 548b3 + 76b4 + 40b5 as before In retrospect, the table (and column multiplication below) will be easier to do if we arrange the powers of b in ascending (that is, increasing) order along the top row.
and align the left column at the left of the page (or page margin), so that we may write from left to right. Here again all possible products appear, but aligned in columns. Finally, let us introduce or switch to the table method for multiplication. Column Method for Products of Polynomials 8 + 6b + 4b2
+ 10b3 Note: The ascending order appears to work best as the position of the intermediate products shifts to the right away from the left margin where the computation begins. Do you see how each entry in each row of the column method corresponds to an entry in the rectangular approach and the two preceding tabular approaches to the computation of the product? All is a consequence of the distributive seen algebraically or geometrically visualized. The case where the variable b is replaced by the number 10 leads to a justification of the column method for multiplication of whole numbers using their decimal representation with powers of 10 written in decreasing order rather than increasing. |
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