|
YOU are better than YOU think. Show yourself how:
|
-/[]\- 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;
|
-/[]\- 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. |
Area Development of Multiplication
|
| 10b3 | 4b2 | 6b | 8 | |
| 50 | ||||
| 6b | ||||
| 4b2 |
and compute the number of subrectangles in each. The latter operation first gives
| 10b3 | 4b2 | 6b | 8 | |
| 50 | 50 x 10b3 | 50 x 4b2 | 50 x 6b | 50 x 8 |
| 6b | 6b x 10b3 | 6b x 4b2 | 6b x 6b | 6b x 8 |
| 4b2 | 4b2 x 10b3 | 4b2 x 4b2 | 4b2 x 6b | 4b2 x 8 |
and then this
| 10b3 | 4b2 | 6b | 8 | |
| 50 | 500b3 | 200b2 | 300b | 300 |
| 6b | 60b4 | 24b3 | 36b2 | 48b |
| 4b2 | 40b5 | 16b4 | 24b3 | 32b2 |
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 P = (50+ 6b +4b2)(8+ 6 b + 4b2+ 10b3)
= 300 + (300 + 48)b + (200 + 36 + 32)b2 + (500+24+24)b3 + (60+16)b4 + 40b5
= 300 + 348b + 268b2 + 548b3 + 76b4 + 40b5
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
P = (50+ 6b +4b2)(8+ 6 b + 4b2+ 10b3) =
300 + 348b + 268b2 + 548b3 + 76b4
+ 40b5
That is computation of the left hand side for a given value of b gives the same result as computation of the right.
The above geometric approach suggests a table method:
| × | 10b3 | 4b2 | 6b | 8 |
| 50 | 500b3 | 200b2 | 300b | 300 |
| 6b | 60b4 | 24b3 | 36b2 | 48b |
| 4b2 | 40b5 | 16b4 | 24b3 | 32b2 |
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.
Here we modify the table approach and dedicate a column to each power of b as follows
| × | 10b3 | 4b2 | 6b | 8 | ||
| 50 | 500b3 | 200b2 | 300b | 300 | ||
| 6b | 60b4 | 24b3 | 36b2 | 48b | ||
| 4b2 | 40b5 | 16b4 | 24b3 | 32b2 |
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.
| × | 10b3 | 4b2 | 6b | 8 | ||
| 50 | 500b3 | 200b2 | 300b | 300 | ||
| 6b | 60b4 | 24b3 | 36b2 | 48b | ||
| 4b2 | 40b5 | 16b4 | 24b3 | 32b2 | ||
| P= | 40b5 | + 76b4 | + 548b3 | + 268b2 | +348b | +300 |
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.
| × | 8 | 6b | 4b2 | 10b3 | ||
| 50 | 300 | 300b | 200b2 | 500b3 | ||
| 6b | 48b | 36b2 | 24b3 | 60b4 | ||
| 4b2 | 32b2 | 24b3 | 16b4 | 40b5 | ||
| P= | 300 | + 348b | + 268b2 | +548b3 | +76b4 | +40b5 |
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 + 6 b + 4b2
+ 10b3
50 + 6b + 4b2
×
300 + 300b + 200b2 + 500b3
(product of top row with 50)
48b + 36b2 + 24b3 +
60b4
(product of first row with 6b)
32b2 + 24b3 + 16b4
+ 40b5 (product of first row with 4b2)
------------------------------------------------------------------------ +
300 + 348b + 268b2 + 548b3 + 76b4
+ 40b5
------------------------------------------------------------------------
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.
In the above calculation, we see how to add polynomials
(300 + 300b + 200b2 + 500b3 ) + (48b
+ 36b2 + 24b3 + 60b4 )
+ (32b2 + 24b3 + 16b4
+ 40b5 )
via an addition methods in which like terms are aligned in columns as follows.
300 + 300b + 200b2 + 500b3
48b + 36b2 + 24b3 +
60b4
32b2 + 24b3 + 16b4
+ 40b5
------------------------------------------------------------------------ +
300 + 348b + 268b2 + 548b3 + 76b4
+ 40b5
------------------------------------------------------------------------
So one pedagogical methods is to developed the column method for multiplication first, and then introduce column methods for addition and subtraction of polynomials, second.
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, Some conversion need to be considered. See Distributive Law Consequences
www.whyslopes.com
Volume 2, Three Skills for Algebra -Preview, starter & further lessons for logic and algebra to (i) improve work & study skills; (ii) to to ease or avoid algebra (math) fears & difficulties; and (iii) to fill gaps in the exposition of mathematics.
Foreword, Chapters and Appendices follow.
Foreword 1. Introduction 2. Implication Rules 3. Chains of Reason 4. Romeo and Juliet 4. Induction Mathematical 5 Knowledge Islands 6 Old Language 7 Arith Skill Check 7. The Next Chapters 8 The Three Skills 8 VNR-Concise-Encyclopedia PS. What is a Variable 9. Algebra Talk 10 Two More Skills 11 Why Shorthand 12 Shorthand Usage 13 What's Next 14 Compound Interest 15 Linear Equations PS I. Distributive Law PS II. Polynomials 16 Painless Proofs 17 Pythagoras 18 Rules of Algebra 19 Functions & Sets 20 Degrees & Radians 21 What's Next 22. Arith & Geometric Sums 23 Summation Notation 24 Your Money 25 Induction & Recursion 26 What's Next 27 Pronouns in Logic 28 Occurrence Tables 29 Contrapositive 30 Truth Tables 31 Indirect Reason A. Advice For Learning
Real Player Videos
Perfect arithmetic skills with whole numbers & fractions after or besides chapters 1 to 14.
Arithmetic Videos Summary Addition with Decimals Subtraction with Decimals Multiplication with Decimals Fraction Arithmetic Recognizing Primes Long Division for Decimals Square Root Simplification Greatest Common Divisors Least Common Multiples Words Before Symbols:
What is a Variable?
Introduction
Variation between Examples
Variation of Letters
A letter denotes a variable
Cases of Double Variation
Three Notions of a Variable
Constants, Parameters
& Variables
Talking about numbers
Dependent or Independent
Variable, a Matter of Choice
Complex number: starter lessonSolving Linear Equations:
A. Letters and Lengths
B. & C. Solving Linear Eq'ns
with stick diagrams.
(i) x + 20 = 29
(ii) 2x + 5 = 20
(iii) 3x + 10 = 32
(iv) 5a + 16 = 3a+ 24
(v) (½)x + 8 = 24½
(vI) (¾)a + 16 = (¼)a+ 24
(vii) (¾)q + 17 = 32
(viii) 13 =[2/3]x +7 twice
(x) Animated Examples
(i) Integral Coefficients (A)
(ii) Integral Coefficients (B)
(iii) Fractional Coefficients
(iv) With Parameters
Problem Solving with Linear
Equations in one or many
unknowns, and in essentially
one unknown - Symbols before
words.
C. Solving Linear Eq'ns
without
Stick Diagrams
D. Problems in
essentially one unknown
E: 2D Systems - Sub Methods.
F. Larger Systems
|
www.whyslopes.com
|