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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. 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. |
Chapter 14
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Here is a numerical examples with i = 5% and P = 1000 to show how or why the formula works. Observe how the amount at the end of a period is the same as the amount at the start of the next period.
| Period n |
Amount at Start of Period |
Amount of Interest | Amount at end of Period | 103(1.05)n |
| 1 | 1000.00 | 50.00 | 1050.00 | 1050.00 |
| 2 | 1050.00 | 52.50 | 1102.50 | |
| 3 | 1102.50 | |||
| 4 | ||||
| 5 | ||||
| Observe how the amout at the end of a period is equals 100% of the initial amount plus 5% of the initial amount. So the amount at the end each period is 105% of 1.05 times the initial amount. | ||||
| Fill in this table with the aid of a calculator to the nearest penny (two decimal places). Observe the formula use shortens the calculation. Note how the amount at the end of one period becomes the amount at the start of the next. If you do not like to work with interest calculations, turn this whole chapter into a compound population growth model using the values of A = P(1+i)n to nearest whole number as an approximation to the whole number of individuals present in the population. | ||||
The compound interest formula gives an example of a calculation described in algebraic shorthand notation. To use the compound interest formula someone has to explain or show to you the role of each piece of the shorthand. That is done next.
The final compounded amount A on the left-hand side of the compound interest formula can be computed when three numbers are given, namely
We could try to describe the compound interest calculation in words alone.
This description might be a good essay assignment in a language course alongside
the essay of describing in words alone how to tie a shoelace. The task is
formidable. The task should persuade you that the algebraic shorthand notation
has a few space-saving advantages, even if it may be difficult to read aloud in
an understandable way. Formulas like pictures need to be seen to be fully
appreciated. Often, mathematics is better written and not spoken.
Examples of how to use the formula directly and indirectly follow. Try to understand both the numerical (arithmetic) and algebraic solutions.
More Chapter Sections:
www.whyslopes.com
2. Three Skills for AlgebraForeword, Chapters
& Appendices
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
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