YOU are better than YOU think. Show yourself how:
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Read logic
chapters 1 to 5 in online volume Three
Skills for Algebra for greater skills & confidence
in work |
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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
Amazing, Amusing, Amorous, Delicious, Delightful, Edifying,
Strengthening Elixir.
It eases work & learning difficulties Makes the hard easier. Opens eyes.
Leads to greater precision.
in reading and
writing
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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Caution: Site advice is approximately correct, for some circumstances, not all. That leaves room for thought |
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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.
For online automated help in senior high school maths & calculus,
visit quickmath.com For Automatic
Calculus and Algebra Help with derivatives, integrals, graphs, linear equations,
matrix algebra, visit calc101.com
With overlap, each site quickmath
& calc101offers a different range of
services, some free, some not, all based on webmathematica. Good luck.
A Cross-Country Skier and Her Trail
[Play Video] 2¼ minutes: Slope Interpretation for a 2D ski hill y = f(x). (Appeared earlier in Why Slopes Appetizer)Meet the cross-country skier, Barbara:
She has only one ski. Alternatively, you can imagine she always travels with both skis parallel. Travel with one ski was the way in which both alpine and cross-country skiing began. Also meet the Jack Rabbit ski trail y = h(x) (see below) which she skis, always in the direction ® from left to right.5 That is, she travels in the direction of increasing x.
5 Footnote: The slope to a curve at point can be approximated by taking the slope of a short line segment which has one end at the point and another end also on the curve. This approximation should get better as the line segment gets shorter. The finite limiting value of this approximation, should it exist, is taken to be the slope. Before discussing this approximation any further, we will make the improper assumption that the slope of a short ski placed on the graph of y = f(x), or the graph of one quantity versus another, is the slope to the graph. This will allow some exploration of why slopes are studied.
Imagine or suppose that the hill is smooth enough, so that, at most points, a ski can lie flat against the hill surface. The slope beneath a foot or ski gives what should be the slope of a tangent line to the hill. The slope of a ski can in principle be measured any time by freezing a skier in place, or equivalently taking a photograph (snapshot) and then measuring the slope from the photograph.6
6 footnote What happens if Barbara goes up and over a sharp peak? As she gets to the top and pivots from the uphill to downhill side, the slope of her ski goes from positive to negative.
The vertical line segment in the above graph represents a jump or cliff. The above diagram strictly speaking consists of the graph of a function y = h(x) plus a vertical portion to represent a ski jump.
- Above the point x = a on the horizontal axis, the height above the x axis of her ski midpoint is y = h(a). We will call h(x), the height function.
- The height function h(x) might be measured or computed from a formula, a map, or a graph, such as the one shown above. Or, in speaking of a height function h(x), we could leave its values unmeasured, not computed or unknown.
| The slope m is | For Motion |
| > 0 (or positive) | Uphill |
| < 0 (or negative) | Downhill |
| = 0 (or zero) | Horizontal |
The shorthand for the word positive is +ve. The shorthand for negative is similarly -ve.
A Skier in Motion
Immediately below are a few snapshots of Barbara on another portion of the skill trails and hills y = h(x). In each snapshot, the slope of the ski is assumed to be equal to the slope of the hill at the ski midpoint.
In the diagram observe:
- At x = c, the slope m > 0 and she is moving uphill: her height is increasing.
- At x = b, the slope m < 0 and she is moving downhill: her height is decreasing.
- At x = a, the skier is approaching the top of the hill. What is the sign of the ski slope before, at and after the top of this smooth hill?
www.whyslopes.com
Volume 3, Why Slopes and More Math - Preview, starter & further lessons for calculus to ease or avoid algebra shock in instruction & self-instructionForeword, One Calculus preview and Online Chapters: (V) signals video (RealPlayer Format) to watchChapters 2 to 6: offer a very simple preview of calculus and a context for earlier study of slopes and factored polynomials
Area Entrance & Hub Foreword Chapter Descriptions 1. Introduction 2. Calculus Starter Lesson 2. Second Preview Begins 2 Skier in Motion (V) 2 The Skier (V) 2. Position Dependent (V) 3 Slope & Extrema (V) 4 Single Factor Analysis (V) 4 Two Factor (V) 4 More Factors (V) 4 With Divisors (V) 5 Maxima & Minima Tests 6 Jumps & Discontinuities 8 Review (optional) 9 On Calculus Studies 11 Slope of Slope 13 Acceleration 14 Limits & Error Control (V) 14 Limit of a Fn. 14. Limited Error Control 14 Signif. Digits 14 Cauchy Limits 14 Sequence Limits 14 Decimal Arith. 15 What is Slope (V) 15 Slope Calculation (V) 15 Slope, a Limit 15 Tangent Lines 15 Linear Approx., 15 Limits via Algebra (V) 15 Recap. PS.Chain Rule for Polys PS Chain Rule- General (V) - PS More Chain Rule (V) PS - Sign Analysis (V) 16 What is Velocity 17 What is Area 18 Integration 18 Area Calculation 18 Fn DefN, 6 Ways 19 Logs & Powers 19 Natural Log. 19 Exponential Fn. 20 What's Next 21 Add Vectors 22 Complex #'s 23 Complex #'s 23 Trig Identity 23 Proofs of. 24 Complex Logs etc
Units in Calculations:
7 Velocity 7 Varying Velocity Example 7. Velocity Calculation 7 Changing Units 7 Same Velocity Motions 10 Slopes without Units. 10 Units & Slopes 10 Units in Cost vs. Quantity 10 How Units Appear 10 Unit Elimination 10 Partial Elimination 10 Interest & Units 12 More on Units Content Guide
Enriched material: The Appendices of Volume 3 are located in the Real Analysis Area.
Pigeon Hole Principle
Constant Difference Thm
Continuous Functions
Rational Functions
Mean Value Theorem
One Side Range Theorem
Range On One Side Theorem
Integration & Lipschitz
Continuity
These appendices continue the
decimal viewpoint of limits, error
control and continuity begun
in Chapter 14. The One Sided
Range Theorem is a postscript,
not in printed version.
Online Volume 2, Three Skills for Algebra, Chapters 1 to 25 - skip 18., verbalizes and explains key skills and concepts, those needed in calculus, again to make the hard easier. A visual understanding of complex numbers may help - serve as back ground info, in partial fraction decomposition.