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.
Slopes and Velocity
Distance Versus Time
Signed Distance along a Road
Suppose in traveling along a road, position at time t is given say by d = f(t). The coordinate d gives a signed distance to the origin or point of reference. Assume positions on one side of this origin have a positive d-coordinate and the positions on the other side of this origin have a negative d coordinate. The absolute value or magnitude of d, that is |d|, gives the unsigned distance to the origin or point of reference. The coordinate d will just be called the distance or signed distance hereafter.
Varying Velocity Example
Problem: Graph the distance d to the origin of a path versus time t for the following journey of Harry Snail.
- At two o'clock in the afternoon, he is 50 km west of the origin, he travels further west at 100 km/hr. He drives at this speed for 1[1/2] hours.
- At half past three in the afternoon, he stops for one-half hour.
- He then drives eastward at 75 km/hr for the next two hours and then stops for another 2 hours.
Solution.
The trip has five segments. Comments on each segment or portion follow.
1. Before his trip begins, he could be stationary, that is, not moving. This possibility, a suspicion which cannot be confirmed, is represented by the horizontal dashed line. The slope of this speculative dashed line is
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Footnote: When in doubt leave out, is a rule to follow in solutions of problems. Or, when in doubt say so, to show what is certain and what is not. Your credibility is at stake. Indicating precisely where you are guessing in a solution, identifies a question to be answered later by yourself or your instructor. And in marking assignments or tests, I would be less severe with mistakes explicitly identified as guesses than I would be with guesses deceptively presented as sure knowledge. Caution: Not all instructors will have this opinion.
2. The first described portion of the trip starts at the point A = (2 hrs,50km). He reaches the point B = ([3½] hr, 200km) after traveling at 100 kilometer per hour for one and a half hours. The slope of this portion of the trip m = 100[ km/hr] = 100 km per hour.
3. The second described portion of the trip lasts for one half hour. By remaining stopped (stationary) for [1/2] hour, his (t,d) coordinate changes from B = (3[1/2]hr,200km) to C = (4hr,200km). The slope or speed m here is again zero.
4. By traveling at 75 kilometers per hour back towards the origin for two hours, his position coordinates (t,d) change from C = (4hr,200km) to D = (6hr,50km). The slope
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5. Finally, he does not move for 2 hours. This gives the last portion of the graph with d = 50km and slope m = 0.
www.whyslopes.com
Fractions, Ratios, Units, Rates & ProportionalityFraction Starter Lesson
(simplify, multiply, divide & then add or subtract)
An Alternative Starter Lesson
(take your pick, or try both)
Area Map & Intro Fraction Starter Lesson A Fraction Starter Lesson B 1 What is a Fraction 2 Multiplication I 3 Multiplication II 4 Multiplication III 5 Equivalent Fractions 6. Mixed Numbers 7 Comparison 8 Addition I 9 Addition II 10 Addition III 11 Multiplication IV 12 Division 13 Two Term Ratios 14 Implied Ratios 15 Multiple Ratios 16 Units in Arithmetic 16 Longer Explanation 16 Change Units 16 Products of Quantities 16. Fractions with Units 16. Division+Reciprocals 17 Proportionality 17 Examples 18 Rates & Slopes EGs 18 Constant Rate 18 Varying Rate 18 Velocity Calc., EGs 18 Changing Units 18 Slopes and Units 18 Slopes, No Units 19 RealPlayer Videos Links Arithmetic Videos - Real Player Format
Decimal Addition
Methods
Decimal Subtraction Methods
Decimal Multiplication Methods
Decimal Division
Methods
Fractions
Primes
Greatest Common
Divisors
Least Common Multiples
Square Root
Simplification
Area Content Summary
- Fraction Starter Lesson
- Real Player Videos on Operations with Primes and Fractions
- Continuous Ruler & Line Segment
model for fractions and operations on fractions - Number Theory Area points to the general model. - Distinction between Ratios and Fractions, a nuance: While binary ratios a:b may be identified with a fraction, triple ratios a:b:c and further multiple ratios cannot.
- Saying how to add and subtract like monomials in units and their powers, and saying how multiply and divide like and unlike monomials leads to fraction like expressions involving units and a framework for discussion rates - ratios of quantities - a framework for handling proportionality constants, and framework for carrying units through calculation in quantitative disciplines
Hint: See site area on solving linear equations to strengthen fraction sense and algebra skills together. Good luck.