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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
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
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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.
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6. Exponents and Radicals
Students have seen the definition of an for integers n and real
numbers a in working with polynomials and rational functions. The next
step in the development of their algebraic and computational skills is to
provide a coherent exposition of (i) powers ax where the base a is
positive and the exponent x is real, and (ii) n-th roots (radicals) for n
even or odd, along with properties of these computations or functions.
Examples: Two pages Compound
Growth and Decay and Compound
Growth and Decay -More show how exponents and radical occur in the direct
and indirect use of compound growth and decay formulas involving money (the
compound interest formula) and biology. One of the indirect uses uses
and introduces a need for logarithms. Related topics in physics or secondary V
mathematics include decay and growth rates, half-lives and doubling for
continuous compounding in biology, radioactivity, investments and loans.
Saying how to calculate an number or quantity defines it.
And if the calculation of a number or quantity is described in different
ways, two or more, all the different ways have to agree, or else there is an
inconsistency, and the number or quantity in question is not defined.
Alternate (Higher Level Development): Two
pages logs and exponentials and
on Logs and Exponentials
Summary describe how to calculate powers and radicals with the
natural logs and the exponential function. Showing how to
calculate powers and radicals with the aid of the natural
logarithm and exponential function makes the domain of definition of
powers and radical clear and obvious. That simplifies the treatment of
exponents and radicals at the expense of assuming the properties of natural
logarithm and the exponential function. It provides or agrees with the pure
mathematics (analysis or advanced calculus) alternative to the calculator
version.
Related Topics: (i) Rationalization of radical expressions
without and the with the use of conjugate expressions Use of latter is
related to the Difference of Two Squares, and almost represents an
inverse operation to factoring via the difference of two squares.
First Version - a calculator viewpoint
Here we may rely on calculator use except in some numerical circumstances
where the n-th root or power has an exact representation. See exercise 4 below.
- Try section 2.4 in this set of Arithmetic
Review Problems. from site Volume 2, Three Skills for Algebra.
- In numerical exercises or illustrations, first table of values
can be used to plot points on the graphs y = x2 and y
= x3 and second the rest of the graphs can be obtained by
interpolation. The horizontal line method for solving equations from their
graphs implies (i) that the equation x2 = a has one
or two solutions for a > 0 and none for a < 0; and (ii) that
the equation x3 = a has one solution for each real
number a, and that further if a < 0 and X3
= |a| then x = - X.
Teachers: Earlier exercises with graphs of y = x2
and y = x3 over the intervals [-2,2] and [-3,3] may be useful
here. If we put exponents and radicals before functions and relations, there
would be need to discuss graphing and the mixed or impure mathematics
assumptions about the interpolation of calculated points in order to obtain
the full graphs for formulas y = x2 and y = x3,
etc.
- Numerical exercises or illustrations in the aforementioned Arithmetic
Review Problems. imply the n-th root of b > 0 be written
in the power form
x = b(1/n) = exp((1/n) ln(b))
is a positive solution of the equation xn = b
Now for n real and n an odd natural number, the n-th root of a nonzero
real number b is given by
b(1/n) = sign(b) exp((1/n) ln(|b| ))
The latter agrees with b(1/n) = exp((1/n) ln(b)) when b
> 0, and extends the formula when b < 0.
- Square
Roots - How to do exact calculations with Webvideos.. Examples here show
how to simplify or represent square and cube roots of whole numbers
with and without the aid of prime number factorization of the latter. The
simplification here may be a cosmetic convention or fashion in mathematics
that leads students and teachers to answers of the same form, the so-called
simplified form.
- Problem: Show a link between the graph of y = ln(x) and y = ex
via the following steps - an illustration of the reflection across a certain
line between the graph of a function and its inverse.
(2 points) (i) Use the ln(x) button on your
calculator to fill in the table (or a copy of it)
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x
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1/6 |
0.2 |
0.25 |
1/3 |
0.5 |
1 |
2 |
3 |
4 |
5 |
6 |
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ln(x)
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0
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(3 points) (ii) Use the ex button on
your calculator to complete in the table (or a copy of it).
| x |
-1.792
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-1.61 |
-1.39 |
-1.099 |
-0.693 |
0 |
0.693 |
1.099 |
1.39 |
1.61 |
1.79 |
| exp(x) |
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1
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(5 points) (iii) Draw the straight
line y = x from [-2,-2] to [6,6] on graph paper. Then on the same
graph, use the above points to sketch graph of y = ln(x) for x in the
interval [1/6, 6] and y = exp(x) = ex for x in
the interval [-1.79, 1.79]. Label all curves.
Links:
The next page: 6. More on
Exponents & Radicals continues the numerical viewpoint (and so provides
motivation for theory below
More Pages on exponents and radicals : [ Compound Growth & Decay ] [ More Growth & Decay ] [ Logs and Exponents ] [ Logs & Exponentials - Summary ]
Bon Appetit.
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www.whyslopes.com
Lesson & Lesson Plans for
Sec IV (Maths 436)
a reference for learning and teaching functions, polynomials, solving linear
systems,
powers + exponents + bases + radicals (roots) , quadratic formulas, equations
of straight lines
1A. Master Logic
1B. Problems Solving Method
2A Solve Linear Equations i
2B.Solve Linear Equation II
2C Use Equal Sign Properly
2D. Perfect Arithmetic Skills
3 Words & Symbols
3 Goals to Set for Students
4 Use Equations Backwardly
5. Master Functions & Relations
6. Exponents & Radicals I
6 Exponents & Radicals II
7. Straight Lines
8. Polynomials (x,/,+/-)
9. Quadratics
10 Prove it
13 Similarity Scale Factors
12 Trig & Triangles
14 Statistics
MEQ Intermediate Objectives
Remarks for Teachers
Sit down and study - no one else can do that for you.
Advice and Directions
What to do in School & Why
How
to Study Maths & Why
Preparing
for Science
Good News: If you can learn to follow a multi-step
methods in any subject precisely, you should be able to do so in other
subjects, as well. Hint: Start with arithmetic
Words Before Symbols:
What is a Variable?
Level: Secondary II to VI, or Grades 7 to 12)
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 lesson
Arithmetic Videos
Fractions
Primes
Greatest Common Divisors
Least Common Multiples
Square Root Simplification
Arithmetic Videos
Decimal Addition Methods
Decimal
Subtraction Methods
Decimal
Multiplication Methods
Decimal Division Methods
Fraction
Starter Lesson
(simplify, multiply, divide &
then add or subtract)
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