YOU are better than YOU think. Show yourself  how:  

      |      
//  _   _ \\
/\             /\
  <|  (o)   (o)   |> 
 \     | |      / 

 -/[]\- 
||
   / \_ 
 ||||||||||||||||||||||||||||

 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;

      |      
//  _   _ \\
/\             /\
<|   (o)   (o)  |> 
|      | |     |
   \             /   
\    =   /

 -/[]\- 
||
  _ / \     
 ||||||||||||||||||||||||||||

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.

6. Exponents and Radicals

Questions to provide an Arithmetic Based Introduction

In this introduction, 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.

1. Try section 2.4 in this set of  Arithmetic Review Problems. from site Volume 2, Three Skills for Algebra.
   
2. 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.
   
3. 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.
   
4. 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.
   
5. Problem:  Show a link between the graph of y = ln(x) and y = e^x  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) 

   x	1/6	0.2	0.25	1/3	0.5	1	2	3	4	5	6
   ln(x)						0

    (3 points)  (ii) Use the e^x button on your calculator to complete in the table (or a copy of it). 

   x	-1.792	-1.61	-1.39	-1.099	-0.693	0	0.693	1.099	1.39	1.61	1.79
   exp(x)						1

    (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) =  e^x  for x in the interval [-1.79, 1.79]. Label all curves.  

More Pages on exponents and radicals : Bon Appetit.

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
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)

e || Algèbre || Arithmetique || Logique | | 

All trademarks and copyrights on this page are owned by their respective owners.
Copyright to comments & contributions are owned by the Poster. 
The Rest © 1995 onward by site author,   Alan Selby, a 1983 McGill. Ph. D. in mathematics
All Rights Reserved.