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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
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.
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Exercises
Instruction: If your method is correct, but your check says your
answer is wrong, writes OOPs besides your check, and go on. Correct your work
later. The error will be somewhere between the start of your solution and the
end of the check.
1A. (4 points) Express A and B as simplified, improper
fraction: For the calculation of A, cancel common factors
in numerators and denominators before multiplication. For the calculation of B,
use the least common denominator.
A = |
25
27 |
. |
81
100 |
B = |
3
26 |
+ |
7
10 |
|
For A and B, there is 1 point for right answer and 4 points for following
instruction on how to arrive at the answer. Show work! If you use a
calculator (not recommended), show the reasoning needed to arrive at your
solution without its use.
2 . (6 points) Express as a single, simplified, improper
fraction:
B= |
4
5 |
[ |
5
4 |
+ |
35
24 |
+ |
2
7 |
] |
C = |
8
15 |
[ |
25
4 |
+ |
75
8 |
+ |
15
2 |
] |
3A. (4 points) Solve the (scrambled) triangular system
3x + 2y + z = 65
3y = 45
6x + y = 75
3B. (4 points) Solve the (scrambled) triangular system
2x + 3y + 4z = 21
8y + 2z = 26
5y = 15
4. (Bonus 2 points) John has twice as many apples as Harry.
Harry and John together have 18 apples. Introduce notation for the number
of apples each has. Write down a system of two equations in two unknowns that
would allow you to find the number of apples each has. Do not solve the
system.
5. (a) (8 points) Solve for x and y - obtain
a literal solution.
ax = p
bx + cy = q
(b) (2 points) Verify (check) your solution.
(c)(2 points) Use your formulas for the solution in (a) to solve
4x = 10
2x + 2y = 8
6. (12 points) Solve
3x + 2y = 17
2x - y = 2
7. (10 points) Solve
x - 2y = 4
-3x + y = 3
8. (12 points) Solve
x + y + z= 12
2x + 3y + 5z = 32
6x + 7y + z =64
9. (10 points) Solve - note minus sign in
each equation.
-x + y + z = 10
x - y + z = 8
x + y - z = 6
x + y + z - w =14
10. (4 points) (i) Show by substitution
that the formulas
x = |
dg
ad-bc |
and y = |
-cg
ad-bc |
|
satisfy (solve) the system
ax + by = g
cx + dy = 0
(2 points) (ii) Give values
for coefficients a, b, c, d, and g to make the system in (i) the same
as
4x + y = 10
x + 4y = 0
(4 points) (iii) Use the formulas in (i) to solve
the system in (ii)
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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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