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whyslopes.com
Entrance Level
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For Montreal Students: Head
Start Math Tutoring is available from the site author.
YOU are better than YOU think. Show yourself how:
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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
Do not leave here without it - 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
linear2007 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.
George Orwell: Is it
nonsense for arts and disciplines based on and respected for carefully
mastery of rules and methods, alone and combined, to face education reforms
based on the supposition that mastery of rules and methods is not a sign of
intelligence. Would you like to rewrite 1984 to include that angle?
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 twiddla.com
to set up whiteboards to work with the webpage of your choice.
Precalculus sites mathsisfun
& purplemath are
visually more appealling than this one. Do not go.
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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Fraction Skill and Concept Check List
Teachers and Parents: Your students or charges should be able to do the
following. One way to check mastery is to provide one to several exercise sets
that cover the skills and concepts required below to see what learners can do,
and then go over the corrections in class.
- Give or recognize examples of unit fractions (fractions of the form
1/N): 1/2, 1/3, 1/4, 1/5, 1/6, 1/10, 1/12 of
collections of like objects, say pennies, marbles, coins; and of
geometric shapes such as circles and rectangles; and of heaps (amounts) of
material such as piles of sand or sugar or flour. The context here may
division of collections, food or areas filling a geometric shape, or heaps
of material into equal shares or division.
- Give or recognize examples of simple fractions (fractions of the form M/N)
that is whole number multiples of unit fractions using the same collections,
geometric shapes and heaps as for unit fractions.
- Know that the shorthand for M times 1/N is M/N
- Give or recognize whole number multiples of examples of simple fractions
(fractions of the form M/N), so that P x (M/N) = (PxM)/N
- See that N times an N-th of a collection, divisible geometric shape
or heap is the whole collection geometric shape or heap. Here is the first
illustration of how a simple fraction M/N may be equivalent to or have the
same value a whole number.
- Give and recognize how to form a unit fraction of a unit
fraction: 1/2 of 1/2, 1/2 of 1/3, 1/3 of 1/4 and so on. So 1/M-th of
1Nth is one 1/(MN)-th. The foregoing can be illustrated through the
division of rectangles, circles and line segments.
- Equivalent Fractions (II): Students should learn that M times 1/M-th of an
N-th is an N-th.
- Equivalent Fractions (III): Students should learn that A/N = A times
1/N = A times M times 1/M th of 1/N-th = (A x M)/(M x N)
- Equivalent Fractions (IV): Should use the property A/N = (A x
M)/(M x N) forwards and backwards to raise and lower terms in fractions, and
thus obtain sequences of equivalent fractions (sequences of fractions with
the same value).
- Learn how to use equivalent fractions, raising or lowering terms, to
compare different fractions
- Cosmetic Operation; Learn how to simplify fractions by lowering terms.
- Learn how to add and subtract fractions by recognizing and using a common
denominator.
- Favour the rule of thumb: Addition and subtraction with the
aid of a least common denominator in involving smaller numerators and
denominators in intermediate calculations may be more efficient than methods
using larger common denominators.
- Learn how to improper fractions have the same value as a whole number plus
a fraction that may be put in simplified form..
- Learn how that conversion of an improper fraction to a mixed number with
proper fractional part in simplified form done with simplification of the
improper first involves larger numbers that simplification of the fractional
part after conversion. Rule of Thumb: Conversion first to whole
number plus fraction to be simplified can be quicker or more efficient than
simplification first and conversion second.
- Learn learn how to compare mixed numbers by first comparing with their
whole number parts and if the latter are equal, their fractional
parts.
Rule of Thumb: Conversion of mixed numbers into improper fractions
with the same denominators is another way to compare, a way that involves
larger numbers and so may be less efficient.
- Master Length Arithmetic: Comparison, Addition, Subtraction,
Multiplication and two types of division: (i) Division with Remainder, (ii)
Division where the remainder is a fractional multiple of the dividend.
- Learn how to lengths can be measured if a unit length is choosen.
- See how to express length arithmetic in terms of operations on whole
numbers, mixed numbers and fractions, once a unit length is chosen.
- Learn how to divide by a fraction.
- Learn how to divide mixed numbers.
- Review how to add, subtract, compare, multiply and divide mixed numbers
with multiplication and division done by conversion into improper fractions.
- For multiplication of mixed numbers, use Distributive Law in place of
Conversion to Improper Fractions
- For division by improper fractions, convert the divisor into an improper
fraction and then multiply by reciprocal. Use distributive law.
- Use the equal sign = as shorthand for the phrase "has the same
value as". That seems to fit more circumstances than other
phrases.
- Learn that writing 3 x (5 x 2) = 10 = 30 misuses the equal sign
since 10 and 30 do not have the same value, and since the value of the full
expression 3 x (5 x 2) = A means the quantity A has the same value as
the full expression 3 x (5 x 2). Learn to write a = b when and only
when a and b are supposedly lengths, numbers, arithmetic expressions
or algebraic expressions with the same value. Students should be instructed
to write 3 x (5 x 2) = 3 x 10 = 30 instead. Anything else is an unacceptable
abuse of the equal sign.
One or two exercise sets may be best for students expected to familiar with
the skills and concepts while several exercise sets may be best for others less
familiar. If you see your charges are getting bored or restless with
explanations in advance of what to do, give them the exercise sets, walk around
the classroom to help students and to spot common difficulties, and
correct their answers later. Students may have more patience with explanations,
and more curiosity about how you would do the exercises, after they have
invested time and effort in doing the exercise sets.
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whyslopes.com
Entrance Level
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Tutors - All Subjects
(use at your own risk)
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Pages For Teachers
Site Entrance & Hub
Permissions for Instructors
Lesson Plans - Sec I
Lesson Plan, Sec II
Lesson Plans - Sec III
Secondary Maths, Core Elements
About Site Books & Areas
Site History/Content
Site Reviews
Vol 1. Elements of Reason
Maps Plans Drawings
Quantitative_Skills/index.html
Order Site Books
HIP, HIP, HIP, Hooray for
site
content & history. Hype, Hype,
Hype, Hoorary, for deception.
Your IP Address & how to use
it
Pages for Students
Site Entrance & Hub
Head Start Page
25 hours per tear
More Advice & Directions
Aims to adopt to aid
Arithmetic Check List
Fraction Skill and Concept Check List
Site History and Content
Books to Read
Complex No.s Intro.,.
Calculus Motivation
Calculus. Guide Short
Calculus. Guide-Long
Calculus Guide - Longest
Links - Many Subjects
Links - Games/Activities
Long Site Intro
Still More Advice
Logos Cafe
Logic Check List
Mathematics Cafe
Math CheckList
Site Areas by Age and Subject
A Site Map
Advice for Secondary I Students
Three Ways to be a Better Student
Reason for HS Mathematics
Montreal Tutors
Three Links for Teachers:
(i) First
Year High School Math - Lesson Plans with Fraction Focus
(ii) Second
Year High School Math - Lesson Plans with an algebra focus
(iii) Algebra
Lesson Plans
Help U Learn/ Teach
- Algebra
words before symbols
- direct & indirect
use of formula, numerical versus algebraic solutions - what
is a variable (more words)
- Arithmetic
- exercises
- with fractions
-
videos on primes, lcm, gcm,lcd, square roots etc
- Calculus - geometric
preview, algebraic
preview,
3 study guides,
much more
- Complex numbers
-starter lesson with java applet - easy
consequences for trig & vectors in the plane
- Education
- Empirical Course Design
& Delivery
- Fractions
- alone
- by rote
- with
algebra
- videos
- Functions - introduction
hindsight
- composition aka
substitution -
- Geometry, Euclidean - Correspondence
of triangles, Triangle
construciton, duplication & Isometry - Failure
of ASA & the // line postulate - angle
sum in triangles -//
grams - Triangle
Similarity
- Geometry-
Analytic - functions, polynomials, complex numbers, unit circle
trigonometry
- Logic
- First Steps -
Symbols in Logic
-
Occurrence &
Truth Tables - Indirect
Reason -Indirect
Reason More
- Proportionality
- Definition -
Direct & Indirect Use - Numerical versus Algebraic Solutions
- Real Analysis
- Decimal View of concepts
and of proofs
- Rules &Patterns in Science, Technology & Society
- Pattern Based Reason
- Mathematical Reasoning, empirical, inductive or deductive
- Units
- in rates & slopes &
(?) derivatives
- in ratios
& proportions - slopes & rates included
- Complex Numbers & Vectors & Trig
- trig expression for dot
& cross - cosine law
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