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HIP,
HIP, HIP, Hooray
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
liinear2007 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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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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Lesson Plans
Help U Learn/ Teach
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words before symbols
- direct & indirect
use of formula, numerical versus algebraic solutions - what
is a variable (more words)
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- exercises
- with fractions
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videos on primes, lcm, gcm,lcd, square roots etc
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preview, algebraic
preview,
3 study guides,
much more
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hindsight
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sum in triangles -//
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trigonometry
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Reason More
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Direct & Indirect Use - Numerical versus Algebraic Solutions
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and of proofs
- Rules &Patterns in Science, Technology & Society
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- in rates & slopes &
(?) derivatives
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& proportions - slopes & rates included
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& cross - cosine law
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