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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 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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-/[]\- 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. |
11. Efficient ways to Multiply Fractions[Play Video] 2-3 minutes. Multiplying Fractions with cancellation of common factors done first (recommended) or not, with more simplification to be done later. In general, we may multiply fractions as follows:.
In the resulting fraction, the the numerator (top) is a product of the numerators of the factors and the denominator (bottom) is a product of the denominator of the factors. The foregoing describes the first method for multiplying fractions. After that, we would simplify the resulting fraction by canceling common factors in the products numerator and denominator. The rule here is multiply first and cancel second. But this order can be changed. Cancellation first leads to smaller numbers and a quicker way (usually) to get the simplified form of the product. Example:
Now instead of compute the products of the numerators and denominators (and
then factoring the products to cancel common factors), we take advantage of the
situation that the original numerators and denominators provide factors of the
product numerators, and factor further to locate common factors that will
cancel. Cancelled factors are
Here we kept the original numerators and denominators and then factored them in a way that would help simplifcation (lowering terms) in the product fraction. So we cancelled the 25 and 11 after factorization. Then after no further factors could be cancelled, computed the decimal representation of the product numerator and denominator in reduced form. Here is the above product computation revisited with in place cancellation - the same calculation with a cosmetic change.
The first way we did the cancellation (that is, multiplying the fractions together and then factoring to reduce) provides justification for the cancellation of common factors in the original fractions before multiplication is done. Algebraic Shorthand Description (rather complicated, can be ignored. None the less, the challenge is to understand what is says or suggests, good luck).
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