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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. |
Role of Letters in formulas for Areas of Rectangles, Triangles and CirclesWelcome. In some formulas for areas and perimeters, letters denote and serve as shorthand for lengths of sides or heights. We can describe calculations with or without knowing the measures of the lengths until the last possible moment. 1.1 RectanglesRecall the area of a rectangle is given by its length times its width. We can write this as
The computation of the area of the rectangle can be rewritten with shorthand notation as follows. To introduce shorthand notation, we say the area A of a rectangle is given by its width W times its length L. Here, we use A as shorthand for the area of a rectangle, L as shorthand for its length and W as shorthand for its width. The formula (recipe) for calculating the area A of a rectangle can be written more briefly as
Shorthand notation provides a code for the description of calculations. Formula decoding is required. The shorthand formula A = L ·W is more compact (takes less room) than the word-only description. This formula is meaningless for us when the role of the letters in this shorthand description is not explained. To understand and to use the shorthand description or formula, you need information. You need to know or find what numbers or quantities the symbols mean or represent. In the above formula, L stood for the length of a rectangle. This has to be said to you or you have to ask. To anyone without this information, the formula remains mysterious. Talking about and describing computations almost gives us the power to do them. In the area calculation, the area A is obtained from the recipe A = L×W provided the length L and W are given or can be found. Without this information, we can describe or understand a calculation but not use it. The above rectangle example reminds us of the following:
1.2 TrianglesIn words, the area of a triangle is given by one half the length of a base of
the triangle multiplied by the height of the triangle. This formula can be
justified but at this moment we will not worry about why it holds. We may also
write more briefly
We have used single letters in this shorthand
description of the calculation. Any mark or squiggle or symbol you can draw and
name can serve as shorthand for some number or quantity.
Perhaps, we should use Atriangle or another symbol, since
we have already used the letter A in the previous rectangle example.
Alternatively, we adopt the following rule: while you are reading this triangle
example, we use the letter A here as shorthand for the area of the
triangle only. More will be said on using and reusing (recycling) shorthand
symbols (for example, letters) and the roles they take. Think of them as actors
which can perform many parts. They may take only one role in any scene, except
for stories and scenes involving identical twins or cases of mistaken
identities.
The symbol for the Greek letter called Pi is p. In
words, the area of a circle is given by the number p
times the square of the circle's radius. The square of a number or quantity
refers to the number or quantity times itself. The square10 of 5 for instance is 52 = 5 ×5 =
25. We can also more briefly write
To rewrite or encode this formula in shorthand form, we will first describe
the code. Let A be shorthand for the area of a circle.11 Let r be our
shorthand for the radius of the same circle. Then the previous word-only formula
for the area of a circle is written A = p·r
·r or as
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Site books and further webpages on learning and teaching mathematics and pattern based reason may develop critical thinking, improve reading and writing, and give a base for learning or teaching high school and college mathematics. Great_Expectations: If you can learn to follow a multi-step methods in any subject precisely, you can do so in other subjects, as well. Good news: Site pages identify what you need to study. Bad news: Site pages do not explain everything Worse news: Learning takes time, yours
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[Top] Favourite Sites: BBC
News and the Mathematics portion of English
National Curriculum
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