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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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.

Compound Units

Mastery of formal operation with expression involving units is needed for proportionality computations or relationships between quantities or between numbers and quantities. We  do operations on (formal) expressions without requiring those expression have any meaning or physical interpretation.  That provide on paper exercises with marks or expression on paper.  In a following section on proportionality, multiples of simple and compound units will serve as proportionality constants and rates  in exercises that may some physical or social meaning.

Longer Explanation - More Details

Whole numbers and fractions (pure numbers without units) may appear in counting how many items are present.  Counts may be fractional. Units of measurement may appear to identify what we a counting.  

5 apples,  10 oranges,  10.5 centimeters, 5.34 kilograms,  10 degrees Celsius (temperature measure),  90 degrees (angle measure)

A number times (or written besides) a unit of measurement is called a quantity. In daily life, science and technology, there are systems of measurements for length, mass, time, money, 

Addition and Subtraction of Quantities

When we have  5 apples and 6 bananas and 1 orange in a bag, the expression

5 apples + 6 bananas + 1 orange

represent this collection of objects or fruit. The units here apples, bananas and oranges.  We write units in singular or plural form in accordance with language rules. But in writing expressions, we do not care or distinguish units written in singular or plural form. So in our calculation with units,  we write  5 penny means the same as 5 pennies.   The expression looks like a sum. Now from bag of 5 apples and 6 bananas and 1 orange in a bag, we may remove  2 apples, 3 bananas and 1 orange.  The result would be 3 apples, 3 bananas and zero oranges.  We may write the foregoing in shorthand form  (algebraic or symbolic form) as 

(5 apples + 6 bananas + 1 orange) - (2 apples + 3 bananas + 1 orange) 

= (5-2) apples + (6-3) bananas + (1-1) oranges
= 3 apples + 3 bananas + 0 oranges
= 3 apples + 3 bananas  

In this subtraction we are combining like terms: those involving apples, bananas and oranges, respectively.

On the other hand if I have 10 dimes  and 4 pennies and you have  6 dimes and 20 pennies, together we have

(10 dimes + 4 pennies) + (6 dimes + 20 pennies)  = 16 dimes + 24 pennies.

This addition combines like terms - terms with the same units.  A dime is coin worth ten pennies. Changing dimes in pennies or vice-versa is optional here. 

In general,    for a units of a quantity plus another b units of the same quantity together give  (a+b) units of the same quantity. Symbollically, we write

a units + b units = (a+b) units

provided of course the unit of measurement in all terms are identical. In the same circumstances,

a units - b units = (a-b) units

Examples with like units

• 5 kilogram + 4.5 kilograms = 9.5 kilograms
  
• 4 hours + 8 hours = 12 hours
  
• 20 seconds - 16 seconds  = 4 seconds
  
• 8 centimeters + 6 centimeters + 2 centimeters = (8+6 +2) centimeters = 16 centimeters
  
• 10 meters - 18 meters  = - 8 meters

The last makes sense if  1 meter represented one step to the right and -1 meter represented one step to the left. 

Examples with unlike units (Read + as and)

• (8 apples + 4 pennies) + ( 3 pennies + 2 apples)  = (8+2) apples + (4+2) pennies = 10 apples + 7 pennies
  
• (4 oranges + 3 bananas + 2 lemons) + (2 oranges + 3 lemons) = 6 oranges + 3 bananas + 5 lemons 

Repeated Addition and Multiplication by Real Numbers

Three times the amount  7 dishes is  7 dishes + 7 dishes + 7 dishes.  So we write

3 x  (7 dishes)  = (3 x 7) dishes = 21 dishes.

Here multiplication by 3 is just repeated addition. 

In general, if a and b are whole number, fraction or decimals, we assume

 a (b units)  = (ab) units

and use this equality to compute the left hand side a (b units)  -- read as a times b units. 

Moreover, if a is nonzero, we assume

For instance a quarter of 12 cups, all identical, is given by 3 cups where 3 is a quarter of 12.  In symbols

(¼) 12 cups =  (12/4) cups = 3 cups.

Like wise 12 cups divided by 4 is again three cups:

Division by four (4) is the same as multiplying by a quarter (¼)

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