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.

Review of Operations - Fraction Know-How

Students: You may need a tutor, parent or teacher to guide you through this lesson, and to provide more examples. Except for that, the path below will provide a quick path for developing your fraction know-how. When time permit, I will post more online to make this material readable without a guide, or less need for one.

Teachers and Tutors: This multi-step starter lesson  points to  fraction know-how.  Mastery of  simplification, cross-cancellation in multiplication (an exercise in simplification), division of fractions (another exercise in efficient multiplication and simplification), and then addition and subtraction with least common denominators and more simplification. Simplification may employ rules for recognizing multiples of 2, 3, 5 and 10, and exploit or emphasize 10 or 12 times table. Simplification and more simplification (lowering terms) is the theme. However, raising terms appears in the addition and subtraction of fractions with unlike denominators as an aid to these operations and via the choice of least common denominators, to simplification.  (These methods and rules can be applied to mixed numbers as well since each mixed number, a whole number plus a proper fraction, is equivalent to, that is has the same value as a improper fraction.)

Teachers and Tutors: See too the end notes below and for details, the site lesson plans for Secondary I - fractions & allied concepts (decimals, percentages)

Students: Practice the following rules in your fraction calculations at school to obtain repeatable and reproducible, and so verifiable results. Later, for explanations of why these methods work, explore the further pages on fraction in this area.

Fraction sense and skills are required for solving linear equations and for all the calculations we do in this course. So you should be able to add and subtract fractions with like and unlike denominators efficiently, and you should be able to multiply and divide fractions efficiently. While you may use a calculator to aid your calculations with the whole numbers that appear in the numerator and denominators of fractions, I expect you can do arithmetic with whole numbers and fractions efficiently and exactly. Decimal approximations should be avoided as much as possible.

Example of simplification

From cancellation of the common factor 2, we see the fractions 

3    and   6
4             8

are equivalent. They have the same value.  The equal sign is use to indicate two numbers or fractions have the same value above and below.

The common factor cancellation property, algebraic description: 

is a way to recognize equivalent fractions. To apply it, we look for a common factor N (like the number 2 above) to cancel.

The left and right hand side in foregoing equation are said to be equivalent fractions.

Replacing the left hand side by the right hand side in a calculation is called a simplification, a reduction, a cancellation or a lowering of terms.  On the other hand, replacing the right hand side by the left hand side is called raising terms.  Raising of terms is useful in the addition and multiplication of fractions. 

Some Video Examples of raising terms and of simplification (lowering terms) follow.

1. [Play Video] 2-3 minutes A few examples of Simplifying Fractions - lowering terms by canceling common factors until there are no more common factors, so that the numerator and denominator are relatively prime, that is there prime decompositions have no primes in common.
2. [Play Video] 3-4 minutes. Equivalent fractions - Lowering and raising terms (the values of numerators and denominators) to obtain equivalent fractions. Simplification involves lowering terms - cancelling common factors or divisors on top and bottom. Addition & subtraction of fractions may involve raising terms to obtain a common denominators. See below.
   

For text or html examples, visit www.purplemath.com and www.mathsisfun.com

Rules for multiplication

But before you calculate the products A*C ad B*D of the numerators and denominators, factor A, B, C and D to cancel common divisors.

For example

In this example, we did not calculate 12*15 nor 25*16. Instead we factored further the factors 12, 15, 25 and 16 in these products in order to apply cancellation rules for simplifying fractions. To apply the formula efficiently

factor A, C, B and D, so that you can start fraction simplification process (reducing terms) before you do multiplication.

1. [Play Video] 2-3 minutes. Multiplying Fractions with  cancellation of  common factors done first (recommended) or not. If not,  more simplification needs to be done.
   
2. [Play Video] 3-4 minutes. Equivalent fractions - Lowering and raising terms (the values of numerators and denominators) to obtain equivalent fractions. Simplification involves lowering terms - cancelling common factors or divisors on top and bottom. Addition & subtraction of fractions may involve raising terms to obtain a common denominators. See below.
   
3. [Play Video] 2-3 minutes A few examples of Simplifying Fractions - lowering terms by canceling common factors until there are no more common factors, so that the numerator and denominator are relatively prime, that is there prime decompositions have no primes in common.
   
4. [Play Video] 2 minutes - Fraction Simplification using Prime Decomposition (factorization) to identify common factors for  cancellations.
   
5. [Play Video] 5 minutes - Product Simplification using Prime Decomposition by Canceling Common Primes, thus avoiding some denominator and numerator multiplication. An alternative common factors as they appear, more opportunistic, is given and is to be recommended.

Rule for Division of Fractions

 This rule converts division by a fraction C/D into multiplication by its reciprocal D/C. Then you apply the previous rule for multiplication of fractions.  For Example

In the previous example, there is a compound fractions

Division of one fraction by another, that is can be written as compound fraction.

A compound fraction has a numerators and denominator given by a proper or improper fraction. The latter may be written as a mixed number.

Remember: A mixed number is given by a whole number and a fractional part. The fractional part is proper. The above rules for multiplication and division of fractions can also be applied to mixed numbers after converting the latter into improper fractions with the same value. With time and experience, you may also use the distributive law a(b+c) = ab +ac in order to modify the above rules and do arithmetic more efficiently. But we keep our initial rules as a simple as possible in the first instance.

Rules for addition and subtraction

Here the denominator M can be any common multiple of B and D. To apply these formula efficiently, remember the smallest common multiple usually gives less work in the simplification of the right hand sides.

Example of Addition:

In this example, M = 24 = the least common multiple of the the two denominators 8 and D = 12 while A = 5 and C = 7. So M/A = 24/8 = 3 and M/D = 24/12 = 2.

[Play Video] 3 minutes  Another example of how to add fractions with and without the least common denominators with an explanation that not using the LCD (least common denominator)  leads to ratios that can be simplified. So use of LCDs is advised.

How to Choose M:

Method 1: List the first B multiples of D, and list the first D multiples of B.  The number B x D = D x B is the last number in each list. Let M  < B x D be the smallest number in both lists. That number will be the smallest common multiple of B and D.

Subexample: Let B = 8 and D = 12 as above.

1	2	3	4	5	6	7	8	9	10	11	12
8	16	24	32	40	48	56	64	72	80	88	96
12	24	36	48	60	72	84	96

The number 24 is the smallest in both lists. So 24 = the least common multiple of 8 and 12.  Shortcut: Calculate the first D multiples of the smallest number B until the first multiple of D appears.

Method 2: From the prime factorizations of B and D form a product of primes where each prime in the product appears to the greatest power that occurs in the prime decomposition of B and D.

Subexample:  B = 8 = 2³ and D = 12 = 3*2².  Then M = 3*2³3 = 8 * 4 = 24 as before.

Method 1  works best with pairs of numbers < 15. Each list is then  < 15 numbers long. Method 2 works best if you know how to obtain the prime factorization of a number.

Real Player Videos

1. [Play Video] 5 minutes. How to add fractions using common denominators. Here the common dominators is the lowest or least common denominator (LCD) and its given by the least common multiple (LCM) of the denominators in the fractions added together.  Here the listing multiples method is used to compute the LCM. The alternative of not using the LCD for the fractions is explored to show what happens when the LCD is not used.
   
2. [Play Video] 3 minutes - Another example of the listing multiples method to find the LCM and thus the LCD for the sum of two fractions.
   
3. [Play Video] 4 minutes - Factorization method to obtain  a common denominator, here the LCM and thus the LCD for the sum of two fractions. See if you can recognize the GCD of the denominators here. It is not mentioned here. In this example,  the LCD is given by a product that does not have to be evaluated explicity due to cancellation of common terms after addition of fractions.
   
4. [Play Video] 5 minutes - How to use Prime Factorization or Decomposition for LCM and LCD for a pair of denominators, an example.

[Play Video] 3 minutes - Comparison of Fractions Size or Magnitude, and (?) more examples of the use of common denominators in addition and subtraction.

To compare two fractions with unlike denominators

express both over a common denominator M. Then

A B

=

A(MB) M

and

C D

=

C(MD) M

Then compare the numerators.

The case M = BD gives

The first fraction A/B is then

1. less than the second fraction C/D if AD < BD. 
2. greater than the second fraction C/D if AD > BD, and
3. has the same value as the second fraction C/D if AD = BD

In case III, we may also say the two fractions are equivalent.

In general, when

A B

=

A(MB) M

and

C D

=

C(MD) M

The first fraction A/B is then

1. less than the second fraction C/D
       if A(MB) < C(MD) 
2. greater than the second fraction C/D
       if A(MB) > C(MD), and
3. has the same value as the second fraction C/D
       if  A(MB) = C(MD)

In case III, we may also say the two fractions are equivalent.

End Notes for Teachers and Tutors:

For the sake of an operational command of fractions:  Students who have seen fractions before can be given an operation command of fractions through the following steps:  (i)  Learn how to simplify fractions by canceling common factors in enumerators and denominators; (ii) Learn how to multiply fractions but with an emphasis on postponing multiplication in favor of  factoring the numerator and denominators of products in order to cancel and simplify; (iii)  Learn how to divide fractions by turning divisions into multiplication by a reciprocal, and then applying the efficient product simplification methods in step; (iv) learn how to add and subtract fractions with like denominators and how to simplify the sum; (v) learn how to add and subtract fractions with unlike denominators and the role of least common denominators in reducing the amount of simplification needed in sums.

In the foregoing,  prime decompositions can be introduced to aid simplification and to aid the computation of least common denominators and greatest common divisors. Teach students to look for factors of whole number among those primes whose square is less than or equal to the whole number in question. If none those of those primes are factors, the whole number in question is prime. Calculators and knowledge of all primes less than 50 are sufficient to quickly generate the prime number decomposition of all numbers < 2500.  The link to prime numbers can be postponed.)

The foregoing program provides fraction skills but does not develop develop fraction sense. The site area Solving Linear Equations with and without Stick Diagrams develops fractions skills and sense by illustrating and demanding fractional operations on line segments - the sticks - while also introducing and/or consolidating algebra skills and sense.

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