Online Volumes (Book Orders)
1,  Elements of Reason. 1996
1A. Pattern Based Reason  1995
1B. Math Curriculum Notes 1996
2. Three Skills for Algebra  1995
3._Why_Slopes_&_More_Math_1995

Links To Tutoring Services

       Fraction Webvideos

1. [Play Video] 5 minutes - A Times Table (10 x 10) and how a number is not prime (composite) if it is in the interior of the table, that is if it is a product of smaller natural numbers. Some where in here is a Definition for Primes. A Natural number is composite if it is not prime.
   
2. [Play Video] 9½ minutes - Digit- Based Rules for recognizing divisibility by the divisors 2, 3, 5, 9, 10 and 11 or  calculating the remainders on division by these divisors. These rules follow from  10 = 0 mod 2 or 5, and 10 = 1 mod 3 or 9, and 10 = -1 mod 11.  Exercise: (1) Use  100 = 2 mod 49 to develop a digit-based rule for division by 49 or 7. (2) Give digit-based rules for division by 2, 3,5, 7, 11 and 13 that apply to the hexadecimal representation of whole numbers.
   
3. Square Root Rule: A number N is prime if it is not divisible by all primes p whose square p² is less than or equal to N.  On the other hand if a number N is not prime, it will be divisible by a prime p with p² less than N+1. With a calculator, the best bet is check where all primes p < sqrt(N) starting with the smallest.  Here if N = Mq where all primes < p are not divisor of the prime N then all primes < p will not be divisors of M. With the aid of a calculators and rules for divisibility by 2,3, 5, and 11, you can quickly get the prime decomposition of a whole number N.
   
4. [Play Video] 10 minutes - Recognizing Primes in the interval to 100 by eliminating all numbers that are multiples of primes < 11 = the first prime with square 11² = 121 > 100. (The Sieve of Erasothenes)
   
   If a first number N is a product of two factors, the square of the  larger factor will be greater than or equal  to the first number, and the square of the smaller will be less than or equal the first number N. So if the first number N can be factored, there will be a divisor, the smallest factor in a product with square < the first number N. That in turn implies there will be a prime <  the smallest factor which divides N and whose square is  <  N. From the study of logic (the contrapositive of an implication rule), if all primes with square < N do not divide N, N cannot be written as a product of factors - natural numbers smaller than N.
   
5. [Play Video] 2½  minutes -  Prime Factorizations (also called decomposition) for numbers in the interval 2 to 15.
   
6. [Play Video] 3     minutes - Prime Factorizations  for numbers in the interval 16 to 30.
   
7. [Play Video] 4½  minutes - Prime Factorizations for numbers in the interval 31 to 49.
   
8. [Play Video] 4     minutes - Prime Factorizations  for numbers in the interval 50 to 66. Note: 51 = 3 x 17 is not prime as stated in video. Oops.
   
9. [Play Video] 5½  minutes - Prime Factorizations for numbers in the interval 67 to 82.
   Note: 76 = 2 x 38 = 2 x 2 x 19. Video shows 17 instead of 19. Oops
   
10. [Play Video] 5½  minutes -  Prime Factorizations  for numbers in the interval 83 to 100.
    Note: 90 = 6 x 15 = 2 x 3 x 3 x 5 = 2 3² 5 Video write 4 x 15 instead of 6 x 15. Oops

1. [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.
   
2. [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.
   
3. [Play Video] 2-3 minutes. Multiplying Fractions with  cancellation of  common factors done first (recommended) or not, with more simplification to be done later.
   
4. [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.
   
5. [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 promoted.
   
6. [Play Video] 3 minutes - Comparison of Fractions Size or Magnitude, and more examples of the use of common denominators in addition and subtraction.
   
7. [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.
   
8. [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.
   
9. [Play Video] 2 minutes - Fraction Simplification using Prime Decomposition (factorization) to identify common factors for  cancellations.
   
10. [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.
    
11. [Play Video] 5 minutes - How to use Prime Factorization or Decomposition for LCM and LCD for a pair of denominators, an example.

1. [Play Video] 7 minutes. Finding All Divisors of a Natural number from its Prime Factorization/Decomposition
   
2. [Play Video] 6 minutes. Computing GCD for pairs of Natural Numbers from their Prime Factorizations /Decompositions)
   
3. [Play Video] 2½ minutes Computing GCD  from  Prime Factorizations /Decompositions, another example.
   
4. [Play Video] 3 minutes. Computing GCDs using Primes, yet another example.
   
5. [Play Video] 6½ minutes. Euclid Algorithm computes GCDs not using Prime Factorization.
   
6. [Play Video] 3 minutes. Another Euclid Algorithm GCD example  with result confirmed using Prime Decomposition.
   
7. [Play Video] 1½ minutes. Two numbers are relatively prime  when and only when they have GCD =1 when and only when the numbers have no prime divisors in common. Euclid algorithm leads to a quick identification of relatively prime whole numbers in the numerators and denominators of fractions by themselves or products.
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8. [Play Video] 4 minutes. Two Ways to Find the GCD of a pair of numbers. Both lead to the same result.
   

1. [Play Video] 2¼ minutes.  Common Multiples and Least Common Multiples for a par of natural numbers,  finding by listing mutliples of first and second number - the list method.
   
2. [Play Video]2¼ minutes.   Least Common Multiple for a pair of Natural numbers from Prime factorizations of each, and then by list method.
   
3. [Play Video]1 minute. Least Common Multiple for a pair of Natural numbers by computing the GCD divisor with the aid of Prime Factorization of each.
   
4. [Play Video] 4 minutes. Least Common Multiple for a pair of Natural numbers by computing the GCD divisor with the aid of Euclid's Algorithm, 1st Example.
   
5. [Play Video] 3 minutes. Least Common Multiple for a pair of Natural numbers by computing the GCD divisor with the aid of Euclid's Algorithm,  2nd Example. Note use of calculator.

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