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C Decimal Multiplication Methods

Notes

Six Practical Lesson Description

1. Theory: Explanation of why we put 3 × 5 = 15. In essence, m × n is the number of squares units in a m by n units. Here is the multiplication analogue of why we put 3 + 5 = 8. Includes a dot counting exercise where 3 × 5 is given by the number of dots in a 3 row of 5 dots. Here is elementary school level approach to understanding and explaining multiplication and a times tables.
   
2. Between Practice and Theory: Single Digit Multiplier Examples which begin with the optional view or a hint it, that multiplication is related to repeated addition. Includes exercises.
   
3. Practice: Single Digit Multiplier Examples and Exercises in which multiple digit number are multiplied by a single digit 2 to 9. Multiple conversions (carries) appear. Includes examples and exercises with one digit multipliers. Includes exercises.
   
4. Practice: Multiplication Examples and Exercises with 2 and 3 digit multipliers. Includes exercises.
   
5. Practice and Theory: Multiplication when factors include a decimal point. The multiplication method is illustrated. An explanation of why is based on the properties of fractions. You may need to study fractions before understanding the included explanations of why the methods work. Includes exercises.
   
6. Between Practice and Theory. How or why the order of a pair of factors in a product can be changed while still giving the same result.
   

Five More Lessons - theorectical

1. Theory: A quick review of elementary school ideas to explain how addition and multiplication are possible (both are a form of counting), and to set the stage for justifying decimal methods for multiplications. Here is an echo or extension of lesson 1.
   
   Many people want an empirical, plug-and-play approach to mathematics in which they want a rule and numbers to use in it, in which they are satisfied with a method if it works, and do not need to understand how - ouch. But there is more to mathematics that. There is a thought-based development of arithmetic, algebra and geometry which provides richer and deeper understanding. Mathematics is one of the few arts and discipline in which a full-thought based development is possible. Enjoy it if you can for its own sake or as a means to test and develop your skills. Good luck.
   

2. Practice: Single Digit Multiples of Powers of ten as factors, their representation as decimal counts, and calculating their products. This and the next lesson represent preparation for lesson 10.
   

3. Practice: Area Calculation with Single Digit Multiples of Powers of ten appearing to give the lengths of sides. This sets the stage for the next lesson. It also provides a preview of a geometric view of the distributive law.
   

4. Theory: An explanation or justification for Decimal multiplication methods for decimal counts - the decimal representation of whole numbers. The justification is (nominally) based on area calculations. Accept that if you like and go no further. That being said, the area calculations themselves are based on counting principles.
   
   Teachers and Tutors: While the theory on the surface relies on area considerations, the area considerations represent counting principles and the definition of products of whole numbers in disguise.

Arithmetic - Ages 10+
1. Deciml Place Value - fun
2. Decimals for Tutors
3. Prime Factors - quickly
4. Fractions + Ratios
5. Arith with units - science