Ch. 5 of Number Theory

 Multiplication is repeated addition…wasn’t until 1650 BC that methods for multiplication began appearing.  Egyptians used a method for multiplication that required the doubling of successive numbers, followed by the addition of appropriate multiples.  This method works because every whole number can be written as the sum of “doubling numbers” (or numbers of the form 2 n ).  Example: 142 x 28  1 x 142 = 142  2 x 142 = = 28,  *4 x 142 = 568 so 28 x 142 =  *8 x 142 = = 3976  *16 x 142 = 2272

 Using the Egyptian Multiplication Method, solve the following problems:  34 x x 190  46 x x 111  22 x 9138 x 56

 A book entitled Introductio Arithmetiea written by Nicomachus of Gerasa around 100 AD provided a multiplication table up to 10 x 10. No rules of multiplication or division were included.  The Hindu-Arabic system of numbers (place value and zero included) began to be seen in Europe towards the end of the 13 th century.  Luca Pacioli described some methods of multiplication in his work Summa de arithmetica, geometrica, proportioni et proportionolita (usually referred to as the Suma) in  Included in his work was the method of multiplication known as Lattice Multiplication.

  Solve the following using the Line Multiplication:  234 x x 32

 Students learn to multiply through a stepwise progression of multiplication methods that is similar to that of addition.  Repeated addition and skip-counting are examples of how students will initially learn multiplication.  Many methods of multiplication developed by students are supported by neither textbooks nor instruction.

 Every composite number N can be factored uniquely into prime factors.  Example:30 / \ 15 2 / \ 3 5 So 30 = 3 x 5 x 2

 Material from the text, Number Theory for Elementary School Teachers By: Edward Wall  Youtube video