Sabrina Jalbert
MAT 130 Elements of Math 1
Thursday April 9th 2015
Multiplication of Whole Numbers Packet

Component 1
Digits 1, 2, 3, 4, and 5
12 x 345 = 4,140
23 x 145 = 3,335
13 x 245 = 3,185
14 x 235 = 3,290
15 x 234 = 3,510
21 x 345 = 7,245
24 x 135 = 3,240
25 x 134 = 3,350
31 x 245 = 7,595
32 x 145 = 4,640
34 x 125 = 4,250
35 x 124 = 4,340
41 x 235 = 9, 635
42 x 135 = 5,670
43 x 125 = 5,375
45 x 123 = 10,910
51 x 234 = 11, 934
52 x 134 = 6,968
53 x 124 = 6,572
54 x 123 = 6,642
1) The arrangement that results in the greatest possible product is: 51 x 234 = 11,934

2) The arrangement that results in the least possible product is: 13 x 245 = 3,185

3) Given any five-digit non-zero digits, the placement of the two digits will determine the greatest or lowest products. In that two digit number, the tens place determines it.

Component 2
1) Patterns noticed on the multiplication table:
Arithmetic Patterns with the common differences of: 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10.
The product of 6x6 forms a pattern on the table by itself, as does 7x7, 8x8, 9x9, 2x2, 1x1 10x10.
All the squared numbers form a diagonal on the table.
The four table is doubled to make the 8.
The products for columns 1, 3, 5, 7, and 9, alternate between even and odd numbers.
The sum of the digits in each product for the multiples of nine is nine.

2) Every odd number on the multiplication table is surrounded by even numbers, but the reverse is not true. This could be because, an odd product can only have odd factors, but an even number can have two even factors, or one even factor and one odd factor. Another possibility is that with the multiplication of two numbers, the only way to get an odd is to multiply odd with odd, therefore every odd number is surrounded by even numbers.

Component 3
1 – 2)
a) The Russian Peasant Multiplication Algorithm
No multiplication facts are needed.
Students only need to know how to add and cut the numbers in half.
Steps to follow for Algorithm:
1. Put each number into a column, one for halves one for doubles – try keeping the odd numbers in the halves column.
2. Halve the number in the halve column until you are left with the number 1 –disregard any remainders, simply place them into quotient.
3. Double the number from that column as many times as it takes to match the halved side.
4. Cross out any rows that have evens on both sides.
5. Add remaining numbers in the doubles column and there is your answer.

b) Lattice Multiplication
Method of multiplying numbers on a grid. It is relatively equivalent to long multiplication, but it breaks down the steps so students find it easier.