Math 365 Lecture Notes © S. Nite 8/18/2012
Section 3-4
Page 1 of 5
3.4 Algorithms for Whole-Number Multiplication and Division
In grade 4 Focal Points, “They [students] select appropriate methods and apply them
accurately to estimate products or calculate them mentally, depending on the context
and numbers involved. They develop fluency with efficient procedures, including the
standard algorithm for multiplying whole numbers, understand why the procedures
work, and use them to solve problems” (p. 16).
Multiplication Algorithms
Examining Simpler Computations First
Example: 313
Tens
1
×
Ones
3
3
13
10 + 3
×3
13
×
9
30
×3
39
3
30 + 9
39
Example: 3(x + 3)
Example: 32 14
and (3x + 2)(1x + 4)
Math 365 Lecture Notes © S. Nite 8/18/2012
Section 3-4
Page 2 of 5
Multiplication by 10n
Example: 10 12
12 10 = (110 + 2) 10
= (1 10) 10 + 2 10
= 1 102 + 2 10
= 1 102 + 2 10 + 0 1
= 120
If a is a natural number and m and n are whole numbers, aman is given by

 

a m ⋅ a n =  a⋅ a⋅ a ⋅
a⋅ a ⋅
⋅ a
...
...
⋅ a  ⋅  a⋅ 
 

 m factors   n factors 
= a⋅ a⋅ a ⋅
⋅a
...
m + n factors
=a
m+n
Math 365 Lecture Notes © S. Nite 8/18/2012
Section 3-4
Theorem 3-9
For every whole number a and natural numbers m and n,
aman = am+n
Theorem 3-10
For every whole number a, and natural numbers m and n,
(am)n = amn
Theorem 3-11
For every whole number a and natural number n,
an bn = (ab)n
Theorem 3-12
If a, m, and n are natural numbers with m > n, then
an am = am+n
Definition of a0 for natural numbers a
If a is a natural number, then
a0 = 1
Multiplication with Two-Digit Factors
13 24
Page 3 of 5
Math 365 Lecture Notes © S. Nite 8/18/2012
Section 3-4
Page 4 of 5
Lattice Multiplication
1
3
2
4
Division Algorithms
Using Repeated Subtraction to Develop the Standard Division Algorithm
6 726
6 726
Using Base-Ten Blocks to Develop the Standard Division Algorithm
Math 365 Lecture Notes © S. Nite 8/18/2012
Section 3-4
Short Division
5 2880
Division by a Two-Digit Divisor
32 2618
Multiplication and Division in Different Bases
Example:
4 five 1442 five
Example:
3 five 20312 five
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