Chapter Whole Numbers and Their Operations 3 3 Copyright © 2013, 2010, and 2007, Pearson Education,...

Chapter

Whole Numbers and Their Operations

33

Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

3-4 Algorithms for Whole-Number Multiplication and Division

Multiplication Algorithms Division Algorithms Division by a Two-Digit Divisor Multiplication and Division in

Different Bases


Algorithms for Whole-Number Multiplication and Division

They select appropriate methods and apply them accurately to estimate products or calculate them mentally, depending on the context and numbers involved.

NCTM grade 4 Focal Points, p. 16

They develop fluency with efficient procedures, including the standard algorithm, for multiplying whole numbers, understand why the procedures work (on the basis of place value and properties of operations), and use them to solve problems.


Properties of Exponents

Definition of an for whole numbers

If a, the base and n, the exponent, are whole numbers and n 0, then


1

factors

... and n

n

a a a a a a


Theorem 3-9

For any whole number a and natural numbers m and n:

Theorem 3-10



m n m na a a

nm mna a


Theorem 3-11


Theorem 3-12

If a, m, and n are natural numbers with m > n, then


( )n n na b ab

m n m na a a

Example 3-4

Write each of the following with only one exponent.

a. b.

a.

b.


6 5 32 8 16 4 5 189 36 3

6 5 32 8 16 6 3 5 4 32 (2 ) (2 ) 6 15 122 2 2 6 15 12 332 2

4 5 189 36 3 2 4 2 5 2 53 (2 ) (3 )

8 10 103 2 3

18 103 2

18 10 183 2 3

102

Multiplication Algorithms



To multiply by 10, replace each piece with a base-ten piece that represents the next higher power of 10. Replace each unit with a long, and replace each long with a flat.

Multiplication by 10n



Multiplication using expanded addition



Multiplication with Two-Digit Factors



Lattice Multiplication


Division Algorithms Using Repeated Subtraction to Develop the Standard Division Algorithm

or


Division Algorithms

or


Division Algorithms

Using Base-Ten Blocks to Develop the Standard Division Algorithm

1. Represent 726 with base-ten blocks.


2. Determine how many sets of 6 flats (hundreds) there are. There is one set of 6 flats with 1 flat, 2 longs, and 6 units left over.


3. Convert the one leftover flat to 10 longs.


4. Determine how many sets of 6 longs there are in the 12 longs. There are 2 set of 6 longs and 6 units left over.


5. Now determine how many sets of 6 units there are. There is 1 set of 6 units with no units left over.


Division Algorithms

Short Division

Decide where to start.

Divide the hundreds. Write the

remainder by the tens.

Divide the tens. Write

the remainder by

the ones.

Divide the ones.


Division by a Two-Digit Divisor

Four step method:

1. Estimate

2. Multiply

3. Subtract

4. Compare (check)


Divide

1. Estimate the quotient. The quotient is between 10 and 100.

2. Find the number of tens in the quotient.

Division by a Two-Digit Divisor


3. Find the number of units in the quotient.

4. Check. 32 · 81 + 26 = 2618


Multiplication and Division in Different Bases

Base-Five Multiplication Table



Multiply 21five · 3five.

Fives Ones

2 1

3→

→ →


Example 3-4a

Multiply:


two

two

101

11

two

two

two

101

11

101

101

1111

Chapter Whole Numbers and Their Operations 3 3 Copyright © 2013, 2010, and 2007, Pearson Education,...

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