Transcript Multiplication Algorithms

Chapter
3
Whole Numbers and
Their Operations
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
Copyright © 2013, 2010, and 2007, Pearson Education, Inc.
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. 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.
NCTM grade 4 Focal Points, p. 16
Copyright © 2013, 2010, and 2007, Pearson Education, Inc.
Properties of Exponents
Definition of an for whole numbers
If a, the base and n, the exponent, are whole
numbers and n  0, then
a n  a  a...  a and a1  a
n factors
Copyright © 2013, 2010, and 2007, Pearson Education, Inc.
Properties of Exponents
Theorem 3-9
For any whole number a and natural numbers m
and n:
am  an  am n
Theorem 3-10
For any whole number a and natural numbers m
and n:
n
m
mn
a

a
 
Copyright © 2013, 2010, and 2007, Pearson Education, Inc.
Properties of Exponents
Theorem 3-11
For any whole number a and natural numbers m
and n:
an  bn  (ab)n
Theorem 3-12
If a, m, and n are natural numbers with m > n, then
a m  a n  a m n
Copyright © 2013, 2010, and 2007, Pearson Education, Inc.
Example 3-4
Write each of the following with only one exponent.
6
5
3
4
5
18
2

8

16
9

36

3
a.
b. 

a. 26  85  163  26  (23 )5  (24 )3  26  215  212
 261512  233


4
5
18
24
2 5
2 5
9

36

3

3

(2
)

(3
)
b.
 38  210  310
 318  210
Copyright © 2013, 2010, and 2007, Pearson Education, Inc.
 318  210  318
 210
Multiplication Algorithms
Copyright © 2013, 2010, and 2007, Pearson Education, Inc.
Multiplication Algorithms
Multiplication by 10n
To multiply by 10, replace each piece with a baseten piece that represents the next higher power of
10. Replace each unit with a long, and replace
each long with a flat.
Copyright © 2013, 2010, and 2007, Pearson Education, Inc.
Multiplication Algorithms
Copyright © 2013, 2010, and 2007, Pearson Education, Inc.
Multiplication Algorithms
Multiplication using expanded addition
Copyright © 2013, 2010, and 2007, Pearson Education, Inc.
Multiplication Algorithms
Multiplication with Two-Digit Factors
Copyright © 2013, 2010, and 2007, Pearson Education, Inc.
Multiplication Algorithms
Lattice Multiplication
Copyright © 2013, 2010, and 2007, Pearson Education, Inc.
Division Algorithms
Using Repeated Subtraction to Develop the Standard
Division Algorithm
or
Copyright © 2013, 2010, and 2007, Pearson Education, Inc.
Division Algorithms
or
Copyright © 2013, 2010, and 2007, Pearson Education, Inc.
Division Algorithms
Using Base-Ten Blocks to Develop the Standard
Division Algorithm
1. Represent 726 with base-ten blocks.
Copyright © 2013, 2010, and 2007, Pearson Education, Inc.
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.
Copyright © 2013, 2010, and 2007, Pearson Education, Inc.
3. Convert the one leftover flat to 10 longs.
Copyright © 2013, 2010, and 2007, Pearson Education, Inc.
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.
Copyright © 2013, 2010, and 2007, Pearson Education, Inc.
5. Now determine how many sets of 6 units there
are. There is 1 set of 6 units with no units left
over.
Copyright © 2013, 2010, and 2007, Pearson Education, Inc.
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.
Copyright © 2013, 2010, and 2007, Pearson Education, Inc.
Divide the
ones.
Division by a Two-Digit Divisor
Four step method:
1. Estimate
2. Multiply
3. Subtract
4. Compare (check)
Copyright © 2013, 2010, and 2007, Pearson Education, Inc.
Division by a Two-Digit Divisor
Divide
1. Estimate the quotient. The quotient is
between 10 and 100.
2. Find the number of tens in the quotient.
Copyright © 2013, 2010, and 2007, Pearson Education, Inc.
3. Find the number of units in the quotient.
4. Check. 32 · 81 + 26 = 2618
Copyright © 2013, 2010, and 2007, Pearson Education, Inc.
Multiplication and Division in
Different Bases
Base-Five Multiplication Table
Copyright © 2013, 2010, and 2007, Pearson Education, Inc.
Multiplication and Division in
Different Bases
Multiply 21five · 3five.
Fives
2

Ones
1
3
→
→
→
Copyright © 2013, 2010, and 2007, Pearson Education, Inc.
Multiplication and Division in
Different Bases
Copyright © 2013, 2010, and 2007, Pearson Education, Inc.
Example 3-4a
Multiply: 101two
11two
101two
11two
101
101
1111two
Copyright © 2013, 2010, and 2007, Pearson Education, Inc.