Download Multiplication Algorithms

Chapter
3
Whole Numbers and
Their Operations
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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
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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
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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
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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
 
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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
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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
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 318  210  318
 210
Multiplication Algorithms
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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.
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Multiplication Algorithms
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Multiplication Algorithms
Multiplication using expanded addition
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Multiplication Algorithms
Multiplication with Two-Digit Factors
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Multiplication Algorithms
Lattice Multiplication
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Division Algorithms
Using Repeated Subtraction to Develop the Standard
Division Algorithm
or
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Division Algorithms
or
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Division Algorithms
Using Base-Ten Blocks to Develop the Standard
Division Algorithm
1. Represent 726 with base-ten blocks.
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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.
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3. Convert the one leftover flat to 10 longs.
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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.
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5. Now determine how many sets of 6 units there
are. There is 1 set of 6 units with no units left
over.
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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.
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Divide the
ones.
Division by a Two-Digit Divisor
Four step method:
1. Estimate
2. Multiply
3. Subtract
4. Compare (check)
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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.
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3. Find the number of units in the quotient.
4. Check. 32 · 81 + 26 = 2618
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Multiplication and Division in
Different Bases
Base-Five Multiplication Table
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Multiplication and Division in
Different Bases
Multiply 21five · 3five.
Fives
2

Ones
1
3
→
→
→
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Multiplication and Division in
Different Bases
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Example 3-4a
Multiply: 101two
11two
101two
11two
101
101
1111two
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