Download I. Whole Number Multiplication Multiplication is repeated addition

Section 2.4 Multiplication and Division of Whole Numbers
I. Whole Number Multiplication
Multiplication is repeated addition.
vs.
3⋅4 =
Example: 4⋅3 =
Help students by reading “4 of 3” and “3 of 4”
Definition: Multiplication of Whole Numbers as
Repeated Addition: The product of a and b,
writtena ⋅b , is defined by a ⋅b = b
+ b +
...+b and 0⋅b = 0 .
a addends
Also write: a ⋅b = a×b = a*b = ab = a(b) = (a)b = (a)(b) .
Words to know:
1) a and b are factors, a ⋅b is the product.
2) Multiplication is a binary operation.
Properties of Whole Number Multiplication
Closure Property: If a and b are any whole numbers,
then a ⋅b is a unique whole number.
Commutative Property: If a and b are any whole
numbers, then a ⋅b = b⋅a .
Associative Property: If a, b, and c are any whole
numbers, then a(bc) = (ab)c .
Multiplicative Identity Property of One: If a is a
whole number, then a ⋅1=1⋅a = a .
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Section 2.4 Multiplication and Division of Whole Numbers
Multiplication, Repeated Addition Context, Set
Model
Example: Jamie is buying boxes of 3 chocolate eggs
each. She buys 4 boxes. If she gives each friend one
egg, to how many friends can she give eggs?
?
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4⋅3 = 3+ 3+ 3+ 3 =12
Multiplication, Repeated Addition Context, Number
line Model
Example: Sue hiked 2 miles each day for 3 days. How
far did she hike in all?
2
0
1
2
2
3
2
4
5
6
7
8
9
3⋅2 = 2 + 2 + 2 = 6
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Section 2.4 Multiplication and Division of Whole Numbers
Array Context – Set Model (horizontal rows,
vertical columns)
Example: Jim plants 2 rows of three bean seeds. How
many seeds did he plant?
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2 rows of 3
2 ⋅3
vs.
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3 rows of 2
3⋅2
Rectangular Area Context – Number Line Model
Example: How many tiles are needed to tile a 2 feet by
3 feet area with 1 square foot tiles?
2
1
0
1
2
3
4
row × column = 2 × 3
Example: Use C-rods to show 5 × 3 with a repeated
addition context. Use C-rods to show 5 × 3 with a
rectangle context.
Use C-rods and an area model to illustrate (4+2)(3+5).
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Section 2.4 Multiplication and Division of Whole Numbers
Cartesian Product Model
The Cartesian Product of sets A and B, written A× B ,
is the set of all ordered pairs whose first coordinate is
an element of the set A and second coordinate is an
element of B: {(a,b)| a∈ A and b∈B}.
Example: Find the Cartesian product of the sets
A = {r, y, b} and B = {+, –,×, /}
How many members in the Cartesian Product A× B ?
Tree Diagram Model (See Figure 2.21, page 123)
A = {r, y, b} and B = {+, –,×, /}
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Section 2.4 Multiplication and Division of Whole Numbers
II. Whole Number Division
Definition: Division of Whole Numbers
If a and b ≠ 0 are whole numbers, their quotient,
writtena ÷b , is the unique whole number c such
thata = b⋅c . That is, a ÷b = c if and only if there is a
whole number c such that a = b⋅c .
Vocabulary to know:
1) a is the dividend, b is the divisor, c is the quotient.
2) Division is a binary operation.
Properties of Whole Number Division?
If a, b, and c are any whole numbers, then
Closure Property?
Commutative Property?
Associative Property?
Identity property of 1?
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Division: Repeated Subtraction (Measurement)
Context with Set Model
Example: Mr. Milton puts 15 students into groups of
3. How many groups of 3 will he have?
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15 ÷ 3 = 5 groups
Division: The Partition (Fair Share) Context
Example: Mr. Milton has 3 sets of blocks with 15
students. How many students are assigned to each set
of blocks if he wants equal size groups?
1 set 2 set 3 set
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15 ÷ 3 = 5 students in one group
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Section 2.4 Multiplication and Division of Whole Numbers
Repeated Subtraction Context with Number Line
Model
Example: Ruby has 15 yards of ribbon with which to
make bows. Each bow requires 3 yards of ribbon.
How many bows can she make?
Fact Families – Write the fact family for3⋅4 =12 .
Division with zero:
0÷2 = ?
2 ÷0 = ?
0 ÷0 = ?
Theorem: The Division Algorithm If a and b ≠ 0 are
whole numbers, there is a unique whole number q called
the quotient and a unique whole number called the
remainder such that a = q ⋅b + r, 0 ≤ r < b .
Example: Use repeated subtraction to divide 43 by 9.
Use long division to divide 403 by 9.
Definition: The Power Operation for Whole Numbers:
If a and m are whole numbers the exponential
expression am or a to the mth power is given by
am = a ⋅a ⋅a
⋅...⋅a ⋅a . Number a is the base; m is the
m times
exponent.
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