§ 3-3 Multiplication and Division of WholeNumbers

Why We Need This

The 3rd Grade Common CoreIn the overview for grade 3, the Common Core Standards list:Operations and Algebraic Thinking

Represent and solve problems involving multiplication anddivision

Understand the properties of multiplication and the relationshipbetween multiplication and division

Multiply and divide within 100

Solve problems involving the four operations, and identify andexplain patterns in arithmetic

Why We Need This

The 4th Grade Common CoreIn the overview for grade 4, the Common Core Standards list:Operations and Algebraic Thinking

Use the four operations with whole numbers to solve problems

Gain familiarity with factors and multiples

Numbers and Operations in Base TenUse place value understanding and properties of operations toperform multi-digit arithmetic

What is Multiplication?

Can we define multiplication?

DefinitionThe multiplication of two whole numbers is equivalent to the additionof one of them with itself as many times as the value of the other one.

The two numbers being multiplied are called factors.

The answer when multiplying is called the product.

DefinitionFor every whole number a and n 6= 0,

n · a = a + a + a + . . .+ a︸ ︷︷ ︸n terms

If n = 0, then 0 · a = 0.

What is Multiplication?

Can we define multiplication?

DefinitionThe multiplication of two whole numbers is equivalent to the additionof one of them with itself as many times as the value of the other one.

The two numbers being multiplied are called factors.

The answer when multiplying is called the product.

DefinitionFor every whole number a and n 6= 0,

n · a = a + a + a + . . .+ a︸ ︷︷ ︸n terms

If n = 0, then 0 · a = 0.

What is Multiplication?

Can we define multiplication?

DefinitionThe multiplication of two whole numbers is equivalent to the additionof one of them with itself as many times as the value of the other one.

The two numbers being multiplied are called

factors.

The answer when multiplying is called the product.

DefinitionFor every whole number a and n 6= 0,

n · a = a + a + a + . . .+ a︸ ︷︷ ︸n terms

If n = 0, then 0 · a = 0.

What is Multiplication?

Can we define multiplication?

DefinitionThe multiplication of two whole numbers is equivalent to the additionof one of them with itself as many times as the value of the other one.

The two numbers being multiplied are called factors.

The answer when multiplying is called the product.

DefinitionFor every whole number a and n 6= 0,

n · a = a + a + a + . . .+ a︸ ︷︷ ︸n terms

If n = 0, then 0 · a = 0.

What is Multiplication?

Can we define multiplication?

DefinitionThe multiplication of two whole numbers is equivalent to the additionof one of them with itself as many times as the value of the other one.

The two numbers being multiplied are called factors.

The answer when multiplying is called the

product.

DefinitionFor every whole number a and n 6= 0,

n · a = a + a + a + . . .+ a︸ ︷︷ ︸n terms

If n = 0, then 0 · a = 0.

What is Multiplication?

Can we define multiplication?

DefinitionThe multiplication of two whole numbers is equivalent to the additionof one of them with itself as many times as the value of the other one.

The two numbers being multiplied are called factors.

The answer when multiplying is called the product.

DefinitionFor every whole number a and n 6= 0,

n · a = a + a + a + . . .+ a︸ ︷︷ ︸n terms

If n = 0, then 0 · a = 0.

What is Multiplication?

Can we define multiplication?

DefinitionThe multiplication of two whole numbers is equivalent to the additionof one of them with itself as many times as the value of the other one.

The two numbers being multiplied are called factors.

The answer when multiplying is called the product.

DefinitionFor every whole number a and n 6= 0,

n · a = a + a + a + . . .+ a︸ ︷︷ ︸n terms

If n = 0, then 0 · a = 0.

Repeated-Addition Model

One of the first ways we teach multiplication is using therepeated-addition model because it relates the third basic operation tothe first one they learned.

To use this, we think of the multiplication of whole numbers in termsof the quantities we have in multiple sets of the same size.

Repeated-Addition Model

One of the first ways we teach multiplication is using therepeated-addition model because it relates the third basic operation tothe first one they learned.

To use this, we think of the multiplication of whole numbers in termsof the quantities we have in multiple sets of the same size.

Repeated-Addition Model

ExampleIllustrate 5× 3.

5 5 5+ +

5× 3

Repeated-Addition Model

ExampleIllustrate 5× 3.

5 5 5+ +

5× 3

Repeated-Addition Model

ExampleIllustrate 5× 3.

5 5 5

+ +

5× 3

Repeated-Addition Model

ExampleIllustrate 5× 3.

5 5 5+ +

5× 3

Repeated-Addition Model

ExampleIllustrate 5× 3.

5 5 5+ +

5× 3

Repeated-Addition Model

We can also use number lines with this type of model.

0 5 10 15

5× 3

Repeated-Addition Model

We can also use number lines with this type of model.

0 5 10 15

5× 3

Repeated-Addition Model

We can also use number lines with this type of model.

0 5 10 15

5× 3

Repeated-Addition Model

We can also use number lines with this type of model.

0 5 10 15

5× 3

Repeated-Addend Model

With smaller children, physical manipulatives are a great way to getacross this connection between addition and multiplication.

Repeated-Addend Model

With smaller children, physical manipulatives are a great way to getacross this connection between addition and multiplication.

Rectangular Array Model

This method helps student relate multiplication to real life. Can youthink of any places an elementary school student might see arrays?

ExampleIllustrate 5× 3

Rectangular Array Model

This method helps student relate multiplication to real life. Can youthink of any places an elementary school student might see arrays?

ExampleIllustrate 5× 3

Rectangular Array Model

This method helps student relate multiplication to real life. Can youthink of any places an elementary school student might see arrays?

ExampleIllustrate 5× 3

Rectangular Array Model

This method helps student relate multiplication to real life. Can youthink of any places an elementary school student might see arrays?

ExampleIllustrate 5× 3

Rectangular Array Model

This method helps student relate multiplication to real life. Can youthink of any places an elementary school student might see arrays?

ExampleIllustrate 5× 3

Area Model

We can use a similar idea for students who are a bit older when weemploy the area model.

ExampleIllustrate 3× 5

Area Model

We can use a similar idea for students who are a bit older when weemploy the area model.

ExampleIllustrate 3× 5

Area Model

We can use a similar idea for students who are a bit older when weemploy the area model.

ExampleIllustrate 3× 5

Area Model

We can use a similar idea for students who are a bit older when weemploy the area model.

ExampleIllustrate 3× 5

Area Model

We can use a similar idea for students who are a bit older when weemploy the area model.

ExampleIllustrate 3× 5

Cartesian Products

When the students are still a little older, we can use the idea ofCartesian products to explain multiplication. This relatesmultiplication back to the idea of sets and cardinality.

The benefit of this method is the practical application we can apply tothe situation in a natural way.

ExampleSuppose you were getting dressed this morning and had to decidewhat to wear. You narrowed your choices between 3 pairs of shoesthat each match with 5 different shirts. How many options do youhave?

Cartesian Products

When the students are still a little older, we can use the idea ofCartesian products to explain multiplication. This relatesmultiplication back to the idea of sets and cardinality.

The benefit of this method is the practical application we can apply tothe situation in a natural way.

ExampleSuppose you were getting dressed this morning and had to decidewhat to wear. You narrowed your choices between 3 pairs of shoesthat each match with 5 different shirts. How many options do youhave?

Cartesian Products

When the students are still a little older, we can use the idea ofCartesian products to explain multiplication. This relatesmultiplication back to the idea of sets and cardinality.

The benefit of this method is the practical application we can apply tothe situation in a natural way.

ExampleSuppose you were getting dressed this morning and had to decidewhat to wear. You narrowed your choices between 3 pairs of shoesthat each match with 5 different shirts. How many options do youhave?

Cartesian Products

We think of each of the options as an ordered pair:

1

2

3

1

2

3

1

2

3

1

2

3

1

2

3

a b c d e

a b c d e

a b c d e

Alternately we could use tree diagrams to model this situation. Howwould we end up with our product in that case?

Cartesian Products

We think of each of the options as an ordered pair:

1

2

3

1

2

3

1

2

3

1

2

3

1

2

3

a b c d e

a b c d e

a b c d e

Alternately we could use tree diagrams to model this situation. Howwould we end up with our product in that case?

Cartesian Products

We think of each of the options as an ordered pair:

1

2

3

1

2

3

1

2

3

1

2

3

1

2

3

a b c d e

a b c d e

a b c d e

Alternately we could use tree diagrams to model this situation. Howwould we end up with our product in that case?

Cartesian Products

We think of each of the options as an ordered pair:

1

2

3

1

2

3

1

2

3

1

2

3

1

2

3

a b c d e

a b c d e

a b c d e

Alternately we could use tree diagrams to model this situation. Howwould we end up with our product in that case?

Cartesian Products

We think of each of the options as an ordered pair:

1

2

3

1

2

3

1

2

3

1

2

3

1

2

3

a b c d e

a b c d e

a b c d e

Alternately we could use tree diagrams to model this situation. Howwould we end up with our product in that case?

Properties of Multiplication

Are the whole numbers closed under multiplication?

Closure property of multiplication of whole numbersFor whole numbers a and b, a · b is a unique whole number.

Does the commutative property hold for multiplication of wholenumbers?

Commutative property of multiplication of whole numbersFor whole numbers a and b, a · b = b · a.

Does the associative property hold for multiplication of wholenumbers?

Associative property of multiplication of whole numbers

For whole numbers a, b and c, (a · b) · c = a · (b · c).

Properties of Multiplication

Are the whole numbers closed under multiplication?

Closure property of multiplication of whole numbersFor whole numbers a and b, a · b is a unique whole number.

Does the commutative property hold for multiplication of wholenumbers?

Commutative property of multiplication of whole numbersFor whole numbers a and b, a · b = b · a.

Does the associative property hold for multiplication of wholenumbers?

Associative property of multiplication of whole numbers

For whole numbers a, b and c, (a · b) · c = a · (b · c).

Properties of Multiplication

Are the whole numbers closed under multiplication?

Closure property of multiplication of whole numbersFor whole numbers a and b, a · b is a unique whole number.

Does the commutative property hold for multiplication of wholenumbers?

Commutative property of multiplication of whole numbersFor whole numbers a and b, a · b = b · a.

Does the associative property hold for multiplication of wholenumbers?

Associative property of multiplication of whole numbers

For whole numbers a, b and c, (a · b) · c = a · (b · c).

Properties of Multiplication

Are the whole numbers closed under multiplication?

Closure property of multiplication of whole numbersFor whole numbers a and b, a · b is a unique whole number.

Does the commutative property hold for multiplication of wholenumbers?

Commutative property of multiplication of whole numbersFor whole numbers a and b, a · b = b · a.

Does the associative property hold for multiplication of wholenumbers?

Associative property of multiplication of whole numbers

For whole numbers a, b and c, (a · b) · c = a · (b · c).

Properties of Multiplication

Are the whole numbers closed under multiplication?

Closure property of multiplication of whole numbersFor whole numbers a and b, a · b is a unique whole number.

Does the commutative property hold for multiplication of wholenumbers?

Commutative property of multiplication of whole numbersFor whole numbers a and b, a · b = b · a.

Does the associative property hold for multiplication of wholenumbers?

Associative property of multiplication of whole numbers

For whole numbers a, b and c, (a · b) · c = a · (b · c).

Properties of Multiplication

Are the whole numbers closed under multiplication?

Closure property of multiplication of whole numbersFor whole numbers a and b, a · b is a unique whole number.

Does the commutative property hold for multiplication of wholenumbers?

Commutative property of multiplication of whole numbersFor whole numbers a and b, a · b = b · a.

Does the associative property hold for multiplication of wholenumbers?

Associative property of multiplication of whole numbers

For whole numbers a, b and c, (a · b) · c = a · (b · c).

More Properties

What is the multiplicative identity element for the whole numbers?

Identity property of multiplication of whole numbersThere is a unique whole number 1 such that for all a ∈W,a · 1 = a = 1 · a.

What is the multiplicative inverse of an element a ∈W?

Zero multiplication property of whole numbersFor every a ∈W, 0 · a = 0 = a · 0.

More Properties

What is the multiplicative identity element for the whole numbers?

Identity property of multiplication of whole numbersThere is a unique whole number 1 such that for all a ∈W,a · 1 = a = 1 · a.

What is the multiplicative inverse of an element a ∈W?

Zero multiplication property of whole numbersFor every a ∈W, 0 · a = 0 = a · 0.

More Properties

What is the multiplicative identity element for the whole numbers?

Identity property of multiplication of whole numbersThere is a unique whole number 1 such that for all a ∈W,a · 1 = a = 1 · a.

What is the multiplicative inverse of an element a ∈W?

Zero multiplication property of whole numbersFor every a ∈W, 0 · a = 0 = a · 0.

More Properties

What is the multiplicative identity element for the whole numbers?

Identity property of multiplication of whole numbersThere is a unique whole number 1 such that for all a ∈W,a · 1 = a = 1 · a.

What is the multiplicative inverse of an element a ∈W?

Zero multiplication property of whole numbersFor every a ∈W, 0 · a = 0 = a · 0.

Distributive Property

Can anyone state the distributive property of multiplication overaddition?

Distribution of multiplication over addition of the whole numbers

For all whole numbers a, b and c, a(b + c) = ab + ac.

Distribution of multiplication over subtraction of the whole numbers

For all whole numbers a, b and c, a(b− c) = ab− ac.

Distributive Property

Can anyone state the distributive property of multiplication overaddition?

Distribution of multiplication over addition of the whole numbers

For all whole numbers a, b and c, a(b + c) = ab + ac.

Distribution of multiplication over subtraction of the whole numbers

For all whole numbers a, b and c, a(b− c) = ab− ac.

Distributive Property

Can anyone state the distributive property of multiplication overaddition?

Distribution of multiplication over addition of the whole numbers

For all whole numbers a, b and c, a(b + c) = ab + ac.

Distribution of multiplication over subtraction of the whole numbers

For all whole numbers a, b and c, a(b− c) = ab− ac.

Modeling Distribution

Example

Illustrate 2(3 + 4) = 2(3) + 2(4)

Modeling Distribution

Example

Illustrate 2(3 + 4) = 2(3) + 2(4)

Modeling Distribution

Example

Illustrate 2(3 + 4) = 2(3) + 2(4)

Modeling Distribution

Example

Illustrate 2(3 + 4) = 2(3) + 2(4)

Modeling Distribution

Example

Illustrate 2(3 + 4) = 2(3) + 2(4)

Division of Whole Numbers

How can we define division considering we only have whole numbersat our disposal?

DefinitionFor any whole numbers a and b, with b 6= 0, a÷ b = c if and only if cis the unique whole number such that b · c = a.

This idea is also known as the Missing Factor Approach.

Are the whole numbers closed under division?

Division of Whole Numbers

How can we define division considering we only have whole numbersat our disposal?

DefinitionFor any whole numbers a and b, with b 6= 0, a÷ b = c if and only if cis the unique whole number such that b · c = a.

This idea is also known as the Missing Factor Approach.

Are the whole numbers closed under division?

Division of Whole Numbers

How can we define division considering we only have whole numbersat our disposal?

DefinitionFor any whole numbers a and b, with b 6= 0, a÷ b = c if and only if cis the unique whole number such that b · c = a.

This idea is also known as the Missing Factor Approach.

Are the whole numbers closed under division?

Division of Whole Numbers

How can we define division considering we only have whole numbersat our disposal?

DefinitionFor any whole numbers a and b, with b 6= 0, a÷ b = c if and only if cis the unique whole number such that b · c = a.

This idea is also known as the Missing Factor Approach.

Are the whole numbers closed under division?

Division Visually

ExampleModel 20÷ 5

To show division with the younger students, we often begin with aset partition model.

Division Visually

ExampleModel 20÷ 5

To show division with the younger students, we often begin with aset partition model.

Division Visually

ExampleModel 20÷ 5

To show division with the younger students, we often begin with aset partition model.

Division Visually

ExampleModel 20÷ 5

To show division with the younger students, we often begin with aset partition model.

Division Visually

ExampleModel 20÷ 5

To show division with the younger students, we often begin with aset partition model.

Division Visually

ExampleModel 20÷ 5

To show division with the younger students, we often begin with aset partition model.

Division Visually

ExampleModel 20÷ 5

To show division with the younger students, we often begin with aset partition model.

Division Visually

ExampleModel 20÷ 5

To show division with the younger students, we often begin with aset partition model.

The Division Algorithm

Once we get students a little older so that they don’t need the visualsas much, we can turn to one of the most important formulas inarithmetic.

The Division AlgorithmFor any whole numbers a and b with b 6= 0, there exists wholenumbers q and r such that

a = bq + r 0 ≤ r < b

We call q the quotient.

We call r the remainder.

The Division Algorithm

Once we get students a little older so that they don’t need the visualsas much, we can turn to one of the most important formulas inarithmetic.

The Division AlgorithmFor any whole numbers a and b with b 6= 0, there exists wholenumbers q and r such that

a = bq + r 0 ≤ r < b

We call q the quotient.

We call r the remainder.

The Division Algorithm

Once we get students a little older so that they don’t need the visualsas much, we can turn to one of the most important formulas inarithmetic.

The Division AlgorithmFor any whole numbers a and b with b 6= 0, there exists wholenumbers q and r such that

a = bq + r 0 ≤ r < b

We call q the

quotient.

We call r the remainder.

The Division Algorithm

Once we get students a little older so that they don’t need the visualsas much, we can turn to one of the most important formulas inarithmetic.

The Division AlgorithmFor any whole numbers a and b with b 6= 0, there exists wholenumbers q and r such that

a = bq + r 0 ≤ r < b

We call q the quotient.

We call r the remainder.

The Division Algorithm

Once we get students a little older so that they don’t need the visualsas much, we can turn to one of the most important formulas inarithmetic.

The Division AlgorithmFor any whole numbers a and b with b 6= 0, there exists wholenumbers q and r such that

a = bq + r 0 ≤ r < b

We call q the quotient.

We call r the

remainder.

The Division Algorithm

Once we get students a little older so that they don’t need the visualsas much, we can turn to one of the most important formulas inarithmetic.

The Division AlgorithmFor any whole numbers a and b with b 6= 0, there exists wholenumbers q and r such that

a = bq + r 0 ≤ r < b

We call q the quotient.

We call r the remainder.

The Division Algorithm

The key here is the uniqueness assertion. There is exactly one q thatsatisfies this equation.

ExampleUse the Division Algorithm to express 55÷ 7.

55 = 7q + r

55 = 7(7) + 6

The Division Algorithm

The key here is the uniqueness assertion. There is exactly one q thatsatisfies this equation.

ExampleUse the Division Algorithm to express 55÷ 7.

55 = 7q + r

55 = 7(7) + 6

The Division Algorithm

The key here is the uniqueness assertion. There is exactly one q thatsatisfies this equation.

ExampleUse the Division Algorithm to express 55÷ 7.

55 = 7q + r

55 = 7(7) + 6

The Division Algorithm

The key here is the uniqueness assertion. There is exactly one q thatsatisfies this equation.

ExampleUse the Division Algorithm to express 55÷ 7.

55 = 7q + r

55 = 7(7) + 6

The Division Algorithm

ExampleWhich of the following violate the Division Algorithm?

37 = 5 · 6 + 7

45 = 5 · 9150 = 11 · 13 + 7

The Division Algorithm

ExampleWhich of the following violate the Division Algorithm?

37 = 5 · 6 + 7

45 = 5 · 9

150 = 11 · 13 + 7

The Division Algorithm

ExampleWhich of the following violate the Division Algorithm?

37 = 5 · 6 + 7

45 = 5 · 9150 = 11 · 13 + 7

The Division Algorithm

ExampleIf 221 is divided by some whole number, the remainder is 33. Whatare the possible divisors?

221 = bq + 33

188 = bq

So, we need to find 2 whole numbers whose product is 188.

188 = 1 · 188

188 = 2 · 94

188 = 4 · 47

So which are the possible divisors? 47, 94, 188

The Division Algorithm

ExampleIf 221 is divided by some whole number, the remainder is 33. Whatare the possible divisors?

221 = bq + 33

188 = bq

So, we need to find 2 whole numbers whose product is 188.

188 = 1 · 188

188 = 2 · 94

188 = 4 · 47

So which are the possible divisors? 47, 94, 188

The Division Algorithm

ExampleIf 221 is divided by some whole number, the remainder is 33. Whatare the possible divisors?

221 = bq + 33

188 = bq

So, we need to find 2 whole numbers whose product is 188.

188 = 1 · 188

188 = 2 · 94

188 = 4 · 47

So which are the possible divisors? 47, 94, 188

The Division Algorithm

ExampleIf 221 is divided by some whole number, the remainder is 33. Whatare the possible divisors?

221 = bq + 33

188 = bq

So, we need to find 2 whole numbers whose product is 188.

188 = 1 · 188

188 = 2 · 94

188 = 4 · 47

So which are the possible divisors? 47, 94, 188

The Division Algorithm

ExampleIf 221 is divided by some whole number, the remainder is 33. Whatare the possible divisors?

221 = bq + 33

188 = bq

So, we need to find 2 whole numbers whose product is 188.

188 = 1 · 188

188 = 2 · 94

188 = 4 · 47

So which are the possible divisors? 47, 94, 188

The Division Algorithm

ExampleIf 221 is divided by some whole number, the remainder is 33. Whatare the possible divisors?

221 = bq + 33

188 = bq

So, we need to find 2 whole numbers whose product is 188.

188 = 1 · 188

188 = 2 · 94

188 = 4 · 47

So which are the possible divisors?

47, 94, 188

The Division Algorithm

ExampleIf 221 is divided by some whole number, the remainder is 33. Whatare the possible divisors?

221 = bq + 33

188 = bq

So, we need to find 2 whole numbers whose product is 188.

188 = 1 · 188

188 = 2 · 94

188 = 4 · 47

So which are the possible divisors? 47, 94, 188

Other Properties

Division with 0For any whole number c,

0÷ c =

0

c÷ 0 = undefined

0÷ 0 = undefined

Division with 1For any whole number c,

c÷ 1 =c

1÷ c = ???

Other Properties

Division with 0For any whole number c,

0÷ c = 0

c÷ 0 = undefined

0÷ 0 = undefined

Division with 1For any whole number c,

c÷ 1 =c

1÷ c = ???

Other Properties

Division with 0For any whole number c,

0÷ c = 0

c÷ 0 =

undefined

0÷ 0 = undefined

Division with 1For any whole number c,

c÷ 1 =c

1÷ c = ???

Other Properties

Division with 0For any whole number c,

0÷ c = 0

c÷ 0 = undefined

0÷ 0 = undefined

Division with 1For any whole number c,

c÷ 1 =c

1÷ c = ???

Other Properties

Division with 0For any whole number c,

0÷ c = 0

c÷ 0 = undefined

0÷ 0 =

undefined

Division with 1For any whole number c,

c÷ 1 =c

1÷ c = ???

Other Properties

Division with 0For any whole number c,

0÷ c = 0

c÷ 0 = undefined

0÷ 0 = undefined

Division with 1For any whole number c,

c÷ 1 =c

1÷ c = ???

Other Properties

Division with 0For any whole number c,

0÷ c = 0

c÷ 0 = undefined

0÷ 0 = undefined

Division with 1For any whole number c,

c÷ 1 =

c

1÷ c = ???

Other Properties

Division with 0For any whole number c,

0÷ c = 0

c÷ 0 = undefined

0÷ 0 = undefined

Division with 1For any whole number c,

c÷ 1 =c

1÷ c = ???

Other Properties

Division with 0For any whole number c,

0÷ c = 0

c÷ 0 = undefined

0÷ 0 = undefined

Division with 1For any whole number c,

c÷ 1 =c

1÷ c =

???

Other Properties

Division with 0For any whole number c,

0÷ c = 0

c÷ 0 = undefined

0÷ 0 = undefined

Division with 1For any whole number c,

c÷ 1 =c

1÷ c = ???

Order of Operations

What are the order of operations?

Multiplication and division from left to right

Addition and subtraction from left to right

What can affect just going from left to right? parentheses ...

Order of Operations

What are the order of operations?

Multiplication and division

from left to right

Addition and subtraction from left to right

What can affect just going from left to right? parentheses ...

Order of Operations

What are the order of operations?

Multiplication and division from left to right

Addition and subtraction from left to right

What can affect just going from left to right? parentheses ...

Order of Operations

What are the order of operations?

Multiplication and division from left to right

Addition and subtraction

from left to right

What can affect just going from left to right? parentheses ...

Order of Operations

What are the order of operations?

Multiplication and division from left to right

Addition and subtraction from left to right

What can affect just going from left to right? parentheses ...

Order of Operations

What are the order of operations?

Multiplication and division from left to right

Addition and subtraction from left to right

What can affect just going from left to right?

parentheses ...

Order of Operations

What are the order of operations?

Multiplication and division from left to right

Addition and subtraction from left to right

What can affect just going from left to right? parentheses ...

Order of Operations

When solving equations, we have to think about order of operations inreverse, which means we have to think about which operations areinverses of each other.

The inverse of addition is subtraction.

The inverse of division is multiplication.

Order of Operations

When solving equations, we have to think about order of operations inreverse, which means we have to think about which operations areinverses of each other.

The inverse of addition is

subtraction.

The inverse of division is multiplication.

Order of Operations

When solving equations, we have to think about order of operations inreverse, which means we have to think about which operations areinverses of each other.

The inverse of addition is subtraction.

The inverse of division is multiplication.

Order of Operations

When solving equations, we have to think about order of operations inreverse, which means we have to think about which operations areinverses of each other.

The inverse of addition is subtraction.

The inverse of division is

multiplication.

Order of Operations

When solving equations, we have to think about order of operations inreverse, which means we have to think about which operations areinverses of each other.

The inverse of addition is subtraction.

The inverse of division is multiplication.