Operations Involving Integers Blaine Bernard Intermediate Mathematics In-Service Tuesday, August 24,...

Operations Involving Operations Involving IntegersIntegers

Blaine BernardBlaine BernardIntermediate Mathematics In-ServiceIntermediate Mathematics In-Service

Tuesday, August 24, 2010Tuesday, August 24, 2010

Algebra TilesAlgebra Tiles

Are used to enhance student Are used to enhance student understanding of mathematics traditionally understanding of mathematics traditionally taught at the symbolic level.taught at the symbolic level.

Provide access to symbol manipulation for Provide access to symbol manipulation for students with weak number sense.students with weak number sense.

Provide a geometric interpretation of Provide a geometric interpretation of symbol manipulation.symbol manipulation.

Algebra TilesAlgebra Tiles

Support cooperative learning and improve Support cooperative learning and improve discourse in the classroom by giving students discourse in the classroom by giving students objects to think with and to talk about.objects to think with and to talk about.

When I listen, I hear.When I listen, I hear.When I see, I remember.When I see, I remember.But when I do, I understand.But when I do, I understand.

Algebra TilesAlgebra Tiles

Algebra tiles can be used to model Algebra tiles can be used to model operations involving integers.operations involving integers.

Let the small red square represent +1 and Let the small red square represent +1 and the small white square (the flip-side) the small white square (the flip-side) represent -1.represent -1.

The red and white squares are additive The red and white squares are additive inverses of each other.inverses of each other.

Zero PairsZero Pairs

Called zero pairs because they are Called zero pairs because they are additive inverses of each other.additive inverses of each other.

When put together, they cancel each other When put together, they cancel each other out to model zero.out to model zero.

Addition of IntegersAddition of Integers

Addition can be viewed as combining. Addition can be viewed as combining.

Combining involves the forming and Combining involves the forming and removing of all zero pairs.removing of all zero pairs.

Addition of IntegersAddition of Integers

(+3) + (+1) =(+3) + (+1) =

(-2) + (-1) =(-2) + (-1) =

Addition of IntegersAddition of Integers

(+3) + (-1) = (+3) + (-1) =

(+4) + (-4) = (+4) + (-4) =

Addition PracticeAddition Practice

(+2) + (+3) = (+2) + (+3) =

(-3) + (-4) =(-3) + (-4) =

(+1) + (-5) =(+1) + (-5) =

(-4) + (+2) =(-4) + (+2) =

Subtraction of IntegersSubtraction of Integers

Subtraction can be interpreted as take Subtraction can be interpreted as take away.away.

Subtraction can also be thought of as Subtraction can also be thought of as adding the opposite.adding the opposite.

Subtracting IntegersSubtracting Integers

Take AwayTake Away

(+5) – (+2) =(+5) – (+2) =

(-4) – (-3) =(-4) – (-3) =

Subtracting IntegersSubtracting Integers

Take AwayTake Away

(+3) – (-5) =(+3) – (-5) =

(-4) – (+1) =(-4) – (+1) =

Subtracting IntegersSubtracting Integers

Take AwayTake Away

(+3) – (-3) =(+3) – (-3) =

Subtracting IntegersSubtracting Integers

Adding the OppositeAdding the Opposite

(+3) – (-2) =(+3) – (-2) =

(+1) – (+4) =(+1) – (+4) =

Subtraction PracticeSubtraction Practice

(+5) – (+3) = (+5) – (+3) =

(-4) – (-2) =(-4) – (-2) =

(+1) – (-3) =(+1) – (-3) =

(0) – (+2) =(0) – (+2) =

Next StepsNext Steps

It is hoped that through the use of algebra It is hoped that through the use of algebra tiles to model integers, students will tiles to model integers, students will develop on their own, and understand, the develop on their own, and understand, the rules that we commonly use when adding rules that we commonly use when adding and subtracting integers.and subtracting integers.

It is at that point that they will be able to It is at that point that they will be able to work with integers at the symbolic level.work with integers at the symbolic level.

Multiplying IntegersMultiplying Integers

Consider a greenhouse whose Consider a greenhouse whose temperature is regulated by using hot temperature is regulated by using hot coals and ice cubes.coals and ice cubes.

An ice cube reduces the temperature of An ice cube reduces the temperature of the greenhouse by 1the greenhouse by 100C and a hot coal C and a hot coal increases the temperature of the increases the temperature of the greenhouse by 1greenhouse by 100C.C.

Multiplying IntegersMultiplying Integers

Multiplying IntegersMultiplying Integers

Ex) Consider: (+3) x (-7) = ?Ex) Consider: (+3) x (-7) = ?

Multiplying IntegersMultiplying Integers

Ex) Consider: (+3) x (-7) = ?Ex) Consider: (+3) x (-7) = ?

Think of our Model:Think of our Model:

Multiplying IntegersMultiplying Integers

Ex) Consider: (+3) x (-7) = ?Ex) Consider: (+3) x (-7) = ?

Think of our Model:Think of our Model:

Make 3 trips into the greenhouse to add . . .Make 3 trips into the greenhouse to add . . .

seven ice cubes each time.seven ice cubes each time.

Multiplying IntegersMultiplying Integers

Ex) Consider: (+3) x (-7) = ?Ex) Consider: (+3) x (-7) = ?

Think of our Model:Think of our Model:

Make 3 trips into the greenhouse to add . . .Make 3 trips into the greenhouse to add . . .

seven ice cubes each time.seven ice cubes each time.

What is the overall effect?What is the overall effect?

Multiplying IntegersMultiplying Integers

Ex) Consider: (+3) x (-7) = ?Ex) Consider: (+3) x (-7) = ?

Think of our Model:Think of our Model:

Make 3 trips into the greenhouse to add . . .Make 3 trips into the greenhouse to add . . .

seven ice cubes each time.seven ice cubes each time.

What is the overall effect? The temperature drops by 21What is the overall effect? The temperature drops by 21°.°.

Multiplying IntegersMultiplying Integers

Ex) Consider: (+3) x (-7) = ?Ex) Consider: (+3) x (-7) = ?

Think of our Model:Think of our Model:

Make 3 trips into the greenhouse to add . . .Make 3 trips into the greenhouse to add . . .

seven ice cubes each time.seven ice cubes each time.

What is the overall effect? The temperature drops by 21What is the overall effect? The temperature drops by 21°.°.

So, (+3) x (-7) = So, (+3) x (-7) =

Multiplying IntegersMultiplying Integers

Ex) Consider: (+3) x (-7) = ?Ex) Consider: (+3) x (-7) = ?

Think of our Model:Think of our Model:

Make 3 trips into the greenhouse to add . . .Make 3 trips into the greenhouse to add . . .

seven ice cubes each time.seven ice cubes each time.

What is the overall effect? The temperature drops by 21What is the overall effect? The temperature drops by 21°.°.

So, (+3) x (-7) = (-21)So, (+3) x (-7) = (-21)

Multiplying IntegersMultiplying Integers

Ex) (-5) x (-4) = Ex) (-5) x (-4) =

Multiplying IntegersMultiplying Integers

Ex) (-5) x (-4) = Ex) (-5) x (-4) =

We need to make 5 trips into the greenhouse to remove . . .We need to make 5 trips into the greenhouse to remove . . .

four ice cubes each time.four ice cubes each time.

Multiplying IntegersMultiplying Integers

Ex) (-5) x (-4) = Ex) (-5) x (-4) =

We need to make 5 trips into the greenhouse to remove . . .We need to make 5 trips into the greenhouse to remove . . .

four ice cubes each time.four ice cubes each time.

The temperature will rise by 20The temperature will rise by 20° C.° C.

Multiplying IntegersMultiplying Integers

Ex) (-5) x (-4) = Ex) (-5) x (-4) =

We need to make 5 trips into the greenhouse to remove . . .We need to make 5 trips into the greenhouse to remove . . .

four ice cubes each time.four ice cubes each time.

The temperature will rise by 20The temperature will rise by 20° C.° C.

So, (-5) x (-4) = (+20)So, (-5) x (-4) = (+20)

Multiplying IntegersMultiplying Integers

Ex) (-3) x (+8) = Ex) (-3) x (+8) =

Multiplying IntegersMultiplying Integers

Ex) (-3) x (+8) = Ex) (-3) x (+8) =

We need to make 3 trips into the greenhouse to remove . . .We need to make 3 trips into the greenhouse to remove . . .

8 hot coals each time.8 hot coals each time.

Multiplying IntegersMultiplying Integers

Ex) (-3) x (+8) = Ex) (-3) x (+8) =

We need to make 3 trips into the greenhouse to remove . . .We need to make 3 trips into the greenhouse to remove . . .

8 hot coals each time.8 hot coals each time.

Overall Effect:Overall Effect:

Multiplying IntegersMultiplying Integers

Ex) (-3) x (+8) = Ex) (-3) x (+8) =

We need to make 3 trips into the greenhouse to remove . . .We need to make 3 trips into the greenhouse to remove . . .

8 hot coals each time.8 hot coals each time.

Overall Effect: A decrease in temperature of 24Overall Effect: A decrease in temperature of 24° C.° C.

Multiplying IntegersMultiplying Integers

Ex) (-3) x (+8) = Ex) (-3) x (+8) =

We need to make 3 trips into the greenhouse to remove . . .We need to make 3 trips into the greenhouse to remove . . .

8 hot coals each time.8 hot coals each time.

Overall Effect: A decrease in temperature of 24Overall Effect: A decrease in temperature of 24° C.° C.

So, (-3) x (+8) = (-24)So, (-3) x (+8) = (-24)

Multiplying IntegersMultiplying Integers

Ex) (+5) x (+6) = Ex) (+5) x (+6) =

Multiplying IntegersMultiplying Integers

Ex) (+5) x (+6) =Ex) (+5) x (+6) =

We are making 5 trips into the greenhouse to add . . .We are making 5 trips into the greenhouse to add . . .

6 hot coals each time. 6 hot coals each time.

Multiplying IntegersMultiplying Integers

Ex) (+5) x (+6) =Ex) (+5) x (+6) =

We are making 5 trips into the greenhouse to add . . .We are making 5 trips into the greenhouse to add . . .

6 hot coals each time. 6 hot coals each time.

Overall Effect:Overall Effect:

Multiplying IntegersMultiplying Integers

Ex) (+5) x (+6) =Ex) (+5) x (+6) =

We are making 5 trips into the greenhouse to add . . .We are making 5 trips into the greenhouse to add . . .

6 hot coals each time. 6 hot coals each time.

Overall Effect: The temperature increases by 30Overall Effect: The temperature increases by 30° C.° C.

Multiplying IntegersMultiplying Integers

Ex) (+5) x (+6) =Ex) (+5) x (+6) =

We are making 5 trips into the greenhouse to add . . .We are making 5 trips into the greenhouse to add . . .

6 hot coals each time. 6 hot coals each time.

Overall Effect: The temperature increases by 30Overall Effect: The temperature increases by 30° C.° C.

So, (+5) x (+6) = (+30)So, (+5) x (+6) = (+30)

Multiplying IntegersMultiplying Integers

We have:We have:

Multiplying IntegersMultiplying Integers

We have:We have:

(+3) x (-7) = (-21)(+3) x (-7) = (-21)

Multiplying IntegersMultiplying Integers

We have:We have:

(+3) x (-7) = (-21)(+3) x (-7) = (-21)

(-5) x (-4) = (+20)(-5) x (-4) = (+20)

Multiplying IntegersMultiplying Integers

We have:We have:

(+3) x (-7) = (-21)(+3) x (-7) = (-21)

(-5) x (-4) = (+20)(-5) x (-4) = (+20)

(-3) x (+8) = (-24)(-3) x (+8) = (-24)

Multiplying IntegersMultiplying Integers

We have:We have:

(+3) x (-7) = (-21)(+3) x (-7) = (-21)

(-5) x (-4) = (+20)(-5) x (-4) = (+20)

(-3) x (+8) = (-24)(-3) x (+8) = (-24)

(+5) x (+6) = (+30)(+5) x (+6) = (+30)

Multiplying IntegersMultiplying Integers

We have:We have:

(+3) x (-7) = (-21)(+3) x (-7) = (-21)

(-5) x (-4) = (+20)(-5) x (-4) = (+20)

(-3) x (+8) = (-24)(-3) x (+8) = (-24)

(+5) x (+6) = (+30)(+5) x (+6) = (+30)

Can we write some rules?Can we write some rules?

Multiplying IntegersMultiplying Integers

RULES:RULES:

Multiplying IntegersMultiplying Integers

RULES:RULES:

1) When multiplying integers with the same signs, the 1) When multiplying integers with the same signs, the answer will be answer will be

Multiplying IntegersMultiplying Integers

RULES:RULES:

1) When multiplying integers with the same signs, the 1) When multiplying integers with the same signs, the answer will be positive.answer will be positive.

Multiplying IntegersMultiplying Integers

RULES:RULES:

1) When multiplying integers with the same signs, the 1) When multiplying integers with the same signs, the answer will be positive.answer will be positive.

2) When multiplying integers with different signs, the answer 2) When multiplying integers with different signs, the answer will bewill be

Multiplying IntegersMultiplying Integers

RULES:RULES:

1) When multiplying integers with the same signs, the 1) When multiplying integers with the same signs, the answer will be positive.answer will be positive.

2) When multiplying integers with different signs, the answer 2) When multiplying integers with different signs, the answer will be negative.will be negative.

Multiplying IntegersMultiplying Integers

RULES:RULES:

1) When multiplying integers with the same signs, the 1) When multiplying integers with the same signs, the answer will be positive.answer will be positive.

2) When multiplying integers with different signs, the answer 2) When multiplying integers with different signs, the answer will be negative.will be negative.

Will this be true for division?Will this be true for division?

Dividing IntegersDividing Integers

If:If:

(+3) x (-7) = (-21)(+3) x (-7) = (-21)

(-5) x (-4) = (+20)(-5) x (-4) = (+20)

(-3) x (+8) = (-24)(-3) x (+8) = (-24)

(+5) x (+6) = (+30)(+5) x (+6) = (+30)

Dividing IntegersDividing Integers

If:If: Then:Then:

(+3) x (-7) = (-21)(+3) x (-7) = (-21) (-21) ÷ (+3) = (-7)(-21) ÷ (+3) = (-7)

(-5) x (-4) = (+20)(-5) x (-4) = (+20)

(-3) x (+8) = (-24)(-3) x (+8) = (-24)

(+5) x (+6) = (+30)(+5) x (+6) = (+30)

Dividing IntegersDividing Integers

If:If: Then:Then:

(+3) x (-7) = (-21)(+3) x (-7) = (-21) (-21) ÷ (+3) = (-7)(-21) ÷ (+3) = (-7)

(-5) x (-4) = (+20)(-5) x (-4) = (+20) (+20) ÷ (-5) = (-4)(+20) ÷ (-5) = (-4)

(-3) x (+8) = (-24)(-3) x (+8) = (-24)

(+5) x (+6) = (+30)(+5) x (+6) = (+30)

Dividing IntegersDividing Integers

If:If: Then:Then:

(+3) x (-7) = (-21)(+3) x (-7) = (-21) (-21) ÷ (+3) = (-7)(-21) ÷ (+3) = (-7)

(-5) x (-4) = (+20)(-5) x (-4) = (+20) (+20) ÷ (-5) = (-4)(+20) ÷ (-5) = (-4)

(-3) x (+8) = (-24)(-3) x (+8) = (-24) (-24) ÷ (-3) = (+8)(-24) ÷ (-3) = (+8)

(+5) x (+6) = (+30)(+5) x (+6) = (+30)

Dividing IntegersDividing Integers

If:If: Then:Then:

(+3) x (-7) = (-21)(+3) x (-7) = (-21) (-21) ÷ (+3) = (-7)(-21) ÷ (+3) = (-7)

(-5) x (-4) = (+20)(-5) x (-4) = (+20) (+20) ÷ (-5) = (-4)(+20) ÷ (-5) = (-4)

(-3) x (+8) = (-24)(-3) x (+8) = (-24) (-24) ÷ (-3) = (+8)(-24) ÷ (-3) = (+8)

(+5) x (+6) = (+30)(+5) x (+6) = (+30) (+30) ÷ (+5) = (+6)(+30) ÷ (+5) = (+6)

Dividing IntegersDividing IntegersIf:If: Then:Then:

(+3) x (-7) = (-21)(+3) x (-7) = (-21) (-21) ÷ (+3) = (-7)(-21) ÷ (+3) = (-7)

(-5) x (-4) = (+20)(-5) x (-4) = (+20) (+20) ÷ (-5) = (-4)(+20) ÷ (-5) = (-4)

(-3) x (+8) = (-24)(-3) x (+8) = (-24) (-24) ÷ (-3) = (+8)(-24) ÷ (-3) = (+8)

(+5) x (+6) = (+30)(+5) x (+6) = (+30) (+30) ÷ (+5) = (+6)(+30) ÷ (+5) = (+6)

Are the rules the same?Are the rules the same?

Dividing IntegersDividing IntegersIf:If: Then:Then:

(+3) x (-7) = (-21)(+3) x (-7) = (-21) (-21) ÷ (+3) = (-7)(-21) ÷ (+3) = (-7)

(-5) x (-4) = (+20)(-5) x (-4) = (+20) (+20) ÷ (-5) = (-4)(+20) ÷ (-5) = (-4)

(-3) x (+8) = (-24)(-3) x (+8) = (-24) (-24) ÷ (-3) = (+8)(-24) ÷ (-3) = (+8)

(+5) x (+6) = (+30)(+5) x (+6) = (+30) (+30) ÷ (+5) = (+6)(+30) ÷ (+5) = (+6)

Are the rules the same?Are the rules the same? Yes!Yes!

Dividing IntegersDividing Integers

RULES:RULES:

1) When multiplying or dividing integers with the same signs, 1) When multiplying or dividing integers with the same signs, the answer will be positive.the answer will be positive.

Dividing IntegersDividing Integers

RULES:RULES:

1) When multiplying or dividing integers with the same signs, 1) When multiplying or dividing integers with the same signs, the answer will be positive.the answer will be positive.

2) When multiplying or dividing integers with different signs, 2) When multiplying or dividing integers with different signs, the answer will be negative.the answer will be negative.