Operations Involving Integers Blaine Bernard Intermediate Mathematics In-Service Tuesday, August 24, 2010

Algebra Tiles Are used to enhance student understanding of mathematics traditionally taught at the symbolic level. Provide access to symbol manipulation for students with weak number sense. Provide a geometric interpretation of symbol manipulation.

Algebra Tiles Support cooperative learning and improve discourse in the classroom by giving students objects to think with and to talk about. When I listen, I hear. When I see, I remember. But when I do, I understand.

Algebra Tiles Algebra tiles can be used to model operations involving integers. Let the small red square represent +1 and the small white square (the flip-side) represent -1. The red and white squares are additive inverses of each other.

Zero Pairs Called zero pairs because they are additive inverses of each other. When put together, they cancel each other out to model zero.

Addition of Integers Addition can be viewed as combining. Combining involves the forming and removing of all zero pairs.

Addition of Integers (+3) + (+1) = (+3) + (+1) = (-2) + (-1) = (-2) + (-1) =

Addition of Integers (+3) + (-1) = (+3) + (-1) = (+4) + (-4) = (+4) + (-4) =

Addition Practice (+2) + (+3) = (+2) + (+3) = (-3) + (-4) = (-3) + (-4) = (+1) + (-5) = (+1) + (-5) = (-4) + (+2) = (-4) + (+2) =

Subtraction of Integers Subtraction can be interpreted as take away. Subtraction can also be thought of as adding the opposite.

Subtracting Integers Take Away (+5) – (+2) = (+5) – (+2) = (-4) – (-3) = (-4) – (-3) =

Subtracting Integers Take Away (+3) – (-5) = (+3) – (-5) = (-4) – (+1) = (-4) – (+1) =

Subtracting Integers Take Away (+3) – (-3) = (+3) – (-3) =

Subtracting Integers Adding the Opposite (+3) – (-2) = (+1) – (+4) =

Subtraction Practice (+5) – (+3) = (+5) – (+3) = (-4) – (-2) = (-4) – (-2) = (+1) – (-3) = (+1) – (-3) = (0) – (+2) = (0) – (+2) =

Next Steps It is hoped that through the use of algebra tiles to model integers, students will develop on their own, and understand, the rules that we commonly use when adding and subtracting integers. It is at that point that they will be able to work with integers at the symbolic level.

Multiplying Integers Consider a greenhouse whose temperature is regulated by using hot coals and ice cubes. An ice cube reduces the temperature of the greenhouse by 1 0 C and a hot coal increases the temperature of the greenhouse by 1 0 C.

Multiplying Integers

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

Multiplying Integers Ex) Consider: (+3) x (-7) = ? Think of our Model:

Multiplying Integers Ex) Consider: (+3) x (-7) = ? Think of our Model: Make 3 trips into the greenhouse to add... seven ice cubes each time.

Multiplying Integers Ex) Consider: (+3) x (-7) = ? Think of our Model: Make 3 trips into the greenhouse to add... seven ice cubes each time. What is the overall effect?

Multiplying Integers Ex) Consider: (+3) x (-7) = ? Think of our Model: Make 3 trips into the greenhouse to add... seven ice cubes each time. What is the overall effect? The temperature drops by 21°.

Multiplying Integers Ex) Consider: (+3) x (-7) = ? Think of our Model: Make 3 trips into the greenhouse to add... seven ice cubes each time. What is the overall effect? The temperature drops by 21°. So, (+3) x (-7) =

Multiplying Integers Ex) Consider: (+3) x (-7) = ? Think of our Model: Make 3 trips into the greenhouse to add... seven ice cubes each time. What is the overall effect? The temperature drops by 21°. So, (+3) x (-7) = (-21)

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

Multiplying Integers Ex) (-5) x (-4) = We need to make 5 trips into the greenhouse to remove... four ice cubes each time.

Multiplying Integers Ex) (-5) x (-4) = We need to make 5 trips into the greenhouse to remove... four ice cubes each time. The temperature will rise by 20° C.

Multiplying Integers Ex) (-5) x (-4) = We need to make 5 trips into the greenhouse to remove... four ice cubes each time. The temperature will rise by 20° C. So, (-5) x (-4) = (+20)

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

Multiplying Integers Ex) (-3) x (+8) = We need to make 3 trips into the greenhouse to remove... 8 hot coals each time.

Multiplying Integers Ex) (-3) x (+8) = We need to make 3 trips into the greenhouse to remove... 8 hot coals each time. Overall Effect:

Multiplying Integers Ex) (-3) x (+8) = We need to make 3 trips into the greenhouse to remove... 8 hot coals each time. Overall Effect: A decrease in temperature of 24° C.

Multiplying Integers Ex) (-3) x (+8) = We need to make 3 trips into the greenhouse to remove... 8 hot coals each time. Overall Effect: A decrease in temperature of 24° C. So, (-3) x (+8) = (-24)

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

Multiplying Integers Ex) (+5) x (+6) = We are making 5 trips into the greenhouse to add... 6 hot coals each time.

Multiplying Integers Ex) (+5) x (+6) = We are making 5 trips into the greenhouse to add... 6 hot coals each time. Overall Effect:

Multiplying Integers Ex) (+5) x (+6) = We are making 5 trips into the greenhouse to add... 6 hot coals each time. Overall Effect: The temperature increases by 30° C.

Multiplying Integers Ex) (+5) x (+6) = We are making 5 trips into the greenhouse to add... 6 hot coals each time. Overall Effect: The temperature increases by 30° C. So, (+5) x (+6) = (+30)

Multiplying Integers We have:

Multiplying Integers We have: (+3) x (-7) = (-21)

Multiplying Integers We have: (+3) x (-7) = (-21) (-5) x (-4) = (+20)

Multiplying Integers We have: (+3) x (-7) = (-21) (-5) x (-4) = (+20) (-3) x (+8) = (-24)

Multiplying Integers We have: (+3) x (-7) = (-21) (-5) x (-4) = (+20) (-3) x (+8) = (-24) (+5) x (+6) = (+30)

Multiplying Integers We have: (+3) x (-7) = (-21) (-5) x (-4) = (+20) (-3) x (+8) = (-24) (+5) x (+6) = (+30) Can we write some rules?

Multiplying Integers RULES:

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

Multiplying Integers RULES: 1) When multiplying integers with the same signs, the answer will be positive.

Multiplying Integers RULES: 1) When multiplying integers with the same signs, the answer will be positive. 2) When multiplying integers with different signs, the answer will be

Multiplying Integers RULES: 1) When multiplying integers with the same signs, the answer will be positive. 2) When multiplying integers with different signs, the answer will be negative.

Multiplying Integers RULES: 1) When multiplying integers with the same signs, the answer will be positive. 2) When multiplying integers with different signs, the answer will be negative. Will this be true for division?

Dividing Integers If: (+3) x (-7) = (-21) (-5) x (-4) = (+20) (-3) x (+8) = (-24) (+5) x (+6) = (+30)

Dividing Integers If:Then: (+3) x (-7) = (-21)(-21) ÷ (+3) = (-7) (-5) x (-4) = (+20) (-3) x (+8) = (-24) (+5) x (+6) = (+30)

Dividing Integers If:Then: (+3) x (-7) = (-21)(-21) ÷ (+3) = (-7) (-5) x (-4) = (+20)(+20) ÷ (-5) = (-4) (-3) x (+8) = (-24) (+5) x (+6) = (+30)

Dividing Integers If:Then: (+3) x (-7) = (-21)(-21) ÷ (+3) = (-7) (-5) x (-4) = (+20)(+20) ÷ (-5) = (-4) (-3) x (+8) = (-24)(-24) ÷ (-3) = (+8) (+5) x (+6) = (+30)

Dividing Integers If:Then: (+3) x (-7) = (-21)(-21) ÷ (+3) = (-7) (-5) x (-4) = (+20)(+20) ÷ (-5) = (-4) (-3) x (+8) = (-24)(-24) ÷ (-3) = (+8) (+5) x (+6) = (+30)(+30) ÷ (+5) = (+6)

Dividing Integers If:Then: (+3) x (-7) = (-21)(-21) ÷ (+3) = (-7) (-5) x (-4) = (+20)(+20) ÷ (-5) = (-4) (-3) x (+8) = (-24)(-24) ÷ (-3) = (+8) (+5) x (+6) = (+30)(+30) ÷ (+5) = (+6) Are the rules the same?

Dividing Integers If:Then: (+3) x (-7) = (-21)(-21) ÷ (+3) = (-7) (-5) x (-4) = (+20)(+20) ÷ (-5) = (-4) (-3) x (+8) = (-24)(-24) ÷ (-3) = (+8) (+5) x (+6) = (+30)(+30) ÷ (+5) = (+6) Are the rules the same?Yes!

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

Dividing Integers RULES: 1) When multiplying or dividing integers with the same signs, the answer will be positive. 2) When multiplying or dividing integers with different signs, the answer will be negative.