Grade 8 Number Sense and Numeration Multiplying and Dividing Integers

Overall Expectation Solve problems involving integers using a variety of computational strategies Solve problems involving integers using a variety of computational strategies

Specific Expectations Represent the multiplication and division of integers, using a variety of tools Represent the multiplication and division of integers, using a variety of tools Solve problems involving operations with integers, using a variety of tools Solve problems involving operations with integers, using a variety of tools

Adding Integers Recall: Recall: When adding two positive integers the result is positive When adding two positive integers the result is positive When adding two negative integers, the result is negative When adding two negative integers, the result is negative

Subtracting Integers Recall: Recall: When subtracting two integers, we actually add the opposite When subtracting two integers, we actually add the opposite Ex. (+2) – (-5) = (+2) + (+5) = (+7) Ex. (+2) – (-5) = (+2) + (+5) = (+7) Think Out Loud: When we remove a negative, things become more positive Think Out Loud: When we remove a negative, things become more positive Note: We include our positive/negative sign inside brackets so that we don’t confuse it with our operation Note: We include our positive/negative sign inside brackets so that we don’t confuse it with our operation

Multiplying Integers Recall: When we first learned to multiply we learned that multiplication was simply repeated addition Recall: When we first learned to multiply we learned that multiplication was simply repeated addition Ex = 2 x 3 = 6 Ex = 2 x 3 = 6 We also learned that multiplication is reversible We also learned that multiplication is reversible Ex. 2 x 3 = 3 x 2 Ex. 2 x 3 = 3 x 2

Multiplying Integers Finally, we learned that multiplication means, “groups of” Finally, we learned that multiplication means, “groups of” Ex. 2 x 3 means two “groups of” 3 Ex. 2 x 3 means two “groups of” 3 We will use this “groups of” idea when we multiply integers We will use this “groups of” idea when we multiply integers Let’s use: Let’s use: blue circles to represent (-1) blue circles to represent (-1) red circle to represent (+1) red circle to represent (+1) Just like cold and hot Just like cold and hot

Multiplying Integers Positive x Positive Positive x Positive This is just the multiplication we’ve always done This is just the multiplication we’ve always done Ex. (+3) x (+2) = (+6) Ex. (+3) x (+2) = (+6) We have 3 groups of 2 red dots We have 3 groups of 2 red dots Thus we have 6 red dots altogether Thus we have 6 red dots altogether

Multiplying Integers Positive x Negative Positive x Negative Now we have groups of blue dots Now we have groups of blue dots Ex. (+3) x (-2) = (-6) Ex. (+3) x (-2) = (-6) We have 3 groups of 2 blue dots We have 3 groups of 2 blue dots Thus we have 6 blue dots altogether Thus we have 6 blue dots altogether

Multiplying Integers If we switch signs so the question were to become (-3) x (+2) we could say we had 2 groups of 3 blue dots but our answer would be the same If we switch signs so the question were to become (-3) x (+2) we could say we had 2 groups of 3 blue dots but our answer would be the same When multiplying one negative and one positive integer, the positive number represents how many groups there are, and the negative number represents the number of blue dots in each group When multiplying one negative and one positive integer, the positive number represents how many groups there are, and the negative number represents the number of blue dots in each group

Multiplying Integers Negative x Negative Negative x Negative How can we have a negative number of groups of negative numbers? How can we have a negative number of groups of negative numbers? We can’t, so here’s the trick… We can’t, so here’s the trick…

Multiplying Integers When we changed (+3) x (-2)into (-3) x (+2)we still got (-6) as long as we switched both signs to its opposite When we changed (+3) x (-2)into (-3) x (+2)we still got (-6) as long as we switched both signs to its opposite We do the same with two negative integers We do the same with two negative integers We switch both signs We switch both signs

Multiplying Integers So a question like (-3) x (-2)becomes (+3) x (+2) and we already know how to solve Positive x Positive So (-3) x (-2) = (+3) x (+2) = (+6) So a question like (-3) x (-2)becomes (+3) x (+2) and we already know how to solve Positive x Positive So (-3) x (-2) = (+3) x (+2) = (+6)

In Summary When multiplying integers: When multiplying integers: Positive x Positive = Positive Positive x Positive = Positive Positive x Negative = Negative Positive x Negative = Negative Negative x Positive = Negative Negative x Positive = Negative Negative x Negative = Positive Negative x Negative = Positive To further summarize: To further summarize: When signs match, the answer is Positive When signs match, the answer is Positive When signs do not match, the answer is negative When signs do not match, the answer is negative

Dividing Integers Our summary for division is the same: Our summary for division is the same: Positive ÷ Positive = Positive Positive ÷ Positive = Positive Positive ÷ Negative = Negative Positive ÷ Negative = Negative Negative ÷ Positive = Negative Negative ÷ Positive = Negative Negative ÷ Negative = Positive Negative ÷ Negative = Positive To further summarize: To further summarize: When signs match, the answer is Positive When signs match, the answer is Positive When signs do not match, the answer is negative When signs do not match, the answer is negative

Practice Recall: If no sign is present, assume the digit is positive: Recall: If no sign is present, assume the digit is positive: 6 x (-2) 6 x (-2) 4 x 2 4 x 2 (-8) x 2 (-8) x 2 3 x (-3) 3 x (-3) (-4) x (-2) (-4) x (-2) Now repeat these questions by inserting a division sign Now repeat these questions by inserting a division sign