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SEEING HOW INTEGERS WORK Adding positive integers is like the temp.going up on a thermometer – a rise. Adding negative integers is like the temp. going down on a thermometer – a drop. Examples: Start at zero on the thermometer ~ a) (+5) + (+3) = go up 5 degrees, and then go up 3 more. Result equals +8. b) (+7) + (-10) = go up 7 degrees, and then go down 10 degrees. Result equals -3. Integer Addition: If the signs are the same, you find the sum of the numbers and use their sign. Examples: (+ 3) + (+ 2) Find the sum (add) and then use the positive sign. Answer = + 5 (- 5) + (- 5) Find the sum (add) and then use the negative sign. Ans. Is neg - 10 If the signs are different, you find the difference of the numbers and use the sign of the larger number (not including the numbers’ sign, + or -). (- 10) + ( + 4) Find the difference (subtract: 10 - 4 = 6) and use the sign of the larger number (-10), which is negative. So the answer becomes negative 6, (-6). Integer Subtraction: When working with subtraction, it’s wise to rewrite the question as an Addition question before continuing with the calculations: Subtracting a positive integer, takes away its positive direction, so it becomes the same as adding a negative integer: - (+ 4) becomes + (- 4). Subtracting a negative integer, takes away its negative direction, so it becomes the same as adding a positive integer: - (- 4) becomes + (+ 4) Examples: Start at the place of the first number on the thermometer ~ a) (- 3) – (+ 4) = -3, take away a rise of 4 becomes (- 3) + (- 4) = -3 plus a drop of 4 degrees = puts the answer at 7 below zero, or at (- 7) ~ Page 2 ~ b) (+5) – (-5) = a rise of 5 degrees, take away a drop of 5 degrees becomes (+5) +(+5) = a rise of 5 degrees plus a rise of 5 degrees = a rise of 10 degrees or (+10) Integer Multiplication/Division: Multiplication of integers can be seen as repeated addition of a certain number: 5 times +3, 5 X +3, is 5 moves of +3, moving up on the thermometer 5 times., = +15. The same way, 5 times -3, 5 X -3 is moving down on the thermometer 5 times = -15 Using the same logic, -5 times -3, -5 X -3, would give you the opposite of 5 X -3 (-15), or +15. So… we can see that when any two integers are being multiplied, if only one of the numbers has a negative sign in the expression, the result (answer to the question) becomes negative. If both integers in the expression have negative signs, the result becomes positive. pos. X neg. = neg. neg. X pos. = neg. neg. X neg. = pos. These same rules apply to division. Examples: Multiplication - 4 X +5 = -20 Division -20 divided by +5 = -4, +20 divided by -5 = -4 + 3 X -7 = -21 -21 divided by -7 = +3, -21 divided by +3 = -7 - 2 X -3 = +6 +6 divided by -3 = -2, +6 divided by -2 = -3 Practice: a) -7 + -3 = _____ k) +5 - +3 = ______ b) +6 + +2 = _____ l) +8 - ( -2) = ______ c) -8 + -1 = ______ m) -6 - (-7) = ______ d) +9 + +2 = ______ n) +10 - (-5) = ______ e) +7 + - 5 = ______ o) - 17 – (+ 4) = ______ f) -4 + +5 = ______ p) -50 - (-30) = ______ g) +12 + -20 = ______ q) +24 – (-40) = ______ h) -25 + -13 = ______ r) -30 - (+2) = ______ i) -10 + -5 = ______ s) -10 - (10) = ______ j) +14 + -14 = ______ t) +13 – (+5) = ______ a) +4 X +3 = ______ j) +4 div. by -2 = ______ b) -2 X +5 = ______ k) -10 div. by +5 = ______ c) +6 X -3 = ______ l) -6 div. by -3 = ______ d) – 8 X -2 = ______ m) -14 div. by +2 = ______ e) + 4 X -5 = ______ n) +20 div. by -10 = ______ f) -3 X -4 = ______ o) -100 div. by -2 = ______ g) -2 X -3 X -4 = ______ Use order of operations for mixed expressions: Complete Multiplication & Division in order from Left to Right, then continue to do the Addition and Subtraction in order from L to R: Example: -12 div. by -3 X -2 = ______ becomes +4 X -2 = - 8 h) +5 X -3 X -2 = ______ p) -10 div. by -2 X -1 = ______ i) -10 X +2 X -3 = ______ q) +15 div. by -3 X -5 = ______ For multiple number questions, calculate two at a time: Example: -3 X +7 X - 1 = ______ becomes -21 X -1 = +21 r) - 4 X -8 div. by -2 = ______ Devise your own questions and answer them on another sheet, then challenge a partner! ______________________________________ ______________________________________ ______________________________________ ______________________________________ ______________________________________ ______________________________________ ______________________________________ ______________________________________ ______________________________________ ______________________________________ ______________________________________ ______________________________________ ______________________________________ ______________________________________