#### Maintaining Mathematical Proficiency

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##### Sections
###### Exercises
Exercise name Free?
Exercises 1 Adding a negative number is the same as subtracting a positive number. That means we can calculate -5−2 in place of of -5+(-2). -5+(-2)a+(-b)=a−b-5−2Subtract term-7
Exercises 2 Adding a negative number is the same thing as subtracting a positive number. That means we can calculate 0−13 in place of 0+(-13). 0+(-13)a+(-b)=a−b0−13Subtract term-13
Exercises 3 To add 14 to -6, we start at -6 on the number line and then move 14 steps to the right.The number we end on is our answer. -6+14=8​
Exercises 4 Subtracting a negative number is the same as adding its positive counterpart, so 19−(-13) is the same as calculating 19+13. 19−(-13)a−(-b)=a+b19+13Add terms32
Exercises 5 To subtract 6 from -1, we start on -1 on a number line and move 6 steps to the left.The number we end on is our answer. -1−6=-7​
Exercises 6 Subtracting a negative number is the same thing as adding its positive counterpart, so -5−(-7) is the same as calculating -5+7. -5−(-7)a−(-b)=a+b-5+7Add terms2
Exercises 7 To add 17 to 5, we start at 5 on the number line and then move 17 steps to the right.The number we end on is our answer. 5+17=22​
Exercises 8 Adding a negative number is the same thing as subtracting a positive number. That means we can calculate 8−3 in place of 8+(-3). 8+(-3)a+(-b)=a−b8−3Subtract term5
Exercises 9 To subtract 15 from 11, we start on 11 on a number line and move 15 steps to the left.The number we land on is the answer, which is -4. 11−15=-4​
Exercises 10 Remember that the product of a negative number and a positive number is negative. -3(8)(-a)b=-ab-3⋅8Multiply-24 The product is -24.
Exercises 11 Remember that the product of two negative numbers is positive. -7(-9)-a(-b)=a⋅b7⋅9Multiply63 The product is 63.
Exercises 12 The product of a positive number and a negative number is negative. 4⋅(-7)a(-b)=-a⋅b-4⋅7Multiply-28 The product is -28.
Exercises 13 For this exercise, both the divisor and dividend are negative. When calculating, remember that the quotient of two negative numbers is always positive. -24÷-6a÷b=ba​-6-24​-b-a​=ba​624​Calculate quotient4
Exercises 14 Notice that the divisor and dividend have opposite signs. The quotient of a negative and a positive number is always negative. -16÷2a÷b=ba​2-16​Put minus sign in front of fraction-216​Calculate quotient-8
Exercises 15 Notice that the divisor and dividend have opposite signs. The quotient of a positive and a negative number is always negative. 12÷-3a÷b=ba​-312​Put minus sign in front of fraction-312​Calculate quotient-4
Exercises 16 We can calculate this by thinking about the fact that 6×8 means that we want to add 6 to itself 8 times. 6⋅8=8 times6+6+6+6+6+6+6+6​​=48​
Exercises 17 We can use that 36 can be factorized into 6×6 to help us find the quotient. Recall that the expression a÷b is the same as ba​. 36÷6a÷b=ba​636​Split into factors66⋅6​ba​=b/6a/6​16⋅1​a⋅1=a16​1a​=a6
Exercises 18 When calculating the product, remember that multiplying two negative numbers results in a positive number. -3(-4)-a(-b)=a⋅b3⋅4Multiply12
Exercises 19 aThe addition of integers depends heavily on the signs, positive or negative, of the numbers. There are two situations we need to consider.The signs are the same. The signs are different.In the situation where the signs are the same, we can add the number values and keep the existing sign. Let's look at an example for both possible cases. ​(+)+(+)=(+)⇒-3+5=8(−)+(−)=(−)⇒-3+(-5)=-8​ When the signs are different, we subtract the number values and keep the sign of the larger number. ​(+)+(−)=(+)⇒5+(-3)=2(+)+(−)=(−)⇒3+(-5)=-2​bSubtraction can be thought of as "adding the opposite." Therefore, we have all the same rules as in Part A. 5−3-5−35−(-3)​⇒-5+(-3)=2⇒-5+(-3)=-8⇒-5+3=8​cThe multiplication of integers has two possible situations, but the numeric value does not change. Only the sign changes.The signs are the same. The signs are different.In the situation where the signs are the same, the product will always be positive. ​(+)×(+)=(+)⇒(-)5×3=15(−)×(−)=(−)⇒(-5)×(-3)=15​ In the situation where the signs are opposite, the product will always be negative. ​(+)×(−)=(−)⇒5×(-3)=-15(−)×(+)=(−)⇒(-5)×3=-15​dDivision of integers follows the same rules as multiplication. ​(+)÷(+)=(+)⇒15÷3=5(−)÷(−)=(+)⇒(-15)÷(-3)=5(+)÷(−)=(−)⇒15÷(-3)=-5(−)÷(+)=(−)⇒(-15)÷3=-5​