NAME DATE PERIOD Lesson 1...
Transcript of NAME DATE PERIOD Lesson 1...
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NAME ___________________________________________________ DATE _____________________ PERIOD __________
Lesson 1 ReteachIntegers and Absolute Value
Example 1 Write an integer for each situation. Then identify its opposite and describe what it means.
a. 16 feet below the surface b. 5 strokes over par
The integer is -16. The integer is +5 or 5.The opposite is 16. The opposite is -5. It means 16 feet above the surface. It means 5 strokes below par.
Numbers on opposite sides of zero and the same distance from zero have the same absolute value.
The symbol for absolute value is two vertical bars on either side of the number. |2| = 2 and |-2| = 2
Example 2 Evaluate each expression.
a. |-4| b. |-3| + |6|
|-4| = 4 On the number line, -4 is 4 units from 0.
Exercises Write an integer for each situation. Then identify its opposite and describe what it means. 1. 2 inches less than normal 2. 13°F above average
3. a deposit of $50 4. a loss of 8 yards
Evaluate each expression if x = 8 and y = - 3. 5. 12 + |y| 6. x - |y| 7. 2|x| + 3|y|
8. x + |y| 9. 6|y| 10. 3x - 4|y|
A negative number is a number less than zero. A positive number is a number greater than zero. The set of integers can be written {…, -3, -2, -1, 0, 1, 2, 3, …} where … means continues indefinitely. Two integers can be compared using an inequality, which is a mathematical sentence containing < or >.
|- 3| + |6| = 3 + 6 = 9
6-6 420-4 -2
4 units |-3| = 3, |6| = 6
Simplify.
Program: Pre-Algebra
Vendor: Aptara
Component: ANC_C02_L1
Grade: AMPDF Pages
Math Accelerated • Chapter 2 Operations with Integers
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NAME ___________________________________________________ DATE _____________________ PERIOD __________
Lesson 2 ReteachAdding Integers
Add Integers
with the Same Sign
Add their absolute values. The sum is:
· positive if both integers are positive.
· negative if both integers are negative.
Example 1 Find the sum -3 + (-4).-3 + (-4) = -7 Add ⎪-3⎥ and ⎪-4⎥ . The sum is negative.
Add Integers
with Different Signs
Subtract their absolute values. The sum is:
· positive if the positive integer’s absolute value is greater.
· negative if the negative integer’s absolute value is greater.
Example 2 Find the sum -5 + 4.-5 + 4 = ⎪-5⎥ - ⎪4⎥ Subtract ⎪4⎥ from ⎪-5⎥ .
= 5 - 4 or 1 Simplify.
= -1 The sum is negative because ⎪-5⎥ > ⎪4⎥ .
Two numbers with the same absolute value but different signs are opposites. An integer and its opposite are also called additive inverses. This property is useful when adding 2 or more integers.
Example 3 Find the sum 12 + (- 4) + 9 + (-7).12 + (- 4) + 9 + (-7) = 12 + 9 + (- 4) + (-7) Commutative Property
= (12 + 9) + [- 4 + (-7)] Associative Property
= 21 + (-11) or 10 Simplify.
Exercises Find each sum. 1. 6 + (-3) 2. -3 + (-5) 3. 7 + (-3)
4. -4 + (-4) 5. -8 + 5 6. -12 + (-10)
7. 7 + (-18) 8. -12 + (-15) 9. 10 + (-14)
10. -33 + 19 11. -20 + (-5) 12. -15 + (-20)
13. -15 + 4 14. -34 + 29 15. 46 + (-32)
16. 6 + (-14) + (-5) + (-6) 17. -18 + 9 + (-7) + 18
18. 5 + 13 + (-11) + 6 19. -20 + 15 + (- 10) + 3
20. -33 + (-7) + 20 + 4 21. 16 + (-12) + 21 + (-25)
Program: Pre-Algebra
Vendor: Aptara
Component: ANC_C02_L2
Grade: AMPDF Pages
Math Accelerated • Chapter 2 Operations with Integers
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NAME ___________________________________________________ DATE _____________________ PERIOD __________
Lesson 3 ReteachSubtracting Integers
Subtract
IntegersTo subtract an integer, add its additive inverse.
Example 1 Find each difference.
a. 9 - 17 b. -7 - 3 9 - 17 = 9 + (-17) To subtract 17, add -17. -7 - 3 = -7 + (-3) To subtract 3, add -3.
= -8 Simplify. = -10 Simplify.
Example 2 Find each difference.
a. 4 - (-5) b. -6 - (-2) 4 - (-5) = 4 + 5 To subtract -5, add 5. -6 - (-2) = -6 + 2 To subtract -2, add 2.
= 9 Simplify. = -4 Simplify.
To find the distance between two integers on a number line, you can count the units on the number line or use absolute value.
Example 3 Find the distance between 4 and -9 on a number line. ⎪4 - (-9)⎥ = ⎪13⎥ Find the absolute value of the difference of 4 and -9.
= 13 Simplify.
Exercises Find each difference. 1. 9 - 16 2. 7 - 19 3. 12 - 21
4. -5 - 3 5. -8 - 9 6. -13 - 17
7. -24 - 8 8. 18 - (-9) 9. 26 - 49
10. -45 - (-26) 11. -15 - (-25) 12. 29 - (-6)
Find the distance between the integers on a number line. 13. -2 and -6 14. 9 and -9 15. 0 and -5
16. -12 and 15 17. -1 and -11 18. -4 and -16
Program: Pre-Algebra
Vendor: Aptara
Component: ANC_C02_L3
Grade: AMPDF Pages
Math Accelerated • Chapter 2 Operations with Integers
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NAME ___________________________________________________ DATE _____________________ PERIOD __________
Lesson 4 ReteachMultiplying Integers
Multiply Integers
with Different SignsThe product of two integers with different signs is negative.
Example 1 Find each product.
a. 4(-3) b. -8(5) 4(-3) = -12 -8(5) = -40
Multiply Integers
with the Same SignThe product of two integers with the same sign is positive.
Example 2 Find each product.
a. 6(6) b. -7(-4) 6(6) = 36 -7(-4) = 28
Example 3 Evaluate 4xy if x = 3 and y = -5.4xy = 4(3)(-5) Replace x with 3 and y with -5.
= [4(3)](-5) Associative Property of Multiplication
= 12(-5) The product of 4 and 3 is positive.
= -60 The product of 12 and -5 is negative.
Exercises Find each product.
1. -5(7) 2. 6(-9) 3. -10 · 4
4. -12 · -2 5. 5(-11) 6. -15(-4)
7. 11(3)(-2) 8. -5(-6)(7) 9. -2(-5)(-9)
Evaluate each expression if x = -4 and y = 8.
10. 4x 11. 3y 12. -12x
13. -2xy 14. 5xy 15. -3x(-y)
Program: Pre-Algebra
Vendor: Aptara
Component: ANC_C02_L4
Grade: AMPDF Pages
Math Accelerated • Chapter 2 Operations with Integers
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NAME ___________________________________________________ DATE _____________________ PERIOD __________
Lesson 5 ReteachDividing Integers
Divide Integers
with Different SignsThe quotient of two integers with different signs is negative.
Example 1 Find each quotient.
a. 36 ÷ (-4) The signs are different. 36 ÷ (-4) = -9 The quotient is negative.
b. - 42 − 6 The signs are different.
- 42 − 6 = -7 The quotient is negative.
Divide Integers
with the Same SignThe quotient of two integers with the same sign is positive.
Example 2 Find each quotient.
a. 14 ÷ 2 The signs are the same.
14 ÷ 2 = 7 The quotient is positive.
b. -25 − -5
-25 − -5 = -25 ÷ (-5) The signs are the same.
= 5 The quotient is positive.
To find the mean, or average, of a set of numbers, find the sum of the numbers and then divide by the number of items in the set.
Example 3 The diving depths in feet of 7 scuba divers studying schools of fish were -12, -9, -15, -8, -20, -17, and -10. Find the mean diving depth.
-12 + (-9) + (-15) + (-8) + (-20) + (-17) + (-10) −−− 7 = -91 − 7 Find the sum of the diving depths.
Divide by the number of divers.
= -13 Simplify.
The mean diving depth is -13 feet, or 13 feet below sea level.
Exercises Find each quotient.
1. 40 ÷ (-5) 2. -18 ÷ (-2) 3. -24 ÷ 6
4. -28 − 2 5. 36 − -4 6. -150 −
-25
7. The low temperatures in degrees Fahrenheit for a week were -3, 5, -9, 2, 6, -11, and -4. Find the mean temperature.
8. During 5 rounds of golf, James had scores of 2, -1, 0, -2, and -4. Find the mean of his golf scores.
Program: Pre-Algebra
Vendor: Aptara
Component: ANC_C02_L5
Grade: AMPDF Pages
Math Accelerated • Chapter 2 Operations with Integers
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NAME ___________________________________________________ DATE _____________________ PERIOD __________
Lesson 6 ReteachGraphing in Four Quadrants
Example 1 Graph and label each point on a coordinate plane. Name the quadrant in which each point lies.
a. M(-2, 5) Start at the origin. Move 2 units left. Then move 5 units up and draw a dot. Point M(-2, 5) is in Quadrant II.
b. N(4, -4) Start at the origin. Move 4 units right. Then move 4 units down and draw a dot. Point N(4, -4) is in Quadrant IV.
Exercises Graph and label each point on the coordinate plane. Name the quadrant in which each point is located.
1. A(2, 6) 2. B(-1, 4)
3. C(0, -5) 4. D(-4, -3)
5. E(2, 0) 6. F(3, -2)
7. G(-4, 4) 8. H(2, -5)
9. I(6, 3) 10. J(-5, -8)
11. K(3, -5) 12. L(-7, -3)
The coordinates are (negative, positive).
The coordinates are (negative, negative).
The coordinates are (positive, positive).
The coordinates are (positive, negative).
y
O1-1-2-3-4
234
1
-1-2-3-4
2 3 4 x
(+, +)(-, +)
(-, -) (+, -)
M
N
x
y
O
1-1-2-3-4
2345
1
-1-2-3-4
2 3
2 x
y
O1-1-3-2-4-5-6-7-8
34
12
5678
-1-2-3-4-5-6-7-8
3 4 5 6 7 8
Program: Pre-Algebra
Vendor: Aptara
Component: ANC_C02_L6
Grade: AMPDF Pages
Math Accelerated • Chapter 2 Operations with Integers
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NAME ___________________________________________________ DATE _____________________ PERIOD __________
Lesson 1 ReteachFractions and Decimals
Example 3 Replace � with < , > , or = to make -0.37 � - 4 − 11 a true sentence.
-0.37 � - 4 − 11
-0.37 � -0. −− 36 Write 4 −
11 as a decimal.
-0.37 < -0. −− 36 -0.37 is to the left of -0. −−
36 on the
number line, so -0.37 < -0. −− 36 .
Exercises Write each fraction as a decimal. Use a bar to show a repeating decimal.
1. 7 − 20 2. 2 − 11 3. 5 − 9
4. - 7 − 9 5. 27 − 40 6. - 2 − 3
Replace each � with < , > , or = to make a true sentence.
7. 5 − 8 � 6 − 9 8. 4 − 5 � 0.8 9. 7 − 8 � 4 − 5
10. 0.16 � 4 − 25 11. - 11 − 40 � -0.02 12. 7 − 8 � 0.88
Some fractions can be written as decimals by making equivalent fractions with denominators of 10, 100, or 1,000. All fractions can be written as decimals by dividing the numerator by the denominator. Repeating decimals have a pattern in their digits that repeats without ending. If the repeating digit is zero, then the decimal is a terminating decimal.
It may be easier to compare numbers when they are written as decimals.
Example 1 Write 3 − 8 as a decimal.
3 − 8 0.375 8 � ���������������������� 3.000 2 4 60 56 40 40 0
0.375 is a terminating decimal.
Example 2 Write 4 − 9 as a decimal.
4 − 9 0.444 9 � ���������������������� 4.000 3 6 40 36 40 36 4
0.444... is a repeating decimal. You can indicate that a decimal repeats by writing a bar or line over the repeating digit(s): 4 − 9 = 0. − 4 .
-0.37 -0.36
-0.39 -0.37 -0.35 -0.33 -0.31-0.40 -0.38 -0.36 -0.34 -0.32 -0.30
Program: Pre-Algebra
Vendor: Aptara
Component: ANC_C03_L1
Grade: AMPDF Pages
Math Accelerated • Chapter 3 Operations with Rational Numbers
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NAME ___________________________________________________ DATE _____________________ PERIOD __________
Lesson 2 ReteachRational Numbers
Example 2 Identify all sets to which each number belongs.
a. -0.08 This is neither a whole number nor an integer.
Since -0.08 can be written as - 8 −
100 , it is rational.
b. 8.282282228... This is a nonterminating and nonrepeating decimal. So, it is irrational.
ExercisesWrite each number as a fraction.
1. 1 1 − 5 2. -2 3. 0.7
4. -9.08 5. -0.0 − 6 6. 6 8 − 11
Identify all sets to which each number belongs. 7. -12 8. 8.5 9. 582
10. 0 11. -68 12. 1 − 5
A number that can be written as a fraction is a rational number. Mixed numbers, integers, terminating decimals, and repeating decimals can all be written as fractions. Any number that can be expressed as a −
b ,
where a and b are integers and b ≠ 0 is a rational number.
Numbers can be classified into a variety of different sets. The diagram at the right illustrates the relationships among the sets of natural numbers, whole numbers, integers, and rational numbers.
Decimal numbers such as π = 3.141592... and 6.767767776... are infinite and nonrepeating. They are called irrational numbers.
a. 3 2 − 5
3 2 − 5 = 17 − 5 Write the mixed number as an improper fraction.
b. 0.14
0.14 is 14 hundredths. 0.14 = 14 − 100 or 7 − 50 Simplify.
Example 1 Write each number as a fraction.
RationalNumbers
3.777...
-
-1.5Integers
Whole0
Natural7
-100 -8
13
5 78
12
34
6.32
Program: Pre-Algebra
Vendor: Aptara
Component: ANC_C03_L2
Grade: AMPDF Pages
Math Accelerated • Chapter 3 Operations with Rational Numbers
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NAME ___________________________________________________ DATE _____________________ PERIOD __________
Lesson 3 ReteachMultiplying Rational Numbers
Example 1 Find 7 1 − 2 · 2 2 − 3 . Write in simplest form.7 1 − 2 · 2 2 − 3 = 15 − 2 · 8 − 3 Rename mixed numbers as improper fractions.
= 15 − 2 · 8 − 3 Divide 15 and 3 by 3, and 8 and 2 by 2.
= 5 · 4 − 1 · 1 Multiply.
= 20 − 1 or 20 Simplify.
To multiply fractions, multiply the numerators and multiply the denominators: a − b
· c − d
= a · c − b · d
, where
b, d ≠ 0. Fractions may be simplified before or after multiplying. For negative fractions, assign the negative sign to the numerator.
Algebraic expressions are expressions which contain one or more variables. Variables can represent fractions in algebraic expressions.
Example 2 Evaluate 2 − 3 ab if a = 3 3 − 7 and b = - 5 − 12 . Write the product in simplest form.
2 − 3 ab = 2 − 3 (3 3 − 7 ) (-
5 − 12 ) Replace a with 3 3 − 7
and b with - 5 −
12 .
= 2 − 3 ( 24 − 7 ) ( -5 − 12 ) Rename 3 3 −
7 as 24 −
7 .
= 2 − 3 ( 24 − 7 ) ( -5 − 12 ) The GCF of 24 and 12 is 12.
= - 20 − 21 or -
20 − 21 Simplify.
ExercisesFind each product. Write in simplest form.
1. 1 − 2 · 3 − 5 2. - 8 − 9 · 5 − 16 3. 4 − 5 · 5 − 8
4. 3 − 10 · (- 1 − 4 ) 5. -2 1 − 8 · (- 4 4 − 7 ) 6. 2 4 − 9 · (- 3 6 − 11 )
Evaluate each expression if x = 7 − 10 , y = -4 2 − 5 , and z = - 4 − 7 . Write the product in simplest form.
7. xy 8. yz 9. xyz
10. 5 5 − 6 xz 11. 2 − 5 (-x) 12. 9 − 10 y
5 4
2
1
Program: Pre-Algebra
Vendor: Aptara
Component: ANC_C03_L3
Grade: AMPDF Pages
Math Accelerated • Chapter 3 Operations with Rational Numbers
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NAME ___________________________________________________ DATE _____________________ PERIOD __________
Lesson 4 ReteachDividing Rational Numbers
Example 1 Find 3 − 4 ÷ 5 − 8 . Write in simplest form.
3 − 4 ÷ 5 − 8 = 3 − 4 · 8 − 5 Multiply by the multiplicative inverse of 5 − 8
, 8 − 5
.
= 3 − 4 · 8 − 5 Divide 4 and 8 by their GCF, 4.
= 6 − 5 or 1 1 − 5 Simplify.
Algebraic fractions are fractions which contain one or more variables. You can divide algebraic fractions just as you would divide numerical fractions.
Example 2 Find 4 − qrs ÷ 10 − qs . Write in simplest form.
4 − qrs ÷ 10 − qs = 4 − qrs · qs
− 10 Multiply by the reciprocal of 10 − qs , qs
− 10
.
= 4 − qrs · qs
− 10 Divide out common factors.
= 2 − 5r Simplify.
Exercises Find each quotient. Write in simplest form.
1. 5 − 16 ÷ 5 − 8 2. 7 − 9 ÷ 2 − 3 3. 16 − 21 ÷ (- 2 − 7 )
4. - 4 − 5 ÷ 3 − 10 5. 18 − 21 ÷ 3 6. -4 5 − 8 ÷ (-3 1 − 3 )
7. 2x − y ÷ 3 − y 8. c − 4d ÷ 3 − 8d 9. 4a − b ÷ 2ac − b
10. m − 9 ÷ mn 2 − 3 11. ab − 9 ÷ bc − 12 12. 2st − q ÷ 4t − q
13. 15yz
− 6x ÷ 10z − 3x 14. de − 20f ÷ e − 2f 15. 6i − 5gh ÷ 8i − 3h
2
1
5
1
Two numbers whose product is 1 are called multiplicative inverses or reciprocals.
For any fraction a − b
, where a, b ≠ 0, b − a is the multiplicative inverse and a − b
· b − a = 1.
This means that 2 − 3
and 3 − 2
are multiplicative inverses because 2 − 3
· 3 − 2
= 1.
To divide by a fraction, multiply by its multiplicative inverse: a − b
÷ c − d
= a − b
· d − c = ad − bc
, where b, c, d ≠ 0.
2
1 1
Program: Pre-Algebra
Vendor: Aptara
Component: ANC_C03_L4
Grade: AMPDF Pages
Math Accelerated • Chapter 3 Operations with Rational Numbers
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NAME ___________________________________________________ DATE _____________________ PERIOD __________
Lesson 5 ReteachAdding and Subtracting Like Fractions
Example 1 Find 3 − 8 + (- 7 − 8 ) . Write in simplest form.
3 − 8 + (- 7 − 8 ) = 3 + (-7) − 8 The denominators are the same. Add the numerators.
= -4 − 8 or - 1 − 2 Simplify.
Example 2 Find 1 2 − 9 + 3 4 − 9 . Write in simplest form.
1 2 − 9 + 3 4 − 9 = (1 + 3) + ( 2 − 9 + 4 − 9 )
= 4 + 2 + 4 − 9
= 4 6 − 9 or 4 2 − 3
To subtract fractions with like denominators, subtract the numerators and write the difference over the
denominator. So, a − c - b − c = a - b − c , where c ≠ 0.
Example 3 Find 3 − 8 - 5 − 8 . Write in simplest form.
3 − 8 - 5 − 8 = 3 - 5 − 8 The denominators are the same. Subtract the numerators.
= - 2 − 8 or - 1 − 4 Simplify.
ExercisesFind each sum. Write in simplest form.
1. 11 − 12 + 9 − 12 2. 13 − 15 + 9 − 15 3. 4 − 9 + 8 − 9
4. -6 7 − 12 + (-8 11 − 12 ) 5. -4 9 − 14 + 3 5 − 14 6. 2 3 − 5 + (- 1 − 5 )
Find each difference. Write in simplest form.
7. 19 − 20 - 17 − 20 8. 23 − 25 - 8 − 25 9. 5 − 9 - 2 − 9
10. 3 − 7 - 5 − 7 11. 4 − 12 - 7 − 12 12. 14 − 15 - 9 − 15
Add the whole numbers and fractions separately or write as
improper fractions.
Add the numerators.
Simplify.
To add fractions with the same denominators, called like denominators, add the numerators and write the sum over the denominator.
So, a − c + b − c = a + b − c , where c ≠ 0.
Program: Pre-Algebra
Vendor: Aptara
Component: ANC_C03_L5
Grade: AMPDF Pages
Math Accelerated • Chapter 3 Operations with Rational Numbers
021_025_AM_ANC_C03_L5_664489.indd Page 21 24/07/12 1:50 AM s-019021_025_AM_ANC_C03_L5_664489.indd Page 21 24/07/12 1:50 AM s-019 /Volumes/104/GO01101_del/Accelerated_Maths_2014/PRE_ALGEBRA/ANCILLARIES.../Volumes/104/GO01101_del/Accelerated_Maths_2014/PRE_ALGEBRA/ANCILLARIES...
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NAME ___________________________________________________ DATE _____________________ PERIOD __________
Lesson 6 ReteachAdding and Subtracting Unlike Fractions
Example 1 Find 4 − 7 + 1 − 3 . Write in simplest form.
4 − 7 + 1 − 3 = 4 − 7 · 3 − 3 + 1 − 3 · 7 − 7 Use 7 · 3 or 21 as the common denominator.
= 12 − 21 + 7 − 21 Rename each fraction with the common denominator.
= 19 − 21 Add the numerators.
To subtract fractions with unlike denominators, rename the fractions with a common denominator. Then subtract and simplify.
Example 2 Find 9 2 − 9 - 8 5 − 6 . Write in simplest form.
9 2 − 9 - 8 5 − 6 = 83 − 9 - 53 − 6 Write the mixed numbers as improper fractions.
= 83 − 9 · 2 − 2 - 53 − 6 · 3 − 3 Rename fractions using the LCD, 18.
= 166 − 18 - 159 − 18 Simplify.
= 7 − 18 Subtract the numerators.
ExercisesFind each sum. Write in simplest form.
1. 8 − 9 + 2 − 5 2. - 2 − 3 + 1 − 4 3. 7 − 8 + 1 − 4
4. 7 4 − 9 + 9 1 − 6 5. -7 1 − 2 + (-3 2 − 9 ) 6. -10 1 − 7 + 6 1 − 4
Find each difference. Write in simplest form.
7. 3 − 8 - 1 − 12 8. - 7 − 9 - 4 − 5 9. 5 − 12 - (- 3 − 8 )
10. 5 1 − 10 - 3 2 − 3 11. -6 3 − 5 - (-2 1 − 4 ) 12. 10 5 − 6 - (-5 2 − 3 )
Fractions with different denominators are called unlike fractions. To add fractions with unlike denominators, rename the fractions with a common denominator. Then add and simplify.
Program: Pre-Algebra
Vendor: Aptara
Component: ANC_C03_L6
Grade: AMPDF Pages
Math Accelerated • Chapter 3 Operations with Rational Numbers
026_030_AM_ANC_C03_L6_664489.indd Page 26 24/07/12 1:46 AM s-019026_030_AM_ANC_C03_L6_664489.indd Page 26 24/07/12 1:46 AM s-019 /Volumes/104/GO01101_del/Accelerated_Maths_2014/PRE_ALGEBRA/ANCILLARIES.../Volumes/104/GO01101_del/Accelerated_Maths_2014/PRE_ALGEBRA/ANCILLARIES...