review of 6th grade - pc\|macimages.pcmac.org/sisfiles/schools/nj/southampton/school3... ·...

$: REVIEW OF 6th GRADE - PC\|MACimages.pcmac.org/SiSFiles/Schools/NJ/Southampton/School3... · 2019-09-25 · Advanced Unit 1: REVIEW OF 6TH GRADE 7 2 Find the GCF of 72 and 75. Pull$
Advanced Unit 1: REVIEW OF 6TH GRADE

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www.njctl.org20120731

Name: _____________________

Advanced Unit 1 REVIEW OF 6th GRADE

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Table of Contents

Fractions

Number System

Decimal Computation

Click on the topic to go to that section

Expressions

Equations and Inequalities

Ratios and Proportions

Geometry

Statistics

3

Fractions

Return to Table of Contents


4

List what you remember about fractions.

Hint

5

We can use prime factorization to find the greatest common factor (GCF).

1. Factor the given numbers into primes.

2. Circle the factors that are common.

3. Multiply the common factors together to find the greatest common factor.

Greatest Common Factor

6

1 Find the GCF of 18 and 44.

Pull

Pull


7


Pull

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8


Pull

Pull

9

A multiple of a whole number is the product of the number and any nonzero whole number.

A multiple that is shared by two or more numbers is a common multiple.

Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, ...

Multiples of 14: 14, 28, 42, 56, 70, 84,...

The least of the common multiples of two or more numbers is the least common multiple (LCM). The LCM of 6 and 14 is 42.


10

There are 2 ways to find the LCM:

1. List the multiples of each number until you find the first one they have in common.

2. Write the prime factorization of each number. Multiply all factors together. Use common factors only once (in other words, use the highest exponent for a repeated factor).

11

EXAMPLE: 6 and 8

Multiples of 6: 6, 12, 18, 24, 30Multiples of 8: 8, 16, 24

LCM = 24

Prime Factorization:

2 3 2 4

2 2 2

2 3 23 LCM: 23 3 = 8 3 = 24

6 8

12

4 Find the least common multiple of 10 and 14.

A 2

B 20C 70D 140

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13

5 Find the least common multiple of 6 and 14.

A 10B 30C 42D 150

Pull

Pull

14

6 Find the LCM of 24 and 60.

Pull

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15

Which is easier to solve?

28 + 42 7(4 + 6)

Do they both have the same answer?

You can rewrite an expression by removing a common factor. This is called the Distributive Property.


15

The Distributive Property allows you to:

1. Rewrite an expression by factoring out the GCF.

2. Rewrite an expression by multiplying by the GCF.

EXAMPLE

Rewrite by factoring out the GCF:

45 + 80 28 + 635(9 + 16) 7(4 + 9)

Rewrite by multiplying by the GCF:3(12 + 7) 8(4 + 13) 36 + 21 32 + 101

17

7 In order to rewrite this expression using the Distributive Property, what GCF will you factor?

56 + 72

Pull

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18

8 In order to rewrite this expression using the Distributive Property, what GCF will you factor?

48 + 84

Pull

Pull


19

9 Use the distributive property to rewrite this expression:

36 + 84

A 3(12 + 28)B 4(9 + 21)C 2(18 + 42)D 12(3 + 7)

Pull

Pull

20

10 Use the distributive property to rewrite this expression:

88 + 32

A 4(22 + 8)

B 8(11 + 4)

C 2(44 + 16)

D 11(8 + 3)

Pull

Pull

21

Adding Fractions...

1. Rewrite the fractions with a common denominator.2. Add the numerators.3. Leave the denominator the same.4. Simplify your answer.

Adding Mixed Numbers...

1. Add the fractions (see above steps).2. Add the whole numbers.3. Simplify your answer. (you may need to rename the fraction)

Link Backto List


22

11 3 10 2 10

+

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23

12 5 8 1 8

+

Pull

Pull

24

13 Find the sum.

5 3 10

+ 7 5 10

Pull

Pull


26

14 Is the equation below true or false?

True False

1 8 12

+ 1 5 12

3 1 12

Pull

Pull

Don't forget to regroup to the whole number if you end up with the numerator

larger than the denominator.

ClickFor reminder

26

A quick way to find LCDs...

List multiples of the larger denominator and stop when you find a common multiple for the smaller denominator.

Ex: and

Multiples of 5: 5, 10, 15

Ex: and

Multiples of 9: 9, 18, 27, 36

2 5

1 3

3 4

2 9

27

Common DenominatorsAnother way to find a common denominator is to multiply the two denominators together.

Ex: and 3 x 5 = 15

= =

2 5

1 3

1 3

x 5

x 5 5 15

2 5

6 15

x 3

x 3


28

15 2 5 1 3

+

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29

16 3 10 2 5

+

Pull

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30

17 5 8 3 5

+

Pull

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31

18

A

5 3 4

+ 2 7 12

=

7 1612

B 8 4 12

C

7 5 8

D

8 1 3

Pull

Pull

32

19

A

2 3 8

+ 5 5 12

=

7 1924

7 8 20

B

7 8 12

C

8 7 12

D

Pull

Pull

33

20

5 2 10

5 5 12

A

3 1 4

+ 2 1 6

=

B

5 1 2

C

6 5 12

D

Pull

Pull


34

Subtracting Fractions...

1. Rewrite the fractions with a common denominator.2. Subtract the numerators.3. Leave the denominator the same.4. Simplify your answer.

Subtracting Mixed Numbers...

1. Subtract the fractions (see above steps..). (you may need to borrow from the whole number)2. Subtract the whole numbers.3. Simplify your answer. (you may need to simplify the fraction)

Link Backto List

35

21 7 8 4 8

Pull

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36

22 6 7

4 5 P

ull

Pull


37

23 2 3

1 5 P

ull

Pull

38


True False

4 5 9

3 9

3 2 9

Pull

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39


True False

2 7 9

1 9

1 2 3

1

Pull

Pull


40

26 Find the difference.

4 7 8 2 3

8

Pull

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41

27 6 7 3 5

Pull

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42

A Regrouping Review

When you regroup for subtracting, you take one of your whole numbers and change it into a fraction with the same denominator as the fraction in the mixed number.

3 3 5

= 2 5 5

3 5

= 2 8 5

Don't forget to add the fraction you regrouped from your whole number to the fraction already given in the problem.


43

5 1 4

3 7 12

5 3 12

3 7 12

41212

3 7 12

3 12

41512

3 7 12

1 8 12

1 2 3

44

28 Do you need to regroup in order to complete this problem?

Yes or No

3 1 2

1 4

Pull

Pull

45

29 Do you need to regroup in order to complete this problem?

7 2 3

3 46

Pull

Pull


46

30 What does 17 become when regrouping? 3 10 P

ull

Pull

47

31 What does 21 become when regrouping? 5 8 P

ull

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48

32

2 1 12

A

1 2224

B

4 1 6 2 1

4=

1 1112

C

1 1 12

D

Pull

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49

33

A

3 1321

B

6 2 7 3 2

3=

3 8 21 2 2

3C

2 1321

D

Pull

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50

34

A

6 1 6

B

15 8 1012

=

7 5 6 7 1

6C

6 2 12

D

Pull

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51

Multiplying Fractions...

1. Multiply the numerators.2. Multiply the denominators.3. Simplify your answer.

Multiplying Mixed Numbers...

1. Rewrite the Mixed Number(s) as an improper fraction. (write whole numbers / 1)2. Multiply the fractions.3. Simplify your answer.

Link Backto List


52

35 1 5

x 2 3

= Pull

Pull

53

36 2 3

x 3 7

= Pull

Pull

54

37 = 4

9 3 8( )

Pull

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55

38

True

False

x 1 2

=5 5 1

x 1 2

Pull

Pull

56

39

A

x 4 73

B

C

3 5 7

D

1221

12 7

1 5 7

Pull

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57

40

True

False

x =2 1 4 3 1

8 6 3 8

Pull

Pull


58

41

15 1 4

A

18 1 8

B

20 3 8

C

19 1 8

D

5 8( )5 2

5(3 ) Pull

Pull

59

Dividing Fractions...

1. Leave the first fraction the same.2. Multiply the first fraction by the reciprocal of the second fraction.3. Simplify your answer.

Dividing Mixed Numbers...

1. Rewrite the Mixed Number(s) as an improper fraction(s). (write whole numbers / 1)2. Divide the fractions.3. Simplify your answer.

60

To divide fractions, multiply the first fraction by the reciprocal of the second fraction. Make sure you simplify your answer!

Some people use the saying "Keep Change Flip" to help them remember the process.

3 5

x 8 7

= 3 x 8 5 x 7

= 2435

3 5

7 8

=

1 5

x 2 1

= 1 x 2 5 x 1

= 2 5

1 5

1 2

=


61

42

True

False

8 10

= 5 4

x 8 10

4 5 P

ull

Pull

62

43

True

False

2 7

= 3 4 2 7

8

Pull

Pull

63

44

1A

3940

B

C

8 10

= 4 5

4042

Pull

Pull


64

45

Pull

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65

To divide fractions with whole or mixed numbers, write the numbers as an improper fractions. Then divide the two fractions by using the rule (multiply the first fraction by the reciprocal of the second).

Make sure you write your answer in simplest form.

5 3

x 2 7

= 1021

2 3

=1 1 2

3 5 3

7 2

=

6 1

x 2 3

= 12 3

=6 1 2

1 6 1

3 2

= = 4

66

46

= 1 2 2 2

31

Pull

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67

47

= 1 2 2 2

31

Pull

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68

48

= 1 2 52

Pull

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69

Decimal Computation



70

List what you remember about decimals.

71

Some division terms to remember....

• The number to be divided into is known as the dividend

• The number which divides the other number is known as the divisor

• The answer to a division problem is called the quotient divisor 5 20 dividend

4 quotient

20 ÷ 5 = 420__5= 4

72

When we are dividing, we are breaking apart into equal groups

EXAMPLE 1

Find 132 3

Step 1: Can 3 go into 1, no so can 3 go into 13, yes

4

12 1

3 x 4 = 1213 12 = 1Compare 1 < 3

3 132

3 x 4 = 1212 12 = 0Compare 0 < 3

12 0

2

Step 2: Bring down the 2. Can 3 go into 12, yes

4

Click for step 1

Click for step 2


73

EXAMPLE 2(change pages to see each step)

Step 1: Can 15 go into 3, no so can 15 go into 35, yes

2

30 5

15 x 2 = 3035 30 = 5Compare 5 < 15

15 357

74

2

30 5

15 35715 x 3 = 4557 45 =12Compare 12 < 15

7 45 12

Step 2: Bring down the 7. Can 25 go into 207, yes

3


75

2

30 5

15 357.0

7 45 120 120 0

3

Step 3: You need to add a decimal and a zero since the division is not complete. Bring the zero down and continue the long division.

15 x 8 = 120120 120 = 0Compare 0 < 15

.8

Click for step 3



76

49 Compute.

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77

50 Compute.

Pull

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78

51 Compute.

Pull

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79

If you know how to add whole numbers then you can add decimals. Just follow these few steps.

Step 1: Put the numbers in a vertical column, aligning the decimal points.

Step 2: Add each column of digits, starting on the right and working to the left.

Step 3: Place the decimal point in the answer directly below the decimal points that you lined up in Step 1.

80

C

52 Add the following:

0.6 + 0.55 =

A 6.1

B 0.115click

C 1.15

D 0.16

81

53 Find the sum

1.025 + 0.03 + 14.0001 =

15.0551click


82

54 Find the sum:

5 + 100.145 + 57.8962 + 2.312 = 165.3532click

83

What do we do if there aren't enough decimal places when we subtract?

4.3 2.05

Don't forget...Line 'em Up!

4.32.05

What goes here?

4.302.05

2.25

2 1

84

55

5 0.238 =4.762click


85

56

12.809 4 =8.809click

86

57

4.1 0.094 = 4.006click

87

58

17 13.008 = 3.992click


88

If you know how to multiply whole numbers then you can multiply decimals. Just follow these few steps.

Step 1: Ignore the decimal points.

Step 2: Multiply the numbers using the same rules as whole numbers.

Step 3: Count the total number of digits to the right of the decimal points in both numbers. Put that many digits to the right of the decimal point in your answer.

89

23.2x 4.04

928

92800 0000

93.728

There are a total of three digits to the right of the decimal points.

There must be three digits to the right of the decimal point in the answer.

EXAMPLE

90

59 Multiply 0.42 x 0.032 0.1344click


91

60 Multiply 3.452 x 2.1 7.2492click

92

4.7383661 Multiply 53.24 x 0.089 click

93

DividendDivisor

Step 1: Change the divisor to a whole number by multiplying by a power of 10.

Step 2: Multiply the dividend by the same power of 10.

Step 3: Use long division.

Step 4: Bring the decimal point up into the quotient.

Divide by Decimals

Quotient


94

15.6 6.24

Multiply by 10, so that 15.6 becomes 1566.24 must also be multiplied by 10

156 62.4

.234 23.4

Multiply by 1000, so that .234 becomes 23423.4 must also be multiplied by 1000

234 23400

Try rewriting these problems so you are ready to divide!

95

62 Divide

0.78 ÷ 0.02 = 39click

96

63

10 divided by 0.25 = 40click


97

64

12.03 ÷ 0.04 = 300.75click

98

There are two types of decimals terminating and repeating.

A terminating decimal is a decimal that ends.All of the examples we have completed so far are terminating.

A repeating decimal is a decimal that continues forever with one or more digits repeating in a pattern.

To denote a repeating decimal, a line is drawn above the numbers that repeat. However, with a calculator, the last digit is rounded.

99

Examples:

6600 2342 2200 14200 13200 10000

8800 12000 11000 10000 8800 12000 11000

63 48 45 39 36 32 27 51

45 60 54 6


100

65

click

101

66

click

102

67

click


103

Statistics


104

List what you remember about statistics.

105

Measures of Center Vocabulary:

• Mean The sum of the data values divided by the number of items; average

• Median The middle data value when the values are written in numerical order

• Mode The data value that occurs the most often


106

68 Find the mean

14, 17, 9, 2, 4,10, 5, 3

Pull

Pull

107

69 Find the median: 5, 9, 2, 6, 10, 4

A 5B 5.5

C 6D 7.5

Pull

Pull

108

70 Find the mode(s): 3, 4, 4, 5, 5, 6, 7, 8, 9

A 4

B 5

C 9

D No mode

Pull

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109

71 Consider the data set: 78, 82, 85, 88, 90. Identify the data values that remain the same if "79" is added to the set.

A mean

B median

C mode

D range

E minimum

Pull

Pull

110

Measures of Variation Vocabulary:

Minimum The smallest value in a set of data

Maximum The largest value in a set of data

Range The difference between the greatest data value and the least data value

Quartiles are the values that divide the data in four equal parts.

Lower (1st) Quartile (Q1) The median of the lower half of the data

Upper (3rd) Quartile (Q3) The median of the upper half of the data.

Interquartile Range The difference of the upper quartile and the lower quartile. (Q3 Q1)

Outliers Numbers that are significantly larger or much smaller than the rest of the data

111

72 Find the range: 4, 2, 6, 5, 10, 9

A 5B 8C 9D 10

Pull

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112

73 Find the range for the given set of data: 13, 17, 12, 28, 35

Pull

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113

QuartilesThere are three quartiles for every set of data.

LowerHalf

UpperHalf

10, 14, 17, 18, 21, 25, 27, 28

Q1 Q2 Q3

The lower quartile (Q1) is the median of the lower half of the data which is 15.5.

The upper quartile (Q3) is the median of the upper half of the data which is 26.

The second quartile (Q2) is the median of the entire data set which is 19.5.

The interquartile range is Q3 Q1 which is equal to 10.5.

114

74 The median (Q2) of the following data set is 5.

3, 4, 4, 5, 6, 8, 8

True False

Pull

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115

75 What are the lower and upper quartiles of the data set

3, 4, 4, 5, 6, 8, 8?

A Q1: 3 and Q3: 8

B Q1: 3.5 and Q3: 7

C Q1: 4 and Q3: 7

D Q1: 4 and Q3: 8

Pull

Pull

116

76 What is the interquartile range of the data set

3, 4, 4, 5, 6, 8, 8?

Pull

Pull

117

77 What is the median of the data set

1, 3, 3, 4, 5, 6, 6, 7, 8, 8?

A 5B 5.5

C 6D No median

Pull

Pull


118

78 What is the interquartile range of the data set

1, 3, 3, 4, 5, 6, 6, 7, 8, 8?

Pull

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119

Outliers Numbers that are relatively much larger or much smaller than the data

Which of the following data sets have outlier(s)?

A. 1, 13, 18, 22, 25

B. 17, 52, 63, 74, 79, 83, 120

C. 13, 15, 17, 21, 26, 29, 31

D. 25, 32, 35, 39, 40, 41

Pull

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120

79 The data set: 1, 20, 30, 40, 50, 60, 70 has an outlier which is ________ than the rest of the data.

A higher

B lower

C neither

Pull

Pull


121

80 In the following data what number is the outlier? 1, 2, 2, 4, 5, 5, 5, 13

Pull

Pull

122

81 Find the maximum value: 15, 10, 32, 13, 2

A 2B 15

C 13

D 32

Pull

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123

The mean absolute deviation of a set of data is the average distance between each data value and the mean.

Steps

1. Find the mean.2. Find the distance between each data value and the mean. That is, find the absolute value of the difference between each data value and the mean.3. Find the average of those differences.

*HINT: Use a table to help you organize your data.


124

Let's continue with the "Phone Usage" example.Step 1 We already found the mean of the data is 56.Step 2 Now create a table to find the differences.

48

52

54

55

58

59

60

62

Data Value

Absolute Value of the Difference|Data Value Mean|

125

Step 3 Find the average of those differences.

8 + 4 + 2 + 1 + 2 + 3 + 4 + 6 = 3.75 8

The mean absolute deviation is 3.75.

The average distance between each data value and the mean is 3.75 minutes.

This means that the number of minutes each friend talks on the phone varies 3.75 minutes from the mean of 56 minutes.

126

82 Find the mean absolute deviation of the given set of data.

Zoo Admission Prices$9.50 $9.00 $8.25$9.25 $8.00 $8.50

A $0.50

B $8.75

C $3.00

D $9.00

Pull

Pull


127

83 Find the mean absolute deviation for the given set of data.

Number of Daily Visitors to a Web Site

112 145 108 160 122

Pull

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128

FREQUENCY

8

6

4

2

030 40 50 60 70 80 9039 49 59 69 79 89 99

GRADE

Grade Tally Frequency3039 I 14049 05059 06069 I 17079 IIII 48089 IIII III 89099 III 3

TEST SCORES95 85 9377 97 7184 63 8739 88 8971 79 8382 85

SAMPLES:

Data

TEST SCORES87 53 9585 89 5986 82 8740 90 7248 68 5764 85

FREQUENCY

8

6

4

2

040 50 60 70 80 9049 59 69 79 89 99

GRADE

Grade Tally Frequency4049 II 25059 III 36069 II 27079 I 18089 IIII II 79099 II 2

FrequencyTable

Histogram

129

A box and whisker plot is a data display that organizes data into four groups

10 2 3 4 5 6 7 8 9 1012345678910 80 90 100 110 120 130 140 150

The median divides the data into an upper and lower half

The median of the lower half is the lower quartile.

The median of the upper half is the upper quartile.

The least data value is the minimum.

The greatest data value is the maximum.


130

10 2 3 4 5 6 7 8 9 1012345678910 80 90 100 110 120 130 140 150

median

25% 25%25%25%

The entire box represents 50% of the data. 25% of the data lie in the box on each side of the median

Each whisker represents 25% of the data

131

84 The minimum is

A 87

B 104

C 122

D 134

10 2 3 4 5 6 7 8 9 1012345678910 80 90 100 110 120 130 140 150

Pull

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132

85 The median is

A 87

B 104

C 122

D 134

10 2 3 4 5 6 7 8 9 1012345678910 80 90 100 110 120 130 140 150

Pull

Pull


133

86 The lower quartile is

A 87

B 104

C 122

D 134

10 2 3 4 5 6 7 8 9 1012345678910 80 90 100 110 120 130 140 150

Pull

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134

87 The upper quartile is

A 87

B 104

C 122

D 134

10 2 3 4 5 6 7 8 9 1012345678910 80 90 100 110 120 130 140 150

Pull

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135

88 In a box and whisker plot, 75% of the data is between

A the minimum and median

B the minimum and maximum

C the lower quartile and maximum

D the minimum and the upper quartile

10 2 3 4 5 6 7 8 9 1012345678910 80 90 100 110 120 130 140 150

Pull

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136

89 In a box and whisker plot, 50% of the data is between

A the minimum and median

B the minimum and maximum

C the lower quartile and upper quartile

D the median and maximum

10 2 3 4 5 6 7 8 9 1012345678910 80 90 100 110 120 130 140 150

Pull

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137

A dot plot (line plot) is a number line with marks that show the frequency of data. A dot plot helps you see where data cluster.

Example:

35 40 45 5030

xxxxxx

xxx

xxx

xxxx

xx

xxx

xxxxx

Test Scores

The count of "x" marks above each score represents the number of students who received that score.

138

90 How many more students scored 75 than scored 85?

Pull

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139

91 What is the median score?

Pull

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140

92 What are the mode(s) of the data set?

A 75

B 80

C 85

D 90

E 95

F 100

Pull

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141

93 Which measure of center appropriately represents the data?

A Mean

B Median

C Mode

Paper Plane Competition

Distance (ft)

FREQUENCY

4

3

2

1

0 04 59 1014 1519 2024

Pull

Pull


142

Number System


143

List what you remember about the number system.

144

...6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7...

Definition of Integer:

The set of natural numbers, their opposites, and zero.

Define Integer

Examples of Integers:


145

1 02345 1 2 3 4 5

Integers on the number line

NegativeIntegers

PositiveIntegers

Numbers to the left of zero are less than zero

Numbers to the right of zero are greater than zero

Zero is neitherpositive or negative

`

Zero

146

94 Which of the following are examples of integers?

A 0B 8C 4.5D 7

E

Pull

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147

95 Which of the following are examples of integers?

A

B 6C 4D 0.75E 25%

Pull

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148

96 What is the opposite of 5?

Pull

Pull

149

97 What is the opposite of 0?

Pull

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150

Absolute Value of Integers

The absolute value is the distance a number is from zero on the number line, regardless of direction.

Distance and absolute value are always nonnegative (positive or zero).

10 2 3 4 5 6 7 8 9 1012345678910

What is the distance from 0 to 5?


151

798 Find

Pull

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152

2899 Find

Pull

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153

3100 Find

Pull

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154

101 Which numbers have 50 as their absolute value?

A 50B 25C 0D 25E 50

Pull

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155

To compare integers, plot points on the number line.

The numbers farther to the right are greater.

The numbers farther to the left are smaller.

Use the Number Line

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156

102 The integer 7 is ______ 7.

A =B <C >

Pull

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157

103 The integer 20 is ______ 14.

A =B <C >

Pull

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158

104 The integer 4 is ______ 6.

A =B <C >

Pull

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159

105

A

B

C

10 2 3 4 5 6 7 8 9 1012345678910

What is the position of the dot on the number line below?

Pull

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160

106

A 5.5B 6.5C 5.2

10 2 3 4 5 6 7 8 9 1012345678910

What is the position of the dot on the number line below?

Pull

Pull

161

Comparing Rational NumbersSometimes you will be given fractions and decimals that you need to compare.

It is usually easier to convert all fractions to decimals in order to compare them on a number line.

To convert a fraction to a decimal, divide the numerator by the denominator.

4 3.0028 020 20 0

0.75

162

107

A =B <C >

10 2 3 4 5 6 7 8 9 1012345678910

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163

108

A =B <C >

10 2 3 4 5 6 7 8 9 1012345678910

Pull

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164

109

A =B <C >

10 2 3 4 5 6 7 8 9 1012345678910

Pull

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165

The coordinate plane is divided into four sections called quadrants.

The quadrants are formed by two intersecting number lines called axes.

The horizontal line is the xaxis.

The vertical line is the yaxis.

The point of intersection is called the origin. (0,0)

0 x axisy axis

origin

(+, )

(, +)

(, )

(+, +)


166

To graph an ordered pair, such as (3,2):• start at the origin (0,0)• move left or right on the xaxis depending on the first number• then move up or down from there depending on the second number • plot the point

(3,2)

167

110 The point (5, 4) is located in quadrant_____.

A I

B II

C III

D IV

Pull

Pull

168

111 The point (7, 2) is located in quadrant _____.

A I

B II

C III

D IV

Pull

Pull


169

112 The quadrant where the x & y coordinates are both negative is quadrant ___.

A I

B II

C III

D IV

Pull

Pull

170

113 When plotting points in the Cartesian Plane, you always start at ____.

A the x axis

B the origin

C the yaxis

D the Coordinate Plane

E (0,0)

Pull

Pull

171

114 Point A is located at (5, 1)

True False P

ullPull


172

115 Point A is located at (2, 3)

True False A P

ullPull

173

Expressions


174

List what you remember about expressions.


175

Exponents

Exponents, or Powers, are a quick way to write repeated multiplication, just as multiplication was a quick way to write repeated addition.

These are all equivalent:

24 Exponential Form2∙2∙2∙2 Expanded Form16 Standard Form

In this example 2 is raised to the 4th power. That means that 2 is multiplied by itself 4 times.

176

Powers of IntegersBases and Exponents

When "raising a number to a power",

The number we start with is called the base, the number we raise it to is called the exponent.

The entire expression is called a power.

You read this as "two to the fourth power."

24

177

116 What is the base in this expression?

32

Pull

Pull


178

117 What is the exponent in this expression?

32

Pull

Pull

179

118 Evaluate 32.

Pull

Pull

180

119 Evaluate 43.

Pull

Pull


181

120 Evaluate 24.

Pull

Pull

182

What does "Order of Operations" mean?

The Order of Operations is an agreed upon set of rules that tells us in which "order" to solve a problem.

183

The P stands for Parentheses: Usually represented by ( ). Other grouping symbols are [ ] and . Examples: (5 + 6); [5 + 6]; 5 + 6/2

The E stands for Exponents: The small raised number next to the larger number. Exponents mean to the ___ power (2nd, 3rd, 4th, etc.) Example: 23 means 2 to the third power or 2(2)(2)

The M/D stands for Multiplication or Division: From

left to right. Example: 4(3) or 12 ÷ 3

The A/S stands for Addition or Subtraction: From left to right. Example: 4 + 3 or 4 3

What does P E M/D A/S stand for?


184

Watch Out!

When you have a problem that looks like a fraction but has an operation in the numerator, denominator, or both, you must solve everything in the numerator or denominator before dividing.

453(72)

453(5)

4515

3

185

121

1 + 5 x 7

Pull

Pull

186

122 40 ÷ 5 x 9

Pull

Pull


187

123

6 5 + 2

Pull

Pull

188

124 18 ÷ 9 x 2

Pull

Pull

189

125

5(32)

Pull

Pull


190

[ 6 + ( 2 8 ) + ( 42 9 ) ÷ 7 ] 3

Let's try another problem. What happens if there is more than one set of grouping symbols?

[ 6 + ( 2 8 ) + ( 42 9 ) ÷ 7 ] 3

When there are more than 1 set of grouping symbols, start inside and work out following the Order of Operations.

[ 6 + ( 16 ) + ( 16 9 ) ÷ 7 ] 3[ 6 + ( 16 ) + ( 7 ) ÷ 7 ] 3

[ 6 + ( 16 ) + 1 ] 3[ 22 + 1 ] 3

[ 23 ] 369

191

126

4 2[5 + 3] + 7

Pull

Pull

192

127

42 + 9 + 3[2 + 5]

Pull

Pull


193

128

62 ÷ 3 + (15 7)

Pull

Pull

194

129 Which expression with parenthesis added in changes the value of: 5 + 4 7

A (5 + 4) 7 B 5 + (4 7)

C (5 + 4 7)

D none of the above change the value

Pull

Pull

195

130 Which expression with parenthesis added in changes the value of: 36 ÷ 2 + 7 + 1

A (36 ÷ 2) + 7 + 1B 36 ÷ (2 + 7) + 1C (36 ÷ 2 + 7 + 1)D none of the above change the value

Pull

Pull


196

What is a Constant?A constant is a fixed value, a number on its own, whose value does not change. A constant may either be positive or negative.

Example: 4x + 2

In this expression 2 is a constant.click to reveal

Example: 11m 7

In this expression 7 is a constant.click to reveal

197

What is a Variable?

A variable is any letter or symbol that represents a changeable or unknown value.

Example: 4x + 2

In this expression x is a variable.

click to reveal

198

What is a Coefficient?

A coefficient is the number multiplied by the variable. It is located in front of the variable.

Example: 4x + 2

In this expression 4 is a coefficient.click to reveal


199

If a variable contains no visible coefficient, the coefficient is 1.

Example 1: x + 4 is the same as 1x + 4

x + 4 is the same as

1x + 4

Example 2:

x + 2has a coefficient of

Example 3:

200

131 In 3x 7, the variable is "x"

True

False

Pull

Pull

201

132 In 4y + 28, the variable is "y"

True

False

Pull

Pull


202

133 In 4x + 2, the coefficient is 2

True

False

Pull

Pull

203

134 What is the constant in 6x 8?

A 6B xC 8D 8

Pull

Pull

204

135 What is the coefficient in x + 5?

A none

B 1C 1

D 5

Pull

Pull


205

136 Evaluate 3h + 2 for h = 3

Pull

Pull

206

137 Evaluate 2x2 for x = 3

Pull

Pull

207

138 Evaluate 4a + a for a = 8, c = 2 c

Pull

Pull


208

139 Use the distributive property to rewrite the expression without parentheses (x + 6)3

A 3x + 6

B 3x + 18

C x + 18D 21x

Pull

Pull

209

140 Use the distributive property to rewrite the expression without parentheses 3(x 4)

A 3x 4

B x 12

C 3x 12

D 9x

Pull

Pull

210

141 Use the distributive property to rewrite the expression without parentheses 2(w 6)

A 2w 6

B w 12

C 2w 12

D 10w

Pull

Pull


211

Equations and Inequalities


212

List what you remember about equations and inequalities.

213

A solution to an equation is a number that makes the equation true.

In order to determine if a number is a solution, replace the variable with the number and evaluate the equation.

If the number makes the equation true, it is a solution.If the number makes the equation false, it is not a solution.

Determining the Solutions of Equations


214

142 Which of the following is a solution to the equation:

x + 17 = 21 2, 3, 4, 5

Pull

Pull

215


m 13 = 28 39, 40, 41, 42

Pull

Pull

216


3x + 5 = 32 7, 8, 9, 10

Pull

Pull


217

Why are we moving on to Solving Equations?

First we evaluated expressions where we were given the value of the variable and had which solution made the equation true.

Now, we are told what the expression equals and we need to find the value of the variable.

When solving equations, the goal is to isolate the variable on one side of the equation in order to determine its value (the value that makes the equation true).

This will eliminate the guess & check of testing possible solutions.

218

To solve for "x" in the following equation... x + 7 = 32

Determine what operation is being shown (in this case, it is addition). Do the inverse to both sides.

x + 7 = 32 7 7

x = 25

To check your value of "x"...

In the original equation, replace x with 25 and see if it makes the equation true.

x + 7 = 3225 + 7 = 32 32 = 32

219

145 What is the inverse operation needed to solve this equation?

7x = 49

A Addition

B Subtraction

C Multiplication

D Division

Pull

Pull


220


x 3 = 12

A Addition

B Subtraction

C Multiplication

D Division

Pull

Pull

221


A Addition

B Subtraction

C Multiplication

D Division

Pull

Pull

222


A Addition

B Subtraction

C Multiplication

D Division

Pull

Pull


223

To solve equations, you must use inverse operations in order to isolate the variable on one side of the equation.

Whatever you do to one side of an equation, you MUST do to the other side!

+5+5

224

149 Solve.

x + 6 = 11

Pull

Pull

225

150 Solve.

x 13 = 54

Pull

Pull


226

151 Solve.

j + 15 = 27

Pull

Pull

227

152 Solve.

x 9 = 67

Pull

Pull

228

153 Solve.

115 = 5x

Pull

Pull


229

154 Solve.

33 = 11m

Pull

Pull

230

155 Solve.

48 = 12y

Pull

Pull

231

156 Solve.

n = 136

Pull

Pull


232

An inequality is a statement that two quantities are not equal. The quantities are compared by using one of the following signs:

Symbol Expression Words

< A < B A is less than B

> A > B A is greater than B

< A < B A is less than orequal to B

> A > B A is greater than orequal to B

233

157 Write an inequality for the sentence:

m is greater than 9

A m < 9

B m < 9

C m > 9

D m > 9

Pull

Pull

234


12 is less than or equal to y

A 12 < y

B 12 < y

C 12 > y

D 12 > y

Pull

Pull


235


The grade, g, on your test must exceed 80%

A g < 80

B g < 80

C g > 80

D g > 80

Pull

Pull

236


y is not more than 25

A y < 25

B y < 25

C y > 25

D y > 25

Pull

Pull

237

Remember: Equations have one solution.

Solutions to inequalities are NOT single numbers. Instead, inequalities will have more than one value for a solution.

10 2 3 4 5 6 7 8 9 1012345678910

This would be read as, "The solution set is all numbers greater than or equal to negative 5."

Solution Sets


238

Let's name the numbers that are solutions to the given inequality.

r > 10 Which of the following are solutions? 5, 10, 15, 20

5 > 10 is not trueSo, 5 is not a solution

10 > 10 is not trueSo, 10 is not a solution

15 > 10 is trueSo, 15 is a solution

20 > 10 is trueSo, 20 is a solution

Answer:15, 20 are solutions to the inequality r > 10

239

161 Which of the following are solutions to the inequality:

x > 11 9, 10, 11, 12

Select all that apply.

A 9B 10

C 11

D 12

Pull

Pull

240


m < 15 13, 14, 15, 16


A 13B 14

C 15

D 16

Pull

Pull


241


x > 34 32, 33, 34, 35


A 32B 33

C 34

D 35

Pull

Pull

242

Since inequalities have more than one solution, we show the solution two ways.

The first is to write the inequality. The second is to graph the inequality on a number line.

In order to graph an inequality, you need to do two things:

1. Draw a circle (open or closed) on the number that is your boundary.

2. Extend the line in the proper direction.

243

Remember!

Closed circle means the solution set includes that number and is used to represent ≤ or ≥.

Open circle means that number is not included in the solution set and is used to represent < or >.

Extend your line to the right when your number is larger than the variable.

# > variable variable < #

Extend your line to the left when your number is smaller than the variable.

# < variable variable > #


244

164 This solution set graphed below is x > 4?

True

False

10 2 3 4 5 6 7 8 9 1012345678910

Pull

Pull

245

1 02345 1 2 3 4 5165

A x > 3

B x < 3

C x < 3

D x > 3

Pull

Pull

246

1 02345 1 2 3 4 5

166

A x > 1

B x < 1

C x < 1

D x > 1

Pull

Pull


247

167

A x > 0

1 02345 1 2 3 4 5

B x < 0

C x < 0

D x > 0

Pull

Pull

248

Geometry


249

List what you remember about geometry.


250

A = length(width)A = lw

A = side(side)A = s2

The Area (A) of a rectangle is found by using the formula:

The Area (A) of a square is found by using the formula:

251

168 What is the Area (A) of the figure?

13 ft

7 ft

Pull

Pull

252

169 Find the area of the figure below.

8

Pull

Pull


253

A = base(height)A = bh

The Area (A) of a parallelogram is found by using the formula:

Note: The base & height always form a right angle!

254

170 Find the area.

10 ft 9 ft

11 ft

Pull

Pull

255

171 Find the area.

8 m

13 m 13 m

8 m

12 m

Pull

Pull


256

172 Find the area.

13 cm

12 cm

7 cm

Pull

Pull

257

The Area (A) of a triangle is found by using the formula:


258

173 Find the area.

8 in

6 in

10 in 9 in

Pull

Pull


259

174 Find the area

14 m

9 m10 m 12 m

Pull

Pull

260

The Area (A) of a trapezoid is also found by using the formula:


10 in

12 in

5 in

261

175 Find the area of the trapezoid by drawing a diagonal.

Pull

Pull

9 m

11 m

8.5 m


262

176 Find the area of the trapezoid using the formula.

20 cm

13 cm

12 cm

Pull

Pull

263

Area of Irregular Figures

1. Divide the figure into smaller figures (that you know how to find the area of)

2. Label each small figure and label the new lengths and widths of each shape

3. Find the area of each shape

4. Add the areas

5. Label your answer

264

Example:Find the area of the figure.

12 m

8 m

4 m2 m

12 m6 m

4 m2 m #1

#2

2 m


265Pull

177 Find the area.

4'

3'

1'

2'

10'

8'

5'

266

178 Find the area.

12

101320

25

10 Pull

267

179 Find the area.

8 cm 18 cm

9 cm

Pull


268

Area of a Shaded Region

1. Find area of whole figure.

2. Find area of unshaded figure(s).

3. Subtract unshaded area from whole figure.

4. Label answer with units2.

269

Example

Find the area of the shaded region.

8 ft

10 ft

3 ft3 ft

Area Whole Rectangle

Area Unshaded Square

Area Shaded Region

270

180 Find the area of the shaded region.

11'

8'

3'4'

Pull


271

181 Find the area of the shaded region.

16"

17"

15"7"

5"

Pull

272

3Dimensional SolidsCategories & Characteristics of 3D Solids:

Prisms1. Have 2 congruent, polygon bases which are parallel to one another2. Sides are rectangular (parallelograms)3. Named by the shape of their base

Pyramids1. Have 1 polygon base with a vertex opposite it2. Sides are triangular3. Named by the shape of their base

Cylinders1. Have 2 congruent, circular bases which are parallel to one another2. Sides are curved

Cones1. Have 1 circular bases with a vertex opposite it2. Sides are curved

273

3Dimensional Solids

Vocabulary Words for 3D Solids:

Polyhedron A 3D figure whose faces are all polygons(Prisms & Pyramids)

Face Flat surface of a Polyhedron

Edge Line segment formed where 2 faces meet

Vertex (Vertices) Point where 3 or more faces/edges meet

Solid a 3D figure

Net a 2D drawing of a 3D figure(what a 3D figure would look like if it were unfolded)


274

182 Name the figure.

A rectangular prismB triangular prismC triangular pyramidD cylinderE coneF square pyramid

Pull

275

183Name the figure.

A rectangular prismB triangular prismC triangular pyramidD cylinderE coneF square pyramid Pu

ll

276

184Name the figure.

A rectangular prismB triangular prismC triangular pyramidD pentagonal prismE coneF square pyramid Pu

ll


277

185Name the figure.

A rectangular prismB triangular prismC triangular pyramidD pentagonal prismE coneF square pyramid Pu

ll

278

186Name the figure.

A rectangular prismB cylinderC triangular pyramidD pentagonal prismE coneF square pyramid Pu

ll

279

187 How many faces does a cube have?

Pull


280

188 How many vertices does a triangular prism have?

Pull

281

189 How many edges does a square pyramid have?

Pull

282

6 in

2 in7 in

7 in2 in

2 in6 in

A net is helpful in calculating surface area.

Simply label each section and find the area of each.

#2 #4

6 in

#1

#3

#5

#6

Surface Area

The sum of the areas of all outside faces of a 3D figure.

To find surface area, you must find the area of each face of the figure then add them together.


283

7 in2 in

2 in6 in

#2 #4

6 in

#1

#3

#5

#6

#1 #2 #3 #4 #5 #6

Example

284

190Find the surface area of the figure given its net.

7 yd

7 yd

7 yd

7 yd

Pull

Since all of the faces are the same, you can find the area of one face and multiply it by 6 to calculate the surface area of a cube.

What pattern did you notice while finding the surface area of a cube?

285

191Find the surface area of the figure given its net.

9 cm

25 cm

12 cm

Pull


286

Volume FormulasFormula 1

V= lwh, where l = length, w = width, h = height

Multiply the length, width, and height of the rectangular prism.

Formula 2

V=Bh, where B = area of base, h = height

Find the area of the rectangular prism's base and multiply it by the height.

287

Example

Each of the small cubes in the prism shown have a length, width and height of 1/4 inch.

The formula for volume is lwh.

Therefore the volume of one of the small cubes is:

Multiply the numerators together, then multiply the denominators. In other words, multiply across.

Forget how to multiply fractions?

288

192Find the volume of the given figure.

Pull


289


Pull

290


Pull

291

Ratios and Proportions



292

List what you remember about the ratios and proportions.

293

Ratio A comparison of two numbers by division

Ratios can be written three different ways:

a to b a : b a b

Each is read, "the ratio of a to b." Each ratio should be in simplest form.

294

195 There are 27 cupcakes. 9 are chocolate, 7 are vanilla and the rest are strawberry. What is the ratio of vanilla cupcakes to strawberry cupcakes?

A 7 : 9

B 7 27

C 7 11

D 1 : 3


295

196 There are 27 cupcakes. 9 are chocolate, 7 are vanilla and the rest are strawberry. What is the ratio of chocolate & strawberry cupcakes to vanilla & chocolate cupcakes?

A 20 16

B 11 7

C 5 4

D 16 20

296

197 There are 27 cupcakes. 9 are chocolate, 7 are vanilla and the rest are strawberry. What is the ratio of total cupcakes to vanilla cupcakes?

A 27 to 9

B 7 to 27

C 27 to 7

D 11 to 27

297

Equivalent ratios have the same value

3 : 2 is equivalent to 6: 4

1 to 3 is equivalent to 9 to 27

5 35 6 is equivalent to 42


298

4 125 15

x 3 Since the numerator and denominator were multiplied by the same value, the ratios are equivalent

There are two ways to determine if ratios are equivalent.1.

4 125 15

x 3

4 125 15

Since the cross products are equal, the ratios are equivalent.4 x 15 = 5 x 12 60 = 60

2. Cross Products

299

198 4 is equivalent to 8 9 18

True

False

300

199 5 is equivalent to 30 9 54

True

False


301

Rate: a ratio of two quantities measured in different units

Examples of rates:4 participants/2 teams

5 gallons/3 rooms

8 burgers/2 tomatoes

Unit rate: Rate with a denominator of one Often expressed with the word "per"

Examples of unit rates:

34 miles/gallon

2 cookies per person

62 words/minute

302

Finding a Unit RateSix friends have pizza together. The bill is $63. What is the cost per person?

Hint: Since the question asks for cost per person, the cost should be first, or in the numerator.

$63 6 people

Since unit rates always have a denominator of one, rewrite the rate so that the denominator is one. $63 6 6 people 6 $10.50 1 person

The cost of pizza is $10.50 per person

303

200 Sixty cupcakes are at a party for twenty children. How many cupcakes per person?


304

201 John's car can travel 94.5 miles on 3 gallons of gas. How many miles per gallon can the car travel?

305

202 The snake can slither 24 feet in half a day. How many feet can the snake move in an hour?

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