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Buckle Down North Carolina
EOG 5 Mathematics
Number and Operations
Lesson 1: Rational Numbers
Lesson 2: Converting and Comparing Rational Numbers
Lesson 3: Computing with Rational Numbers
Lesson 4: Estimation and Problem Solving
Unit 2
Patterns, Relationships, and Algebra
Lesson 5: Patterns and Relationships
Lesson 6: Algebra
Unit 3
Geometry and Spatial Sense
Lesson 7: Geometric Figures
Lesson 8: Geometric Concepts
Unit 4
Measurement
Lesson 9: Units of Measurement
Lesson 10: Geometric Measurement
Unit 5
Data Analysis and Probability
Lesson 11: Data Analysis
Lesson 12: Probability
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Catalog # 4BDNC05MM01
4TH EDITION
5 MATHEMATICS
Check out our complete line of EOG/Comprehensive materials for Grades 3–8 and 10
READING • WRITING • MATHEMATICS
North Carolina
North Carolina EOG
The cover image combines geometry
and art to illustrate the properties
of polygons, such as the sum of the
measures of interior angles and
the parallelism or perpendicularity
of sides.
Unit 1
5
Mathematics
EOG
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Page iii
TABLE OF CONTENTS
Introduction ..................................................................................... 1
Testwise StrategiesTM ........................................................ 2
Unit 1 – Number and Operations ............................................... 3
Lesson 1: Rational Numbers.............................................. 4
EOG Standards: 1.01a, 1.01b
Lesson 2: Converting and Comparing Rational
Numbers ........................................................... 21
EOG Standard: 1.01c
Lesson 3: Computing with Rational Numbers................ 29
EOG Standard: 1.02a
Skills to Maintain: whole number computation
Lesson 4: Estimation and Problem Solving .................... 43
EOG Standards: 1.01d, 1.02b, 1.02c, 1.03
Unit 2 – Patterns, Relationships, and Algebra....................... 55
Lesson 5: Patterns and Relationships............................. 56
EOG Standards: 5.01, 5.03
Lesson 6: Algebra.............................................................. 68
EOG Standard: 5.02
Unit 3 – Geometry and Spatial Sense....................................... 77
Lesson 7: Geometric Figures............................................ 78
EOG Standards: 2.02, 3.01, 3.02a–c, 3.04a–c
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Lesson 8: Geometric Concepts ......................................... 99
EOG Standard: 3.03
Skills to Maintain: transformations, coordinate grid
Unit 4 – Measurement ................................................................ 109
Lesson 9: Units of Measurement ................................... 110
EOG Standard: 2.01
Lesson 10: Geometric Measurement ............................. 133
Skills to Maintain: perimeter and area
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Table of Contents
Unit 5 – Data Analysis and Probability ................................. 139
Lesson 11: Data Analysis ............................................... 140
EOG Standards: 4.01, 4.02, 4.03
Skills to maintain: line graphs, median, mode,
and range
Lesson 12: Probability .................................................... 164
iv
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Unit 1 – Number and Operations
Lesson 1
Rational Numbers
Rational numbers include the whole numbers, some decimal numbers, and all
the fractions and mixed numbers. In this lesson, you will read, write, estimate,
order, and compare rational numbers.
Place Value
A number such as 820,305 might look hard to work with, but it’s easy if you
understand place value. The following table helps you see the values of the
digits so you can write the number in standard form.
Hundred
Thousands
Ten
Thousands
8
2
Thousands Hundreds
0
3
Tens Ones
0
5
When you write large numbers in standard form, put a comma to the left of
every third digit, starting at the ones place and moving to the left.
8 2 0,3 0 5
2
1
This number can also be written in expanded form and word form.
Expanded Form: 800,000 20,000 300 5
Word Form: eight hundred twenty thousand, three hundred five
Notice that the place values with zero as a placeholder are not written out in
the expanded form of the number. They are also not written out in the word
form of the number.
TIP: Sometimes it is hard to remember where to put commas when you
are writing in word form. Look at the commas in standard form. Keep
them in the same place when you write the words out.
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3
start
here
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Lesson 1: Rational Numbers
Practice
Directions: Write the following numbers in the table. Then write the
numbers in expanded and word form in Numbers 1 and 2.
710,051
164,980
Hundred
Ten
Thousands Thousands Thousands Hundreds Tens Ones
1. 710,051
Expanded Form: _______________________________________________________
Word Form: ___________________________________________________________
2. 164,980
Expanded Form: _______________________________________________________
Word Form: ___________________________________________________________
3. Which digit is in the ten thousands place in the number 271,839? _______
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4. Which digit is in the hundreds place in the number 510,436?
A 0
B 4
C 5
D 6
5. Which of the following is the standard form of forty-two thousand, one
hundred ten?
A
4,211
B
42,110
C 420,110
D 421,010
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Unit 1 – Number and Operations
Comparing and ordering whole numbers
Understanding place value helps you compare and order whole numbers. Start
at the far left place value of the numbers, adding zeros as placeholders if
necessary, and compare the digits of each place value from left to right. Stop at
the place value where the digits are different. The way those digits compare is
the way the whole numbers compare. Use (greater than), (less than), or
(equal to) when comparing numbers.
Example
Compare 23,948 and 23,961.
Both numbers’ first digits are in the ten-thousands place. Compare the
digits of each place value from left to right using a place-value table
until you notice a different digit. Both numbers’ digits are the same
until the tens place.
Ten
Thousands Thousands Hundreds Tens Ones
2
3
9
4
8
2
3
9
6
1
Compare 4 and 6: 4 6.
Another way to compare and order whole numbers is to use a number line. A
number to the right of another number on a number line is greater in value. A
number to the left of another number is lesser (or smaller) in value.
Example
Order 938, 916, 960, 899, and 927 from least to greatest.
Write the numbers in their correct places on a number line. The order
of the numbers from least to greatest will be from left to right on the
number line.
899
890
900
916
910
920
927
930
938
940
960
950
960
The order from least to greatest is 899, 916, 927, 938, and 960.
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Therefore, 23,948 23,961.
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Lesson 1: Rational Numbers
Practice
Directions: For Numbers 1 through 4, write the correct symbol (, , or )
to compare the numbers.
1. 121,150 __________ 121,105
3. 30,544 __________ 300,455
2. 61,219 __________ 61,219
4. 801,760 __________ 810,760
5. Write the following numbers in their correct places on the number line.
Then write the numbers in order from greatest to least.
86,573
83,000
87,635
84,000
87,356
85,000
86,735
86,000
87,000
83,765
88,000
6. Write the following numbers in order from least to greatest.
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210,798
220,376
208,139
201,934
7. Write the following numbers in order from greatest to least.
538,312
539,082
8. Which of these numbers is
greater than 83,621 but less
than 83,802?
541,293
538,311
9. Which list of numbers is in order
from least to greatest?
A 74,429 75,013 74,216 73,912
A 83,548
B 75,013 74,429 73,912 74,216
B 83,912
C 74,216 73,912 75,013 74,429
C 83,795
D 73,912 74,216 74,429 75,013
D 83,617
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Unit 1 – Number and Operations
Decimals
A decimal expresses a whole as divided into ten equal parts (tenths), into one
hundred equal parts (hundredths), and so on.
A decimal point separates the whole-number part of the decimal from the part
that is less than 1. Read the number from left to right. The decimal point is
read and. When you get to the decimal side, read the number as a whole
number, followed by the last digit’s place value.
Ones
Decimal
Point
Tenths
Hundredths
Thousandths
5
•
7
3
4
5.734 is read . . .
“five and seven hundred thirty-four
thousandths.” The last digit is 4, which
is in the thousandths place.
0.07 is read . . .
“seven hundredths.”
1.5 is read . . .
“one and five tenths.”
2.654 is read . . .
“two and six hundred fifty-four thousandths.”
Practice
Directions: Write the decimal for Numbers 1 through 5.
1. thirty-two hundredths: _______________________________
2. six tenths: _______________________________
3. one hundred fifty-four thousandths: _______________________________
4. two tenths: _______________________________
5. ninety-nine thousandths: _______________________________
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Examples
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Lesson 1: Rational Numbers
Comparing and ordering decimals
Just like whole numbers, decimals can be compared and ordered by looking at
their place values.
Example
Which of the following decimals is the greatest?
3.458
3.526
3.5
3.59
The comparison is easier if you write the numbers in a place-value
table. Put zeros in as placeholders where necessary, and compare as
you would whole numbers, looking at each value from left to right.
Ones
Decimal
Point
Tenths
Hundredths
Thousandths
3
•
4
5
8
3
•
5
2
6
3
•
5
0
0
3
•
5
9
0
3.59 is the greatest.
Practice
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Directions: For Numbers 1 through 4, compare the decimals. (Use , , or .)
1. 0.76 ______________ 0.763
3. 8.2 ______________ 8.12
2. 5.99 ______________ 5.9
4. 43.38 ______________ 42.96
Directions: For Numbers 5 through 7, order the decimals from greatest to
least.
5. 3.08, 3.9, 3.75, 3.801 ___________________________________
6. $38.50, $38.06, $39.99, $37.75 ___________________________________
7. 19.88, 18.98, 19.8, 19.08 ___________________________________
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Unit 1 – Number and Operations
Fractions
A fraction expresses a whole divided into a number of equal parts. The
numerator, the top number, tells you how many equal parts of the whole you
have. The denominator, the bottom number, tells you how many equal parts
the whole is divided into.
numerator
’ 1
2
{ denominator
The fraction above tells you that a whole is divided into 2 parts and that you
have 1 of those parts.
Example
Don’s family has 7 pets. They have 3 dogs, 2 cats, and 2 birds. What
fraction of the family’s pets are dogs?
In this problem, the whole is the total number of pets the family has.
Therefore, the denominator of the fraction is 7.
The numerator of the fraction is the number of dogs the family has,
which is 3.
3
Three out of the family’s 7 pets are dogs. The fraction form is .
7
1. Mrs. O’Leary bought 12 boxes of cereal at the grocery store, 5 of which are
Choco-bran Monster Pops. What fraction of the cereal boxes Mrs. O’Leary
bought are Choco-bran Monster Pops?
__________
2. Fred threw a total of 9 pitches, and 6 of them were strikes. What fraction of
the pitches were not strikes?
__________
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Practice
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Lesson 1: Rational Numbers
Comparing and ordering fractions with the same denominator
Fractions with the same denominator can be compared and ordered by looking
at their numerators. The fraction with the lesser numerator will have the
lesser value. The fraction with the greater numerator will have the greater
value.
Example
Greg and Heather ordered two 10-inch pizzas. They cut each pizza
into 6 equal slices. Greg has 2 slices remaining from his pizza
Heather has 3 slices remaining from her pizza
. Who has
3
6
.
2
6
the smaller amount of pizza remaining?
Compare the numerators: 2 3.
2
6
3
6
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Greg has the smaller amount remaining from his pizza:
2
6
3
.
6
Practice
Directions: For Numbers 1 through 4, compare the fractions. (Use , , or .)
1.
1
8
2.
7
10
______________
2
8
______________
6
10
3.
2
3
4.
5
12
______________
1
3
______________
5
12
Directions: For Numbers 5 and 6, order the fractions from least to greatest.
5.
6 4 7 2
, , , 8 8 8 8
________________________________
6.
21
20
23
19
, , , 100 100 100 100
________________________________
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Unit 1 – Number and Operations
Comparing and ordering unit fractions
You can compare and order fractions with the same numerator by looking at
their denominators. The fraction with the lesser denominator will have the
greater value. The fraction with the greater denominator will have the
lesser value.
Example
Arthur and Agatha ordered two 10-inch pizzas. They cut 1 pizza into
2 equal slices and the other pizza into 6 equal slices. Which is larger,
a slice from the pizza cut into 2 slices
into 6 slices
?
1
6
or a slice from the pizza cut
1
2
Compare the denominators: 2 6.
1
6
A slice from the pizza cut into 2 slices is larger:
1
2
1
.
6
Practice
Directions: For Numbers 1 through 4, compare the fractions. (Use , , or .)
1.
1
6
2.
1
10
______________
1
5
______________
1
12
3.
4.
1
4
______________
1
100
1
4
______________
1
1,000
Directions: For Numbers 5 and 6, order the fractions from greatest to least.
5.
1 1 1 1
, , , 10 6 3 12
________________________________
6.
1
1
1
1
, , , 100 10 1,000 5
________________________________
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1
2
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Lesson 1: Rational Numbers
Improper Fractions
An improper fraction has a numerator that is equal to or greater than its
denominator. You can write an improper fraction as a whole number or a mixed
number by dividing its numerator by its denominator. When the numerator can
be divided evenly by the denominator, the quotient is a whole number.
When the numerator cannot be divided evenly by the denominator, the improper
fraction is written as a mixed number. The quotient becomes the whole number
part of the mixed number. The remainder becomes the numerator of the fraction
part. The denominator of the fraction part is the same as the denominator of the
original improper fraction. Make sure the fraction part is in lowest terms.
Example
How is
28
8
written as a mixed number?
Divide 28 by 8.
3
8
82
24
4
The quotient is 3. This is the whole number part of the mixed number.
The remainder is 4 and the denominator of the improper fraction is 8.
4
Therefore, the fraction part of the mixed number is . This can be
8
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simplified to
1
.
2
The mixed number for
28
8
1
is 3
.
2
Practice
Directions: For Numbers 1 through 3, write the improper fraction as a
whole number or a mixed number.
1.
13
6
__________
2.
40
8
__________
3.
17
5
__________
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Unit 1 – Number and Operations
Directions: For Numbers 4 and 5, make a model to represent each improper
fraction.
4.
8
5
5.
11
4
submarine sandwiches
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cherry pies
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Lesson 1: Rational Numbers
Mixed Numbers
A mixed number is the sum of a whole number and a fraction. Both parts
are written together.
Example
What mixed number represents the shaded parts of these figures?
How many whole figures are shaded? 3
What fraction of the fourth figure is shaded?
5
6
5
The mixed number 3
represents the shaded parts of the figures.
6
Practice
Directions: Use the following information to answer Numbers 1 and 2.
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The pictures represent how much is left of 3 pies after Cindy’s party.
1. How many whole pies can be made with the slices that are left? __________
2. Write a mixed number that shows how much pie is left altogether.
__________
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Unit 1 – Number and Operations
Directions: For Numbers 3 and 4, make a model to represent each mixed
number.
6
3. 2 10
1
4. 4 5
6. Which mixed number represents
the shaded parts of these figures?
1
A 2
4
B
1
2
3
1
C 3
3
D
3
3
4
1
A 3
2
2
B 3
3
1
C 4
3
1
D 4
2
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5. Which mixed number represents
the shaded parts of these figures?
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Lesson 1: Rational Numbers
Equivalent Fractions
Equivalent fractions are different fractions that represent the same value.
If Tom ate
1
2
of a 10-inch pizza and Bonita ate
3
6
of a 10-inch pizza, who ate
more? Neither of them—they ate the same amount. The fractions
1
2
and
3
6
are
equivalent.
1
2
3
6
One way to find out whether fractions are equivalent is to draw pictures of them.
1
5
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1
10
3
5
1
5
1
10
1
10
1
5
1
10
1
10
1
10
6
10
When both the numerator and the denominator are the same number, the
fraction is equivalent to 1.
7
7
1
1
1
1
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Unit 1 – Number and Operations
Practice
Directions: For Numbers 1 and 2, write equivalent fractions to describe the
shaded parts of the figures.
1.
2.
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Lesson 1: EOG Practice
EOG Practice
1. What digit is in the hundredthousands place in the following
number?
4. What is the value of the 4 in the
number 245,991?
A
4
B
400
A 2
C
40,000
B 5
D 400,000
254,079
C 7
D 9
2. For track practice, Michael ran
2.32 miles, Jenn ran 2.23, Kayla
ran 3.02 miles, and Gabe ran
2.14 miles. Who ran the shortest
distance?
A Jenn
B Gabe
C Kayla
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D Michael
3. Five of the 12 students in Mr.
Franklin’s class are boys. What
fraction of the students are girls?
A
5
17
B
5
12
C
7
12
D
12
17
5. During the summer, Worlds of
Fun Amusement Park had a
total of two hundred fourteen
thousand, fifty-eight visitors.
How is this number written in
standard form?
A 240,580
B 240,058
C 214,580
D 214,058
6. Which list of fractions is in order
from least to greatest?
A
7
1
8
4
, , , 12 12 12 12
B
1
4
7
8
, , , 12 12 12 12
C
7
4
8
1
, , , 12 12 12 12
D
8
7
4
1
, , , 12 12 12 12
7. How is thirteen-hundredths
written as a decimal?
A 1,300
B
1.3
C
0.13
D
0.013
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Unit 1 – Number and Operations
8. Which set of figures shows
7
4
shaded?
A
11. Which of the following decimals is
the greatest?
A 0.63
B 0.52
C 0.60
B
C
D
D 0.59
12. Which of the following is the
standard form of one hundred
nineteen and twenty-five
hundredths?
A 119.025
B 119.25
C 190.025
9. What is the expanded form of
300,609?
A 300,000 600 9
B 300,000 600 90
C 300,000 6,000 90
D 190.25
13. Which symbol correctly compares
the following two fractions?
11
4
D 300,000 6,000 900
_____
5
4
A A 0.25, 0.025, 0.52, 0.525
B C D B 0.25, 0.52, 0.025, 0.525
C 0.025, 0.525, 0.25, 0.52
D 0.025, 0.25, 0.52, 0.525
14. Which fraction is equivalent
to
15
?
100
A
1
85
B
3
50
C
3
20
D
1
10
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10. Which list shows the decimals in
order from least to greatest?