Lesson 4.1 Assignment
Name_________________________________________________________ Date__________________________
Math Football
Using Models to Understand Integers
Negative integers cannot be represented in concrete reality, but they are very useful for representing
some sort of progress from a starting quantity or position. Therefore, common applications of negative
integers often involve money, temperature, or direction. Problems that deal with sequential events of a
story can often be modeled by a number sentence involving negative integers.
1. A small dog that can jump 5 feet off the ground chases a chipmunk across a yard towards a tree.
The chipmunk runs 8 feet up the tree trunk, and then cautiously walks back down 5 feet to see
how close the dog is. Seeing the dog closing in, the chipmunk then scurries up 3 feet before the
dog reaches the tree.
a. Write a number sentence for the situation.
(18) 1 (25) 1 (13) 5 16
b. Is the dog able to reach the chipmunk? Explain.
The chipmunk ends up 6 feet up the tree trunk. Because the dog can only jump 5 feet high,
the dog is not able to reach the chipmunk.
2. While spending the day at the beach, Keeley tries to dig a deep hole in the sand. She digs down
20 inches before the walls of the hole collapse, filling the hole with 7 inches of sand. She then
continues to dig down 10 inches before the sides of the hole collapse again, filling the hole with
4 inches of sand.
© 2011 Carnegie Learning
a. Write a number sentence for the situation.
(220) 1 (17) 1 (210) 1 (14) 5 219
b.Keeley decides to place her sand pail that is 18 inches tall into the hole in the sand. Is the hole
deep enough for her entire pail to fit? Explain.
Because Keeley’s pail is 18 inches tall and the hole is 19 inches deep, the hole is deep
enough for her entire pail to fit.
3. Write a story problem that can be modeled by a number sentence that includes negative integers.
Then write a number sentence for the situation.
Answers will vary.
Chapter 4 Assignments • 65
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66 • Chapter 4 Assignments
Lesson 4.2 Assignment
Name_________________________________________________________ Date__________________________
Walk the Line
Adding Integers, Part I
The atom is the basic unit of matter. Everything is made up of atoms. Each atom is made up of protons,
neutrons, and electrons. The protons carry a positive (1) charge and make up the nucleus of an atom
with the neutrons. Neutrons do not carry a charge. The electrons carry a negative (2) charge and circle
the nucleus.
Atom
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Proton
Neutron
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Electron
Atoms have no positive or negative charge. This means that they must have the same number of
protons and electrons.
1. A partial model of a Nitrogen atom is shown.
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© 2011 Carnegie Learning
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a.How many electrons should be drawn on the model of a Nitrogen atom so that it has the same
number of protons and electrons?
There are 7 protons in a Nitrogen atom. So, 7 electrons need to be drawn on the model.
Chapter 4 Assignments • 67
Lesson 4.2 Assignment
page 2
b. Complete the model of the Nitrogen atom by drawing in the electrons.
See model.
c.Write an expression that represents the sum of the number of protons and electrons in a
Nitrogen atom.
7 1 (27)
d. Use the number line to show that the sum of the number of protons and electrons in the
Nitrogen atom is zero.
+7
–7
–10
–5
0
5
10
15
© 2011 Carnegie Learning
–15
68 • Chapter 4 Assignments
Lesson 4.2 Assignment
page 3
Name_________________________________________________________ Date__________________________
2. Atoms contain protons (1), neutrons (no charge), and electrons (2). Atoms must have the same
number of protons and electrons. Sometimes an atom gains electrons. It then becomes an anion.
The model shows a Nitrogen anion.
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a.Write an expression that represents the sum of the number of protons and electrons
in the Nitrogen anion.
7 1 (210)
b. Use the number line to determine the sum of the protons and electrons in the
Nitrogen anion.
+7
–10
© 2011 Carnegie Learning
–15
–10
–5
0
5
10
15
c. What is the charge of the Nitrogen anion?
The charge of the Nitrogen anion is 23.
Chapter 4 Assignments • 69
Lesson 4.2 Assignment
page 4
3. A Magnesium anion is shown. Anions contain more electrons (2) than protons (1).
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a. Write an expression that represents the sum of the number of protons and electrons
in the Magnesium anion.
12 1 (214)
b. Use the number line to determine the sum of the protons and electrons in the
Magnesium anion.
12
–14
–15
–10
–5
0
5
10
15
c. What is the charge of the Magnesium anion?
The charge of the Magnesium anion is 22.
© 2011 Carnegie Learning
70 • Chapter 4 Assignments
Lesson 4.2 Assignment
page 5
Name_________________________________________________________ Date__________________________
d.Suppose that the Magnesium anion gains 7 more electrons. Complete the model shown
by adding the 7 electrons.
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e. Write an expression that represents how the charge of the Magnesium anion
changed in part (d).
22 1 (27)
f. What is the charge of the Magnesium anion in part (d)? Use a number line to verify
your answer.
The charge of the Magnesium anion is 29.
–2
–7
–10
–5
0
5
10
15
© 2011 Carnegie Learning
–15
Chapter 4 Assignments • 71
© 2011 Carnegie Learning
72 • Chapter 4 Assignments
Lesson 4.3 Assignment
Name_________________________________________________________ Date__________________________
Two-Color Counters
Adding Integers, Part II
Ruth’s family is on a road trip from San Francisco, California, to Orlando, Florida. During the first day
of the trip, the family drives from San Francisco to Phoenix, Arizona. On the second day, they drive
from Phoenix to Las Cruces, New Mexico. On the third day, they drive to San Antonio, Texas, to visit
Ruth’s grandmother. After a few days, they drive to Mobile, Alabama. Finally, on the last day of the trip,
the family drives to Orlando, Florida. Ruth’s little sister, Josie, is 3 years old. When she isn’t sleeping in
the car, she is usually asking, “How much longer?” Ruth’s father is a mathematician and likes to keep
his children thinking.
1. When the family drives from San Francisco to Phoenix, Arizona, they get up and leave very early
in the morning. Josie takes a morning nap, and when they stop for lunch she asks, “How much
longer?” Ruth’s father’s answer is, “The number of hours we have left is equal to the sum of
25 and 13. Ruth, how much longer?”
a.Write an expression to show how much longer Ruth’s family has to drive before they get
to Phoenix, Arizona.
25 1 13
b.Simplify the expression in part (a) to determine the amount of time they have left.
25 1 13 5 8
The family has 8 hours left before they arrive in Phoenix, Arizona.
© 2011 Carnegie Learning
c. Draw a model to verify your answer in part (b).
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Chapter 4 Assignments • 73
Lesson 4.3 Assignment
page 2
2. When Ruth’s family drives from Phoenix, Arizona to Las Cruces, New Mexico, it takes them 6 hours
to make the trip. Then, they drive to San Antonio, Texas to visit Ruth’s grandmother. After driving a
few hours, Ruth’s sister, Josie asks, “How much longer?” Ruth’s father says, “The number of hours
we have left is equal to the absolute value of the sum of 24 and 27.”
a.Write an expression to show how much longer Ruth’s family has to drive before they get to
San Antonio, Texas.
​| 24 1 (27) |​
b. Simplify the expression in part (a) to determine the amount of time they have left.
​| 24 1 (27) |​ 5 ​| 211 |​
5 11
The family has 11 hours left before they arrive in San Antonio, Texas.
3. After spending a couple of days in San Antonio, Ruth’s family travels to Mobile, Alabama.
As soon as her little sister Josie is buckled in her car seat, she asks, “How much longer?” Ruth’s
father replies, “The number of hours we must drive today is the sum of 23 and 212.”
a.Write an expression to show how long Ruth’s family has to drive to get to Mobile, Alabama.
23 1 (212)
23 1 (212) 5 11
The family has 11 hours left before they arrive in Mobile, Alabama.
74 • Chapter 4 Assignments
© 2011 Carnegie Learning
b. Simplify the expression in part (a) to determine the length of their drive.
Lesson 4.3 Assignment
page 3
Name_________________________________________________________ Date__________________________
c. Draw a model to verify your answer in part (b).
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© 2011 Carnegie Learning
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Chapter 4 Assignments • 75
Lesson 4.3 Assignment
page 4
4. On the last day of a long road trip, Ruth’s family is packing the car and getting ready to leave for
Orlando, Florida. Before Ruth’s sister Josie can ask how long they must drive, Ruth’s father says,
“The sum of 219 and the number of hours we must drive today is 211.”
a.Write an expression to show how long Ruth’s family has to drive to get to Orlando, Florida.
219 1 ______5 211
b. Simplify the expression in part (a) to determine the length of their drive.
219 1 8 5 211
The family has 8 hours left before they arrive in Orlando, Florida.
c. Draw a model to verify your answer in part (b).
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© 2011 Carnegie Learning
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76 • Chapter 4 Assignments
Lesson 4.4 Assignment
Name_________________________________________________________ Date__________________________
What’s the Difference?
Subtracting Integers
1. Some people have to wear glasses because they are nearsighted, which means they can see
close objects clearly. This condition is called myopia. Others have to wear glasses because they
are farsighted, meaning they can see far away objects clearly. This condition is called hyperopia.
Strength of optical power is measured in a unit known as diopters. The table shows the ranges of
optical power strength for each condition.
Vision
Range of Strength of Optical Power (diopters)
Low myopia
23 to 0
Medium myopia
26 to 23
High myopia
26 or higher
Low hyperopia
0 to 2
Medium hyperopia
2 to 4
High hyperopia
4 or higher
a.When a person needs glasses, they go to an optometrist to get fitted for lenses that will help
correct their vision so that they can see everything clearly. The table shows the strength of
© 2011 Carnegie Learning
optical power for several students. Using the table above, complete the vision column.
Student
Strength of Optical Power
(diopters)
Vision
Anastasia
22
Low myopia
Danica
5
High hyperopia
Juan
27
High myopia
Bernard
28
High myopia
Frank
24
Medium myopia
Peter
1
Low hyperopia
Chapter 4 Assignments • 77
Lesson 4.4 Assignment
page 2
b.Danica says that the difference between her strength of optical power and Frank’s strength of
optical power can be calculated using the following expression.
5 2 (24)
Draw a counter model and determine the different between Danica’s strength of optical power
and Frank’s strength of optical power using her method.
+
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The difference between Danica’s strength of optical power and Frank’s strength of optical
power is 9 diopters.
c.Frank says the difference between his strength of optical power and Danica’s strength of optical
power can be calculated using the following expression.
24 2 5
Draw a counter model and determine the different between Danica’s strength of optical power
and Frank’s strength of optical power using his method.
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The difference between Frank’s strength of optical power and Danica’s strength of optical
power is 29 diopters.
78 • Chapter 4 Assignments
© 2011 Carnegie Learning
Lesson 4.4 Assignment
page 3
Name_________________________________________________________ Date__________________________
d. Who is correct, Frank or Danica? Use a number line to help explain your answer.
9
–15
–4
–5
–10
0
5
10
15
The actual difference between Frank’s strength of optical power and Danica’s strength of
optical power is the absolute value of either expression. So, neither Frank nor Danica is entirely correct. If you plot each student’s strength of optical power on a number line, you
will notice that the distance between them is 9.
e. Write an expression that represents the difference between Juan’s strength of optical power
and Peter’s strength of optical power. Then, use the number line to calculate the difference and
explain your answer.
|27 2 1| or |1 2 (27)|
8
–15
–10
–7
–5
0
1
5
10
15
The difference between Juan’s strength of optical power and Peter’s strength of optical
power is 8 diopters. If you plot each student’s strength of optical power on a number line,
you will notice that the distance between them is 8.
f. Write an expression that represents the difference between Bernard’s strength of optical power
and Anastasia’s strength of optical power. Then, use the number line to calculate the difference
© 2011 Carnegie Learning
and explain your answer.
​| 28 2 (22) |​ or ​| 22 2 (28) |​
6
–15
–10
–8
–5
–2
0
5
10
15
The difference between Bernard’s strength of optical power and Anastasia’s strength of
optical power is 6 diopters. If you plot each student’s strength of optical power on a number
line, you will notice that the distance between them is 6.
Chapter 4 Assignments • 79
Lesson 4.4 Assignment
page 4
g. Determine the difference between the strength of optical power for the student with the highest
myopia and the student with the highest hyperopia. Show your work. Use a number line to
verify your answer.
13
–15
–10
–8
–5
0
5
10
15
​| 28 2 5 |​ 5 ​| 213 |​
5 13
The student with the highest myopia is Bernard and the student with the highest hyperopia
is Danica. The difference between Bernard’s and Danica’s strengths of optical power is 13 diopters. If you plot each student’s strength of optical power on a number line, you will
notice that the distance between them is 13.
h.Frank says that the difference between his strength of optical power and that of another student
is 2 diopters.
Write a number sentence to represent Frank’s statement and determine which student Frank is
referring to. Explain your answer.
​| 24 2 __ |​ 5 2
The other student is Anastasia. Her strength of optical power is 22 and
© 2011 Carnegie Learning
|242(22)| 5 ​| 22 |​ or 2.
80 • Chapter 4 Assignments
Lesson 4.5 Assignment
Name_________________________________________________________ Date__________________________
What Do We Do Now?
Adding and Subtracting Rational Numbers
© 2011 Carnegie Learning
1. The table shows the freezing points of some of the elements in the periodic table.
Element
Freezing Point (°F)
Helium
2458
Hydrogen
2434
Oxygen
2368.77
Nitrogen
3
2345​ __  ​
4
Chlorine
2149.51
Mercury
7
237​ ___  ​ 
10
Phosphorus
111.7
Potassium
146.03
Sodium
208
Sulfur
239.65
Lead
7
621​ ___  ​ 
10
Zinc
787.51
Chapter 4 Assignments • 81
Lesson 4.5 Assignment
page 2
a. What is the difference between the freezing points of Zinc and Potassium? Show your work.
787.51 2 146.03 5 641.48
The difference between the freezing points of Zinc and Potassium is 641.48 degrees.
b. What is the difference between the freezing points of Chlorine and Helium? Show your work.
2149.51 2 (2458) 5 308.49
The difference between the freezing points of Chlorine and Helium is 308.49 degrees.
c. How much greater is the freezing point of Lead than the freezing point of Nitrogen?
Show your work.
( 
( 
)
)
3
15 ​  ​
14  ​ 2 ​ 2345​ ___
7  ​ 2 ​
  2345​ __ ​   ​ 5 621​ ___
621​ ___
10
20
20
4
9  ​ degrees
 
The freezing point of Lead is 967​ ___
greater than the freezing point of Nitrogen.
20
9  ​ 
5 967​ ___
20
d. How much less is the freezing point of Chlorine than the freezing point of Zinc?
787.51 2 (2149.51) 5 937.02
The freezing point of Chlorine is 937.02 degrees less than the freezing point of Zinc.
82 • Chapter 4 Assignments
© 2011 Carnegie Learning
Show your work.
Lesson 4.5 Assignment
page 3
Name_________________________________________________________ Date__________________________
e. Patricia and Elliott are trying to figure out how much the temperature would have to increase
from the freezing point of Hydrogen to reach the freezing point of Phosphorus. Patricia says the
temperature would have to increase 545.7 degrees, and Elliott says the temperature would have
to increase 322.3 degrees. Who is correct? Show your work and explain your reasoning.
2434 1 545.7 5 111.7
2434 1 322.3 5 2111.7
Patricia is correct. The temperature would have to increase 545.7 degrees from the freezing
point of Hydrogen to reach the freezing point of Phosphorus. If the temperature increases
only 322.3 degrees, it will still be a negative temperature. The freezing point of Phosphorus
is 111.7 degrees Fahrenheit.
f. Francine and Lisa are trying to figure out how much the temperature would have to increase
© 2011 Carnegie Learning
from the freezing point of Nitrogen to reach the freezing temperature of Mercury. Francine says
1  ​ degrees, and Lisa says the temperature would
the temperature would have to increase 308​ ___
20
9  ​ degrees. Who is correct? Show your work and explain your reasoning.
have to increase 383​ ___
20
15 ​ 1 308​ ___
3  ​ 1 308​ ___
1  ​ 
1  ​ 5
  2345​ ___ 
2345​ __
4
20
20
20
7  ​ 
5 237​ ___
10
3  ​ 1 383​ ___
15  383​ ___
9  ​ 
9  ​ 5
  2345​ ___ ​ 1
2345​ __
4
20
20
20
7  ​ 
5 37​ ___
10
1  ​ degrees
 
Francine is correct. The temperature would have to increase 308​ ___
from
20
the freezing point of Nitrogen to reach the freezing point of Mercury. If the temperature 9  ​ degrees,
 
it will be a positive temperature. The freezing point of Mercury
increases 383​ ___
20
7
___
 
Fahrenheit.
is 237​    ​ degrees
10
Chapter 4 Assignments • 83
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84 • Chapter 4 Assignments