Download Integers and Rational Number

Integers and Rational Number··
Chapter Focus
You will describe situations using opposites. You will add, subtract,
multiply, and divide rational numbers. You will also solve equations
containing integers. Rational numbers are numbers that can be written
as a ratio of two integers a and b, where b is not zero. You will convert
rational numbers to decimals. Being able to convert from fractions to
decimals will help you compare and order rational numbers.
Chapter at a Glance
Lesson
COMMON
CORE
Standards for Mathematical Content
2-1
Integers
CC.7.NS.1a
2-2
Adding Integers
CC.7.NS.1a, CC.7.NS.1b, CC.7.NS.1c
2-3
Subtracting Integers
CC.7.NS.1c
2-4
Multiplying and Dividing Integers
CC.7.NS.2a, CC.7.NS.2b
2-5
Solving Equations Containing Integers
CC.7.NS.1b, CC.7.EE.3,
CC.7.EE.4
2-6
Equivalent Fractions and Decimals
CC.7.NS.2, CC.7.NS.2d
2-7
Comparing and Ordering Rational
Numbers
PREP FOR
Problem Solving Connections
Performance Task
Assessment Readiness
CC.7 .EE.4
Unpacking the Standards
Understanding the standards and the vocabulary terms in the
standards will help you know exactly what you are expected to learn
in this chapter.
What It Means to You
You will learn how to add and subtract rational numbers with the same
sign and with different signs.
EXAMPLE
4 + (-7)
+(-7)
I·
4
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-5 -4 -3 -2-1
0
1
2
3
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4 + (-7)
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= -3
What It Means to You
You will learn that subtracting a
rational number is the same as
adding its additive inverse.
Find the difference between
3,000 of and -250 oF, the
temperatures the space
shuttle must endure.
3,000 - (- 250)
3,000
+ 250 = 3,250
The difference in
temperatures the shuttle
must endure is 3,250 oF.
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4
5
Start at O.
Move right
4 units. Then
move left 7
units.
What It Means to You
You will solve real-world and mathematical problems involving the four
operations with rational numbers.
EXAMPLE
A football team must
move the ball forward
at least 10 yards from its
starting point to make
a first down. If the team
has 2 losses of 3 yards
each and a gain of
14 yards, does the team
make a first down?
2·(-3)+14
-6
+ 14
8
Multiply -3 by 2 to find the total loss; then add the gain of 14. Use the order of operations. Multiply first. Then add. The team moved the ball forward 8 yards, so it did not make
a first down.
You will convert rational numbers to a decimal using long division.
You will understand that a rational number will result in a terminating
decimal or a decimal that repeats.
EXAMPLE
Write each fraction as a decimal.
1
3
3"
"4
0.75
4h.oo
-28
20
-20
o
3
4= 0.75
0.333 ... 3h.ooo -9 10 -9 10
-9
1
3"1 = 0.333...
= 0.33
Key Vocabulary
additive inverse (inverso aditivo) The opposite of a number. integers (enteros) The set of whole numbers and their opposites. Inverse Property of Addition (propiedad inversa de fa suma) The sum of a number and its opposite, or additive inverse, is O.
opposites (opuestos) Two numbers that are an equal distance from zero on a number line; also called additive inverse. rational number (numero radonal) Any number that can be expressed as a ratio of two integers. Name _ _ _ _ _ _ _ _ _ _ _ _ _ _ Class _______ Date _ _ _~
Integers
Essential question: How do you describe situations using opposites?
Two numbers are
if they are the same distance from 0
on the number line, but on opposite sides of O. The It!~~~ are
the set ofwhole numbers and their opposites.
Identifying Opposites
Use a ruler to draw a number line on a sheet of graph paper. Use the
graph paper's grid to help you make 21 evenly spaced tick marks on
the number line.
Starting from the left, label the tick marks with the integers from -10 to 10.
Fold the number line in half along the tick mark for O. The integers that
come together when the number line is folded are opposites of each other.
file; Explain why the numbers that come together on the folded number line
are opposites.
Write the opposite of each integer.
1a. 4
1b.
-8 _ __
1c.
6 _ __
1d. -3 - - - -
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1e. When you folded the number line in half, which integer was not paired with
another integer? What do you think this means about the opposite of that integer?
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Explain how you know that -5 and 7 are not opposites.
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Conjecture Make a conjecture about the opposite of a negative number.
Give evidence that supports your conjecture.
Chapter 2
43 Lesson 1
that are opposites of each other combine to make O.
Combining Opposite Quantities
On Wednesday, the temperature increased 5 OF and then decreased 5 OF.
Use integers to find the overall change in temperature.
Step 1: Write an integer quantity for the first change. The change is an increase, so it is positive. First change: 5 OF
Step 2: Write an integer quantity for the second change. Second change:
OF
The change is a decrease, so it is negative. Step 3: Combine the quantities to find the overall change. The quantities are _ _ _ _ _ _-1' so
they combine to make O.
The overall change in temperature is
oF.
Complete the table to find the overall change for each situation.
First
Otange . .
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la. A football team loses 3 yards and then
gains 3 yards.
lb. Arthur depOSits $25 in his account and
then withdraws $25 from his account.
le. A plane drops 300 feet in elevation and
then rises 300 feet in elevation.
The overall change in Robert's score in the first two rounds of a video game
is 0 points. If his score changed by -22 points in the first round, by how much
did his score change in the second round? Explain how you know.
Carmen withdraws $15 from her checking account and then deposits $20.
Is the overall change in the amount of money in her account positive,
negative, or $O? Explain your reasoning.
Chapter 2
44 Lesson 1
Name~
_ _ _ _ _ _ _ _ _ _ _ Class _ _ _ _ _ Date _ _ _ _ ,qjI,
Graph each integer and its opposite on a number line.
1.8 -8 -6 -4 -2
2. -7
0
2
4
6
8
-8 -6 -4 -2
0
2
4
6
8
Identify pairs of opposites in the list and graph them on the
number line.
3. -5,-3,1,2,3,5,7
4. -8,-5,-2,0,2,3,8
01(
-8 -6 -4 -2
0
2
4
6
8
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-8 -6 -4 -2
0
2
4
6
8
5. Christy dove to a depth of 12 feet below the surface of the water. She then swam 12 feet back up to the surface. Use integers to find the overall change in feet. 6. The overall change in a game show is 0 points. If the
contestant scored 300 points in the first round, how many
points did the contestant lose in the second round?
Chapter 2 45
Practice and Problem Solving
Write the correct answer.
1. The coldest place on record in the
United States was in Alaska in 1971.
It was 80 OF below zero. Write this
temperature as an integer and then
find its opposite.
2. The temperature outside was -4 OF
at 8 AM. By noon the temperature had
risen 4 of. By 4 PM the temperature
had dropped 4 of. What was the
overall change in temperature?
3. A small business reported a net loss
of $62,500 during its first year. In its
second year, it reported a profit of
$34,100. Write each amount as an
integer and find their opposites.
4. Barbara wins 60 points on a game
show and then loses 60 points. Find
the overall change in points.
Choose the letter for the best answer.
5. Use the table at right. Which
continent has its highest point that is
the opposite of -29,028 feet?
A Asia
C Africa
B South America
D Australia
! Highest
6. Use the table at right. Which
continent has its lowest point that
is the opposite of 92 feet?
F Europe
G Australia
H North America
J Asia
Chapter 2 Point (ft)
I
North America
20,320
-282
I
South America
22,834
-131
J
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Africa
19,340
-512
I
Asia
29,028
-1,339
Australia
7,310
-52
Europe
18,510
-92
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46
Lowest
Point (ft)
Continent
Practice and Problem Solving
Name _ _ _ _ _ _ _ _ _ _ _ _ _ _ Class _ _ _ _ __
Adding Integers
Essential question: How can you add rational numbers?
A mti~~~~~ is a number that can be written as a ratio of two integers a
and b, where b is not zero. All integers are rational numbers.
Adding Rational Numbers with the Same Sign
A
Andrea has 6 cups of fruit punch in a bowl. She adds 3 cups of fruit punch to the bowl. How many cups of punch are there altogether'? Find 6 + 3. Start at 6. o 1 2 3
Move 131 = 3 units to the right because the
second addend is positive. The result is
. There are _ _ _ ~~ cups of punch. 4
567
8
9 10
B' Selina removes
32 pounds of stones from a pile to make a walkway. Then she removes another 28 pounds on her next trip. How many pounds does she remove altogether! Use negative numbers to represent amounts that are removed from the pile. Find -32
+ (-28). Start at -32.
-60
1- 1
Move 28 = 28 units to the left because
the second addend is negative.
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The result is ____ . Selina removes ____ pounds.
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Use a number line to find each sum.
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-20
-40
1a.
3 + 1 = _ _ __ 1b.
-2
+ (-4)
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3
4
5
-7
-6
-5
-4
-3
-2
-1
o
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1(. Conjecture Work with other students to make a conjecture about the sign of the
sum when the addends have the same sign.
Chapter 2
47 Lesson 2
add two rational numbers with the same sign, find the sum of their
values. Then use the same sign as the sign of the two rational numbers.
Adding Rational Numbers with Different Signs
A football team gains 4 yards on their first play. Then they lose 7 yards
on their next play. What is the team's overall gain or loss on the two plays'!
Use a positive number to represent a gain and a negative number to represent a loss.
Find 4 + (-7).
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Start at 4.
-5 -4 -3 -2 -1
0
1
2
3
4
5
Move 1-71 = 7 units to the _ _ _ _ __
because the second addend is _ _ _ _ __
The result is _ _ __
The team gains I loses ____ yards.
A school band decides to hold a bake sale to raise money. They withdrew
$50 from the band's account for ingredients. The deposit they made back
into the account from the sale was $140. What was the overall increase
or decrease in the band's account'!
Use a positive number to represent a deposit and a negative number to represent a
withdrawaL
Find -50 + 140.
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-60 -40 -20 0 20 40 60 80 100120140
Start at -50.
Move 11401 = 140unitstothe _ _ _ _ __
because the second addend is _ _ _ _ __
The result is _ _ __
The account balance increases I decreases by $_ _ __
Use a number line to find each sum.
2a.
2b.2+(-3)= _ __
-8+5=
.1
-8 -7 -6 -5 -4 -3 -2 -1
2e.
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1
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2
-4
0
4
Conjecture Work with other students to make a conjecture about the sign of the
sum when the addends have different signs.
Chapter 2
48
Lesson 2
add two rational numbers with different signs, find the difference of
absolute values. Then use the sign of the rational number with the greater
absolute value.
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Finding the Additive Inverse
:14.:<. Abby takes 5 gallons ofwater out of an aquarium. Later, she adds
5 gallons ofwater to the aquarium. What is the overall increase
or decrease in the amount ofwater in the aquariuml
Use a positive number to represent water added to the aquarium and a negative number to represent water taken out of the aquarium. + 5. Start at _ _ _ _ __
Move 151 = 5units to the _ _ _ __
Find -5
-5 -4 -3 -2 -1
0
2
3
4
5
because the second addend is _ _ _ _ __
The result is _ _ _ _ __
Thismeans _________________
Kendrick adds 2 cups of chicken stock to a pot. Then he takes 2 cups
of stock out of the pot. What is the overall increase or decrease in
the amount of chicken stock in the potl
Use a positive number to represent chicken stock added to the pot and
a negative number to represent chicken stock taken out of the pot.
Find 2 + (-2). Startat _ _ _ __
Move
1-21 = 2unitstothe _ _ _ _ __
because the second addend is _ _ _ _ __ o
2
The result is _ _ _ _ __ Thismeans _________________ Conjecture Work with other students to make a conjecture about the sum
of a number and its opposite.
What is the opposite of 50? What is the opposite of -751
e
or
of a number is the same distance from
on a number line as the original number, but on the other side of O. The sum
of a number and its additive inverse is O. Zero is its own additive inverse.
Chapter 2
49 Lesson 2
PRACTICE
Use a number line to find each sum.
1. 3 + (-8)
'-_/
=
2. -2 + (-2) =
-5 -4 -3 -2 -1
0
1
2
3
4
"'I
-5 -4 -3 -2 -1
4. 5 + (-7)
3. -4+9= "'I
Ilio "'I -5 -4 -3 -2 -1
Ilio
Ilio
5
"'I
0
1
2
3
4
5
0
1
2
3
4
5
=
Ilio
-4
4
0
Tell what sum is modeled on each number line. Then find the sum.
6.
•
5.
-5 -4 -3 -2 -1
o
1
2
lIol
-5 -4 -3 -2 -1
345
•
1. •
0
1
2
3
4
5
•
8.
Ilio
-4
o
-2
4
-1
o
Find each sum without using a number line.
9. -31 + 16 = _ _ _ _ _ 10. -15 + (-12) = _ _
12. -40 + (-18) + 40
=__
11. 24 + (-54) =
13. 15 + (-22) + 9 = _ __ 14. -1
+ 1 + (-25) = ___
15. Describe a real-world situation that can be represented by the expression -10 + (-2). Then find the sum and explain what it represents in terms of the situation. 16. A contestant on a game show has 30 points. She answers a question correctly
to win 15 points. Then she answers a question incorrectly and loses 25 points.
What is the contestant's final score?
11. Error Analysis A student evaluated -4 + x for x
= -9 and got an answer of 5.
What might the student have done wrong?
Chapter 2
50 Lesson 2
Name _ _ _ _ _ _ _ _ _ _ _ _ Class _ _ _ _ _ Date----..: iW
Additional Practice ] Use a number line to find each sum.
2. 4 + (-6)
1. -1 + 5
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-6 -5 -4 -3 -2 -1
I
0
I
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•
••
234
I
I
-6 -5 -4 -3 -2 -1
I
I
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I
•
0
I
2
I
3
I
4
•
Find each sum.
3. -51 + (-9)
4. 27 + (-6)
5. 1 + (-30) 6. 15 + (-25) 7. 50 + (-7) .
8. -19+(-15)
9. (-23) + 9 10. -19+(-21) 11.-17+11
13. (-15) + (-7) 12. 20 + (-8)
14. 12 + (-14) Evaluate e + ffor the given values.
15. e = 9, f= -24
16. e = -17, f=-7
17. e = 32, f= -19
18. e = -15, f= -15
19. e = -20, f= 20
20. e = -30, f= 12
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21. The temperature rose 9 of in 3 hours. If the starting temperature
was -5 of, what was the final temperature?
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22. Matt is playing a game. He gains 7 points, loses 10 points, gains
2 points, and then loses 8 points. What is his final score?
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Chapter 2
51
Practice and Problem Solving
Write the correct answer.
1. The temperature dropped 12 of
in 8 hours. If the final temperature
was -7 of, what was the starting
temperature?
2. At 3 P.M., the temperature was 9 of.
By 11 P.M., it had dropped 31°F.
What was the temperature at 11 P.M.?
3. Tad owes John $23 and borrows
$12 more. How much does Tad
owe John now?
4. New Orleans, Louisiana, is 6 feet
below sea level. The highest point
in Louisiana, Driskill Mountain, is
541 feet higher than New Orleans.
How high is Driskill Mountain?
6. An airplane at 20,000 ft drops
2,500 ft in altitude. What is the
new altitude?
5. A submarine submerged at a depth of -40 ft dives 57 ft more. What is the new depth of the submarine? Choose the letter for the best answer.
7. Last week, Jane made deposits of
$64, $25, and $37 into her checking
account. She then wrote checks for
$52 and $49. What is the overall
change in Jane's account balance?
A -$99
C $126
B $25
D $227
8. In Indianapolis, Indiana, the coldest
recorded temperature was -23 of.
The hottest recorded temperature
was 127 of higher. What was the
hottest temperature in Indianapolis?
F 150°F
H
104°F
J -150 of
9. Helena borrowed $189 from her
parents to buy an electric bass. She
paid back $56 last week and $64 this
week. How much does Helena still
owe her parents?
10. The Aral Sea and the Caspian Sea
are actually lakes. The elevation of
the Caspian Sea is 92 feet below sea
level. The Aral Sea is 217 feet higher.
What is the elevation of the Aral Sea?
A $133
C $69
F -125 ft
H 309 ft
B $120
D $29
G -309 ft
J 125 ft
Chapter 2 52
Practice and Problem Solving
Name____________________________
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Subtracting Integers
Essential question: How do you subtract rational numbers?
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Subtracting Rational Numbers
The temperature on Monday was 5°C. The temperature on Thursday
was 7 degrees less than the temperature on Monday. What was the
temperature on Thursday!
Subtract to find Thursday's temperature.
Find 5 - 7. Start at 5.
Move 171 = 7units to the left because you are
subtracting a positive number.
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-5 -4 -3 -2 -1
0
1
2
3
4
5
3
4
5
The result is _______.
The temperature on Thursday was _____ dc.
B
The temperature on Friday was -7°C. The temperature on Sunday was -4 °C.
How many degrees did the temperature change from Friday to Sunday!
Subtract to find the difference in temperature.
Find -4 (-7). Start at -4.
Move 171 7units to the right because you are
subtracting a negative number.
-5 -4 -3 -2 -1
0
1
2
The result is ______.
The temperature change from Friday to Sunday was
dc.
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Use a number line to find each difference.
1a.
2 = ____
-6
1b.
1-(-2)= _ __
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-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
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2
3
4
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REFLE·CT
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Work with other students to compare addition of negative numbers on
a number line to subtraction of negative numbers on a number line.
Chapter 2
53
Lesson 3
Adding the Opposite
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Joe is diving 2 feet below sea level. He decides to descend 7 more feet. How many feet below sea level is he! Use negative numbers to represent the number of feet below sea level.
Find -2 -7.
Start at -2.
Move 171 = 7unitstothe _ _ _ __ -10 -9 -8 -7 -6 -5 -4 -3 -2 -1
0
because you are subtracting a ______ number. The result is
. Joe is
feet below sea level. <.•~ Marianne wrote a check for $2. She then withdrew $7 from her
checking account at the bank. How much did Marianne take out
of her checking account?
Use negative numbers to represent amounts of money Marianne took out of her checking account. Find -2 + (-7).
Start at -2.
Move 1-71 = 7 units to the _ _ _ __
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
0
because you are adding a ______ number.
The result is ____. Marianne withdrew _" ____
Use a number line to find each difference or sum. 2a.
-3 - 3 = .____
2b.
-I -8 -7 -6 -5 -4 -3 -2 -1
2e.
-3+ (-3)= _ __ I­
0
1
2
-8 -7 -6 -5 -4 -3 -2 -1
0
1
2
Compare the results from 2a and 2b.
2d. Work with other students to make a conjecture about how to change
a subtraction problem into an addition problem.
Chapter 2
54 Lesson 3
To subtract a number, add its opposite. This can also be written
as p ­ q = p + (-q).
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Numbers
Tomas works as an underwater photographer. He starts at a depth
of 20 feet, ascends 9 feet, descends 12 feet, and then descends 15 feet
more. Write and simplify an expression to find his final depth.
A
Write an expression to solve this problem.
Tomas starts at -20 / 20 feet. When the diver ascends, you add / subtract that distance. When the diver descends, you add / subtract that distance. B
Write an expression.
Start
C
Ascends 9 ft
Descends 12 ft
Descends 15 ft
Add or subtract to simplify the expression. ___________ - _ _ __
The diver ends at a depth of _ _ __
f
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3a. How can you tell that the final depth for Tomas will be deeper than - 20 feet
without doing any calculations?
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3b. Rewrite the expression from B using the additive inverse.
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3e. Anna is collecting samples in a cavern and starts 40 feet underground. She ascends
11 feet, descends 13 feet, descends 5 feet, and ascends 18 feet. Write and simplify
an expression to find her final location.
Chapter 2
55 Lesson 3
Finding the Distance between Two Numbers
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A cave explorer climbed from an elevation of -11 meters to an elevation
of -5 meters. What vertical distance did the explorer climb?
There are two ways to find the vertical distance.
Start at _ _ __
o
Count the number of units on the vertical number line up to -5. The explorer climbed _ _ _ _ meters. -3
This means that the vertical distance between -4
-11 meters and -5 meters is _ _ _ _ meters. ~B -1
-2
Find the difference between the two elevations and use absolute value to find the distance. -11
(-5)= _ __
-6
-7
-8
-9
-10
Take the absolute value of the difference because distance traveled is always a nonnegative number. I
-5
-11
11- (-5)1 = - - ­
The vertical distance is _ _ _ _ meters. Does it matter which way you subtract the values when finding
distance? Explain.
Would the same methods work ifboth the numbers were positive? What if one of the numbers were positive and the other negative? The distance between two values a and b on a number line is represented by the absolute value ofthe difference of a and b. Distance between a and b
Chapter 2
la-bl or Ib-al. 56 Lesson 3