Download Math 7A Unit 1

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Math 7A
Unit #1 - The Number System
Chapter #1A – INTEGERS
Unit 1A Goals:
Established Goals:
7.NS.1 Apply and extend previous understandings of addition and subtraction to add and
subtract rational numbers (INTEGERS); represent addition and subtraction on a
horizontal or vertical number line diagram.
7.NS.2 Apply and extend previous understandings of multiplication and division and of fractions
to multiply and divide rational numbers (INTEGERS).
7.NS.3 Solve real-world and mathematical problems involving the four operations with
rational numbers. (Integers)
Student self-assessment checklist:
1) I understand the difference between Counting #’s, Whole #’s and Integers.
2) I can apply the commutative, associative, identity and inverse properties to perform
calculations with integers.
3) I can add, subtract, multiply and divide integers.
4) I can solve real world problems involving integer operations.
1
AIM: SWBAT distinguish between sets of numbers.
Sets of Numbers:
Rational Numbers
Integers
Whole Numbers
Counting Numbers
All Counting Numbers are also ____________________, ____________________ &
____________________.
All Whole Numbers are also ____________________ and ____________________.
All Integers are also ____________________.
Answer each question with yes or no and explain your reasoning.
Are all Whole Numbers Counting Numbers? _______________________________________
Are all Integers Whole Numbers? ______________________________________________
Are all Integers Counting Numbers? ____________________________________________
Are all Rational Numbers Integers? ____________________________________________
2
Place each number in ALL the sets it belongs to.
-7
0.25
12
0
10
2
1
2

3
4
4
1
3
Rational Numbers
Integers
Whole Numbers
Counting Numbers
Answer each question with…
ALWAYS
SOMETIMES
NEVER
1) Whole Numbers are Integers.
A
S
N
2) Counting Numbers are Rational
A
S
N
3) Rational Numbers are Whole Numbers
A
S
N
4) Integers are Counting Numbers
A
S
N
5) Whole Numbers are Rational
A
S
N
3
HOMEWORK – SETS OF NUMBERS
**Use the chart we made in class to help you answer these questions!**
Answer the following with….
SOMETIMES ALWAYS
NEVER
1) Counting Numbers are Whole Numbers.
S
A
N
2) Whole Numbers are Counting Numbers.
S
A
N
3) Counting Numbers are Integers.
S
A
N
4) Integers are Counting Numbers.
S
A
N
5) Counting Numbers are Rational..
S
A
N
6) Whole Numbers are Integers.
S
A
N
7) Integers are Rational Numbers.
S
A
N
8) Rational Numbers are Whole Numbers.
S
A
N
9) Whole Numbers are Rational.
S
A
N
10) Rational Numbers are Counting Numbers.
S
A
N
State
ALL
of the sets of numbers that each of the following belongs to:
Rational
Integer
Whole
Natural
11) 0 __________________________________________________________________
12)
-5 _________________________________________________________________
13)
1
__________________________________________________________________
2
14) 2.56 ________________________________________________________________
15) 20 __________________________________________________________________
16)
-
3
_________________________________________________________________
5
17) 0. 6 _________________________________________________________________
4
AIM: SWBAT compare & order integers and find the absolute value of an integer.
DO NOW:
Sets Of Numbers
Counting Numbers
1, 2, 3, 4, …
Whole Numbers
0, 1, 2, 3, 4, …
Integers
…,-4,-3,-2,-1, 0, 1, 2, 3, 4, …
Counting Numbers - are the Natural Numbers 1, 2, 3, 4 …
Whole Numbers - all the Counting Numbers AND zero
Integers - all the Whole Numbers AND their opposites
1) All Counting Numbers are also ____________________ and ____________________
2) All Whole Numbers are also _______________.
3) Are all Whole Numbers Counting Numbers? _______
4) Are all Integers Whole Numbers? ________
5) Are all Integers Counting Numbers? ________
CLASSWORK:
Opposite Numbers: are the same distance from zero on a number line in opposite directions.
For example 5 and -5 are opposites.
________________ is a special integer because it is neither positive nor negative.
Why is zero an integer? _________________________________________________
Comparing Integers : > and <
The number farther right on the number line is the larger number.
>_______________
<________________
36 > 12 “36 is greater than 12”
15 < 29 “15 is less than 29”
Compare using > or <.
1) 15 _____ 25
2) 92 _____ 63
3) 0 _____ 12
4) -3 _____ 1
5) -5 _____ 0
6) -5 _____ -18
7) -12 _____ 12
8) -2 _____ -5
Ordering Integers:
Order from least to greatest.
9) -5, -9, 0, -3 _________________
10) -2, 7, -5, -1 _________________
5
**The three questions most often missed about Integers**
1) Name a number that is not an integer? __________
2) Name the largest negative integer. __________
3) Name the smallest positive integer. __________
Absolute Value measures the _________________ a number is from zero on the number line.
Distance is always POSITIVE, therefore, Absolute Value is ALWAYS________________.
The symbol for absolute value is “| |.”
|4| “ What is the absolute value of 4?” |4| = _______
|-4| “What is the absolute value of -4?” |-4| = _______
True or False
-4 = 4 ________________
|-4| = |4| _______________
**Absolute value bars are evaluated like parenthesis. Do whatever is inside the bars
first, and then find the absolute value.
Examples:
|-4| + |5|
|0 + -2|
4 + 5
9
|-2|
2
1) |16| + |-16|
2) |-24 + 0|
3) |12| + |-7|
4) |-20| + |-9|
The negative symbol “- “ means opposite. For example the “opposite of 4” is -4.
Simplify the expression. (Start from the inside and work it out)
5) - (-4) _____
6) - (- (-4)) _____
7) - [- (- (-4))] _____
8) -|-4| _____
9) - (- |-4|) _____
10) - - - |-4| _____
6
HOMEWORK – INTRODUCTION TO INTEGERS
Order the integers from least to greatest:
1) 17, -16, -8, 7, 2,
2)
___________________________
-16, -24, 25, 11, -56, -136
___________________________
3) -5, -1, 4, 6, -10, 0
4) 8, -15, 17, -39, 73,
___________________________
___________________________
First write the OPPOSITE and then write the ABSOLUTE VALUE of each integer:
5)
7
______
______
6)
-25
______
______
7)
106 ______
______
8) -241
______
______
Complete the Statement with  or .
9) -6 _____ 4
12) -11 _____ -3
10) -2 _____ -4
11) 0 _____ 8
13) 31 _____ -16
14) -24 _____ -28
Match the integer expression with the verbal expression:
_____ 15) -|12|
A) the opposite of negative twelve
_____ 16) |-12|
B) the absolute value of twelve
_____ 17) - (-12)
C) the opposite of the absolute value of twelve
_____ 18) |12|
D) the absolute value of negative twelve
Simplify the expression.
19)
|-15|
20)
- (-9)
21)
|-16|
22)
-|-16|
23)
- (-|49|)
24)
- (- (-34))
7
The table below shows the distances of the runners from the finish line when the winner won
the race. Use the table to answer Questions 25 – 27.
Runner
Distance (ft)
Sarah
-16
-2
Beth
Juanita
Tamika
0
-9
-36
Ingrid
25) Who won the race? ____________________
26) Who finished further back, Sarah or Tamika?
____________________
27) Arrange the girls’ names in order from first-place to last-place finish.
____________
1st Place
____________
2nd Place
____________
3rd Place
_____________
4th Place
_____________
5th Place
28) Name a number that is not an integer. __________
29) Name the largest negative integer. __________
30) Name the smallest positive integer. __________
31) State the first 5 Counting Numbers ____________________
32) Whole Numbers are all the ____________________ and ____________________.
33) Integers are all the ____________________ and ____________________.
34) All Counting Numbers are also ____________________ and ____________________.
35) All Whole Numbers are also ____________________.
36) Are all Whole Numbers Counting Numbers? _______
37) Are all Integers Whole Numbers? ________
8
AIM: SWBAT identify properties of addition and multiplication and begin adding integers.
DO NOW:
Write the opposite of each integer.
1) 3 _______
2)
-5 _______
3)
-7 _______
4) 9 _______
Evaluate.
5) |-12| _______
6) |9| _______
7) |-2| _______
8) |-12| + |-1| __
Compare using  or .
9)
8 _____
-6
10)
-7 _____ -411)
-9 _____ 5
12)
7 _____ 2
Order from least to greatest.
13)
-1, -6, 0, -3, -5
______________________
14)
-18, -20, -15, -17
______________________
Properties of Addition and Multiplication
1) Commutative Property of Addition and Multiplication: (Commutative, x; Commutative, +)
Changing the order of the numbers without changing the answer. (#’s commute)
Examples:
A) 2 + 3 = 3 + 2
B) 4(5) = 5(4)
2) Associative Property of Addition and Multiplication: (Associative, x; Associative, +)
Moving the grouping symbols without changing the answer.
Examples:
A) 6 + (2 + 3) = (6 + 2) + 3 B) 7(4 ● 6) = (7 ● 4)6
3) Additive Identity Property: (Identity, +) Identity of # does not change
Any number plus zero equals that number. *The identity element of addition is zero.
Examples:
A) 9 + 0 = 9
B)
x+0=x
4) Multiplicative Identity Property: (Identity, x) Identity of # does not change
Any number times one is that number. * The identity element of multiplication is one.
Examples:
A)
4●1=4
B)
x●1=x
5) Additive Inverse Property: (Inverse, +)
For every number, a, a +
Examples:
-a = 0.
A) 9 +
*Remember: Zero is the identity element
-9 = 0
B)
-x + x = 0
9
6) Multiplicative Inverse Property: (Inverse, x)
For every number, a, a ●
Examples:
A)
= 0 *Remember: One is the identity element
4●
=1
B)
x●
=1
7) Multiplicative Property of Zero: (Zero, x) (Everything becomes zero)
Any number times zero is zero
Examples:
A)
10 ● 0 = 0
B)
x●0=0
Name the property for each of the following:
1)
(13 + 7) + 8 = 13 + (7 + 8)
______________________________________
2)
0 ● (x + 3) = 0
______________________________________
3)
9 ● 5=5 ● 9
______________________________________
4)
(62 + 3) + 0 = (62 + 3)
______________________________________
5)
(19 + 8) + 6 = (8 + 19 ) + 6
______________________________________
6)
(2 ● 3) ● 7 = 2 ● (3 ● 7)
______________________________________
7)
56 ● 1 = 56
______________________________________
8)
7●
______________________________________
9)
-6 + (3
=1
● 8) = -6 + (8 ● 3)
______________________________________
-15 + 15 = 0
10)
______________________________________
Adding/Subtracting Integers
Adding integers means adding with both positive and negative numbers (the whole numbers and their
opposites). There is an important fact to remember when adding integers…
●
“positive 1” (+1) and “negative 1” (-1) are opposites so . . . -1 + 1 = 0, meaning opposites cancel
each other out
We’re going to use Integer Chips to investigate the rules of Adding/Subtracting Integers.
will represent positive numbers
will represent negative numbers
2+2
-1 + -2
-5 + 2
-1 + 4
10
What rules can we come up with after using the Integer Chips:
Same Signs - ____________________
Different Signs - ____________________
Evaluate:
-2 + 2 = ______
-4 + 0 = ______
-5 + 5 = ______
-2 + 5 = ______
-5 + 2 = ______
-2 + -5 = ______
-2 + 3 = ______
2 + -3 = ______
-2 + -3 = ______
-6 + 1 = ______
-1 + 6 = ______
-6 + -1 = ______
11
HOMEWORK – PROPERTIES & INTRO TO ADDING INTEGERS
State the name of the property that is shown.
1) (x + 9) + 1 = x + (9 + 1)
1) ____________________________________
2) 1 x = x
2) ____________________________________
3) (12 + 9) + 15 = (9 + 12) + 15
3) ____________________________________
4) 12 ● (7 ● 15) = (12 ● 7) ● 15
4) ____________________________________
5) 0 + (9 + 1) = 9 + 1
5) ____________________________________
6) r ● 1 = r
6) ____________________________________
7) 106 ● 0 = 0
7) ____________________________________
-y + y = 0
8) ____________________________________
8)
9) (2 + y) + 8 = 8 + (2 + y)
9) ____________________________________
10) c ●
10) ___________________________________
=1
Evaluate.
11)
-4 + 4 = ______
12)
-4 + -3 = ______
13)
-4 + 3 = ______
-3 = ______
15)
-5 + 8 = ______
16)
-1 + 7 = ______
-9 + 3 = ______
18)
-10 + 4 = ______
19)
-2 + -8 = ______
14) 4 +
17)
12
AIM: SWBAT add and subtract integers.
DO NOW:
Evaluate.
1)
-8 + 2 = ______
2)
-10 + 5 = ______
3)
-6 + 6 = ______
4)
-3 + 9 = ______
5) 7 + -2 = ______
6)
-10 + 0 = ______
13
ADDING & SUBTRACTING INTEGERS
I) Get rid of DOUBLE SIGNS first!


+ - -
becomes a NEGATIVE (so 7 + -3 becomes 7 – 3)
becomes a POSITIVE (so 6 - -3 becomes 6 + 3)
II) BOX YOUR TERMS!
** The sign IN FRONT of the number goes with the number **
III) When COMBINING INTEGERS with the SAME signs
⇒ ADD the numbers and KEEP the same sign.
Examples:
A) 12 + 4
B) –12 + –4
16
*Basic Addition – adding two positive numbers*
C) 25 – (–16)
25 + 16
= 41
*Basic Addition – adding two positive numbers*
(get rid of double signs)
–12 – 4
(box terms)
–16
D)
(get rid of double signs)
–12 – 4
(Same Signs  Add & Keep)
–25 + – 16
–25 –
16
–25 – 16
–41
(get rid of double signs)
(box terms)
(Same Signs  Add & Keep)
14
IV) When COMBINING INTEGERS with DIFFERENT signs
IGNORE the signs and SUBTRACT numbers. Keep the sign of whatever you have more of
Subtract the absolute values. Keep the sign of the number with the largest absolute value.
⇒ SUBTRACT and THINK
A) 12 + –8
12
–8
12 –8
4
B)
–37 + 16
–37 + 16
(get rid of double signs)
–21
(box terms)
(box terms)
(Different Signs  Subt. & Think)
(Different Signs  Subt. & Think)
*There are more negatives, so the answer
*There are more positives, so the answer
is negative*
is positive*
Same Signs ⇒ ____________________
1)
12 + 20
4)
12 + -20
7) 20 +
-10 + 5
10) 14 + (-8)
13)
-3 + 12 + -14
2)
5)
Different Signs ⇒ _______________________
-12 + -20
-225 + 125
3)
6)
-12 + 20
-225 + -125
8)
-15 + 7 + 8
9)
11)
-27 + -18
12)
-12 + 5
15)
7 + -2 + 14 + -8
14)
-15 + -7 + -13
-14 + -15 + -26
15
HOMEWORK – ADDING/SUBTRACTING INTEGERS
Evaluate.
Same Signs - ____________________
1) 2 + 23
5)
-500 + 100
9) 50 + -10 + 8
13)
-42 + -8
14)
Different Signs - ___________________
2)
-27 + -30
3)
-28 + 6
4) 25 + -30
6)
-350 + -125
7)
-75 + - 11
8)
10)
-18 + 10 + 8
11)
-79 + 8
12)
-80 + 14
15) 79 +
-1
16)
-90 + 90
-52 + -5
-32 + 5
Complete the statement using always, sometimes, or never.
17) The sum of two positive integers is ______________ zero.
18) The sum of zero and a positive integer is _________________ zero.
19) The sum of zero and a negative integer is _________________ zero.
20) The sum of a positive integer and a negative integer is __________________ zero.
21) Name the first 5 Counting Numbers ____________________
22) Whole Numbers are all the ____________________ and ____________________.
23) Integers are all the ____________________ and ____________________.
24) Name a number that is not an integer? __________
25) Name the largest negative integer. __________
26) Name the smallest positive integer. __________
27) Are all Counting Numbers Whole Numbers? _______
28) Are all Integers Whole Numbers? ________
16
AIM: SWBAT continue to add and subtract integers.
DO NOW:
State the property that is shown.
1) (2 + 7) + 10 = 2 + (7 + 10)
____________________________________
2) 5x● 1 = 5x
____________________________________
3) (5 + 8) + 12 = (8 + 5) + 12
____________________________________
4) 700 + 0 = 700
____________________________________
5)
-75 + 75 = 0
____________________________________
6) 8y ● 0 = 0
____________________________________
CLASSWORK:
Additive Inverse Property - For every number, a, a +-a = 0
State the additive inverse of each of the following. Additive inverse ⇒ ______________
1) -2 _______
2) 14 _______
3) -4x _______
4) -18mn _______
5)
-24 ______
ADDING & SUBTRACTING INTEGERS
I) Get rid of DOUBLE SIGNS first!


+ - -
becomes a NEGATIVE
becomes a POSITIVE
II) BOX YOUR TERMS!
** The sign IN FRONT of the number goes with the number **
III) When COMBINING INTEGERS with the SAME signs
⇒ADD the numbers and KEEP the same sign.
IV) When COMBINING INTEGERS with DIFFERENT signs
IGNORE the signs and SUBTRACT numbers. Keep the sign of whatever you have more of
Subtract the absolute values. Keep the sign of the number with the largest absolute value.
⇒SUBTRACT and THINK
Ex: 7 – 4
-7 – 4
(Diff signs → Subt/Think) (Same signs → Add/Keep)
3
-11
17
BEWARE OF DOUBLE NEGATIVES!!!! Remember (-) means opposite so –(-4) = +4
DOUBLE NEGATIVES - ADD
DOUBLE (Diff) SIGNS - SUBTRACT
- -4
Example: 8
Example: 8 + -4
8+4
8-4
= 12
= 4
Compute. Be sure to box off your terms.
Same signs ⇒ _______________
1) 7 +
- 13
5) 27 + -19
9)
13)
-5 - 3
-18 - -18
17) 4 +
21)
-12 + -18
-38 + -19 + -3
2)
Different signs ⇒ _______________
-8 - 5
3)
6) 5 - 8
10)
-6 - -8
14) -70 + -18
18) 8 +
22) (-41) + (-78) + 51
8)
-21 - (-14)
11) 7 - 13
12)
-13 - -18
-575 - (-600)
19) 7 +
23)
4) 9 - -2
7) 0 - -14
15)
-10 + -19
-17 - 9
-11 + 32
-39 + 25 + -12
24)
16) 90 - -60
20) 25 +
-4 + 32
-31 + 31 + 45
18
HOMEWORK – ADDING/SUBTRACTING INTEGERS
1) What is the additive inverse of 19? _______
2) What is the opposite of -24? _______
3) How many numbers have an absolute value of 12? ______ List them. _______________
4) The counting numbers are _____________________.
5) The whole numbers are ALL the ____________________ and ____________.
6) The integers are ALL the _______________ and their _________________.
For 7-12 state whether each of the following is TRUE or FALSE.
7) One-half is NOT an integer. __________
8) If x is a positive integer, then x > 0 __________
9) If x is a negative integer, then x < 0 __________
10) All whole numbers are integers? __________
11) All integers are whole numbers? __________
12) Zero is a positive integer. __________
Simplify each expression.
REMEMBER – BOX OFF YOUR TERMS AND FOLLOW THE RULES.
BEWARE OF DOUBLE NEGATIVES!
Same signs ⇒ ____________________
13)
-21 + 7
16) 5 + (-7)
Different signs ⇒ ___________________
15)
-20 + 8 + 22 + -10
17) 65 - 72
18)
-85 - -42
–15 + 21
21)
-39 + 25
24)
-50 - 10 – 20 + 100
14)
-29 + -15
19)
-32 - 74
20)
22)
-2 - 3 + 6 - 9
23) 47 + -15 + 8
- 10
19
AIM: SWBAT to apply their knowledge of integer rules to real world situations.
DO NOW:
Tell whether the sum of the two integers is always, sometimes or never positive.
1) two negative integers ___________________
2) a positive integer and a negative integer ___________________
3) two positive integers ___________________
4) a positive integer and zero ___________________
5) a negative integer and zero ___________________
CLASSWORK:
Write an integer for each situation.
1) a bank deposit of $250 _________
2) a loss of 12 pounds _________
3) 7 below zero _________
4) a gain of 6 yards _________
5) 78 m below sea level _________
6) a bank withdrawal of $50 _________
7) a loss of 3 yards _________
8) a gain of 5 hours _________
Read each problem carefully, write a number sentence and solve.
9) The temperature one morning in Juneau, Alaska was –12oF. By the afternoon, the
temperature had risen 8oF. What was the temperature in the afternoon?
10) If the high temperature for the day was 19oF and the low temperature was -5oF, what was
the change in temperature?
11) Your bank account currently has a balance of -$480. You make a deposit of $350. What is
your new account balance?
12) A submarine is 250 meters below sea level. The submarine then rises 175 meters. What is
the new position of the submarine?
20
13) If the deepest point in the area is 37,800 feet below sea level and the highest mountaintop
is 29,012 feet above sea level, find the difference in these elevations.
14) You enter an elevator on the eighth floor. The elevator goes up 5 floors and then down 7
floors, where you exit. On what floor did you exit the elevator?
15) At 5 P.M. the temperature read 10o above zero. At midnight it read 4o below zero. Find the
change in temperature.
16) A whale that is 660 feet below sea level rises up out of the water to a height of 32
feet above sea level. How far up did the whale travel?
Write an addition expression to describe each situation, then find the sum and explain its
meaning.
17) A quarterback is sacked for a loss of 7 yards. On the next play, his team loses 13 yards.
Then the team gains 9 yards on the third play.
18) A seagull starts at 70 feet above sea level. It descends 70 feet to catch a fish.
21
HOMEWORK – ADDING INTEGERS
Compute.
Same Signs: _______________
Different Signs:_______________
1) 9 + –7
2) (–17) + 2
3) –11 + –18
4) 67 + (–43)
5) –6 + 6 + –14
6) –15 + 8 + (–7)
7) –21 + (–25) + (–35)
8) –16 + 7 + 19
Read each problem carefully, write a number sentence and solve.
9) At night the average temperature on the surface of the planet Saturn is –150oC. During the
day the temperature rises 27 oC. What is the average temperature on Saturn’s surface
during the day?
10) To get a first down, a football team must gain 10 yards in 4 plays. If the team gained 5
yards on the first play, lost 7 yards on the second play, and gained 8 yards on the third
play. How many yards must they gain on the fourth play to get a first down?
11) A balloon rises 200 feet from the ground, drops 150 feet, and then rises 300 feet. What is
the new height of the balloon?
12) A scuba diver is 85 feet below sea level. The diver rises 12 feet then descends 60 feet.
How far below sea level is the diver?
22
AIM: SWBAT apply their knowledge of integer rules to real world situations.
Read each problem carefully, write a number sentence and solve.
1) The temperature outside was 20oF. The wind chill made it feel like -15oF. Find the
difference between the real temperature and the apparent temperature.
2) Carly has $50 in a bank account. She writes a check for $75 from the account. What is the
new balance in Carly’s bank account?
3) The highest temperature ever recorded on Earth is 136oF and the lowest temperature
recorded is -129oF. What is the range (difference) of temperatures on Earth?
4) The Panthers lost 5 yards on their first play and lost another 7 yards on their next play.
What was their net result in yards after these two plays?
5) A submarine at -28 feet dives 70 feet. What is the submarine’s position after the dive?
6) Ms. Smith is on a diet. The first week she lost 1 pound, second week lost 3 pounds, third
week gained 1 pound and fourth week lost 4 pounds. What was her net gain or loss?
23
HOMEWORK – INTEGER APPLICATIONS
Evaluate.
1) -5 + 3
2) 51 - (-11)
3) -18 - (-18)
4) -29 + 32
5) -13 - (-18)
6) 7 - 25
7) – 6 + 8
8) -11 - 9
Read each problem carefully, write a subtraction number sentence and solve.
9) The record low temperature for Albany, NY was -28oF. The lowest temperature in U.S.
history is 52oF lower than Albany’s record low. What is the lowest temperature in U.S.
history?
10) During the day the moon can reach a high temperature of 265oF. At night, the temperature
can reach a low of -170oF. What is the difference between the high temperature and low
temperature on the moon?
11) Are the expressions x - y and y - x always opposites? Explain your reasoning.
12) Carl had $200 in his checking account. He wrote 3 checks, one for $57 and another for
$103. He did not record the amount of the third check. Carl received a statement stating
that he overdrew his account (meaning he took out more money than he had) by $55. What
was the amount of the third check that Carl wrote?
24
AIM: SWBAT multiply and divide integers.
DO NOW:
Read each word problem carefully, write a number sentence and evaluate. Your final answer must
be in a complete sentence.
1) A person goes from a sauna at 115F to an outside temperature of -30F. What is the
change in temperature?
2) A submarine is 50 meters below sea level. It goes up 15 meters, then goes down 40 meters.
What is the submarine’s new position?
3) The table shows the temperatures and wind-chill temperatures in three towns.
Town
Actual Temperature
Wind-Chill Temperature
Easton
-3F
-13F
Pine Hills
10F
-10F
West Falls
2F
-17F
Which town had the GREATEST difference between the actual temperature and the wind-chill
temperature?
Evaluate:
4)
-9 + -22
5)
- 6 - - 18
6) 34 + - 30
7) 25 - - 25
8) Whenever you have double signs that are + - you change it into a _____________ sign.
9) Whenever you have double signs that are - - you change it into a _____________ sign.
25
CLASSWORK:
When MULTIPLYING and DIVIDING TWO integers with:
I) TWO SAME SIGNS your answer will be POSITIVE.
Ex:
5 ● 4 = 20 AND -5 ● -4 = 20
18 ÷ 3 = 6 AND -18 ÷
-3 = 6
II) TWO DIFFERENT SIGNS your answer will be NEGATIVE.
Ex: -9 ● 2 = -18
AND 9 ● -2 = -18
-18 ÷ 9 = -2 AND 18 ÷ -9 = -2
You can also use the diagram below to help you choose your sign . . .
Cover the two signs you have, the sign that is remaining is the sign of your answer.
+
-
-
Evaluate.
1)
(-10)(-5)
2)
(12)(4)
3)
(-4)(2)
4)
7(-9)
5)
(100)(-7)
6)
(-6)(13)
7)
-9(-11)
8)
(-15)(7)
9)
 144
12
10)
62 ÷ 2
11)
 150
 30
12)
-90 ÷ -15
13)
-51 ÷ 3
14)
9
3
15)
-120 ÷ 6
16)
17)
-9  -8
18)
-360 ÷
19)
-8  0
20)
60
80
5
88
11
26
BEWARE OF MULTIPLYING MORE THAN 2 NUMBERS!!!!!
1)
-1 ● -1 ● -1
2) -1 ● -1 ● -1 ● -1
3) -1 ● -1 ● -1 ●
-1 ● -1
When multiplying an ODD number of negatives your answer will be _______________.
When multiplying an EVEN number of negatives your answer will be _______________.
(-2)9 will be ____________
(-2)100 will be ____________ (-2)203 will be ____________
4) (-6)(-2)(3)(4)
5) (-8)(-6)(-2)
Evaluate the expression when
6) (-4)(-2)(-5)(-10)
x = -9, y = -7 and z = -3.
(Replace the variable with the
given value.)
7) xy
8)
-2yz
9) xyz
10)
x
z
Find the mean of the data: To find the mean of a set of data, you first add up all the numbers, and
then divide your sum by the number of elements)
11) 8, 5, -4, 9, -3, 11, 2
27
HOMEWORK – MULTIPLYING & DIVIDING INTEGERS
Find the product or quotient.
Same Signs - ___________________
Different Signs - ___________________
1) 4(-9)
2)
-5(-7)
3)
5)
 64
8
6)
 32
4
7) 50 ÷
9)
-12  8
10)
-13  -20
11) 6(-5)(-7)
13)
-36 ÷ 3
17) 42(-3)(0)
14)
18)
30
 15
-5(-7)( -3)
15)
-12(0)
-10
 56
7
19) 4(-5)(-7)(-2)
4)
-9(-11)
8) 0 ÷
12)
-29
-5(-7)(-3)
16) 42 ÷ 2
20) (-10)(-10)(4)
Find the mean of the data: To find the mean of a set of data, you first add up all the numbers, and
then divide your sum by the number of elements)
21)
-16, 2, -18, 4, -11, -8, -6, 5
28
AIM: SWBAT multiply and divide integers.
DO NOW:
DO NOT COMPUTE!! State whether the answer will be POSITIVE, NEGATIVE or ZERO.
Rules for Adding/Subtracting
Rules for Multiplying/Dividing two Integers
Same signs - _____________
Same signs - _____________
Different signs - _____________
Different signs - _____________
1) (-19)(-10)
_____________
5) -152 ÷ 4
_____________
9)
 16
4
2) -27 + 96
_____________
6) 53 + 9
_____________
10) 32 + -12
_____________
_____________
13) 15 - 27
14) 57 - 42
_____________
17) -49 ÷ -7
_____________
_____________
18) (-4)23
_____________
3) -24 - (- 19)
_____________
7) (14)(7)
_____________
11) 0 - - 5
_____________
15) (29)(-30)(-20)
_____________
19) -7 + 7
_____________
4) -80 + 80
_____________
8) (0)(9)
_____________
12) -42 - 50
_____________
16) -4 - (-4)
_____________
20) (-5)10
_____________
CLASSWORK:
Evaluate.
1) 5(-7)
2) -6 ● -3
5) 240 ÷ -15
6)
 96
3
3) -4 ● 8
4) -91 ÷ -7
7) (-5)(-3)(7)
8) (7)(4)(-2)(-1)
29
Read each word problem carefully, write a mathematical sentence and solve. Make sure
your number sentence represents whether the situation is positive or negative.
9) A coin is tossed off a boat into the ocean. The coin’s depth changes by -7 inches per second.
Determine the depth of the coin 15 seconds after it hits the water.
10) You have $700 in a savings account. Over a 2 month period, you make 10 withdrawals of $25
each. What is your new balance?
11) The temperature fell 24oF in 6 hours. If the temperature fell at the same rate every hour,
what is the change in temperature each hour?
12) Evan withdrew a total of $500 from an ATM machine over a 5-day period. If he withdrew
the same amount each day, how much money did Evan withdraw each day?
13) A submarine is diving from the surface of the water at a rate of 80 feet per minute. What
is the depth of the submarine after 8 minutes?
14) Wave erosion causes a certain coastline to recede at a rate of 2 cm each year. This occurs
uninterrupted for a period of 7 years. How much will the coastline have receded?
30
HOMEWORK – MULTIPLYING & DIVIDING INTEGERS
Evaluate.
1) -63 ÷ 7
2) -54 ÷ -9
3) 16 ÷ 2
4) -36 ÷ -6
5) -9 ● -13
6) 4 ● -23
7) -5(-12)
8) (-13)(7)
9) -3(2)(7)
10) -3(4)( -7)
11) (-7)(-8)(5)
12) (-2)(-4)(-5)
Read each word problem, write a mathematical sentence and solve.
13) A shoreline is changing -4 cm each year due to erosion. What will the change in the
shoreline be in 5 years?
14) The temperature during a 5-day period in Center City were -19oF, -14oF, -8oF, 13oF and
18oF. What was the average temperature for those 5 days?
15) The price of a stock rose $3 yesterday. If the stock continues to change at the same rate
each day, what will be the total change over 5 days?
31
CLASS WORK/HOMEWORK – REVIEW FOR INTEGER TEST
Fill in the rules.
Adding/Subtracting Integers:
Multiplying/Dividing two Integers:
1) Same Signs: __________________
2) Same Signs: __________________
3) Different Signs: __________________
DO NOT COMPUTE!
4) Different Signs: __________________
State whether the answer will be POSITIVE, NEGATIVE or ZERO.
5)
-27 + 19 _______________
6) (-14)(-23) _______________
7)
100
_______________
20
8)
9)
-17 + 17
10)
 121
11
_______________
-58 - -64 ______________
_______________
- -12
_______________
12)
-125 ÷ -5 _______________
13)
-8 - -8
_______________
14)
-28 - 45
15)
-27  19
_______________
16)
-10  0 _______________
11) 19
17) 19 - 32 _______________
19)
27
9
_______________
_______________
18) 12 + 7 _______________
20) (-3)(-2)(-12) _______________
State whether the answer is ALWAYS, SOMETIMES or NEVER true.
21) Counting Numbers are Rational Numbers.
_______________
22) Counting Numbers are Whole Numbers.
_______________
23) Whole Numbers are Integers.
_______________
24) Rational Numbers are Whole Numbers.
_______________
25) Integers are Whole Numbers.
_______________
State ALL the sets of numbers each belongs in: COUNTING, WHOLE, INTEGER or RATIONAL
26) 0
27)
-90
_________________________________________________________
_________________________________________________________
32
Evaluate.
28)
-9 + 40
29)
-18 - 18
32)
 70
10
33)
-12 - - 18
34) 4 - - 9
37)
-7 - 3 + 12 + 14
38)
36) (-9)(-10)(-1)
30) (-12)(
-5)
-9 - - 8 + 1
31)
-45 ÷ -9
35)
-10 + -19
39) (-5)(3)(-4)
Write a number sentence and evaluate. Your final answer MUST be a complete sentence.
40) The temperature in the morning was 3 degrees below zero. The temperature dropped 11
degrees by night. What was the temperature at night?
41) Mary has $10 in savings. She owes her parents $50. She does some chores and her
parents pay her $12. She also gets $25 for her birthday from her grandparents. Does
Mary have enough money to pay her parents what she owes them? IF not, how much more
money does she need?
42) Adam is scuba diving. He descends 5 feet below sea level. He then descends the same
distance 4 more times. What is Adam’s final elevation?
43) Clark made four of his truck payments late and was fined four late fees. The total change
to his savings from late fees was
-$40.
How much was one late fee?
33
30
7A Review for Integer Test
DO NOW:
Fill in the rules.
Adding/Subtracting Integers:
Multiplying/Dividing two Integers:
1) Same Signs: __________________
2) Same Signs: __________________
3) Different Signs: __________________
4) Different Signs: __________________
DO NOT COMPUTE!
State whether the answer will be POSITIVE, NEGATIVE or ZERO.
5)
-27 + 19 _______________
7)
100
_______________
20
9)
-17 + 17
_______________
6) (-14)(-23) _______________
8)
10)
 121
11
-58 - -64 ______________
_______________
- -12
_______________
12)
-125 ÷ -5 _______________
13)
-8 - -8
_______________
14)
-28 - 45
15)
-27  19
_______________
16)
-10  0 _______________
11) 19
17) 19 - 32 _______________
19)
27
9
_______________
_______________
18) 12 + 7 _______________
20) (-3)(-2)(-12) _______________
CLASSWORK:
1) When adding/subtracting integers with the same signs you _________________________.
2) When adding/subtracting integers with different signs you ________________________.
3) Beware of double negatives! Always change them to a ______________________.
4) When multiplying/dividing 2 integers with the same sign the answer will be ____________.
5) When multiplying/dividing 2 integers with different signs the answer will be ___________.
6) When multiplying an ODD number of negatives your answer will be _______________.
7) When multiplying an EVEN number of negatives your answer will be _______________.
8) What is the only integer that is neither positive nor negative? ______
9) Is 0 an integer? __________ Why or why not? _________________________________
34
Evaluate each of the following.
10)
-8 + 8 = ____
11)
14) -26 ÷ 2 = ____ 15)
18)
12)
-8  9 = _______
16) -4 
-18 - 10 + 16 + 5 = ______________
20) 12 -
- 7 - 8 + -9 = ______________
22) (-9)(7) = ____
26)
-78 -15 = _____
- 18 - 8 = ____
23)
-9 + 26 = ____
-20 + 25 = ____
13)
-2 = ______
17)
-26 - -14 = ____
 18
= ______
6
19)
-25 - 5 + 10 - 2 = ______________
21)
-6 + 16 - 10 + 4 = ______________
24)
27) (-3)(-17) = ____ 28)
-16 - -18 = ____
-9 + 25 = ____
25)
29)
 100
= ____
 25
-56 ÷ 8 = ____
Read each problem carefully, write a number sentence and evaluate.
30) A submarine 140 feet below sea level dives another 300 feet. What is the submarine’s
position after the dive?
31) The temperature outside was 22˚F. The wind chill made it feel like -8˚F. Find the
difference between the real temperature and the apparent temperature.
32) The enrollment at Davis Middle School dropped by 60 students over a 5-year period. What
is the average yearly drop in enrollment?
33) A hiker starts hiking at the beginning of a trail at a point which is 200 feet below sea level.
He hikes to a location on the trail that is 580 feet above sea level and stops for lunch.
What is the difference between the two elevations?
35
34) The top of an iceberg is 15 feet above sea level. The bottom of the iceberg is 105 feet
below sea level. What is the distance from the top to the bottom of the iceberg?
35) An elevator takes passengers from the ground floor down to an underground parking
garage. Where will the elevator be in relation to the ground floor after 5 seconds if it
travels at a rate of 3 feet per second?
36) State the first 5 Counting Numbers. ________________
37) Whole Numbers are all the _____________________ and _____________________.
38) Integers are all the _____________________ and their _____________________.
39) Rational Numbers are any numbers that can be written in ________________ form. This
includes ________________ decimals and ________________ decimals.
40) Are all Counting Numbers Whole Numbers? Answer yes or no. ___________
41) Are all Whole Numbers Integers? Answer yes or no. ___________
42) Are all Whole Numbers Counting Numbers? Answer yes or no. ___________
43) Are all Counting Numbers Rational Numbers? Answer yes or no. ___________
State ALL the sets each of the following belongs to.
COUNTING, WHOLE, INTEGER, RATIONAL
44)
-9
45) 0.25
46)
-
5
8
47) 0
48)
15
3
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
______________________________________________________________
State the additive inverse for each of the following.
49) 24 _________
50)
-44 _________
51) 4 _________
Compare using  or .
36
52)
-5 ______ 0
53)
-15 ______ 15
54) -9 ______
-11
Put the integers in order from least to greatest.
55) -20, 8, -31, -5, 11 ______________
56) -56, -102, 98, -58, 114 _____________
Evaluate.
57) |22| _____
60)
| 5 + -8| = _______
58) |-13| _____
61)
59) |-512| + |102|
| -8 + 10 | = _______ 62) | 7 | + | -7 | =
= _____
______
63) How many numbers have an absolute value of 17? ______ State which number(s). ______
State the property shown below.
64)
-8 + 8 = 0
_______________________
65) (9 + 7) + 3 = 3 + (9 + 7) _______________________
66) 9 + (8 + 4) = (9 + 8) + 4 _______________________
67)
-9 + 0 = -9
_______________________
68)
-19  0 = 0
_______________________
69) 18  1 = 18
70) 9 
1
=1
9
_______________________
_______________________
37