Unit 4 Continued: OOO, Properties, and Sequences 1
Unit 4 Continued
Order of Operations,
Properties, and
Sequences
Notes
No Calculator
SOL 7.2
The student will describe and represent arithmetic and geometric sequences, using variable
expressions.
SOL 7.3 b
Students will simplify numerical expressions involving integers, using order of operations.
SOL 7.16abcde
Students will apply the following properties of operations with real numbers,
o the commutative and associate properties for addition and multiplication,
o the distributive property,
o the additive and multiplicative identity properties,
o the additive and multiplicative inverse properties, and
o the multiplicative property of zero.
Unit 4 Continued: OOO, Properties, and Sequences 2
Properties Notes
Directions: Apply the property to complete the number sentence.
PROPERTY
KEY IDEA
Number Sentence
Changing the
Commutative Property
ORDER of
1.
3 + 2 + 1 = _______________
numbers when
of Addition or
2.
5 ● 8 ●7 = _______________
adding or
Multiplication
multiplying
Associative Property
of Addition or
Multiplication
Identity Property
of Addition or
Multiplication
Inverse Property
of Addition or
Multiplication
Distributive Property
Multiplicative Property of
Zero
Reflexive Property
ReGROUPING the
3.
numbers without
changing the
4.
order
Keeping the value 5.
of the number the
6.
SAME
UNDOING or
CANCELING the
number
BY ZERO
3 + 0 = _______________
5 ● 1 = ______________
3 + -3 = ___________
8.
5●
1
5
= ___________
3(2 + 1) = _______________
5(x + 2) = _______________
11.
3●0=
12.
x ● 8 ●0 = __________
13.
21 = _______
__________
Number = Number
14.
Symmetric Property
5 ● (8 ●7) = _______________
7.
REWRITING the
expression using 9.
multiplication (BE
10.
FAIR, ALWAYS
SHARE)
MULTIPLYING
(3 + 2) + 1 = _______________
x= _______
15.
if x = 2 then _______________
16.
if a=b then _______________
If, then
Unit 4 Continued: OOO, Properties, and Sequences 3
Properties Practice
Name the property for each.
1) 2  2  0
2) if y  4 then 4  y
3)
2 0  0
4)
(a  b)  c  a  (b  c)
5)
2 3
 1
3 2
6)
066
7)
2
9)
18 1  18
11)
1
1
2
x 4 4 x
13) if x  8 , then 8  x
8) (2  1)  0  2  (1  0)
10)
c c
12)
5(x  2)  5x  10
14) 3  xy  3  yx
Answer the following questions true or false. Then justify your answer .
1. 2 + 0 = 2 is an example of multiplicative property of zero
T
F
2. (3x + 4)= 6x + 8 has parenthesis so it is an example of
associative property.
3. 2 =-2 is an example of the reflexive property
T
T
F
F
4. If 2 + 3 = 6 then 3 + 2 = 6 is an example of commutative property
T
F
5. If x = 3 then 3 = x is an example of symmetric property
T
F
6. -3 • 0 = 0 illustrates multiplicative identity
T
F
7. 5(-2 + 6) = -10 + 30 is an example of distributive property
T
F
8. -8 • 1 = -8 is illustrates multiplicative identity
T
F
Unit 4 Continued: OOO, Properties, and Sequences 4
Properties Practice
1)
2)
3)
4)
Unit 4 Continued: OOO, Properties, and Sequences 5
Properties Practice
1) Which property is shown in the following number
sentence?
2 x  (5  0)  2 x  5
2) What is the additive inverse of 
4
?
5
3) Identify each number sentence that illustrates the 4) Identify each equation that represents the
commutative property of multiplication.
multiplicative property of zero.
2  (3  7)  10  2  3  (7 10)
70  0
7 1  7
3 46  3 64
3  3  0
7  7  0
5 80  5  0
05  5
03  0
2  (8  7)  1  2  (7  8)  1
5) Which student correctly applied the additive identity property?
6) Name the property
8  3  2 1  8  3  2
7) Which property is used in the following number
sentence?
4(3  n)  4(3)  4(n)
8) Which student correctly applied the multiplicative identity property?
Unit 4 Continued: OOO, Properties, and Sequences 6
Sequence: a list of numbers in a
.
Arithmetic Sequence: a sequence created by
same number to or from each term in the list.
Geometric Sequence: a sequence created by
in the list by the same number.
the
each term
Common difference: the constant number that is _________________to each term to find the
next term in a _____________________ sequence.
Common ratio: the constant number that is _________________ to each term to find the
next term in a _________________ sequence.
How do I find the pattern?
Step 1: Study the numbers that are shown
Step 2: Ask, “How do you get to the next number in the pattern?” (common difference or ratio)
Step 3: Find the next number(s)
Step 4: Write the sequence as an expression
Let’s try some together!
1)
1
2
14, 16 , 19, _______, ________
2) -4 , 8 , -16 , 32, __________, __________
Common Ratio/Difference:_______________
Type of Sequence:______________________ Common Ratio/Difference:______________
Expression:___________________________ Type of Sequence:_____________________
Expression:___________________________
Unit 4 Continued: OOO, Properties, and Sequences 7
3) 10 , 5 , 0 , -5, __________ , __________
4) 100 , 10 , 1 , 0.1 , ________ , _________
Common Ratio/Difference:_______________ Common Ratio/Difference:______________
Type of Sequence:______________________ Type of Sequence:_____________________
Expression:____________________________ Expression:___________________________
Now you try!
1)
1
, 2 , 8 , 32 , ________, ________
2
2) 1 , 2, 4 , 7 , __________, __________
Common Ratio/Difference:_______________ Common Ratio/Difference:______________
Type of Sequence:______________________ Type of Sequence:_____________________
Expression:___________________________
Expression:___________________________
3) 18 , 15 , 12 , 9 , _________, __________
4) 0.4 , 0.8 , 1.2 , 1.6 , _________, ________
Common Ratio/Difference:_______________ Common Ratio/Difference:______________
Type of Sequence:______________________ Type of Sequence:_____________________
Expression:____________________________ Expression:___________________________
Find the missing number:
1) 4 , _________, 36 , 108
3) _________, 32 , 20 , 8
2) 5 , 20 , _________, 50
4) 1 , 3.3 , 5.6 , _________
Find the seventh term in the sequence:
1) 3 , 6 , 12 , 24…
_________________
Unit 4 Continued: OOO, Properties, and Sequences 8
Patterns and Sequences Practice
1) Which sequence follows the variable expression n 13
x
2) Which sequence follows the variable expression
3
3) Which sequence follows the variable expression x - 99
4) Robin is going to create a sequence with a common difference of -17. She wants to
ensure that the number 22 is included in her sequence. Which number could she
choose to begin her sequence with?
104
105
106
5) What is the 6th term of the geometric sequence?
107
80, 40, 20, …
6) Ginni wrote the following geometric sequence in her notebook: 3, 9, 27, 81, 243, 729…
What would be the 8th term in Ginni’s sequence?
7) Which sequence used the variable expression y + 1.4 to define the next term?
Sequence 1
-3.17, -4.57, -5.97, -7.37, -8.77, …
Sequence 2
5.73, 7.13, 8.53, 9.93, 11.33, …
Sequence 3
2, 2.8, 3.92, 5.488, 7.6832, …
Sequence 4
-1, -1.4, -1.96, -2.744, -3.8416, …
Unit 4 Continued: OOO, Properties, and Sequences 9
Unit 4 RU Ready
SOL 7.3b Order of Operations
1)
 4  2 18  3  10
3) 10  2(5)  9
42  8(3)
5)
8  10
42  32
2) 4  3  10   5(7) 
4)
1
(10  20) 2
2
3 2  32
6)
3
Unit 4 Continued: OOO, Properties, and Sequences 10
7) Circle which expression is equivalent to 0?
4  2  (1  3) 2  3  6
4  (2  42 )  10  3
23  3  (10  5)
2  3  22  3  8
8) Jim simplified an expression incorrectly. Jim’s work is show below.
Original expression:
( 36  5  13)  10  3
Step 1:
(6  5  13)  10  3
Step 2:
(6  18)  10  3
Step 3:
12 10  3
Step 4:
12  30
Step 5:
-0.4
Which statement best represents Jim’s error?
a) Jim’s answer is incorrect because he should have taken the sum of 5 and 13 between step 1
and step 2.
b) Jim’s answer is incorrect because he evaluated 36 incorrectly in step 1.
c) Jim’s answer is incorrect because he should have found the quotient of -12 and 10 between
step 3 and step 4.
d) Jim’s answer is incorrect because he should have found the difference of 6 and 5 in step 2.
Unit 4 Continued: OOO, Properties, and Sequences 11
Unit 4 RU Ready
Part 1: SOL 7.3b Order of Operations with Substitution
2
1) Solve 4x  x , when x = 2
2) If x = 5 and y = -3, what is the value of
( xy )0
3) What is the value of 2a  3b  6 ,
when a = 6 and b = -5
4) What is the value of a  4(b  3) ,
when a = 10 and b = -5
Unit 4 Continued: OOO, Properties, and Sequences 12
5) If n = 4 what is the value of
3n
 5n  2
4
6) If y = 4 what is the value of
y2
 y 2  144
2
7) Circle which expression has a value of 9, when x = 2 and y = 10
y4
x
3x  5
y
x
5
y
4
2
Unit 4 Continued: OOO, Properties, and Sequences 13
Part 2 Sequences
1. Which of the following is/are geometric
sequence(s)?
a. 2, 4, 8, 16….
b. 1, -10, 100, -1000….
c. 19, 17, 16, 10…..
d. -10, -8, -6, -4…..
2. Which of the following is an arithmetic
sequence?
a. 3, 9, 27, 81…
b. -2, -1, -1/2, -1/4….
c. 110, 210, 310, 410…
d. 1, 4, 40, 100…..
3. A(n) ___________ sequence has a
common difference.
4. A(n) __________ sequence has a common
ratio.
5. What variable expression describes the
relationship between consecutive terms
in this sequence? 3, 9, 27, 81…
6. Emily made an arithmetic sequence with
consecutive terms related by this rule: n +
(-5) Which of these is Emily’s sequence?
a. 15, 20, 25, 30…
b. 60, 55, 50, 45…
c. 5, 55, 555, 5555….
7. Michelle wrote this variable expression
to describe the relationship between
consecutive terms in her sequence: 0.5n
Which is Michelle’s sequence?
a. 10, 50, 250, 1250….
b. 0.5, 1.5, 2.5, 3.5…
c. 36, 18, 9, 4.5….
d. 3, 6, 12, 24…..
8. Fill in the missing term in this geometric
sequence:
32, 8, ____, 0.5
9. What will be the next term in this
arithmetic sequence:
-5.5, -4, -2.5, -1, _______
10.Fill in the terms in this sequence that starts
with 2 and has a common ratio of -5:
2, ______, ______, _______, _____, ______
Unit 4 Continued: OOO, Properties, and Sequences 14
11) Complete the table with an expression that describes each sequence.
SEQUENCE
EXPRESSION
p+5
-1, -3, -9, -27 …
15, 20, 25, 30…
4r
3t
b+2
-6, -4, -2, 0 …
¼ , 1, 4, 16 …
12) Which sequence has a common ratio of 3?
13) Which sequence has a common difference of -13?
Unit 4 Continued: OOO, Properties, and Sequences 15
14)
Look at the following sequence.
50, 10, 2,
2
5
…
What variable expression describes the relationship between the terms in the sequence?
15) Ginni wrote the following geometric sequence in her notebook
3, 9, 27, 81, 243, 729, …
What would be the 8th term in Ginni’s sequence?
16)
What is the common difference of the arithmetic sequence shown below?
-4, -8, -12, -16 …
a) 4
b) 2
c) -2
d) -4
Unit 4 Continued: OOO, Properties, and Sequences 16
Part 3 Properties
1) Which of the following equations is an example of the
associative property of addition?
2) Identify each number sentence that
illustrates the additive identity property.
2 x2  4 x  4 x  2 x2
(2 x 2  4 x)  6  2 x 2  (4 x  6)
0 3 6  3 6
8  (8)  4  0  4
3  7  (7)  3  0
3 0  7  3 7
5  (4  0)  5  4
4  60  4  0
3(2 x  4 x)  6 x  12 x
2
2
3) Identify each number sentence that illustrates the distributive property.
4(3  1)  4(3)  4(1)
7  5(4  2)  7  5(4)  5(2)
(4  7)  1  6  4  (7 1)  6
7(9  1)  2  7(9)  7(1)  2
5(3  7  0)  5(3  0)
1(5  2)  1(2  5)
4) Identify each number sentence that illustrates the associative property of addition.
(4  7)  1  6  4  (7 1)  6
267  276
7  (3  8)  (7  3)  8
1  (5  2)  1  (2  5)
(5  (1))  6  5  (( 1)  6)
(4  1)  3  7  4  (1  3)  7
5) Which property justifies this step?
(4 x  2)  x  27
(4 x  x)  2  27
6) What property justifies the work shown?
12 x  5  (14 x  3)  8 x
12 x  5  8 x  (14 x  3)
a)
b)
c)
d)
Commutative property of addition
Associative property of addition
Identity property of addition
Distributive property
Unit 4 Continued: OOO, Properties, and Sequences 17
7) Which property is being illustrated by this number sentence?
a)
b)
c)
d)
(2  8)4  4(2  8)
Associative Property of addition
Commutative Property of addition
Associative property of multiplication
Commutative property of multiplication
8) Paul solved an equation as shown.
What property justifies the work between Step 1 and Step 2?
a)
b)
c)
d)
Inverse property of addition
Distributive property
Addition property of inequality
Commutative property of addition
9) Which property is used in the following number sentence?
a)
b)
c)
d)
Additive Inverse Property
Associative property of addition
Distributive property
Commutative property of addition
10) Which property is illustrated by this number sentence?
a)
b)
c)
d)
7(3  n)  7( n  3)
12 
1
1
12
Associative property of multiplication
Multiplicative Identity Property
Distributive Property
Multiplicative Inverse Property
11) Match the property with their corresponding expression.
A. Additive Identity
_____ (2 ∙ 4) ∙ 1 = 2 ∙ (4 ∙ 1)
B. Distributive Property
_____ 12 + 0 = 12
C. Associative Property of Multiplication
_____ 7 ∙
D. Additive Inverse Property
_____ 4(2 + 5) = 8 + 20
E. Multiplicative Inverse Property
_____ -6 + 6 = 0
1
7
=1
Unit 4 Continued: OOO, Properties, and Sequences 18
Unit 4 Continued: OOO, Properties, and Sequences 19
Unit 4 Continued
Order of Operations,
Properties, and
Sequences
Homework
No Calculator
SOL 7.2
The student will describe and represent arithmetic and geometric sequences, using variable
expressions.
SOL 7.3 b
Students will simplify numerical expressions involving integers, using order of operations.
SOL 7.16abcde
Students will apply the following properties of operations with real numbers,
o the commutative and associate properties for addition and multiplication,
o the distributive property,
o the additive and multiplicative identity properties,
o the additive and multiplicative inverse properties, and
o the multiplicative property of zero.
Unit 4 Continued: OOO, Properties, and Sequences 20
Properties Homework
Identify the property shown.
1) 7a  0 = 0
2) 67 + 0 = 67
3) 126 + -126 = 0
4) 6 • 7 = 7 • 6
5) 4(3a) = (4 • 3)a
6) 1 • mp = mp
7)
5
6
×
6
5
8) 9 + (5 + 35) = ( 9 + 5) + 35
=1
9) 6(3 + 2) = 6(3) + 6(2)
10)
2 + (3 + 5) = (3 + 5) + 2
11)
12)
Circle each property that is NOT illustrated by the equations show below
Commutative Property of Addition
Additive Inverse Property
Additive Identity Property
Multiplicative Inverse Property
Multiplicative Property of Zero
Distributive Property
My score is _______/12
Unit 4 Continued: OOO, Properties, and Sequences 21
Patterns and Sequences Homework
1) Which sequence follows the variable expression
2) Which sequence follows the variable expression
x + 13
2x
3) Adam is going to create a sequence with a
1
common ratio of 4. He wants to ensure that the
number 6 is included in his sequence. Which
number could he choose to begin his sequence?
4) Which variable expression describes the
relationship between the terms in the sequence?
80, 40, 20, 10, 5, …
𝒏
1536
1792
448
320
𝟐
𝒏 + 𝟏𝟎
𝟐𝒏
𝒏−𝟓
5) What is the common ratio for the following
sequence?
81, 27, 9, 3, 1, …
6) Billy is going to create a sequence with a
common ratio of 2. He wants to ensure that the
number 576 is included in his sequence. Which
number could he choose to begin his sequence
with?
8
9
10
11
7) Which sequence has a common difference of 3?
8) Which sequence has a common difference of -9?
9) What is the common ratio for the following
sequence?
648, 108, 18, 3, …
10) What is the 6th term of the geometric sequence
below?
-128, 64, -32, 16, …
11) What is the variable expression that can
determine the next term in the sequence?
12) What is the variable expression that can
determine the next term in the sequence?
729, 243, 81, 27, …
𝟏
𝟑
𝒌
𝟏
−𝟑𝒌
𝟑𝒌
4, -16, 64, -256, …
−𝟑𝒌
−𝟒𝒃
−𝒃 + 𝟏𝟐
My score is _______/12
𝟏
𝟒
𝒃
𝒃 − 𝟏𝟐
Unit 4 Continued: OOO, Properties, and Sequences 22
Properties Study Guide
Commutative Property of Addition - Commutative Property of Multiplicationorder changes
order changes
Associative Property of Addition –
grouping changes
Associative Property of Multiplication –
grouping changes
Additive Identity –
stays the same
Multiplicative Identity –
stays the same
Additive Inverse –
flip/opposite
Multiplicative Inverse –
flip/opposite
Multiplicative Property of Zero –
always zero
Distributive Property – (two operations)
“pass out”…be fairs, always share
*MAKE SURE YOU MEMORIZE THESE!!**
Name the property shown:
1.)
7a  0 = 0 ___________________________________________
2)
8 • 7 = 7 • 8 __________________________________________
3)
-37 + 37 = 0___________________________________________
4)
-12 + 0 = -12 __________________________________________
5)
4(3a) = (4 • 3)a ________________________________________
6)
1 • mp = mp __________________________________________
7)
8)
9)
10)
8
9
×
9
8
= 1 ___________________________________________
9 +(5 + 35) = (9 + 5)+ 35 __________________________________
4(3 + 2) = 4(3) + 4(2) _____________________________________
1 + (9 + 5) = (9 + 5) + 1 ____________________________________
Directions: Fill in the equation to make it true.
11) Commutative Property 15 + 7.15 = __________________________
12) Distributive Property ____________________ = (18 • 4) – (18 • y)
13) Associative Property (3 • -2) • 5 = ___________________________
Unit 4 Continued: OOO, Properties, and Sequences 23
Match the following property to the correct expression:
14) Additive Identity
_____2 + 3 + 4 = 4 + 3 + 2
15) Multiplicative Identity
_____ -2 + 2 = 0
16) Commutative Property of Addition
_____ 1∙ 3 = 3∙ 1
17) Distributive Property
_____ 4 ∙
18) Associative Property of Multiplication
_____ 5 ∙ 1 = 5
19) Additive Inverse Property
_____ (1 ∙ 3) ∙ 5 = 1 ∙ (3∙ 5)
20) Commutative Property of Multiplication
_____4 + 0 = 4
21) Multiplicative Inverse Property
_____ 2(3 + 6) = 6 + 12
1
4
=1
22) Identify each number sentence that illustrates the commutative property of addition.
23)
(4  7)  1  6  4  (7 1)  6
267  276
7  (3  8)  (7  3)  8
1  (5  2)  1  (2  5)
(5  (1))  6  5  (( 1)  6)
(4  1)  3  7  4  (1  3)  7
Unit 4 Continued: OOO, Properties, and Sequences 24
Sequences Study Guide
PATTERNS AND SEQUENCES
+
×
÷
Arithmetic Sequence: A series of numbers formed by adding the same number to the previous
number.
Common Difference: The number that you are adding to each term in an arithmetic sequence.
Geometric Sequence: A series of number formed by multiplying the same number to the
previous number.
Common Ratio: The number that you are multiplying to each term in a geometric sequence.
**REMEMBER: You can think of sequences like functions with an input and output value
24) Which sequence follows the variable expression 𝟑𝒙
25) Which sequence has a common ratio of 2?
26) Billy is going to create a sequence with a
common ration of 2. He wants to ensure that
the number 176 is included in his sequence.
Which number could he choose to begin his
sequence with?
10,
7,
11,
9
Unit 4 Continued: OOO, Properties, and Sequences 25
27.) 1 goes in, 9 comes out
29.) 6 , 24, 96, _____ , _____
a.) Find the sixth term in the
15 goes in, 23 comes out Common Ratio/Difference:
Rule:_____________
28.) 3 goes in, 15 comes out
8 goes in, 40 comes out
30.) Find the term in the sequence:
__________________
Type of Sequence:
sequence:
54, 50, 46…
b.) Find the eighth term in the
___________________
sequence:
3, 9, 27…
Expression:______________
Rule:_____________
ORDER OF OPERATIONS STUDY GUIDE (GEMDAS)
(G)
**REMEMBER TO SOLVE ONLY ONE
THING PER LINE. REWRITE
WHAT WAS NOT SOLVED
AND CONTINUE UNTIL
YOU HAVE AN
ANSWER**
Grouping
Ex
Exponents
[10 + (3 2 + 1)] ● 5 + 2 3
MD
[10 + (9 + 1)] ● 5 + 2 3
Multiplication
[10 + 10] ● 5 + 2 3
or Division
20 ● 5 + 2 3
AS
20 ● 5 + 8
Addition or
Subtraction
100 + 8
31.) 4 – (8 + 2)2 ÷ 4
108
32.)
15 ÷ 3 + 23
10 – 5 + 1
33.) 12 + [ 14 ÷ (15 – 8)]
Unit 4 Continued: OOO, Properties, and Sequences 26
For #34-36, evaluate each expression if x = -1, y = -2 and z = 4
34)
37)
2z – (x + 3y)
6  12(16 10)  (4)
39)
(2)(7)  (1)(5)
3
35)
3(z – 5) ÷ x
36)
2(y 5)  2 y 2
38) 1.5(12  6.5)  3(3.25  2)
40)
8625
(4  2)3
My score is _______/40