Name
Class
Date
Review 25
_
Using Variables
•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••
MATERIALS: None
OBJECTIVE: Using variables as a shorthand way
of expressing relationships
You often hear word phrases such as half as milch or three times as deep.
These phrases descrihe mathematical relationships. You can translate word
phrases like these into mathematical relationships called expressions.
Translate the following word expressions into algebraic expressions.
the sum of x and 15
x
seven times x
7x
Remember that "times" mean to multiply.
+ 15
Remember that "sum" means to add.
Translale the following word sentence into an algebraic equation.
The weight of the truck is two times the weight of the car.
--
=
The weight of the truck is two times the weight of the car.
~
v
,
"
I
II
j
I
2
•
c
=
t
v
=
=.
.#
=.
--
2c
"g
~
~
w
j
g
Represenl the unknown amounts with
,,'ariables.
The translation is complete. Check
to make sure )'OU h3l'e Iranslated all
parts of the equation.
~
£
Write an equal sign under the word ;!J.
Whatever is written to the left of ;!)'
belongs on the left side of the
Whatever is written to the right of;s
belongs on the right side of the
Exercises
Translate the following word expressions and sentences into algebraic
expressions or equations.
1. a number increased by 5
2. 8 subtracted from a number
3. a number divided by ()
4. 3 less than five times a number
5. A number multiplied by 12 is M.
6. 7 less than
7. 8 times a number x is 72.
8. A number divided by 3 is 18.
II
is 22.
.............................................................................
1EIaI
Name
Class
Date
Review 26
Exponents
_
and Order of Operations
•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••
OBJECTIVE: Using the order of operations
MATERIALS:'Ibree index cards or small pieces
of paper
Review the order of operations to help you with this activity.
I.
2.
3.
4.
Order of Operations
Perform any operations inside grouping symbols.
Simplify any term with exponents.
Multiply and divide in order from left to right.
Add and subtract in order from left to right.
Write + on the first index card, - on the second card, and X on the third
card. Shuffle the cards and place them face down on your desk. Randomly
pick cards to fill in the blanks with operation signs. Once you have filled in
the operation signs. simplify the expression.
0_(9_7)_8
oX (9- 7) + 8
oX
2
12
+ 8
-
+ 8
20
Pick cards to fill in the blanks ,,"'ithoperation signs.
Suhtract 7 from 9 inside the grouping s,)'mbols.
Do multiplication and division first. Multipl,)' 6 b}'2,
Do addition and subtraction last. Add 12 and 8 to get
the answer,
-
The answer is 20,
Exercises
Randoml,)' pick cards to fill in the operation s,)'mhols of the follo,,"ing
expressions. Simplif,)' the expressions.
1.7_5_1
2.(3_9)_4
3.8_2_(5_10)
4.(3_7_0)_1
Simplif}' each expression b,)'following the order of operations.
5. (5 . 3) - 18
6. 5 . (3 - 18)
7. 2 . (27 - 13 . 2)
8. 2.27-13'2
9. 18 7 (9 - 15 7 5)
11. 2 . 8 - 62
m
10. tH + 9 - 15 + 5
12. 2 . (8 - 62)
.
Name
~
Class
Date
Review 27
_
Exploring
_
Real Numbers
•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••
OBJECTIVE: Classifying numbers
MATERIALS: None
Review the following chart which shows the different
classifications
of real numbers.
Rational
terminating or
repeating decimals
I Rea'i
(r~ __
'-_~_,_8_,4_2
Integers
Whole
... -3, -2, -1,0,1,2,3, ...
0,1,2,3, ...
~
"... Irrational
infinite, nonrepeating
decimals
!!,
Ii 3.767667666
Given the numbers
~
~
i
-4.4.
j
-If
and 32, tcll which numbers
32
numbers
Whole:
0,32
natural numbers
Integers:
0, -9,32
whole numbers
Rational:
-4.4,
Irrational:
-~
Real:
-4.4,
14
5'
O. -9.
1
14,32
used to count
and zero
and their opposites
integers and tenninating
infinite. nonrepeating
14
5'
0, -9,
1
14, -N, 32
belong to each set.
rational
and irrational
and nonrepeating
decimals
decimals
numbers
Exercises
Name the -et(s) of numhe"
,,-
5,
<>
O. -9, It.
Natural:
1, -5
<
~4.
...
9,
Vs
ViOO
tu whkh each uumher halongs,
2, 35.99
3, 0
4, 41
8
6, -80
7 , 11
5
12
8, "3
11, 3.24
12, 3.
10,
-v4
Give an example of each kind of numher.
13. irrational
~
number
14. whole number
15. negative integer
16. fractional rational number
17. rational decimal
18, natural number
..............................................
iii
Name
Class
Date
_
Review 28
Adding Real Numbers
•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••
OBJECTIVE: Adding integers and decimals
MATERIALS: None
Review the following addition rules.
•
To add two numbers with thc same sign, add their absolute values.
The sum has the same sign as the numbers.
•
To add two numbers with different signs, find the difference of their
absolute values. The sum has the same sign as the numbcr with the
greater absolute value.
The following example shows you step by step how to add two numbers with
different signs.
-6 + 2
6- 2
-
4
-4
Find the difference of their absolute values.
Suhtract.
-
Since -6 has the greater absolute ".aloe, the ans".'er takes
the negative sign.
Exercises
Simplify. Be sure to check the sign of )'our answer.
1. -3 + (~4)
2. 12 + 5
3. ~5 + 8
4. ~8 + (~2)
5. ~2 + (~3)
6. 9 + (~12)
7. ~3 + 5
8. ~4 + 3
9. ~2.3 + (~1.5)
13. 1.3 + (~1.1)
10. 4.5 + 3.1
11. -5.1 + 2.8
12. 13.9 + 7.3
14. ~3.6 + (~6.7)
15. 1.4 + (~21.4)
16. -9.8 + 3.5
Evaluate each expression for a = 5 and b = ~4.
17. ~a + (~b)
18. -a
+
h
19. a
+ b
20. a
+
(~b)
Evaluate each expression for h = 3.4.
21. 2.5 + h
22. ~2.5 + h
23. 2.5 + (~h)
24. ~2.5 + (~h)
25. h + 7.1
26. ~h + 7.1
27. h + (~7.1)
28. ~h + (~7.1)
IEEJI
.
Name
Class
Date
Review 29
_
Subtracting Real Numbers
•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••
OBJECTIVE: Subtracting integers and decimals
MATERIALS: None
Review the following subtraction rules.
•
To subtract a number, rewrite the problem to add the opposite of
the number.
•
Follow the rules for addition of numbers.
The following example shows you step by step how to subtract
two numbers.
5 - 11
5+(-11)
11 - 5
-
Rewrite the problem to add tbe opposite of the number.
-
Find the difference of their absolute values.
6
-6
Subtnu.1.
-
Since -11 has the greater absolute value, the answer takes the
negath'e sign.
Simplify. Be sure 10 check the sign of Jour answer.
1. 7 - 12
2. 6 - 9
3.4-(-5)
4.7-(-3)
5. -6 - 4
6. -7 - 2
7. -5 - (-4)
8. -3 - (-10)
10. 8.3 - 5.1
11. -7.8 - 6.6
12. -4.8 - 2.5
14. -4.6 - (-3)
15. -9.3 - (-8.1)
16. -9.9 - 3.8
19. a - (-b)
20. -a - (-b)
9. -3.1 - (-5.4)
13. 8.7 - 2.5
E".aluale each expression for a = -4 and b = 3.
17. a - b
18. -a - b
21. 3b - a
22.
-Ibl
23.
la - bl
26.
[-~]- U]
Sublracl. (Hint: Sublracl corresponding clemenls.)
25. [-30
-~]- [~~]
24.
lal - 31bl
..................................................................
m
Name
Class
Date
Review 30
_
Multiplying and Dividing RealNumbers
•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••
OBJECTIVE: Multiplying and dividing integers
and decimals
MATERIALS:A number cube
Review the following multiplication and division rules.
•
The product or quotient of two positive numbers is always positive.
•
The product or quotient of two negative numbers is always positive.
•
The product or quolient of a positive and a negative number
is always negative .
..ID!llI
Roll the number cube to determine the signs of the numbers in the following
example. If you roll an even number (2,4, or 6), write + in the blank to
make the number positive. If you roll an odd number (1, 3, or 5), write a - in
the blank to make the number negative. Decide what sign the answer will
have before you calculate the answer.
56 +
Roll the number cube to fill in the blanks.
7
-56 -:- (+7)
-
Suppose Jour first roll was a 3, so 56 is negath'e. Suppose )'our
second roll was 6, so 7 is positive. Now that )'OU have the signs of
the numhers., decide what the sign of the answer will be. Dividing
a negatil'e number by a positive number results in a negative
numher.
•
- 8
-
The answer is -8.
Roll the numher cuhe to determine the signs of the numbeN in the follow'iug
exercises. Rememher to decide what sign the ans,",'er will have hefore )'OU
calculate the answer.
1.
20.
3.
27
5.
120 -:-
7.
1.4 .
8
+
3
12
3
2.
3.2 .
4.
14 .
6.
45 +
9
8.
96 +
8
10
4
Simplif)' each expression.
9. 4(-2)
10. -6(12)
11. -2(-5)
12. -8(11)
13. (-7)'
14. -10(-5)
IEIDI
.
Name
~~.
__ ~.~
..
Class
Date
Review 31
The Distributive
_
Property
•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••
OBJECTIVE: Using the Distributive Property
MATERIALS: None
You can compare the Distributive Property to distributing paper to the class.
Just as you distribute a piece of paper to each person in the class, you
distribute the number immediately outside the parentheses to each term
inside the parentheses by multiplying.
Simplify 3(2x
~
3(2x
3(2x)
6x
+ 3) by using the Distributive Property.
+ 3)
+ 3(3)
+ 9
-
Draw arrows to show that 3 is distributed to the 2x and to the 3.
-
Use the Distributive Propert)'.
-
Simplif:y.
Simplify -(4x + 7) by using the Distributive Property.
~
~
c
~
~
r=
o
-1(4x + 7)
-
Rewrite usin~ the Multiplication Property of -I.
~
-1(4x + 7)
-
Draw arrows to show that -I is distributed to the 4x and to the 7.
-1(4x) + (-1)(7)
-
Use the Distributiw Propert)'.
-4x - 7
-
Simplif):.
Exercises
Draw arrm\'"sto show the Distributh'c Proper1)'. Then simplify
each expression.
~
~
1. 2(5x + 4)
2. %(I2x - X)
3. 4(7x - 3)
I
4. 5(4 + 2x)
5. 6(5 - 3x)
6. 0.1(30x - 50)
~
7. (2x - 4)3
8. (3x + 4)7
9. 8(x + y)
"
10. -(4x + 3)
11. -(-2x
+ 1)
12. -(-6x
13. -(14x - 3)
14. -(-7x
- 1)
15. -(3x + 4)
- 3)
....................................................................
m
Name
Class
Date
Review 32
_
Properties
of Real Numbers
•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••
OBJEaIVE: Recognizing properties
MATERIALS: None
The properties of real numbers allow you to write equivalent expressions.
The Commutative Properties of Addition and Multiplication allow you to
add or to multiply two numbers in any order.
a
+
h
=
+a
h
a.b=b'a
3+6=6+3
12 . 4
=
4 . 12
The Associative Properties of Addition and Multiplication allow you to
regroup numbers.
+
(a + b) + c = a
(1
+ 3) + 6
= 1
(b + c)
(a • b) . ,
=
a . (b . ,)
+ (3 + 6)
(1 • 3) • 6
=
1 . (3 . 6)
The Distributive Property distributes multiplication over addition and
subtraction.
u(b + c)
+
3(4
6)
=
ah + ue
=
(3 . 4)
+
a(b - e) = ab - ae
(3 . 6)
5(9 - 3)
=
(5 • 9) - (5 • 3)
Name the property that cach equation illustrates.
72+56=56+72
--
Commutative Property of Addition: The order of the addends
is changed.
4(5 - 9)
(4 . 5) - (4 . 9)
--
Distributh'e Pro pert)': The 4 is distributed.
(30 • 14) . 5
--
Associative Property of Multiplication: The numben; are regrouped.
=
30 . (14 • 5)
=
Exercises
Name the property that each equation illustrates.
1. (17
+ 4) + 9
=
17
+ (4 + 9)
2. 7(3 + 4) = (7 • 3) + (7 • 4)
3. 84. 26 = 26 . 84
4. (3 . 6) . 7 = 3 . (6 . 7)
5. 8(6 - 3) = (8 . 6) - (8 . 3)
6. 4.2
+
3.4 = 3.4
+
4.2
Write the number that makes each statement true.
7. 27
+
9. 9(8 - 5)
= 12 + 27
= (_
•
8) - (_
11. 3. (9 . 6) = (3 . 9) . _
1E!3I .•.•..•.•.•.
8. (8
. 5)
+ 20) + 9
+ (20 + 9)
= _
10. 8. 10 = 10 .
12. 7(6 + 4) = (_
~••••••••••••••.•.•..•.•.•..•.•.•..•.•....•..•••..•.•.••••.••••.•
. 6)
+ (_
. 4)
Name
Class
Date
Review 33
_
Graphing Data on the Coordinate Plane
•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••
OBJECTIVE: Identifying coordinates on the
coordinate plane
MATERIALS: Graph paper
Coordinates give the location of a point. To locate a point (x, y) on a graph,
start at the origin, (0,0). Move x units to the right or to the left along the
x-axis and y units up or down along the y-axis.
Give the coordinates of points A, B, C, and D.
Point A is 2 units to the left of the origin and
3 units up. The coordinates of A are (- 2,3).
y
A
B
Point B is 2 units to the right of the origin and
2 units up. The coordinates of Bare (2,2).
x
C
-
-
0
Point Cis 4 units to the left of the origin and
up. The coordinates of Care (-4,0).
o units
Point D is 1 unit to the left of the origin and
3 units down. The coordinates of Dare (-1, - 3).
Exercises
Name the coordinates of each point.
1. G
y
2. H
K
~
••
•
3. J
J
4. K
~
~
0
5. L
6. 1"1
~
7. N
8. P
9. Q
10. R
1 1
1.11
G
L
x
H
0
N
0
E
<
~
0
,
~
w
0
•~
~
0
Q
Graph the points on the same coordinate plane.
11. S (3. -5)
12. T (0. 0)
13. U(-I. -2)
14. V(4.5)
15. W (0. 3)
16. Z (-5.0)
p
R
g
.......................................................
m
Name
Class
Date
--~------
Review 34
---------
Solving One-Step Equations
•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••
OBJECTIVE: Solving one-step equations
MATERIALS: Tiles
As you model an equation with tiles. ask yourself what operation has been
performed on the variable_ With the tiles, perform the inverse operation on
each side of the equation. Simplify by removing zero pairs.
• D:r!IllI!m3
Model each equation with tiles and solve.
1. x
+4
=
6
DDD
DDD
~DD
DD
DDD"III
~DDI!lIiiI
DDII" DDDIIIIII
DD
~
x
=
-
Model the equation with tiles.
-
Subtract 4 from each side of the equation.
-
Simplify b~'rernol-ing zero pairs.
-
Model the equation with tiles.
2
2. 3x = 9
~~~
DDDDD
DDDD
I[DDDI[DDI
[DDD)D
Dh-ide each side into three identical groups.
~~~
-
DDD
Solve for x.
~
x = 3
Model each equation with tiles and solve.
1. x
+
3
10
2.y-4=2
3. -6 ~ 3y
4. 2x ~ 6
5.y+l=4
6. 5y ~ 10
7. x - 5 = 4
8. 12 =
=
Solve.
10. 17 = -8 + x
13. 5.2
+ II = 0.3
9. x
4x
11. -0.5 ~
*
14. 14 = x + 7
11I3I .•.•..•.•.•..•.••.•.•.••.•.•..•.•..•.•.•....•.••.•.•..•.•.•..•.•.••.•..•.•..•
+
12.0.0
0
4 = 2
-"5a
_
15.6x~15
Name
Class
Date
Review 35
_
Solving Two-Step Equations
•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••
MATERIALS: None
OBJECTIVE: Solving two-step equations
The order of operations tells you to do multiplication and division before
you do addition and subtraction. However, when solving two-step equations,
you must first do any addition or subtraction necessary to isolate the
variable on one side of the equation. Start by asking yourself, "Has any
adding or subtracting been done to the variable?" If the answer is yes,
perform the inverse operation. Then repeal this step for multiplication
and division,
Write the steps and solve the equation.
3x+4=10
3x
+
-
Think: Is an)' adding or suhtracting being done to the variahle?
4 is being added. What is the inverse of adding 4'!
-
Suhtract 4 from each side.
3x = 6
-
Simplif:,...
3x = 6
-
Think: Is any multipl)'ing or dividing being done to the variable?
It is heing multiplied b)' 3. What is the inverse of multipl)'ing h)"3?
3x = Q
3
--.-
Dil'ide each side b)' 3.
-
Simplif)'.
4 - 4
=
10 - 4
3
x = 2
l
:E1
~
Exercises
Fill in the blanks to eomplete the steps and soh'e the equation.
i - 5 = -8
1.
,
6
5 + 5 = -8 + 5
~ = -3
6
6( - 3)
-
Think: Is any adding or subtracting heing done to the variable?
_____
is heing
• What is the
of
suhtracting 57
_____
510
side.
Simplify.
Think: Is any multipl)'ing or dil'iding heing done to the nriahle?
It is being
h)' 6. What is the innrse of
b)' 6?
Multipl)' each
by
_
Simplif)'.
Solve eaeh equalion.
2. 3x - 4 = 8
3.
t + 3 = 10
4.4y+5=-7
.............••••••••••••••••................................................
~
Name
~ etass
Date
Review 36
Solving Multi-Step
_
Equations
•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••
OBJECTIVE: Combining like terms
MATERIALS: None
Simplify 3a - 6x + 4 - 2d + 5x by combining like terms.
• Ring each term that has the variable a. Draw a rectangle around each
term that has the variable x, and a triangle around each constant term.
@
1- 188
~
6x
Group the like terms by reordering the terms so that all matching shapcs
arc together.
@8 BI+
18
5x
Combine like terms by adding coefficients.
ll-x+4
Exercises
Draw circles, rectangles, and triangles to help you combine like terms and
simplif)' each expression.
1. 3([
+ 5 - x + 7.• - 2([
3. 7b - b - x
+ 5 - 2x - 7b
2. 2x - 5 + 3a - 5x
4. -6m
,
+ 3t + 4 - 4m - 2t
5. 2,+3s-5,
6. 4 - P - 2x
7. 3k -2x+6k+5
8. 3
9. 4a+3-2y-Sa-7+4y
+ lOa
~
+ 3p - 7.•
~
~
.go
+ 2a - 7x + 2.5 + 5x
~
+ 4x
10. c-3+2x-6c
0
E
",
Simplify cach expression.
Q
"
0
11. 2h+2-
.• +4
13- !a-S-!a
14. 1.Sy - 1.5
15. 6a+3h-2([+4
16.
c
+ D.5y + 0.5.::+ 1
~
••
~
ta+S-la-7
17. -8
+ x - 2 + 3x
18. x
19. ~x
+ 5 - ~x - 4
20. lOy - 3x
1EZaI
~
w
12. -S - c -4+3c
@
+ y - z + 4x - 5y + 2z
+ 5 - 8 - 2y
.