Table Of ContentAbout the Authors
Barbara Sandall—Ed.D., Illinois State University; MA, DePaul University;
BA, Illinois Wesleyan University—is currently assistant professor in Curriculum and
Instruction at Western Illinois University teaching Elementary Science Methods,
Physical Science for Elementary Teachers, and other courses for elementary edu-
cation majors.
Dr. Sandall taught middle-school science for 18 years, a fifth-grade self-
contained class for one year, as well as courses in elementary music and basic
computer programming. She has also worked at Illinois State University in the Cen-
ter for Mathematics, Science, and Technology Education; the Eisenhower Midwest
Consortium for Mathematics and Science Education at the North Central Regional Educational Laboratory
(NCREL); and the Teachers Academy for Mathematics and Science in Chicago.
Dr. Sandall is the past treasurer for the Illinois Science Teachers association and is currently a
Regional Director for the National Science Teachers Association.
Melfried Olson—Ed.D., Oklahoma State University; MA, University of Arkan-
sas; BS, Valley City State College—is currently a mathematical researcher at the
Curriculum Research & Development Group at the University of Hawaii.
Dr. Olson taught mathematics content and methods courses for 18 years
at Western Illinois University and 11 years at the University of Wyoming. He taught
grades 7–12 for three years, teaching grade 7, grade 8, Algebra I, Geometry,
Algebra II, and Pre-Calculus.
Dr. Olson has been active at the local, state, and national level in mathemat-
ics education through numerous publications, presentations, and grants. Dr. Olson has been an active
consultant in mathematics teaching and learning. He directed a nationwide staff development project in
geometry as well as extensive staff development efforts with teachers K–12. For three years, Dr. Olson
served as a co-editor for the Problem Solvers section of the NCTM journal Teaching Children Mathematics.
He served as president for the Research Council on Mathematics Learning and on the board of directors
for the Illinois Council of Teachers of Mathematics and School Science and Mathematics.
Travis Olson has a master’s degree in mathematics from Western Illinois
University, has taught mathematics (intermediate algebra through calculus III) two
years at Cottey College, and is currently a doctoral student in mathematics educa-
tion at the University of Missouri—Columbia. He has assisted with numerous staff
development activities for elementary and middle-school teachers of mathematics
and has taught two years in the Johns Hopkins University Center for Talented Youth
program.
Mathematical consultant: Joseph A. Kunicki received his BS (1986) and MS (1988) in mathematics from Youngstown
State University in northeast Ohio as a non-traditional student. His pre-academic career included five years in the
United States Marine Corps, working as an aviation electrician, and 15 years as a blue-collar worker, predominantly
in the Bay Area of northern California. Dr. Kunicki finished his Ph.D. in mathematics education at the Ohio State
University in 1994, where he concentrated on multi-variate statistical analyses, abstract algebra, and college-level
mathematics pedagogy. He has substantive experience working with the National Council of Mathematics Teachers
directed toward the Curriculum & Evaluation Standards and their alignment with various district and state mathemat-
ics standards. Dr. Kunicki is currently an associate professor of Mathematics at the University of Findlay in northwest
Ohio. He resides in Arcadia, Ohio, with his wife Pat St. Onge, a graphic illustrator for Mark Twain Media, Inc., and
their two Siberian Huskies, Star and Sherman.
Pre-Algebra Practice
By
Dr. BarBara SanDall, Ed.D.,
Dr. MElfriED OlSOn, Ed.D.,
and TraviS OlSOn, M.S.
COPYriGHT © 2006 Mark Twain Media, inc.
iSBn 978-1-58037-755-3
Printing no. 404041-EB
Mark Twain Media, inc., Publishers
Distributed by Carson-Dellosa Publishing Company, inc.
The purchase of this book entitles the buyer to reproduce the student pages for classroom
use only. Other permissions may be obtained by writing Mark Twain Media, inc., Publishers.
This product has been correlated to state, national, and Canadian provincial
standards. visit www.carsondellosa.com to search and view its correlations to
your standards.
All rights reserved. Printed in the United States of America.
Pre-Algebra Practice Table of Contents
Table of Contents
Introduction to the Math Practice Series .............................................................................iv
Common Mathematics Symbols and Terms .........................................................................1
Chapter 1: Real Numbers .......................................................................................................5
Real Numbers ................................................................................................................5
Checking Progress: Real Numbers ................................................................................8
Chapter 2: Operations of Numbers and Variables ...............................................................9
Addition and Subtraction of Numbers and Variables .....................................................9
Multiplication of Numbers and Variables ......................................................................11
Division of Numbers and Variables ..............................................................................13
Order of Operations .....................................................................................................15
Checking Progress: Operations of Numbers and Variables .........................................18
Chapter 3: Integers ...............................................................................................................19
Positive and Negative Numbers ...................................................................................19
Addition and Subtraction of Integers ............................................................................19
Multiplication and Division of Integers .........................................................................22
Checking Progress: All Integers ...................................................................................24
Chapter 4: Properties ...........................................................................................................25
Identity and Inverse Properties of Addition and Multiplication .....................................25
Commutative Properties of Addition and Multiplication ...............................................27
Associative Properties of Addition and Multiplication ..................................................29
Distributive Properties..................................................................................................31
Checking Progress: All Properties ...............................................................................33
Chapter 5: Exponents and Exponential Expressions .......................................................35
Exponents and Exponential Expressions ....................................................................35
Multiplying Exponential Expressions With Coefficients................................................38
Addition and Subtraction of Exponential Expressions With Like Terms .......................40
Multiplication and Division of Exponential Expressions ...............................................42
Raising to a Power, Including Negative Exponents......................................................45
Scientific Notation ........................................................................................................48
Checking Progress: Exponents and Exponential Expressions ....................................50
© Mark Twain Media, Inc., Publishers iii
Pre-Algebra Practice Table of Contents
Table of Contents (cont.)
Chapter 6: Square Roots......................................................................................................51
Square Numbers and Roots ........................................................................................51
Rules for Simplifying Radical Expressions ..................................................................52
Adding and Subtracting Radicals .................................................................................57
Checking Progress: Square Roots and Radicals .........................................................59
Chapter 7: Using Algebra to Generalize Patterns ..............................................................60
Patterns in Number Lists and Tables ...........................................................................60
Generalizing Patterns in Algebraic Expressions and Equations ..................................68
Solving Simple Linear Equations .................................................................................70
Checking Progress: Using Algebra to Generalize Patterns .........................................73
Chapter 8: Problem-Solving Strategies ..............................................................................75
Problem-Solving With Tables .......................................................................................75
Problem-Solving With Linear Equations ......................................................................78
Checking Progress: Problem-Solving Strategies .........................................................82
Check-Up Problems..............................................................................................................84
Real Numbers ..............................................................................................................84
Operations of Numbers and Variables .........................................................................85
Integers ........................................................................................................................86
Properties ....................................................................................................................87
Exponents and Exponential Expressions ....................................................................88
Square Roots ...............................................................................................................89
Using Algebra to Generalize Patterns ..........................................................................90
Problem-Solving Strategies .........................................................................................92
Answer Keys .........................................................................................................................94
Practice Answer Keys ..................................................................................................94
Check-Up Answer Keys .............................................................................................119
References ..........................................................................................................................122
© Mark Twain Media, Inc., Publishers iv
Pre-Algebra Practice Introduction
Introduction to the Math Practice Series
The Math Practice series will introduce students in middle school and high school to the
course topics of Pre-Algebra, Algebra, Algebra II, and Geometry. All of the practice books are
aligned with the National Council of Teachers of Mathematics (NCTM) Principles and Standards
for School Mathematics. (NCTM 2000)
This series is written for classroom teachers, parents, families, and students. The
practice books in this series can be used as full units of study or as individual lessons to
supplement textbooks or curriculum programs. Parents and students can use this series as an
enhancement to what is being done in the classroom or as a tutorial at home. Students will be
given a basic overview of the concepts, examples, practice problems, and challenge problems
using the concepts introduced in the section. At the end of each section, there will be a set of
problems to check progress on the concepts and a challenge set of problems over the whole
section. At the end of the book, there will be check-up problems for each section that can be
used for assessment.
According to the Mathematics Education Trust and NCTM, new technologies require
the fundamentals of algebra and algebraic thinking as a part of the background for all citizens.
These technologies also provide opportunities to generate numerical examples, graph data,
analyze patterns, and make generalizations. An understanding of algebra is also important
because business and industry require higher levels of thinking and problem solving. NCTM
also suggests that understanding geometry, including the characteristics and properties of two-
and three-dimensional shapes, spatial relationships, symmetry, and the use of visualization
and spatial reasoning, can also be used in solving problems.
The NCTM Standards suggest that content and vocabulary are necessary, but of equal
importance are the processes of mathematics. The process skills described in the Standards
include: problem solving, reasoning, communication, and connections. The practice books in
this series will address both the content and processes of algebra and algebraic thinking and
geometry. This worktext, Pre-Algebra Practice, will help students transition from arithmetic to
algebra.
© Mark Twain Media, Inc., Publishers v
Pre-Algebra Practice Common Mathematic Symbols and Terms
Common Mathematics Symbols and Terms
Term Symbol/Definition Example
Addition sign + 2 + 2 = 4
Subtraction sign – 4 – 2 = 2
Multiplication sign x 3 x 2
or a dot • 2 • 2
or 2 numbers or letters together 2x
or parentheses 2(2)
Division sign ÷ 6 ÷ 2
or a slash mark ( / ) 4/2
or a horizontal fraction bar, $s
or 2 4
Equals or is equal to = 2 + 2 = 4
Does Not Equal ≠ 5 ≠ 1
Parentheses – symbols ( ) (2 x 5) + 3 =
for grouping numbers
Pi – a number that is π 3.1415926…
SuS
approximately
or ≈ 3.14
Negative number – – -3
to the left of zero on
a number line
Positive number – + +4
to the right of zero on
a number line
Less than < 2 < 4
Greater than > 4 > 2
Greater than or equal to > 2 + 3 ≥ 4
Less than or equal to < 2 + 1 ≤ 4
Is approximately equal to ≈ π ≈ 3.14
Radical sign 9 The square root of 9
3 27 The cube root of 27
The nth power of a an 32 = 9
© Mark Twain Media, Inc., Publishers 1
Pre-Algebra Practice Common Mathematic Symbols and Terms
Common Mathematics Symbols and Terms (cont.)
Variable A letter used for an x + 8 = 12
unknown number x is the letter representing
the unknown number or
variable
Mathematical Sentence Contains two mathematical 2 + 3 = 5
phrases joined by an equals (=) 9 – 3 > 5
or an inequalities (≠, <, >, <, >) 3x + 8 = 20
sign
Equation Mathematical sentence in which 5 + 7 = 12
two phrases are connected with 3x = 12
an equals (=) sign. 1 = 1
Mathematical Mathematics has four basic + sign indicates addition
Operations operations: addition, subtraction, – sign indicates subtraction
multiplication, and division. ÷ indicates division
Symbols are used for each • or x indicates multiplication
operation.
Like Terms Can be all numbers or variables 3, 4, 5
!s
that are the same letter and same 3c, -5c, c the variable is
exponent the same with the same
exponent; they are like terms
Unlike Terms Can be numbers or variables that 5 + a Cannot be added
are different because they are unlike
terms
3x + 4y + 1z cannot be
added because the variables
are different, so they are
unlike terms
Coefficient The number in front of the 5x In this number, 5 is
variable (letter for the unknown the coefficient.
number)
Identity Property Any number or variable added to 0 + 5 = 5 -3 + 0 = -3
of Addition zero is that number or variable. a + 0 = a
Identity Property Any number or variable times 1 12 • 1 = 12
of Multiplication is equal to that number or variable. b • 1 = b
3y • 1 = 3y
© Mark Twain Media, Inc., Publishers 2
Pre-Algebra Practice Common Mathematic Symbols and Terms
Common Mathematics Symbols and Terms (cont.)
Commutative Property No matter the order in which you 4 + 7 = 7 + 4
of Addition add two numbers, the sum is b + c = c + b
always the same.
!s !s
Commutative Property No matter the order in which you 20 x = x 20
of Multiplication multiply two numbers, the answer 5 • 3 = 3 • 5
is always the same. a • b = b • a
Associative Property No matter the order in which you (5 + 6) + 7 = 5 + (6 + 7)
of Addition add three numbers together, the (a + b) + c = a + (b + c)
the sum will always be the same.
Associative Property No matter the order in which you (5 • 4) • 8 = 5 • (4 • 8)
of Multiplication group numbers when you multiply, (a • b) • c = a • (b • c)
the answer will always be the
same product.
Distributive Property of Allows the choice of multiplication 3(5 + 2) = 3 • 5 + 3 • 2
Multiplication Over followed by addition or addition a(b + c) = a • b + a • c
Addition followed by multiplication.
Inverse Operation Operation that cancels another Multiplication and division
operation 5 • x = 5x
5x
5 = x
Addition and Subtraction
n + 5 – 5 = n
Reciprocal or Two numbers are multiplied, and For any non-zero number:
Multiplicative Inverse the product is 1.
1
Property Number x = 1
Number
1
x Number = 1
Number
1
a • = 1
a
1
5 • 5 = 1
© Mark Twain Media, Inc., Publishers 3
Pre-Algebra Practice Common Mathematic Symbols and Terms
Common Mathematics Symbols and Terms (cont.)
Exponents Shorthand for repeated a2 = a • a
multiplication y4 = y • y • y • y
Square Numbers The result of multiplying a number 4 • 4 = 16
or variable by itself a • a = a2
Square Roots A square root indicated by the 9
radical sign is the number What number multiplied
multiplied by itself to get the by itself = 9?
radicand. 3 • 3 = 9
So 9 = 3
Radicand Number under the radical 9
9 is the radicand
#g
Numerator Top number in a fraction In this fraction,
3 is the numerator
#g
Denominator Bottom number in a fraction In this fraction,
5 is the denominator
Integers Natural numbers, their opposites, Set of Integers:
or negative numbers, and zero {…-3,-2,-1,0,1,2,3…}
Additive Inverse The sum of an integer and its a + -a = 0
Property of Addition opposite integer will always be zero. 5 + -5 = 0
Set Specific group of numbers or Set of Integers:
objects {…-3,-2,-1,0,1,2,3…}
Absolute Value |a| Absolute value of a
The absolute value of a number
can be considered as the distance
between the number and zero on a
number line.
The absolute value of every number |a| = a
will be either positive or zero. |-a| = a
Real numbers come in pared If a is 0, |a| = 0.
opposites, a and -a, that are the
same distance from the origin, but If 0 is the origin on the
in opposite directions. number line on the left, 3 is
the absolute value of the
pair -3 and +3 because they
are both 3 marks from 0.
-3 -2 -1 0 1 2 3
© Mark Twain Media, Inc., Publishers 4
