5 Most Challengi
1
s
’
a
i
n
r
o
f
i
l
a
C
g
n
ng Skills
Masteri
California 7 Mathematics
5 Most Challengi
1
s
’
a
i
n
r
o
f
i
l
a
C
g
n
ng Skills
Masteri
a
i
n
r
o
f
i
l
Ca
Master California’s 15 most challenging mathematics skills with
SkillBridge. With lesson topics chosen based on actual state test
data, SkillBridge offers help in the skills that truly are the most
troublesome. As students move through each lesson, they are
equipped with guidance and connections that allow them to show
independent skill mastery by the end of the lesson. California
references in each lesson offer a unique and familiar
point of entry into difficult skills.
Mathema
tics
California’s 15 most challenging skills
in mathematics, grade 7
• Irrational Numbers
• Decimals
• Squares and Square Roots
• Writing Expressions
• Estimating Square Roots
• Writing Equations
• Scientific Notation
• Solving Equations
• Adding and Subtracting
Integers
• Distance, Rate, and Time
• Multiplying and Dividing
Integers
• Surface Area
• Fractions
• Solving Inequalities
• Volume
Golden Gate Bridge in San Francisco, California
Catalog Number CAB2066W1
P.O. Box 2180
Iowa City, Iowa 52244-2180
ISBN 978-0-7836-6324-1
50599
STUDENT NAME
PHONE: 800-776-3454
FAX: 877-365-0111
www.BuckleDown.com
1BRCA07MM01.indd 1-2
9 780783 663241
12/18/08 11:02:48 AM
Table of Contents
Irrational Numbers (NS.1.4)....................................................... 4
Squares and Square Roots (NS.2.4)......................................... 8
Estimating Square Roots (NS.2.4).......................................... 12
Scientific Notation (NS.1.1)...................................................... 16
Adding and Subtracting Integers (NS.1.2)............................. 20
Multiplying and Dividing Integers (NS.1.2)............................. 24
Fractions (NS.1.2)..................................................................... 28
Decimals (NS.1.2). .................................................................... 32
Writing Expressions (AF.1.1).................................................... 36
Writing Equations (AF.1.1)........................................................ 40
Solving Equations (AF.4.1)....................................................... 44
CA7 © 2009 Buckle Down – Options Publishing. COPYING IS FORBIDDEN BY LAW.
Distance, Rate, and Time (AF.4.2)........................................... 48
Solving Inequalities (AF.4.1).................................................... 52
Surface Area (MG.2.1).............................................................. 56
Volume (MG.2.1)........................................................................ 60
Acknowledgments................................................................ 64
NS.1.4
CA
At the beginning of each lesson, you will see a box with the shape of
California and a content standard code in it. This code tells you what
is being covered in the lesson.
3
1BRCA07MM_FM.indd 3
12/17/08 5:04:24 PM
NS.1.4
CA
I
Irrational
Numbers
Integers are whole numbers and their opposites.
For example, 22, 21, 0, 1, and 2 are integers.
A rational number is any number that can be
written as a fraction of two integers.
The denominator of a rational number written as a
fraction cannot be 0. When the fraction is divided,
it will be either a terminating decimal or a repeating
decimal.
Mammoth Lakes, California,
near Mammoth Mountain,
are a popular place any time
of year for hiking, skiing, ice
hockey and more.
Examples of rational numbers are:
.
26, 0.5, ​ 1 ​,  20%, and 0.​3​
4
An irrational number is a number that cannot be
written as a fraction. Irrational numbers are
non-terminating, non-repeating decimals.
Pi (p) is an example of an irrational number. It is
non-terminating and non-repeating.
Example 1
 as a rational or irrational number.
Classify 0.​27​
Build A Bridge 1
The bar notation over the decimal means those
numbers repeat infinitely. It can be expressed as
0.2727272727… You can see a repeating pattern.
Classify each number as
rational or irrational.
This decimal is non-terminating, but is repeating.
An irrational number is non-terminating and nonrepeating.
 is a rational number.
Therefore, 0.​27​
7
A: 2 ​8​

B: 0.92436…
C: 0.61587
CA7 © 2009 Buckle Down – Options Publishing. COPYING IS FORBIDDEN BY LAW.
The square root of a number that is not a perfect
square is also an irrational number.
4
7M_Irrational Numbers.indd 4
12/17/08 5:05:12 PM
For calculations, p is given the approximation of 3.14 or ​ 22  ​.  Although p is an
7

irrational number, 3.14 and ​ 22  ​ are rational numbers. This is because 3.14 is a
7

terminating decimal and ​ 22  ​ is a fraction of two integers.
7

Example 2
Which number is an irrational number?
A​ 2 ​
9

C​ 16 ​
Build A Bridge 2
B 0.868686…
Which number is an
irrational number?

D​ 5 ​

A 0.​35​
Choice A: ​ 2 ​ is a fraction of two integers. It is a
9
rational number.
Choice B: 0.868686… is a repeating decimal.
The ellipsis (…) indicates the decimal does not
end, and a pattern (86) is shown before the
ellipsis. It is a rational number.

B​ 47 ​

C​ 100 ​
D​ 3  ​
16

Choice C: 16 is a perfect square. A perfect square
 5 4.
is the square of an integer. 16 5 42, so ​ 16 ​
CA7 © 2009 Buckle Down – Options Publishing. COPYING IS FORBIDDEN BY LAW.
It is a rational number.
 is not a perfect square. There is no
Choice D: 
​  5 ​
integer that when multiplied by itself equals 5.
It creates a non-terminating and non-repeating
decimal. It is an irrational number.
Guided PractiCE
1 What is a rational number?
2 How is an irrational number different than a rational number?
5
7M_Irrational Numbers.indd 5
12/9/08 4:35:48 PM
3 Explain how some square roots can be rational numbers while others are
irrational numbers.
4​ 22  ​ is an approximation of p. Explain why the approximation of p is a rational
7

number.
5 4.77777…, 1.285347…, ​ 8 ​,  23 9
,  ​ 14  ​,  2.461461…, 0.375 6 ​ 8 ​
5

,  p  , 
7 1 ​ 5  ​,  6.​3​
​  49 ​
16

8 Which number is a rational number?

A 0.​6​
B p
C 9.351678…

D​ 54 ​

C​ 144 ​

D 14.8​5​
C 9
D 8
9 Which number is an irrational number?
A 3.929292…

B​ 20 ​
10 64 is a perfect square of which integer?
A 32
B 14
CA7 © 2009 Buckle Down – Options Publishing. COPYING IS FORBIDDEN BY LAW.
For Numbers 5–7, write the irrational number in each set.
6
7M_Irrational Numbers.indd 6
12/9/08 4:35:49 PM
Practice
For Numbers 1–5, classify each number as rational or irrational.
1 0.162162… 2 24.379105… 3​ 85   ​ 3

4 5.921634… 5​ 22  ​  7

6 Which number is a rational number?
A p
B 2.01753…
C 0.975
D 0.832146…

C​ 32 ​
D 6.277254…
C 0.992992…
D p
7 Which number is a rational number?
A 3.590831…
B 1.747474…
CA7 © 2009 Buckle Down – Options Publishing. COPYING IS FORBIDDEN BY LAW.
8 Which number is an irrational number?
A 12 ​ 37 ​


B 25.08​3​
9 Which number is an irrational number?

A​ 81 ​

B​ 25 ​

C 7.01130113… D​ 72 ​
10 The number 121 is a perfect square of which integer?
A 9
B 10
C 11
D 12
11 Gavin and his family traveled to Mammoth Lakes one summer. They played golf
at the highest golf course in California. The highest hole is about 8,100 feet in
elevation. 8,100 is a perfect square for which integer?
A 40
B 81
C 90
D 405
7
7M_Irrational Numbers.indd 7
12/9/08 4:35:50 PM