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Rainbow Bridge in Niagara Falls, New York
STUDENT NAME
Table of Contents
Irrational Numbers (7.N.2). ....................................................... 4
Scientific Notation (7.N.5, 7.N.6).................................................. 8
Estimating Square Roots (7.N.18)........................................... 12
Multiples (7.N.9)........................................................................ 16
Factors (7.N.8).......................................................................... 20
Adding and Subtracting Integers (7.N.12, 7.N.13). .................... 24
Multiplying and Dividing Integers (7.N.12)............................. 28
Writing Expressions (7.A.1)..................................................... 32
Solving Equations (7.A.4)........................................................ 36
Evaluating Formulas (7.A.6). .................................................. 40
Volume (7.G.2).......................................................................... 44
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Pythagorean Theorem (7.G.8)................................................ 48
Surface Area (7.G.4). ............................................................... 52
Circle Graphs (7.S.6). .............................................................. 56
Probability (7.S.8, 7.S.12). ........................................................... 60
7.N.2
NY
At the beginning of each lesson, you will see a box with the shape of
New York and a Mathematics Core Curriculum code in it. This code
tells you what is being covered in the lesson.
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7.N.2
NY
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.
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.
Whiteface Mountain is the
fifth highest mountain in New
York. It is one of the High
Peaks of the Adirondack
Mountains. When taking
measurements such as slope
height or land area for which
snow must be plowed or
groomed, resort managers
must use rational and
irrational numbers.
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​
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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
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The square root of a number that is not a perfect
square is also an irrational number.
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For calculations, p is given the approximation of 3.14 or ​ 22  ​.  Although p is an
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
irrational number, 3.14 and ​ 22  ​ are rational numbers. This is because 3.14 is a
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
terminating decimal and ​ 22  ​ is a fraction of two integers.
7

Example 2
Which number is an irrational number?
A​ 2 ​ 
9
 
C​ 16 ​
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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 ​
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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?
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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
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
number.
5 4.77777…, 1.285347…, ​ 8 ​,  23 9
,  ​ 14  ​,  2.461461…, 0.375 6 ​ 8 ​
5

,  p ,  ​ 49 ​
7 1 ​ 5  ​,  6.​3​
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
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For Numbers 5–7, write the irrational number in each set.
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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…
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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 Whiteface Mountain. They rode a gondola from
the base lodge to the top of Little Whiteface. The summit is more than 3,600 feet
high. 3,600 is a perfect square of which integer?
A 40
B 60
C 90
D 180
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