This document provides examples and explanations of concepts related to classifying and ordering different types of numbers: 1) It defines rational and irrational numbers, and provides examples of each. Rational numbers can be expressed as ratios of integers, while irrational numbers cannot. 2) Exercises are included for classifying numbers as rational or irrational, comparing irrational numbers on a number line, and ordering sets of numbers from least to greatest. 3) Answering "Why is it helpful to write numbers in different ways?" the document explains that not all numbers can be written the same way, like irrational numbers that cannot be expressed as integers. Being able to classify numbers facilitates understanding and operations involving different types of numbers.

1. Course 3, Lesson 1-10
Estimate to the nearest whole number.
1.
2.
3.
4.
5. Give two numbers that have square roots between 9
and 10.
39
102
71.3
25.9

2. Course3, Lesson 1-10
ANSWERS
1. 6
2. 10
3. 8
4. 5
5. Sample answer: 82, 87

3. WHY is it helpful to write numbers in
different ways?
The Number System
Course 3, Lesson 1-10

4. The Number System
Course 3, Lesson 1-10 Common Core State Standards © Copyright 2010. National Governors Association Center for Best Practices
and Council of Chief State School Officers. All rights reserved.
• 8.NS.1
Know that numbers that are not rational are called irrational. Understand
informally that every number has a decimal expansion; for rational
numbers show that the decimal expansion repeats eventually, and
convert a decimal expansion which repeats eventually into a rational
number.
• 8.NS.2
Use rational approximations of irrational numbers to compare the size of
irrational numbers, locate them approximately on a number line diagram,
and estimate the value of expressions.

5. Course 3, Lesson 1-10 Common Core State Standards © Copyright 2010. National Governors Association Center for Best Practices and Council of
Chief State School Officers. All rights reserved.
The Number System
8.EE.2
Use square root and cube root symbols to represent solutions to
equations of the form x2 = p and x3 = p, where p is a positive rational
number. Evaluate square roots of small perfect squares and cube roots
of small perfect cubes. Know that is irrational.
Mathematical Practices
1 Make sense of problems and persevere in solving them.
3 Construct viable arguments and critique the reasoning of others.
4 Model with mathematics.
6 Attend to precision.
2

6. To
• classify numbers,
• compare and order real numbers
Course 3, Lesson 1-10
The Number System

7. • irrational number
• real number
Course 3, Lesson 1-10
The Number System

8. Course 3, Lesson 1-10
The Number System
Words
Examples
Rational Number
A rational number is a
number that can be
expressed as the ratio ,
where a and b are integers
and b ≠ 0.
Irrational Number
An irrational number is a
number that cannot be
expressed as the ratio,
where a and b are integers
and b ≠ 0.
7
, -12
8
76-2,5,3. 1.414213562....2

9. 1
Need Another Example?
Step-by-Step Example
1. Name all sets of numbers to which the real
number belongs. 0.2525...
The decimal ends in a repeating pattern. It is a
rational number because it is equivalent to .

10. Answer
Need Another Example?
Name all sets of numbers to which
0.090909… belongs.
rational

11. 1
Need Another Example?
Step-by-Step Example
1. Name all sets of numbers to which the real
number belongs. √36
Since √36 = 6, it is a natural number, a whole
number, an integer, and a rational number.

12. Answer
Need Another Example?
Name all sets of numbers to which √25
belongs.
natural, whole, integer, rational

13. 1
Need Another Example?
Step-by-Step Example
1. Name all sets of numbers to which the real
number belongs. –√7
–√7 ≈ –2.645751311… The decimal does not
terminate nor repeat, so it is an irrational number.

14. Answer
Need Another Example?
Name all sets of numbers to which −√12
belongs.
irrational

15. 1
Need Another Example?
2
3
4
Step-by-Step Example
4. Fill in the with <, >, or = to make a true statement.
√7 ≈ 2.645751311…
2 = 2.666666666…
Since 2.645751311… is less than 2.66666666…,
√7 < 2 .

16. Answer
Need Another Example?
Replace the in √15 3 with <, >, or =
to make a true statement.
<

17. 1
Need Another Example?
2
3
4
Step-by-Step Example
5. Fill in the with <, >, or = to make a true statement.
15.7% √0.02
15.7% = 0.157
√0.02 ≈ 0.141
Since 0.157 is greater than 0.141, 15.7% > √0.02.

18. Answer
Need Another Example?
Replace the in 12.3% √0.01 with <, >, or = to
make a true statement.
>

19. 1
Need Another Example?
2
3
4
5
6
Step-by-Step Example
6. Order the set √30 , 6, 5 , 5.36 from least to
greatest. Verify your answer by graphing on a number line.
Write each number as a decimal. Then order the decimals.
√30 ≈ 5.48
5.36 ≈ 5.37
7
6 = 6.00
5 = 5.80
From least to greatest, the order is 5.36, √30, 5 , and 6.

20. Answer
Need Another Example?
3, √15, 4.21, 4
Order the set √15, 3, 4 , 4.21 from least to
greatest. Verify your answer by graphing on a
number line.

21. 1
Need Another Example?
2
3
4
Step-by-Step Example
7. On a clear day, the number of miles a person can see to the horizon is
about 1.23 times the square root of his or her distance from the ground in
feet. Suppose Frida is at the Empire Building observation deck at 1,250 feet
and Kia is at the Freedom Tower observation deck at 1,362 feet. How much
farther can Kia see than Frida?
Use a calculator to approximate the distance each person can see.
Frida: 1.23 • √1,250 ≈ 43.49
Kia can see 45.39 – 43.49 or 1.90 miles farther than Frida.
Kia: 1.23 • √1,362 ≈ 45.39

22. Answer
Need Another Example?
The time in seconds that it takes an object
to fall d feet is . About how many seconds
would it take for a volleyball thrown 32 feet
up in the air to fall from its highest point to
the sand?
1.4 s

23. How did what you learned
today help you answer the
WHY is it helpful to write numbers in
different ways?
Course 3, Lesson 1-10
The Number System

24. How did what you learned
today help you answer the
WHY is it helpful to write numbers in
different ways?
Course 3, Lesson 1-10
The Number System
Sample answer:
• Not all numbers can be written the same way. For
example, an irrational number cannot be written as an
integer.

25. Write one example of each of the following
•a whole number,
•an integer that is not a whole number,
•a rational number that is not an integer,
•an irrational number.
Ratios and Proportional RelationshipsThe Number System
Course 3, Lesson 1-10