Number Realms Guided Notes
Real Number Realms
Field Guide & Guided Notes
Name:
Date:
The Territory: Visualizing the Real Number System
Use the diagram below to label the 4 subsets of Real Numbers as we discuss them. Think of these as "neighborhoods" where some live inside others!
1. _______________________
2. _______________________
3. _______________________
(Examples: 0, 1, 2...)
4. _______________________
Numbers that can't be written as a fraction
Explorer's Field Notes: Definitions
Whole Numbers:
Counting numbers starting at ____________________. No decimals or fractions.
Examples: 0, 1, 15, 42, √25 (which is 5)
Integers:
All whole numbers and their _____________________ (negatives).
Examples: -3, -2, -1, 0, 1, 2, 3
Rational Numbers:
Any number that can be written as a _____________________ (\( \frac{a}{b} \)). This includes: • _____________________ decimals (like 0.25) • _____________________ decimals (like 0.333...)
Irrational Numbers:
Numbers that NEVER _____________________ and NEVER _____________________.
Examples: π, √2, √7, 3.141592...
Hidden Identities: Simplifying Before Classifying
Sometimes numbers wear masks! Always simplify a number before deciding which set it belongs to.
Number
√16
Simplifies to:
Classification: Whole
Number
− (10 / 2)
Simplifies to:
Classification: Integer
The Navigation Quiz: Set Relationships
Decide if each statement is Always, Sometimes, or Never true.
1
A whole number is an integer.
Because whole numbers are located inside the integer set.
2
An integer is a whole number.
What about -5? Is it whole?
3
A rational number is an irrational number.
Look at the map. Are they separate territories?
4
A square root is an irrational number.
Think about √9 vs √10.
Number Realms Practice Worksheet
Real Number Realms
Field Practice & Classification
Name:
Date:
Task 1: The Sorting Station
Simplify each number if needed, then place a checkmark (✓) in ALL categories that apply to that number.
| Number | Simplified | Whole | Integer | Rational | Irrational |
|---|
| -8 | | | | | |
| 25 / 5 | | | | | |
| √10 | | | | | |
| 0.75 | | | | | |
| 0 | | | | | |
| π | | | | | |
Task 2: Map Placement
Write the following numbers in the correct location on the Venn Diagram below: -4, √49, 1/3, √3, -12.5, 0
Rational
Integers
Whole
Irrational
Task 3: Explorer Reasoning
Answer the following questions using complete sentences and mathematical evidence.
1. Why is the number √36 considered a whole number, but √30 is considered an irrational number? Explain the difference.
2. If a number is an integer, is it ALWAYS a rational number? Use the Venn diagram hierarchy to support your answer.
3. A student says that −5.5 is an integer because it is negative. Are they correct? Explain why or why not.
The Mystery Number Puzzle
"I am a real number. I am NOT irrational. I am NOT a whole number. I AM an integer. Who could I be?"
Your Answer:
(List any two possible examples)
Number Realms Answer Key
Answer Key
Real Number Realms Guided Notes
TEACHER ONLY
Venn Diagram Labels
- 1. Rational Numbers (the largest set on the left)
- 2. Integers (the middle set)
- 3. Whole Numbers (the inner set)
- 4. Irrational Numbers (the separate set on the right)
Definition Blanks
Whole Numbers:
Counting numbers starting at zero.
Integers:
All whole numbers and their opposites (negatives).
Rational Numbers:
...written as a ratio or fraction. Includes: Terminating decimals and Repeating decimals.
Irrational Numbers:
Numbers that NEVER end (terminate) and NEVER repeat.
Navigation Quiz
- 1. Always: All whole numbers (0, 1, 2...) are also in the integer set.
- 2. Sometimes: Positive integers like 5 are whole, but negatives like -5 are not.
- 3. Never: A number cannot be both rational (can be a fraction) and irrational (cannot be a fraction).
- 4. Sometimes: √25 = 5 (Rational), but √10 (Irrational).
Answer Key
Real Number Realms Practice Worksheet
TEACHER ONLY
Task 1: Sorting Station
| Number | Simplified | Whole | Integer | Rational | Irrational |
|---|
| -8 | -8 | | ✓ | ✓ | |
| 25/5 | 5 | ✓ | ✓ | ✓ | |
| √10 | √10 | | | | ✓ |
| 0.75 | 0.75 | | | ✓ | |
| 0 | 0 | ✓ | ✓ | ✓ | |
| π | π | | | | ✓ |
Task 2: Map Placement
- Whole: 0, √49 (since it is 7)
- Integer: -4
- Rational: 1/3, -12.5
- Irrational: √3
Task 3: Reasoning
1. Why sqrt(36) vs sqrt(30)? √36 simplifies to exactly 6, which is a whole number. 30 is not a perfect square, so √30 results in a non-terminating, non-repeating decimal.
2. Integer always Rational? Yes. Any integer can be written as a fraction over 1 (e.g., -5 = -5/1). On the Venn diagram, the integer circle is entirely inside the rational circle.
