Number Sense | Fourth Grade

Complete Notes & Formulas

1. Place Values

Definition: Place value is the value of a digit based on its position in a number.

📊 Place Value Chart (up to 1,000,000):

Hundred Thousands	Ten Thousands	Thousands	Hundreds	Tens	Ones
100,000	10,000	1,000	100	10	1

✏️ Examples:

In 5,376: The digit 5 is in the thousands place → Place value = 5,000
In 5,376: The digit 3 is in the hundreds place → Place value = 300
In 5,376: The digit 7 is in the tens place → Place value = 70
In 5,376: The digit 6 is in the ones place → Place value = 6

📐 Formula for Place Value:

Place Value = (Face Value of Digit) × (Position Value)

2. Convert Between Place Values

Definition: Converting between place values means expressing a number in different forms (expanded, standard, word form).

🔄 Three Forms of Numbers:

1. Standard Form:

5,376

2. Expanded Form:

5,000 + 300 + 70 + 6

3. Expanded Notation (with multiplication):

5 × 1,000 + 3 × 100 + 7 × 10 + 6 × 1

✏️ Conversion Examples:

6,000 + 80 + 20 + 3 = 6,103 (Standard Form)
45,678 = 4 × 10,000 + 5 × 1,000 + 6 × 100 + 7 × 10 + 8 × 1
850,000 = 8 × 100,000 + 5 × 10,000

3. Word Names for Numbers

Definition: Word names are the written form of numbers using words instead of digits.

📝 Number Names Reference:

1 - 10:

One, Two, Three, Four, Five, Six, Seven, Eight, Nine, Ten

11 - 20:

Eleven, Twelve, Thirteen, Fourteen, Fifteen, Sixteen, Seventeen, Eighteen, Nineteen, Twenty

Tens:

20 = Twenty, 30 = Thirty, 40 = Forty, 50 = Fifty, 60 = Sixty, 70 = Seventy, 80 = Eighty, 90 = Ninety

Place Value Words:

100 = Hundred, 1,000 = Thousand, 10,000 = Ten Thousand, 100,000 = Hundred Thousand, 1,000,000 = Million

✏️ Examples:

5,376 = Five Thousand Three Hundred Seventy-Six
45,678 = Forty-Five Thousand Six Hundred Seventy-Eight
850,000 = Eight Hundred Fifty Thousand

4. Ordinal Numbers to 100th

Definition: Ordinal numbers show the position or order of things (1st, 2nd, 3rd, etc.).

🔢 Ordinal Number Rules:

Suffix Rules:

Numbers ending in 1 → add "st" (except 11th): 1st, 21st, 31st, 41st
Numbers ending in 2 → add "nd" (except 12th): 2nd, 22nd, 32nd, 42nd
Numbers ending in 3 → add "rd" (except 13th): 3rd, 23rd, 33rd, 43rd
All other numbers → add "th": 4th, 5th, 6th... 11th, 12th, 13th... 20th

📋 Common Ordinal Numbers:

Number	Ordinal	Word Form
1	1st	First
2	2nd	Second
3	3rd	Third
10	10th	Tenth
20	20th	Twentieth
50	50th	Fiftieth
100	100th	Hundredth

5. Rounding

Definition: Rounding means finding the nearest value of a number to a specified place value.

📐 Rounding Rules:

The Rounding Rule:

If the digit to the right is 5 or more (5, 6, 7, 8, 9) → Round UP

If the digit to the right is 4 or less (0, 1, 2, 3, 4) → Round DOWN

Memory Trick:

"Four and below, just let it go. Five and above, give it a shove (up)!"

✏️ Rounding Examples:

643 rounded to nearest 10 = 640 (look at 3 → round down)
4,623 rounded to nearest 100 = 4,600 (look at 23 → round down)
83,623 rounded to nearest 1,000 = 84,000 (look at 623 → round up)
83,623 rounded to nearest 10,000 = 80,000 (look at 3,623 → round down)

📝 Steps to Round:

Step 1: Identify the place value to round to
Step 2: Look at the digit immediately to the right
Step 3: Apply the rounding rule (5+ up, 4- down)
Step 4: Replace all digits to the right with zeros

6. Even or Odd: Arithmetic Rules

Definitions: Even numbers are divisible by 2 (end in 0, 2, 4, 6, 8). Odd numbers are not divisible by 2 (end in 1, 3, 5, 7, 9).

➕ Addition & Subtraction Rules:

Operation	Result	Example
Even ± Even	Even	4 + 6 = 10 (Even)
Odd ± Odd	Even	7 + 3 = 10 (Even)
Even ± Odd	Odd	8 + 5 = 13 (Odd)
Odd ± Even	Odd	9 + 4 = 13 (Odd)

✖️ Multiplication Rules:

Operation	Result	Example
Even × Even	Even	2 × 6 = 12 (Even)
Odd × Odd	Odd	3 × 5 = 15 (Odd)
Even × Odd	Even	6 × 7 = 42 (Even)
Odd × Even	Even	5 × 8 = 40 (Even)

🔑 Key Rule:

Multiplying ANY number by an EVEN number ALWAYS gives an EVEN result!

7. Inequalities with Number Lines

Definition: Inequalities compare two numbers using symbols to show which is greater or lesser.

🔣 Inequality Symbols:

Symbol	Meaning	Example
>	Greater than	7 > 5
<	Less than	3 < 9
≥	Greater than or equal to	8 ≥ 8
≤	Less than or equal to	4 ≤ 6

📏 Number Line Representation:

For x > 5:

• Open circle at 5, arrow pointing right →

(Numbers greater than 5: 6, 7, 8, 9...)

For x < 3:

• Open circle at 3, arrow pointing left ←

(Numbers less than 3: 2, 1, 0, -1...)

For x ≥ 4:

• Filled/closed circle at 4, arrow pointing right →

(Includes 4: 4, 5, 6, 7...)

💡 Memory Trick:

The "mouth" of < or > opens toward the BIGGER number!

8. Compare Numbers Up to Five Digits

Definition: Comparing numbers means determining which number is greater, smaller, or if they are equal.

📝 Steps to Compare Numbers:

Step 1: Count the number of digits - More digits = Greater number
Step 2: If digits are equal, compare from left to right starting with the highest place value
Step 3: Continue comparing each place value until you find a difference
Step 4: Use the correct symbol: >, <, or =

✏️ Examples:

Compare: 5,376 and 5,389

• Same thousands (5), same hundreds (3), same tens (8)

• Compare ones: 6 < 9

Answer: 5,376 < 5,389

Compare: 45,678 and 9,999

• 45,678 has 5 digits, 9,999 has 4 digits

• More digits means larger number

Answer: 45,678 > 9,999

Compare: 85,421 and 84,999

• Same ten thousands (8)

• Compare thousands: 5 > 4

Answer: 85,421 > 84,999

🔍 Ordering Numbers:

Ascending Order: Smallest to Largest (going up)

Example: 512 < 10,511 < 912,814

Descending Order: Largest to Smallest (going down)

Example: 912,814 > 10,511 > 512

Quick Reference Chart

Topic	Key Formula/Rule
Place Value	Place Value = Face Value × Position Value
Expanded Form	5,376 = 5×1,000 + 3×100 + 7×10 + 6×1
Rounding	5+ Round UP \| 4- Round DOWN
Even × Anything	Always = Even
Odd × Odd	Always = Odd
Ordinal Suffixes	1→st, 2→nd, 3→rd, others→th (except 11,12,13→th)
Comparing Numbers	More digits = Larger \| Same digits? Compare left to right