Chapter Notes: Whole Numbers (Term 3)

Whole Numbers

Order, Compare, and Describe Big Numbers

This section teaches students how to work with large whole numbers, focusing on their structure, comparison, and sequencing.

Understanding Place Value

Whole numbers are numbers like 0, 1, 2, 3, and so on, without fractions or decimals.
Large numbers are built using place values: units, tens, hundreds, thousands, ten thousands, hundred thousands, millions, etc.

• Example: In 100:
  There are 10 tens (100 ÷ 10 = 10).
• Example: In 1000:
  There are 10 hundreds (1000 ÷ 100 = 10).
• Example: In 100,000:
  There are 100 thousands (100,000 ÷ 1000 = 100).
  There are 1000 hundreds (100,000 ÷ 100 = 1000).
  There are 10 ten thousands (100,000 ÷ 10,000 = 10).
• Example: In 100,000,000:
  There are 10,000 ten thousands (100,000,000 ÷ 10,000 = 10,000).
  There are 1,000,000 hundreds (100,000,000 ÷ 100 = 1,000,000).
  There are 100,000 thousands (100,000,000 ÷ 1000 = 100,000).
  There are 10,000,000 tens (100,000,000 ÷ 10 = 10,000,000).

Each digit's position determines its value:

• Example: In 738,264,111, the digit 7 is in the hundred millions place (700,000,000), and 3 is in the ten millions place (30,000,000).
• Example: In 264,738,111, the digit 7 is in the hundred thousands place (700,000), and 3 is in the ten thousands place (30,000).

Comparing Numbers
To compare large numbers, compare digits step-by-step from left to right:

• If the number of digits differs, the number with more digits is larger.
• If the number of digits is the same, compare digits in each place value.
• Use symbols: > (greater than), < (less than).
  Example: 99,999,999 < 111,111,111 (8 digits vs. 9 digits).
  Example: 800,000,008 > 288,888,882 (8 in hundred millions > 2 in hundred millions).
  Example: 76,529,456 < 312,763,459 (7 in ten millions < 3 in ten millions).

Ordering and Sequencing

• Ascending order means arranging numbers from smallest to biggest.
  Example: 467,345; 67,539,234; 219,212,303; 875,549,000; 1,000,000,000.
• Number sequences involve adding or subtracting a constant value:
  Example: The sequence 280,000; 370,000; 460,000; ...; 640,000; ... increases by 90,000 each time, so the missing numbers are 550,000 and 730,000.

Represent 6-Digit to 9-Digit Numbers

This section focuses on writing and rounding large numbers in different forms.

Writing Numbers in Words and Symbols

• Numbers can be written as words or symbols (standard form).
  Example: Three hundred sixty-four million two hundred thirty-four thousand five hundred sixty-seven = 364,234,567.
  Example: Eighty-nine million seven hundred five thousand nine hundred fifteen = 89,705,915.
  Example: Six hundred four million nine hundred ninety-seven thousand one hundred twenty-two = 604,997,122.
• Numbers can be represented using expanded notation (sum of place values):
  Example: 10,000,000 + 5,000,000 + 600,000 + 10,000 + 2000 + 900 + 50 + 2 = 15,612,952.
  Example: 300,000,000 + 7,000,000 + 200,000 + 30,000 + 400 + 2 = 307,230,402.
  Example: 40,000,000 + 6,000,000 + 100,000 + 50,000 + 3000 + 500 + 60 + 4 = 46,153,564.
  Example: 4,000,000 + 500,000 + 3000 + 200 + 80 + 7 = 4,503,287.
  Example: 100,000,000 + 60,000,000 + 400,000 + 600,000 + 8000 + 600 + 70 + 8 = 160,408,678.

Expanded Notation
Expanded notation breaks a number into its place value parts:

• Example: 790,538,209 = 700,000,000 + 90,000,000 + 500,000 + 30,000 + 8000 + 200 + 9.
• Example: 32,679,895 = 30,000,000 + 2,000,000 + 600,000 + 70,000 + 9000 + 800 + 90 + 5.
• Example: 435,034,975 = 400,000,000 + 30,000,000 + 5,000,000 + 30,000 + 4000 + 900 + 70 + 5.
• Example: 206,905,196 = 200,000,000 + 6,000,000 + 900,000 + 5000 + 100 + 90 + 6.
• Example: 76,004,781 = 70,000,000 + 6,000,000 + 4000 + 700 + 80 + 1.
• Example: 14,752,893 = 10,000,000 + 4,000,000 + 700,000 + 50,000 + 2000 + 800 + 90 + 3.

Rounding Numbers
Rounding simplifies a number to the nearest multiple of 5, 10, 100, or 1000:
If the digit in the place to the right is 5 or more, round up; if less than 5, round down.

• Example: For 28,387:
  Nearest 5: 28,387 (7 is closer to 5 than 0) → 28,385.
  Nearest 10: 28,387 (7 ≥ 5) → 28,390.
  Nearest 100: 28,387 (87 ≥ 50) → 28,400.
  Nearest 1000: 28,387 (387 < 500) → 28,000.
• Example: For 28,384:
  Nearest 5: 28,384 (4 is closer to 5) → 28,385.
  Nearest 10: 28,384 (4 < 5) → 28,380.
  Nearest 100: 28,384 (84 < 50) → 28,400.
  Nearest 1000: 28,384 (384 < 500) → 28,000.
• Example: For 42,368:
  Nearest 5: 42,368 (8 ≥ 5) → 42,370.
  Nearest 10: 42,368 (8 ≥ 5) → 42,370.
  Nearest 100: 42,368 (68 ≥ 50) → 42,400.
  Nearest 1000: 42,368 (368 < 500) → 42,000.
• Example: For 50,233:
  Nearest 5: 50,233 (3 < 5) → 50,230.
  Nearest 10: 50,233 (3 < 5) → 50,230.
  Nearest 100: 50,233 (33 < 50) → 50,200.
  Nearest 1000: 50,233 (233 < 500) → 50,000.

Multiples and Factors

This section explores how numbers are related through multiplication, focusing on factors and prime numbers.

Factors and Products

• When numbers are multiplied, the result is a product, and the numbers multiplied are factors.
  Example: 3 × 5 = 15 (3 and 5 are factors; 15 is the product).
  Example: 15 × 20 = 300 (15 and 20 are factors; 300 is the product).
• Every number has at least two factors: 1 and itself.
  Example: For 15, factors include 1, 3, 5, 15 (1 × 15 = 15, 3 × 5 = 15).
• A number can have multiple factor pairs:
  Example: For 300, factors include 1, 2, 3, 5, 6, 10, 15, 20, 25, 30, 50, 60, 75, 100, 150, 300.
  Pairs: 1 × 300, 2 × 150, 3 × 100, 5 × 60, 6 × 50, 10 × 30, 12 × 25, 15 × 20.

Example: For 40, factors are 1, 2, 4, 5, 8, 10, 20, 40 (e.g., 4 × 10 = 40).
Example: For 42, factors are 1, EO1 (e.g., 1 × 42 = 42).
Example: For 17, factors are 1, 17.
Example: For 18, factors are 1, 2, 3, 6, 9, 18.
Example: For 19, factors are 1, 19.

Prime Numbers
A prime number has exactly two distinct factors: 

• 1 and itself.
  Example: 17 is prime (factors: 1, 17).
  Example: 19 is prime (factors: 1, 19).
  Example: 13 is prime (factors: 1, 13).
  Example: 31 is prime (factors: 1, 31).
  Example: 23 is prime (factors: 1, 23).
• Non-prime numbers have more than two factors:
  Example: 32 (factors: 1, 2, 4, 8, 16, 32)
  Example: 39 (factors: 1, 3, 13, 39).
  Example: 93 (factors: 1, 3, 31, 93).

Points to Remember

• Place value: Each digit's position in a large number determines its value (e.g., in 738,264,111, 7 is 700,000,000).
• Comparing numbers: Compare digits from left to right; use > or < (e.g., 800,000,008 > 288,888,882).
• Ascending order: Arrange from smallest to largest (e.g., 467,345; 67,539,234; 1,000,000,000).
• Number representation: Write numbers as words (e.g., three hundred sixty-four million), symbols (e.g., 364,234,567), or expanded notation (e.g., 300,000,000 + 64,000,000).
• Rounding: Round to the nearest 5, 10, 100, or 1000 (e.g., 28,387 to nearest 10 is 28,390).
• Factors: Numbers that multiply to give a product (e.g., 3 and 5 are factors of 15).
• Prime numbers: Have only two factors, 1 and itself (e.g., 17, 19, 23).
• Multiples: Products of a number and any whole number (e.g., multiples of 15: 15, 30, 45).

Difficult Words

• Whole number: A number without fractions or decimals (e.g., 0, 1, 2, 100,000).
• Place value: The value of a digit based on its position (e.g., 7 in 700,000 is 7 hundred thousands).
• Expanded notation: Writing a number as the sum of its place values (e.g., 790,538,209 = 700,000,000 + 90,000,000 + 500,000 + 30,000 + 8000 + 200 + 9).
• Rounding: Simplifying a number to the nearest multiple (e.g., 28,387 to nearest 100 is 28,400).
• Factor: A number that divides another number exactly (e.g., 3 is a factor of 15).
• Product: The result of multiplication (e.g., 3 × 5 = 15, 15 is the product).
• Prime number: A number with exactly two factors, 1 and itself (e.g., 17).
• Ascending order: Arranging numbers from smallest to largest.

Summary

This chapter equips Grade 6 students with a comprehensive understanding of whole numbers, particularly large numbers up to 9 digits. Students learn to describe and compare numbers using place value (e.g., 100,000 has 100 thousands), order them (e.g., ascending order), and represent them in words, symbols, or expanded notation (e.g., 364,234,567). They practice rounding numbers to the nearest 5, 10, 100, or 1000 and explore multiples, factors, and prime numbers (e.g., 17 is prime with factors 1, 17). These skills build a strong foundation for working with large numbers and understanding their properties in real-world and mathematical contexts.