Whole Numbers (Term 1) Chapter Notes | Mathematics for Grade 6 PDF Download

Count and Represent Numbers

This section introduces whole numbers, how to count them, and how to represent them visually or in groups.

Whole Numbers

Definition: Whole numbers are numbers starting from 0 and going up (0, 1, 2, 3, ...), used for counting objects like rings or lines.

Explanation:

• Whole numbers include zero and all positive numbers without fractions or decimals.
• They are used to count items in everyday life, such as the number of rings in a grid or lines on a page.

Example: A grid with rings arranged in rows and columns can be counted to find the total number of rings.

Representing Numbers

Concept: Whole numbers can be represented visually using objects (e.g., rings) or grouped into tens, hundreds, or thousands.

Grouping:

• Numbers can be grouped to make counting easier. For example, 100 lines can be grouped into 10 groups of 10 lines each.
• This helps understand large numbers by breaking them into smaller, manageable parts.

Example:

• If a page has 50 lines, 10 pages have 10 × 50 = 500 lines, and 100 pages have 100 × 50 = 5000 lines.
• Grouping 50 lines into 5 groups of 10 lines each shows the structure of the number 50.

The Place Value Parts of Whole Numbers

This section explains how whole numbers are made up of place value parts and how to write them in expanded notation or as number names.

Place Value Parts

Definition: Whole numbers are composed of parts based on their place values (e.g., thousands, hundreds, tens, units).
Explanation:
Each digit in a number has a place value depending on its position.
For example, in 485627:

• 4 is in the hundred thousands place (400000).
• 8 is in the ten thousands place (80000).
• 5 is in the thousands place (5000).
• 6 is in the hundreds place (600).
• 2 is in the tens place (20).
• 7 is in the units place (7).

When written separately, these parts can be stacked to form the number, with zeros “hidden” in the final number symbol.

Example: The number 485627 is made of 400000 + 80000 + 5000 + 600 + 20 + 7.

Number Names

Concept: Numbers can be written as words using their place value parts.
Process:

• Break the number into groups (thousands, hundreds, tens, units).
• Combine the words for each part, using “and” to connect them.

Example: For 485627:

• 400000 = four hundred thousand.
• 80000 = eighty thousand.
• 5000 = five thousand.
• 600 = six hundred.
• 20 = twenty.
• 7 = seven.

Combined: “four hundred and eighty-five thousand six hundred and twenty-seven.”

Expanded Notation

Definition: Expanded notation expresses a number as the sum of its place value parts.
Explanation:

• Write each place value part separately and add them together.
• This shows how the number is built from its parts.

Example: For 485627:
Expanded notation: 485627 = 400000 + 80000 + 5000 + 600 + 20 + 7.

Ordering Numbers

Concept: Numbers can be compared and arranged from smallest to largest (or vice versa) based on their digits.
Process:

• Compare digits step-by-step, starting from the highest place value.
• For example, to compare 299999 and 311111, notice 311111 has a 3 in the hundred thousands place, while 299999 has a 2, so 311111 is larger.

Example: Arrange 124565, 210763, 401807 from smallest to largest:

• Compare digits: 124565 (1 in hundred thousands), 210763 (2), 401807 (4).
• Order: 124565, 210763, 401807.

Arrange Numbers in Order on Number Lines

This section covers how to place whole numbers on number lines and count in specific intervals.

Number Lines

Definition: A number line is a visual tool to show numbers in order, with equal spacing between them.
Explanation:

• Numbers are arranged from smallest to largest (often upward on vertical number lines).
• Each mark represents a number, and the distance between marks shows the counting interval (e.g., 1000s, 500s).

Example: On a number line from 230000 to 240000 with intervals of 1000:

• Marks are at 230000, 231000, 232000, ..., 240000.
• Missing numbers (e.g., 239000) can be filled in by counting in 1000s.

Counting Intervals

Concept: Counting intervals determine the difference between numbers on a number line (e.g., 10, 100, 1000).
Process:

• Identify the interval by subtracting consecutive numbers.
• Fill in missing numbers by adding or subtracting the interval.

Example: For a number line with 785000, 790000, 795000, 800000:

• Interval: 795000 - 790000 = 5000.
• Missing numbers are calculated by adding 5000 each time.

Placing Numbers on a Number Line

Concept: Numbers can be placed on a number line based on their value relative to the endpoints.
Example: On a number line from 1000 to 5000:

• 2000 is closer to 1000 than 5000, so it’s placed nearer the left.
• 3500 is halfway, placed in the middle.

Factors and Multiples

This section explains factors, multiples, and prime numbers, including how to identify them using calculations and a sieve.

Factors

Definition: Factors are whole numbers that divide exactly into a number with no remainder.
Explanation:

• Factors are also called divisors.
• For example, 91 = 7 × 13, so 7 and 13 are factors of 91.
• Every number has at least 1 and itself as factors.

Example: For 35:

• 35 = 1 × 35 and 35 = 5 × 7.
• Factors: 1, 5, 7, 35.

Multiples

Definition: Multiples are the results of multiplying a number by whole numbers (1, 2, 3, ...).
Explanation:

• For example, multiples of 2 are 2 × 1 = 2, 2 × 2 = 4, 2 × 3 = 6, etc.
• A number is a multiple of its factors. For example, 91 is a multiple of 7 and 13.

Example: First four multiples of 3:

• 1 × 3 = 3, 2 × 3 = 6, 3 × 3 = 9, 4 × 3 = 12.
• Next five: 15, 18, 21, 24, 27.

Prime and Composite Numbers

Prime Numbers:

• Numbers with exactly two factors: 1 and itself.
• Example: 2 (factors: 1, 2), 3 (1, 3), 5 (1, 5).
• List of primes less than 100: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

Composite Numbers:

• Numbers with more than two factors.
• Example: 4 (factors: 1, 2, 4), 6 (1, 2, 3, 6).

Special Case: 1 has only one factor (1), so it is neither prime nor composite.

Sieve of Eratosthenes

Concept: A method to find prime numbers by crossing out multiples of primes.
Process:

• List numbers from 2 to 100.
• Cross out multiples of 2 (greater than 2), then multiples of 3 (greater than 3), then 5, then 7, and so on.
• Numbers not crossed out are prime.

Example: After crossing out multiples of 2, 3, 5, and 7, remaining numbers include 2, 3, 5, 7, 11, etc.

Properties of 0 and 1

• Multiplicative Property of 1: Multiplying any number by 1 gives the number itself (e.g., 37 × 1 = 37).
• Additive Property of 0: Adding 0 to any number gives the number itself (e.g., 37 + 0 = 37).

Rounding Off

This section explains how to round whole numbers to the nearest 5, 10, 100, or 1000.

Rounding to the Nearest 10

Concept: Rounding to the nearest 10 means finding the closest multiple of 10.
Process:

• Look at the units digit.
• If it’s 1–4, round down; if it’s 5–9, round up.
• If the units digit is 5, round up to the larger multiple.

Example:

• 373: Units digit is 3, closer to 370 than 380, so rounds to 370.
• 365: Units digit is 5, rounds up to 370.
• 366: Units digit is 6, rounds up to 370.

Rounding to the Nearest 5

Concept: Rounding to the nearest 5 means finding the closest multiple of 5.
Process:

• Check if the number is closer to the lower or higher multiple of 5.
• If it ends in 2.5 or 7.5 (midway), round up.

Example:

• 373: Closer to 375 than 370, so rounds to 375.
• 372: Closer to 370, so rounds to 370.

Rounding to the Nearest 100

Concept: Rounding to the nearest 100 means finding the closest multiple of 100.
Process:

• Look at the tens digit.
• If the last two digits are 01–49, round down; if 50–99, round up.

Example:

• 353: Last two digits are 53, closer to 400, so rounds to 400.
• 349: Last two digits are 49, closer to 300, so rounds to 300.

Rounding to the Nearest 1000

Concept: Rounding to the nearest 1000 means finding the closest multiple of 1000.
Process:

• Look at the hundreds digit.
• If the last three digits are 001–499, round down; if 500–999, round up.

Example:

• 4501: Last three digits are 501, closer to 5000, so rounds to 5000.
• 4499: Last three digits are 499, closer to 4000, so rounds to 4000.

Points to Remember

• Whole numbers start from 0 and include all positive numbers without fractions (0, 1, 2, 3, ...).
• Numbers can be counted and represented by grouping into tens, hundreds, or thousands (e.g., 100 lines = 10 groups of 10).
• Place value parts show how a number is built (e.g., 485627 = 400000 + 80000 + 5000 + 600 + 20 + 7).
• Number names combine place value parts with words (e.g., 485627 is “four hundred and eighty-five thousand six hundred and twenty-seven”).
• Expanded notation writes a number as the sum of its place value parts.
• Number lines show numbers in order, with equal intervals (e.g., counting by 1000s from 230000 to 240000).
• Factors are numbers that divide exactly into a number (e.g., factors of 91 are 1, 7, 13, 91).
• Multiples are results of multiplying a number by whole numbers (e.g., multiples of 2: 2, 4, 6, 8, ...).
• Prime numbers have only two factors (1 and itself); composite numbers have more than two; 1 is neither.
• The Sieve of Eratosthenes finds prime numbers by crossing out multiples of 2, 3, 5, 7, etc.
• Multiplying by 1 gives the number itself; adding 0 gives the number itself.
• Rounding to the nearest 10, 5, 100, or 1000 depends on how close the number is to the nearest multiple (e.g., 373 rounds to 370 for 10, 375 for 5).
• Numbers ending in 5 round up when rounding to the nearest 10 or 5 (e.g., 365 rounds to 370).

Difficult Words

• Whole Numbers: Numbers starting from 0 (0, 1, 2, 3, ...) with no fractions or decimals.
• Place Value: The value of a digit based on its position in a number (e.g., thousands, hundreds).
• Expanded Notation: Writing a number as the sum of its place value parts (e.g., 485627 = 400000 + 80000 + ...).
• Number Line: A line showing numbers in order with equal spacing between them.
• Interval: The difference between numbers on a number line (e.g., 1000s, 500s).
• Factors: Numbers that divide exactly into a number with no remainder (e.g., 7 and 13 for 91).
• Multiples: Results of multiplying a number by whole numbers (e.g., 2, 4, 6 for 2).
• Prime Numbers: Numbers with only two factors: 1 and itself (e.g., 2, 3, 5).
• Composite Numbers: Numbers with more than two factors (e.g., 4, 6, 9).
• Sieve of Eratosthenes: A method to find prime numbers by crossing out multiples of primes.
• Rounding Off: Adjusting a number to the nearest multiple of 5, 10, 100, or 1000 for simplicity.
• Product: The result of multiplying two numbers (e.g., 91 is the product of 7 and 13).
• Divisor: A number that divides another number exactly (same as a factor).

Summary

Unit 1 introduces whole numbers, starting from 0 and used for counting objects like rings or lines. Numbers can be grouped (e.g., into tens or hundreds) and represented on number lines with equal intervals. Place value parts show how numbers are built, written as number names or in expanded notation (e.g., 485627 = 400000 + 80000 + ...). Numbers can be ordered by comparing digits. Factors divide a number exactly, while multiples are the results of multiplication; prime numbers have two factors, composite numbers have more, and 1 is unique. The Sieve of Eratosthenes identifies primes by eliminating multiples. The multiplicative property of 1 and additive property of 0 preserve a number’s value. Rounding off adjusts numbers to the nearest 5, 10, 100, or 1000, with numbers ending in 5 rounding up for 5 or 10. These concepts form the foundation for further number operations.