Solved Exercises on Whole Numbers Review in Grade 6

Master whole numbers: place value, operations, properties, and number systems through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Place Value
Exercise 1
Write the number 543,210 in expanded form using place value. Then write it in word form. What is the value of the digit 4?
Definition:

Place value: The value of a digit based on its position in a number

Place value system:
  1. Right to left: Ones, tens, hundreds, thousands, ten thousands, hundred thousands
  2. Value calculation: Digit × Place value
  3. Expanded form: Express number as sum of place values
  4. Word form: Write number using words
Place Value Breakdown:

543,210

5: Hundred thousands place = 5 × 100,000 = 500,000

4: Ten thousands place = 4 × 10,000 = 40,000

3: Thousands place = 3 × 1,000 = 3,000

2: Hundreds place = 2 × 100 = 200

1: Tens place = 1 × 10 = 10

0: Ones place = 0 × 1 = 0

Step 1: Identify each digit's place value

5 is in hundred thousands place

4 is in ten thousands place

3 is in thousands place

2 is in hundreds place

1 is in tens place

0 is in ones place

Step 2: Write expanded form

543,210 = 500,000 + 40,000 + 3,000 + 200 + 10 + 0

Step 3: Write word form

Five hundred forty-three thousand, two hundred ten

Step 4: Find value of digit 4

Digit 4 is in ten thousands place

Value of 4 = 4 × 10,000 = 40,000

Expanded: 500,000 + 40,000 + 3,000 + 200 + 10 + 0
Final answer:

Expanded form: 500,000 + 40,000 + 3,000 + 200 + 10 + 0

Word form: Five hundred forty-three thousand, two hundred ten

Value of digit 4: 40,000

Applied rules:

• Place value positions: Each position represents a power of 10

• Expanded notation: Sum of each digit × its place value

• Word form: Write number groups separately with commas

2 Order of Operations
Exercise 2
Evaluate: 24 ÷ 4 + 3 × (8 - 5) - 2². Show your work using the order of operations.
Definition:

Order of operations: PEMDAS - Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right)

PEMDAS Breakdown:

24 ÷ 4 + 3 × (8 - 5) - 2²

Step 1: Parentheses (8 - 5) = 3

Step 2: Exponents 2² = 4

Step 3: Division/Multiplication (left to right)

Step 4: Addition/Subtraction (left to right)

Step 1: Solve parentheses first

24 ÷ 4 + 3 × (8 - 5) - 2²

= 24 ÷ 4 + 3 × 3 - 2²

Step 2: Solve exponents

= 24 ÷ 4 + 3 × 3 - 4

Step 3: Perform multiplication and division (left to right)

24 ÷ 4 = 6

3 × 3 = 9

= 6 + 9 - 4

Step 4: Perform addition and subtraction (left to right)

6 + 9 = 15

15 - 4 = 11

Result: 11
Final answer:

24 ÷ 4 + 3 × (8 - 5) - 2² = 11

Applied rules:

• PEMDAS sequence: Follow the order strictly

• Left to right: When operations have same precedence

• Grouping symbols: Solve inside parentheses first

3 Number Properties
Exercise 3
Identify and explain which property is demonstrated by each equation: a) 7 + 0 = 7, b) 5 × 6 = 6 × 5, c) 3 × (4 + 2) = 3 × 4 + 3 × 2.
Definition:

Number properties: Fundamental rules that govern how numbers behave in operations

Property Analysis:

a) 7 + 0 = 7: Identity Property of Addition

b) 5 × 6 = 6 × 5: Commutative Property of Multiplication

c) 3 × (4 + 2) = 3 × 4 + 3 × 2: Distributive Property

Step 1: Analyze equation a) 7 + 0 = 7

Adding zero to any number gives the same number

This demonstrates the Identity Property of Addition

Step 2: Analyze equation b) 5 × 6 = 6 × 5

Changing the order of multiplication doesn't change the result

This demonstrates the Commutative Property of Multiplication

Step 3: Analyze equation c) 3 × (4 + 2) = 3 × 4 + 3 × 2

Multiplying a number by a sum equals multiplying the number by each addend

This demonstrates the Distributive Property

Step 4: Verify each property

Identity: a + 0 = a

Commutative: a × b = b × a

Distributive: a × (b + c) = a × b + a × c

a) Identity, b) Commutative, c) Distributive
Final answer:

a) Identity Property of Addition: Adding zero doesn't change the number

b) Commutative Property of Multiplication: Order doesn't affect product

c) Distributive Property: Multiplication distributes over addition

Applied rules:

• Identity Property: 0 for addition, 1 for multiplication

• Commutative Property: Order can be changed

• Distributive Property: Multiplication over addition

Number Systems and Properties
N = {0, 1, 2, 3, 4, 5, ...}
Whole Numbers Set
Property 1
Commutative
a + b = b + a
Property 2
Associative
(a + b) + c = a + (b + c)
Property 3
Distributive
a(b + c) = ab + ac
Key definitions:

Whole numbers: The set {0, 1, 2, 3, 4, 5, ...} - all positive integers including zero

Place value: The value of a digit based on its position in a number

Order of operations: The sequence in which operations are performed (PEMDAS)

Number properties: Fundamental rules governing number operations

Number System Hierarchy:
  1. Counting numbers: {1, 2, 3, 4, 5, ...}
  2. Whole numbers: {0, 1, 2, 3, 4, 5, ...}
  3. Integers: {..., -3, -2, -1, 0, 1, 2, 3, ...}
  4. Rational numbers: Numbers expressible as fractions
Tip 1: Remember that whole numbers include zero but do not include negative numbers.
Tip 2: When solving order of operations, rewrite the expression after each step to avoid confusion.
Tip 3: Use the acronym PEMDAS to remember the order of operations.
Tip 4: Properties help simplify calculations and verify results.
Common errors: Confusing order of operations, forgetting properties, misidentifying place values.
Success habits: Writing out steps, checking work, practicing regularly, using properties to simplify.
Essential Properties:

• Identity Property of Addition: a + 0 = a

• Identity Property of Multiplication: a × 1 = a

• Commutative Property of Addition: a + b = b + a

• Commutative Property of Multiplication: a × b = b × a

• Associative Property of Addition: (a + b) + c = a + (b + c)

• Associative Property of Multiplication: (a × b) × c = a × (b × c)

• Distributive Property: a × (b + c) = a × b + a × c

Solution: Exercises 4 to 5
4 Comparing Numbers
Exercise 4
Order these numbers from least to greatest: 54,321; 54,132; 54,231; 54,312. Explain your reasoning step by step.
Definition:

Comparing numbers: Determining which number is greater or lesser by comparing digits from left to right

Comparison Process:

Numbers: 54,321; 54,132; 54,231; 54,312

All start with 54 (ten thousands and thousands places)

Compare hundreds, tens, and ones places

Step 1: Compare first two digits (ten thousands and thousands)

All numbers start with 54, so continue to next digit

Step 2: Compare hundreds place

54,321: 3 in hundreds place

54,132: 1 in hundreds place

54,231: 2 in hundreds place

54,312: 3 in hundreds place

Smallest hundreds digit is 1, so 54,132 is smallest

Step 3: Among remaining numbers, compare tens place

54,231: 3 in tens place

54,321: 2 in tens place

54,312: 1 in tens place

So far: 54,132 < 54,312 < 54,321 < 54,231

Wait - let's reconsider 54,231 vs 54,321

Step 4: Correct comparison

54,132: 1 in hundreds place (smallest)

54,231: 2 in hundreds place

54,312: 3 in hundreds place

54,321: 3 in hundreds place

Between 54,312 and 54,321: 1 < 2 in tens place

Step 5: Final ordering

54,132 < 54,231 < 54,312 < 54,321

Least to greatest: 54,132; 54,231; 54,312; 54,321
Final answer:

From least to greatest: 54,132; 54,231; 54,312; 54,321

Applied rules:

• Left to right comparison: Compare digits from highest place value

• Place value importance: Higher place values determine order

• Systematic approach: Move to next place value only when digits are equal

5 Number Patterns
Exercise 5
The sequence begins: 5, 10, 17, 26, 37, ... What is the 10th term in this sequence? Describe the pattern and write the rule for the nth term.
Definition:

Number pattern: A sequence of numbers that follow a specific rule or relationship

Pattern Analysis:

Sequence: 5, 10, 17, 26, 37, ...

Differences: 5, 7, 9, 11 (increasing by 2)

Second differences: 2, 2, 2 (constant)

This suggests a quadratic pattern

Step 1: Calculate first differences

10 - 5 = 5

17 - 10 = 7

26 - 17 = 9

37 - 26 = 11

Step 2: Calculate second differences

7 - 5 = 2

9 - 7 = 2

11 - 9 = 2

Constant second differences indicate a quadratic pattern

Step 3: Find the quadratic formula

Since second difference is 2, the coefficient of n² is 2/2 = 1

Try: an² + bn + c

For n=1: a(1)² + b(1) + c = 5 → a + b + c = 5

For n=2: a(4) + b(2) + c = 10 → 4a + 2b + c = 10

For n=3: a(9) + b(3) + c = 17 → 9a + 3b + c = 17

Step 4: Solve the system of equations

With a=1: 1 + b + c = 5 → b + c = 4

4 + 2b + c = 10 → 2b + c = 6

Subtracting: b = 2, so c = 2

Formula: n² + 2n + 2

Step 5: Verify and find 10th term

Check: n=1: 1+2+2=5 ✓

n=2: 4+4+2=10 ✓

n=3: 9+6+2=17 ✓

10th term: 10² + 2(10) + 2 = 100 + 20 + 2 = 122

10th term: 122
Final answer:

The 10th term is 122. The pattern follows the rule: nth term = n² + 2n + 2.

Applied rules:

• First differences: Identify linear vs nonlinear patterns

• Second differences: Constant second differences indicate quadratic

• Pattern verification: Check formula against known terms

Complete Summary: Whole Numbers Review
W = {0, 1, 2, 3, 4, 5, ...}
Whole Numbers Set
Key definitions:

Whole numbers: The set of all positive integers including zero: {0, 1, 2, 3, 4, 5, ...}

Place value: The value of a digit based on its position in a number

Order of operations: The sequence in which mathematical operations are performed

Number properties: Fundamental rules that govern how numbers behave in operations

Complete methodology:
  1. Understanding place value: Recognize the value of each digit position
  2. Applying order of operations: Follow PEMDAS sequence precisely
  3. Identifying properties: Recognize and apply number properties
  4. Comparing numbers: Use systematic comparison from left to right
  5. Recognizing patterns: Look for relationships between consecutive terms
Tip 1: Practice writing numbers in expanded form to strengthen place value understanding.
Tip 2: Always rewrite expressions after each operation when using order of operations.
Tip 3: Use properties to simplify calculations mentally before doing them on paper.
Tip 4: When comparing large numbers, focus on the first differing digit from left to right.
Key concepts: Place value, operations, properties, comparison, and patterns form the foundation for higher mathematics.
Connections: These concepts connect to decimals, fractions, integers, and algebra.
Fundamental Rules:

• Place value: Each position represents 10 times the value of the position to its right

• PEMDAS: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction

• Properties: Commutative, Associative, Distributive, and Identity properties

• Comparison: Compare digits from left to right, highest place value first

• Patterns: Look for first differences, then second differences if needed

Questions & Answers

Question: I get confused about when to use the distributive property. How do I know when it will help me solve a problem?

Answer: The distributive property is helpful in several situations:

  • Multiplying mixed numbers: 3 × (4 + 2) = 3×4 + 3×2
  • Simplifying expressions: 5(2x + 3) = 10x + 15
  • Mental math: 8 × 19 = 8 × (20 - 1) = 8×20 - 8×1 = 160 - 8 = 152
  • Factoring: Working backwards: 12 + 18 = 6(2 + 3)

Look for expressions with parentheses containing addition or subtraction, especially when the number outside the parentheses doesn't divide evenly into the numbers inside.

The distributive property is also essential when solving equations and working with algebraic expressions. It helps break down complex problems into simpler parts.

Practice recognizing opportunities to use it - the more you use it, the more natural it becomes!

Question: Why do we need to learn the order of operations? Can't we just solve problems from left to right?

Answer: Without order of operations, the same expression could have multiple answers, which would cause chaos in mathematics! Consider this example:

Expression: 2 + 3 × 4

Left to right: (2 + 3) × 4 = 5 × 4 = 20

Order of operations: 2 + (3 × 4) = 2 + 12 = 14

The order of operations ensures that everyone gets the same answer for the same expression. It's like a universal agreement that mathematicians worldwide follow.

Without these rules:

  • Calculators would give different results
  • Scientific formulas wouldn't work consistently
  • Computer programs would produce unpredictable results
  • Mathematical communication would be impossible

The order of operations reflects how mathematical operations naturally interact with each other. Multiplication and division are more "powerful" than addition and subtraction, which is why they come first.

Question: I struggle with large numbers and place value. How can I better understand the value of each digit?

Answer: Here are strategies to master place value:

  1. Use place value charts: Write numbers in a chart with labeled columns
  2. Practice expanded form: Write numbers as sums (4,321 = 4,000 + 300 + 20 + 1)
  3. Read numbers aloud: Say the value of each digit as you read
  4. Use base-10 blocks: Visualize thousands, hundreds, tens, and ones
  5. Compare systematically: Start from the left when comparing numbers

Remember the pattern: ones, tens, hundreds, thousands, ten thousands, hundred thousands, millions...

Each position is 10 times the value of the position to its right. Practice with progressively larger numbers:

  • Start with 3-digit numbers
  • Move to 4-digit numbers
  • Progress to 5- and 6-digit numbers
  • Work with numbers up to millions

The more you practice writing numbers in different forms and identifying digit values, the more automatic it becomes!