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Chapter 1: Whole Numbers, Integers, and Number Sense Review

Learn Grade 8 Math from beginner foundations through advanced Grade 8 problem solving with detailed explanations, at least five examples per topic, practice exercises, and review questions.

Grade 8Beginner Friendly5+ Examples Per TopicPractice30 Q&A
Estimated reading time0% read
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What this chapter covers

This chapter contains 13 topics. Technical terms are followed by plain-language meanings where they first appear. Each topic includes at least five worked examples. Coding is included only where it naturally supports the Grade 8 coding expectations.

1.1 Whole numbers and place value

Whole numbers (0, 1, 2, 3, and so on) use place value to show the value of each digit. In this section, the focus is Whole numbers and place value.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: In 483,726, what is the value of the first digit 4?

  1. Count the places from the right.
  2. The digit 4 is in the 100,000 place.
  3. Multiply the digit by its place value.

Very beginner explanation: A digit's value depends on where it appears in the number.

Answer: 400,000

Worked Example 2

Problem: In 5,904,218, what is the value of the first digit 5?

  1. Count the places from the right.
  2. The digit 5 is in the 1,000,000 place.
  3. Multiply the digit by its place value.

Very beginner explanation: A digit's value depends on where it appears in the number.

Answer: 5,000,000

Worked Example 3

Problem: In 72,050,601, what is the value of the first digit 7?

  1. Count the places from the right.
  2. The digit 7 is in the 10,000,000 place.
  3. Multiply the digit by its place value.

Very beginner explanation: A digit's value depends on where it appears in the number.

Answer: 70,000,000

Worked Example 4

Problem: In 908,004,315, what is the value of the first digit 9?

  1. Count the places from the right.
  2. The digit 9 is in the 100,000,000 place.
  3. Multiply the digit by its place value.

Very beginner explanation: A digit's value depends on where it appears in the number.

Answer: 900,000,000

Worked Example 5

Problem: In 1,250,700,049, what is the value of the first digit 1?

  1. Count the places from the right.
  2. The digit 1 is in the 1,000,000,000 place.
  3. Multiply the digit by its place value.

Very beginner explanation: A digit's value depends on where it appears in the number.

Answer: 1,000,000,000

Worked Example 6

Problem: In 64,999, what is the value of the first digit 6?

  1. Count the places from the right.
  2. The digit 6 is in the 10,000 place.
  3. Multiply the digit by its place value.

Very beginner explanation: A digit's value depends on where it appears in the number.

Answer: 60,000

Worked Example 7

Problem: In 7,305,040, what is the value of the first digit 7?

  1. Count the places from the right.
  2. The digit 7 is in the 1,000,000 place.
  3. Multiply the digit by its place value.

Very beginner explanation: A digit's value depends on where it appears in the number.

Answer: 7,000,000

Worked Example 8

Problem: In 800,080, what is the value of the first digit 8?

  1. Count the places from the right.
  2. The digit 8 is in the 100,000 place.
  3. Multiply the digit by its place value.

Very beginner explanation: A digit's value depends on where it appears in the number.

Answer: 800,000

Worked Example 9

Problem: In 39,640,125, what is the value of the first digit 3?

  1. Count the places from the right.
  2. The digit 3 is in the 10,000,000 place.
  3. Multiply the digit by its place value.

Very beginner explanation: A digit's value depends on where it appears in the number.

Answer: 30,000,000

Worked Example 10

Problem: In 999,999,999, what is the value of the first digit 9?

  1. Count the places from the right.
  2. The digit 9 is in the 100,000,000 place.
  3. Multiply the digit by its place value.

Very beginner explanation: A digit's value depends on where it appears in the number.

Answer: 900,000,000

Practice exercise

Create one new Grade 8 problem involving Whole numbers and place value. Show the important steps, include units when needed, and explain how you checked the answer.

1.2 Reading and writing large numbers

Reading and writing large numbers is an important Grade 8 concept in Whole Numbers, Integers, and Number Sense Review. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: In 483,726, what is the value of the first digit 4?

  1. Count the places from the right.
  2. The digit 4 is in the 100,000 place.
  3. Multiply the digit by its place value.

Very beginner explanation: A digit's value depends on where it appears in the number.

Answer: 400,000

Worked Example 2

Problem: In 5,904,218, what is the value of the first digit 5?

  1. Count the places from the right.
  2. The digit 5 is in the 1,000,000 place.
  3. Multiply the digit by its place value.

Very beginner explanation: A digit's value depends on where it appears in the number.

Answer: 5,000,000

Worked Example 3

Problem: In 72,050,601, what is the value of the first digit 7?

  1. Count the places from the right.
  2. The digit 7 is in the 10,000,000 place.
  3. Multiply the digit by its place value.

Very beginner explanation: A digit's value depends on where it appears in the number.

Answer: 70,000,000

Worked Example 4

Problem: In 908,004,315, what is the value of the first digit 9?

  1. Count the places from the right.
  2. The digit 9 is in the 100,000,000 place.
  3. Multiply the digit by its place value.

Very beginner explanation: A digit's value depends on where it appears in the number.

Answer: 900,000,000

Worked Example 5

Problem: In 1,250,700,049, what is the value of the first digit 1?

  1. Count the places from the right.
  2. The digit 1 is in the 1,000,000,000 place.
  3. Multiply the digit by its place value.

Very beginner explanation: A digit's value depends on where it appears in the number.

Answer: 1,000,000,000

Worked Example 6

Problem: In 64,999, what is the value of the first digit 6?

  1. Count the places from the right.
  2. The digit 6 is in the 10,000 place.
  3. Multiply the digit by its place value.

Very beginner explanation: A digit's value depends on where it appears in the number.

Answer: 60,000

Worked Example 7

Problem: In 7,305,040, what is the value of the first digit 7?

  1. Count the places from the right.
  2. The digit 7 is in the 1,000,000 place.
  3. Multiply the digit by its place value.

Very beginner explanation: A digit's value depends on where it appears in the number.

Answer: 7,000,000

Worked Example 8

Problem: In 800,080, what is the value of the first digit 8?

  1. Count the places from the right.
  2. The digit 8 is in the 100,000 place.
  3. Multiply the digit by its place value.

Very beginner explanation: A digit's value depends on where it appears in the number.

Answer: 800,000

Worked Example 9

Problem: In 39,640,125, what is the value of the first digit 3?

  1. Count the places from the right.
  2. The digit 3 is in the 10,000,000 place.
  3. Multiply the digit by its place value.

Very beginner explanation: A digit's value depends on where it appears in the number.

Answer: 30,000,000

Worked Example 10

Problem: In 999,999,999, what is the value of the first digit 9?

  1. Count the places from the right.
  2. The digit 9 is in the 100,000,000 place.
  3. Multiply the digit by its place value.

Very beginner explanation: A digit's value depends on where it appears in the number.

Answer: 900,000,000

Practice exercise

Create one new Grade 8 problem involving Reading and writing large numbers. Show the important steps, include units when needed, and explain how you checked the answer.

1.3 Expanded form

Expanded form is an important Grade 8 concept in Whole Numbers, Integers, and Number Sense Review. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Write 483,726 in expanded form.

  1. Look at each non-zero digit.
  2. Write the value of each digit.
  3. Join the values with plus signs.

Very beginner explanation: Expanded form shows what each digit is worth because of its place.

Answer: 400,000 + 80,000 + 3,000 + 700 + 20 + 6

Worked Example 2

Problem: Write 5,904,218 in expanded form.

  1. Look at each non-zero digit.
  2. Write the value of each digit.
  3. Join the values with plus signs.

Very beginner explanation: Expanded form shows what each digit is worth because of its place.

Answer: 5,000,000 + 900,000 + 4,000 + 200 + 10 + 8

Worked Example 3

Problem: Write 72,050,601 in expanded form.

  1. Look at each non-zero digit.
  2. Write the value of each digit.
  3. Join the values with plus signs.

Very beginner explanation: Expanded form shows what each digit is worth because of its place.

Answer: 70,000,000 + 2,000,000 + 50,000 + 600 + 1

Worked Example 4

Problem: Write 908,004,315 in expanded form.

  1. Look at each non-zero digit.
  2. Write the value of each digit.
  3. Join the values with plus signs.

Very beginner explanation: Expanded form shows what each digit is worth because of its place.

Answer: 900,000,000 + 8,000,000 + 4,000 + 300 + 10 + 5

Worked Example 5

Problem: Write 1,250,700,049 in expanded form.

  1. Look at each non-zero digit.
  2. Write the value of each digit.
  3. Join the values with plus signs.

Very beginner explanation: Expanded form shows what each digit is worth because of its place.

Answer: 1,000,000,000 + 200,000,000 + 50,000,000 + 700,000 + 40 + 9

Worked Example 6

Problem: Write 64,999 in expanded form.

  1. Look at each non-zero digit.
  2. Write the value of each digit.
  3. Join the values with plus signs.

Very beginner explanation: Expanded form shows what each digit is worth because of its place.

Answer: 60,000 + 4,000 + 900 + 90 + 9

Worked Example 7

Problem: Write 7,305,040 in expanded form.

  1. Look at each non-zero digit.
  2. Write the value of each digit.
  3. Join the values with plus signs.

Very beginner explanation: Expanded form shows what each digit is worth because of its place.

Answer: 7,000,000 + 300,000 + 5,000 + 40

Worked Example 8

Problem: Write 800,080 in expanded form.

  1. Look at each non-zero digit.
  2. Write the value of each digit.
  3. Join the values with plus signs.

Very beginner explanation: Expanded form shows what each digit is worth because of its place.

Answer: 800,000 + 80

Worked Example 9

Problem: Write 39,640,125 in expanded form.

  1. Look at each non-zero digit.
  2. Write the value of each digit.
  3. Join the values with plus signs.

Very beginner explanation: Expanded form shows what each digit is worth because of its place.

Answer: 30,000,000 + 9,000,000 + 600,000 + 40,000 + 100 + 20 + 5

Worked Example 10

Problem: Write 999,999,999 in expanded form.

  1. Look at each non-zero digit.
  2. Write the value of each digit.
  3. Join the values with plus signs.

Very beginner explanation: Expanded form shows what each digit is worth because of its place.

Answer: 900,000,000 + 90,000,000 + 9,000,000 + 900,000 + 90,000 + 9,000 + 900 + 90 + 9

Practice exercise

Create one new Grade 8 problem involving Expanded form. Show the important steps, include units when needed, and explain how you checked the answer.

1.4 Comparing whole numbers

Whole numbers (0, 1, 2, 3, and so on) use place value to show the value of each digit. In this section, the focus is Comparing whole numbers.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Compare 483,726 and 483,405.

  1. Compare the greatest place values first.
  2. If those are equal, move one place to the right until the digits differ.

Very beginner explanation: For positive whole numbers, the first larger digit from the left determines the larger number.

Answer: 483,726 > 483,405

Worked Example 2

Problem: Compare 5,904,218 and 5,903,576.

  1. Compare the greatest place values first.
  2. If those are equal, move one place to the right until the digits differ.

Very beginner explanation: For positive whole numbers, the first larger digit from the left determines the larger number.

Answer: 5,904,218 > 5,903,576

Worked Example 3

Problem: Compare 72,050,601 and 72,049,638.

  1. Compare the greatest place values first.
  2. If those are equal, move one place to the right until the digits differ.

Very beginner explanation: For positive whole numbers, the first larger digit from the left determines the larger number.

Answer: 72,050,601 > 72,049,638

Worked Example 4

Problem: Compare 908,004,315 and 908,003,031.

  1. Compare the greatest place values first.
  2. If those are equal, move one place to the right until the digits differ.

Very beginner explanation: For positive whole numbers, the first larger digit from the left determines the larger number.

Answer: 908,004,315 > 908,003,031

Worked Example 5

Problem: Compare 1,250,700,049 and 1,250,698,444.

  1. Compare the greatest place values first.
  2. If those are equal, move one place to the right until the digits differ.

Very beginner explanation: For positive whole numbers, the first larger digit from the left determines the larger number.

Answer: 1,250,700,049 > 1,250,698,444

Worked Example 6

Problem: Compare 64,999 and 63,073.

  1. Compare the greatest place values first.
  2. If those are equal, move one place to the right until the digits differ.

Very beginner explanation: For positive whole numbers, the first larger digit from the left determines the larger number.

Answer: 64,999 > 63,073

Worked Example 7

Problem: Compare 7,305,040 and 7,302,793.

  1. Compare the greatest place values first.
  2. If those are equal, move one place to the right until the digits differ.

Very beginner explanation: For positive whole numbers, the first larger digit from the left determines the larger number.

Answer: 7,305,040 > 7,302,793

Worked Example 8

Problem: Compare 800,080 and 797,512.

  1. Compare the greatest place values first.
  2. If those are equal, move one place to the right until the digits differ.

Very beginner explanation: For positive whole numbers, the first larger digit from the left determines the larger number.

Answer: 800,080 > 797,512

Worked Example 9

Problem: Compare 39,640,125 and 39,637,236.

  1. Compare the greatest place values first.
  2. If those are equal, move one place to the right until the digits differ.

Very beginner explanation: For positive whole numbers, the first larger digit from the left determines the larger number.

Answer: 39,640,125 > 39,637,236

Worked Example 10

Problem: Compare 999,999,999 and 999,996,789.

  1. Compare the greatest place values first.
  2. If those are equal, move one place to the right until the digits differ.

Very beginner explanation: For positive whole numbers, the first larger digit from the left determines the larger number.

Answer: 999,999,999 > 999,996,789

Practice exercise

Create one new Grade 8 problem involving Comparing whole numbers. Show the important steps, include units when needed, and explain how you checked the answer.

1.5 Ordering whole numbers

Whole numbers (0, 1, 2, 3, and so on) use place value to show the value of each digit. In this section, the focus is Ordering whole numbers.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Compare 483,726 and 483,405.

  1. Compare the greatest place values first.
  2. If those are equal, move one place to the right until the digits differ.

Very beginner explanation: For positive whole numbers, the first larger digit from the left determines the larger number.

Answer: 483,726 > 483,405

Worked Example 2

Problem: Compare 5,904,218 and 5,903,576.

  1. Compare the greatest place values first.
  2. If those are equal, move one place to the right until the digits differ.

Very beginner explanation: For positive whole numbers, the first larger digit from the left determines the larger number.

Answer: 5,904,218 > 5,903,576

Worked Example 3

Problem: Compare 72,050,601 and 72,049,638.

  1. Compare the greatest place values first.
  2. If those are equal, move one place to the right until the digits differ.

Very beginner explanation: For positive whole numbers, the first larger digit from the left determines the larger number.

Answer: 72,050,601 > 72,049,638

Worked Example 4

Problem: Compare 908,004,315 and 908,003,031.

  1. Compare the greatest place values first.
  2. If those are equal, move one place to the right until the digits differ.

Very beginner explanation: For positive whole numbers, the first larger digit from the left determines the larger number.

Answer: 908,004,315 > 908,003,031

Worked Example 5

Problem: Compare 1,250,700,049 and 1,250,698,444.

  1. Compare the greatest place values first.
  2. If those are equal, move one place to the right until the digits differ.

Very beginner explanation: For positive whole numbers, the first larger digit from the left determines the larger number.

Answer: 1,250,700,049 > 1,250,698,444

Worked Example 6

Problem: Compare 64,999 and 63,073.

  1. Compare the greatest place values first.
  2. If those are equal, move one place to the right until the digits differ.

Very beginner explanation: For positive whole numbers, the first larger digit from the left determines the larger number.

Answer: 64,999 > 63,073

Worked Example 7

Problem: Compare 7,305,040 and 7,302,793.

  1. Compare the greatest place values first.
  2. If those are equal, move one place to the right until the digits differ.

Very beginner explanation: For positive whole numbers, the first larger digit from the left determines the larger number.

Answer: 7,305,040 > 7,302,793

Worked Example 8

Problem: Compare 800,080 and 797,512.

  1. Compare the greatest place values first.
  2. If those are equal, move one place to the right until the digits differ.

Very beginner explanation: For positive whole numbers, the first larger digit from the left determines the larger number.

Answer: 800,080 > 797,512

Worked Example 9

Problem: Compare 39,640,125 and 39,637,236.

  1. Compare the greatest place values first.
  2. If those are equal, move one place to the right until the digits differ.

Very beginner explanation: For positive whole numbers, the first larger digit from the left determines the larger number.

Answer: 39,640,125 > 39,637,236

Worked Example 10

Problem: Compare 999,999,999 and 999,996,789.

  1. Compare the greatest place values first.
  2. If those are equal, move one place to the right until the digits differ.

Very beginner explanation: For positive whole numbers, the first larger digit from the left determines the larger number.

Answer: 999,999,999 > 999,996,789

Practice exercise

Create one new Grade 8 problem involving Ordering whole numbers. Show the important steps, include units when needed, and explain how you checked the answer.

1.6 Rounding whole numbers

Whole numbers (0, 1, 2, 3, and so on) use place value to show the value of each digit. In this section, the focus is Rounding whole numbers.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Round 483,726 to the nearest 10.

  1. Find the 10 place.
  2. Look one digit to the right.
  3. 0–4 means keep the digit; 5–9 means increase it by 1.
  4. Replace lower places with zeros.

Very beginner explanation: Rounding replaces a number with a nearby easier number while keeping its size reasonable.

Answer: 483,730

Worked Example 2

Problem: Round 5,904,218 to the nearest 100.

  1. Find the 100 place.
  2. Look one digit to the right.
  3. 0–4 means keep the digit; 5–9 means increase it by 1.
  4. Replace lower places with zeros.

Very beginner explanation: Rounding replaces a number with a nearby easier number while keeping its size reasonable.

Answer: 5,904,200

Worked Example 3

Problem: Round 72,050,601 to the nearest 1,000.

  1. Find the 1,000 place.
  2. Look one digit to the right.
  3. 0–4 means keep the digit; 5–9 means increase it by 1.
  4. Replace lower places with zeros.

Very beginner explanation: Rounding replaces a number with a nearby easier number while keeping its size reasonable.

Answer: 72,051,000

Worked Example 4

Problem: Round 908,004,315 to the nearest 10,000.

  1. Find the 10,000 place.
  2. Look one digit to the right.
  3. 0–4 means keep the digit; 5–9 means increase it by 1.
  4. Replace lower places with zeros.

Very beginner explanation: Rounding replaces a number with a nearby easier number while keeping its size reasonable.

Answer: 908,000,000

Worked Example 5

Problem: Round 1,250,700,049 to the nearest 100,000.

  1. Find the 100,000 place.
  2. Look one digit to the right.
  3. 0–4 means keep the digit; 5–9 means increase it by 1.
  4. Replace lower places with zeros.

Very beginner explanation: Rounding replaces a number with a nearby easier number while keeping its size reasonable.

Answer: 1,250,700,000

Worked Example 6

Problem: Round 64,999 to the nearest 10.

  1. Find the 10 place.
  2. Look one digit to the right.
  3. 0–4 means keep the digit; 5–9 means increase it by 1.
  4. Replace lower places with zeros.

Very beginner explanation: Rounding replaces a number with a nearby easier number while keeping its size reasonable.

Answer: 65,000

Worked Example 7

Problem: Round 7,305,040 to the nearest 100.

  1. Find the 100 place.
  2. Look one digit to the right.
  3. 0–4 means keep the digit; 5–9 means increase it by 1.
  4. Replace lower places with zeros.

Very beginner explanation: Rounding replaces a number with a nearby easier number while keeping its size reasonable.

Answer: 7,305,000

Worked Example 8

Problem: Round 800,080 to the nearest 1,000.

  1. Find the 1,000 place.
  2. Look one digit to the right.
  3. 0–4 means keep the digit; 5–9 means increase it by 1.
  4. Replace lower places with zeros.

Very beginner explanation: Rounding replaces a number with a nearby easier number while keeping its size reasonable.

Answer: 800,000

Worked Example 9

Problem: Round 39,640,125 to the nearest 10,000.

  1. Find the 10,000 place.
  2. Look one digit to the right.
  3. 0–4 means keep the digit; 5–9 means increase it by 1.
  4. Replace lower places with zeros.

Very beginner explanation: Rounding replaces a number with a nearby easier number while keeping its size reasonable.

Answer: 39,640,000

Worked Example 10

Problem: Round 999,999,999 to the nearest 100,000.

  1. Find the 100,000 place.
  2. Look one digit to the right.
  3. 0–4 means keep the digit; 5–9 means increase it by 1.
  4. Replace lower places with zeros.

Very beginner explanation: Rounding replaces a number with a nearby easier number while keeping its size reasonable.

Answer: 1,000,000,000

Practice exercise

Create one new Grade 8 problem involving Rounding whole numbers. Show the important steps, include units when needed, and explain how you checked the answer.

1.7 Integer review

Integers (negative whole numbers, zero, and positive whole numbers) are useful for temperature, elevation, scores, gains, and losses. In this section, the focus is Integer review.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Calculate -7 + (5).

  1. If signs match, add absolute values and keep the sign.
  2. If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -2

Worked Example 2

Problem: Calculate 4 + (-9).

  1. If signs match, add absolute values and keep the sign.
  2. If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -5

Worked Example 3

Problem: Calculate -6 + (-3).

  1. If signs match, add absolute values and keep the sign.
  2. If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -9

Worked Example 4

Problem: Calculate 12 + (-8).

  1. If signs match, add absolute values and keep the sign.
  2. If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: 4

Worked Example 5

Problem: Calculate -15 + (20).

  1. If signs match, add absolute values and keep the sign.
  2. If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: 5

Worked Example 6

Problem: Calculate -11 + (-4).

  1. If signs match, add absolute values and keep the sign.
  2. If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -15

Worked Example 7

Problem: Calculate 9 + (13).

  1. If signs match, add absolute values and keep the sign.
  2. If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: 22

Worked Example 8

Problem: Calculate -2 + (7).

  1. If signs match, add absolute values and keep the sign.
  2. If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: 5

Worked Example 9

Problem: Calculate 18 + (-25).

  1. If signs match, add absolute values and keep the sign.
  2. If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -7

Worked Example 10

Problem: Calculate -30 + (12).

  1. If signs match, add absolute values and keep the sign.
  2. If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -18

Practice exercise

Create one new Grade 8 problem involving Integer review. Show the important steps, include units when needed, and explain how you checked the answer.

1.8 Positive and negative integers

Integers (negative whole numbers, zero, and positive whole numbers) are useful for temperature, elevation, scores, gains, and losses. In this section, the focus is Positive and negative integers.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Calculate -7 + (5).

  1. If signs match, add absolute values and keep the sign.
  2. If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -2

Worked Example 2

Problem: Calculate 4 + (-9).

  1. If signs match, add absolute values and keep the sign.
  2. If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -5

Worked Example 3

Problem: Calculate -6 + (-3).

  1. If signs match, add absolute values and keep the sign.
  2. If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -9

Worked Example 4

Problem: Calculate 12 + (-8).

  1. If signs match, add absolute values and keep the sign.
  2. If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: 4

Worked Example 5

Problem: Calculate -15 + (20).

  1. If signs match, add absolute values and keep the sign.
  2. If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: 5

Worked Example 6

Problem: Calculate -11 + (-4).

  1. If signs match, add absolute values and keep the sign.
  2. If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -15

Worked Example 7

Problem: Calculate 9 + (13).

  1. If signs match, add absolute values and keep the sign.
  2. If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: 22

Worked Example 8

Problem: Calculate -2 + (7).

  1. If signs match, add absolute values and keep the sign.
  2. If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: 5

Worked Example 9

Problem: Calculate 18 + (-25).

  1. If signs match, add absolute values and keep the sign.
  2. If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -7

Worked Example 10

Problem: Calculate -30 + (12).

  1. If signs match, add absolute values and keep the sign.
  2. If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -18

Practice exercise

Create one new Grade 8 problem involving Positive and negative integers. Show the important steps, include units when needed, and explain how you checked the answer.

1.9 Absolute value

Absolute value (distance from zero on a number line) is always zero or positive. In this section, the focus is Absolute value.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Find |-7|.

  1. Absolute value means distance from zero.
  2. -7 is 7 units from zero.

Very beginner explanation: Distance is never negative, so absolute value is zero or positive.

Answer: 7

Worked Example 2

Problem: Find |4|.

  1. Absolute value means distance from zero.
  2. 4 is 4 units from zero.

Very beginner explanation: Distance is never negative, so absolute value is zero or positive.

Answer: 4

Worked Example 3

Problem: Find |-6|.

  1. Absolute value means distance from zero.
  2. -6 is 6 units from zero.

Very beginner explanation: Distance is never negative, so absolute value is zero or positive.

Answer: 6

Worked Example 4

Problem: Find |12|.

  1. Absolute value means distance from zero.
  2. 12 is 12 units from zero.

Very beginner explanation: Distance is never negative, so absolute value is zero or positive.

Answer: 12

Worked Example 5

Problem: Find |-15|.

  1. Absolute value means distance from zero.
  2. -15 is 15 units from zero.

Very beginner explanation: Distance is never negative, so absolute value is zero or positive.

Answer: 15

Worked Example 6

Problem: Find |-11|.

  1. Absolute value means distance from zero.
  2. -11 is 11 units from zero.

Very beginner explanation: Distance is never negative, so absolute value is zero or positive.

Answer: 11

Worked Example 7

Problem: Find |9|.

  1. Absolute value means distance from zero.
  2. 9 is 9 units from zero.

Very beginner explanation: Distance is never negative, so absolute value is zero or positive.

Answer: 9

Worked Example 8

Problem: Find |-2|.

  1. Absolute value means distance from zero.
  2. -2 is 2 units from zero.

Very beginner explanation: Distance is never negative, so absolute value is zero or positive.

Answer: 2

Worked Example 9

Problem: Find |18|.

  1. Absolute value means distance from zero.
  2. 18 is 18 units from zero.

Very beginner explanation: Distance is never negative, so absolute value is zero or positive.

Answer: 18

Worked Example 10

Problem: Find |-30|.

  1. Absolute value means distance from zero.
  2. -30 is 30 units from zero.

Very beginner explanation: Distance is never negative, so absolute value is zero or positive.

Answer: 30

Practice exercise

Create one new Grade 8 problem involving Absolute value. Show the important steps, include units when needed, and explain how you checked the answer.

1.10 Comparing integers

Integers (negative whole numbers, zero, and positive whole numbers) are useful for temperature, elevation, scores, gains, and losses. In this section, the focus is Comparing integers.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Compare -7 and 5.

  1. Imagine both numbers on a number line.
  2. The number farther right is greater.

Very beginner explanation: Negative numbers farther from zero are smaller when they are to the left.

Answer: -7 < 5

Worked Example 2

Problem: Compare 4 and -9.

  1. Imagine both numbers on a number line.
  2. The number farther right is greater.

Very beginner explanation: Negative numbers farther from zero are smaller when they are to the left.

Answer: 4 > -9

Worked Example 3

Problem: Compare -6 and -3.

  1. Imagine both numbers on a number line.
  2. The number farther right is greater.

Very beginner explanation: Negative numbers farther from zero are smaller when they are to the left.

Answer: -6 < -3

Worked Example 4

Problem: Compare 12 and -8.

  1. Imagine both numbers on a number line.
  2. The number farther right is greater.

Very beginner explanation: Negative numbers farther from zero are smaller when they are to the left.

Answer: 12 > -8

Worked Example 5

Problem: Compare -15 and 20.

  1. Imagine both numbers on a number line.
  2. The number farther right is greater.

Very beginner explanation: Negative numbers farther from zero are smaller when they are to the left.

Answer: -15 < 20

Worked Example 6

Problem: Compare -11 and -4.

  1. Imagine both numbers on a number line.
  2. The number farther right is greater.

Very beginner explanation: Negative numbers farther from zero are smaller when they are to the left.

Answer: -11 < -4

Worked Example 7

Problem: Compare 9 and 13.

  1. Imagine both numbers on a number line.
  2. The number farther right is greater.

Very beginner explanation: Negative numbers farther from zero are smaller when they are to the left.

Answer: 9 < 13

Worked Example 8

Problem: Compare -2 and 7.

  1. Imagine both numbers on a number line.
  2. The number farther right is greater.

Very beginner explanation: Negative numbers farther from zero are smaller when they are to the left.

Answer: -2 < 7

Worked Example 9

Problem: Compare 18 and -25.

  1. Imagine both numbers on a number line.
  2. The number farther right is greater.

Very beginner explanation: Negative numbers farther from zero are smaller when they are to the left.

Answer: 18 > -25

Worked Example 10

Problem: Compare -30 and 12.

  1. Imagine both numbers on a number line.
  2. The number farther right is greater.

Very beginner explanation: Negative numbers farther from zero are smaller when they are to the left.

Answer: -30 < 12

Practice exercise

Create one new Grade 8 problem involving Comparing integers. Show the important steps, include units when needed, and explain how you checked the answer.

1.11 Ordering integers

Integers (negative whole numbers, zero, and positive whole numbers) are useful for temperature, elevation, scores, gains, and losses. In this section, the focus is Ordering integers.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Compare -7 and 5.

  1. Imagine both numbers on a number line.
  2. The number farther right is greater.

Very beginner explanation: Negative numbers farther from zero are smaller when they are to the left.

Answer: -7 < 5

Worked Example 2

Problem: Compare 4 and -9.

  1. Imagine both numbers on a number line.
  2. The number farther right is greater.

Very beginner explanation: Negative numbers farther from zero are smaller when they are to the left.

Answer: 4 > -9

Worked Example 3

Problem: Compare -6 and -3.

  1. Imagine both numbers on a number line.
  2. The number farther right is greater.

Very beginner explanation: Negative numbers farther from zero are smaller when they are to the left.

Answer: -6 < -3

Worked Example 4

Problem: Compare 12 and -8.

  1. Imagine both numbers on a number line.
  2. The number farther right is greater.

Very beginner explanation: Negative numbers farther from zero are smaller when they are to the left.

Answer: 12 > -8

Worked Example 5

Problem: Compare -15 and 20.

  1. Imagine both numbers on a number line.
  2. The number farther right is greater.

Very beginner explanation: Negative numbers farther from zero are smaller when they are to the left.

Answer: -15 < 20

Worked Example 6

Problem: Compare -11 and -4.

  1. Imagine both numbers on a number line.
  2. The number farther right is greater.

Very beginner explanation: Negative numbers farther from zero are smaller when they are to the left.

Answer: -11 < -4

Worked Example 7

Problem: Compare 9 and 13.

  1. Imagine both numbers on a number line.
  2. The number farther right is greater.

Very beginner explanation: Negative numbers farther from zero are smaller when they are to the left.

Answer: 9 < 13

Worked Example 8

Problem: Compare -2 and 7.

  1. Imagine both numbers on a number line.
  2. The number farther right is greater.

Very beginner explanation: Negative numbers farther from zero are smaller when they are to the left.

Answer: -2 < 7

Worked Example 9

Problem: Compare 18 and -25.

  1. Imagine both numbers on a number line.
  2. The number farther right is greater.

Very beginner explanation: Negative numbers farther from zero are smaller when they are to the left.

Answer: 18 > -25

Worked Example 10

Problem: Compare -30 and 12.

  1. Imagine both numbers on a number line.
  2. The number farther right is greater.

Very beginner explanation: Negative numbers farther from zero are smaller when they are to the left.

Answer: -30 < 12

Practice exercise

Create one new Grade 8 problem involving Ordering integers. Show the important steps, include units when needed, and explain how you checked the answer.

1.12 Number-line reasoning

Number-line reasoning is an important Grade 8 concept in Whole Numbers, Integers, and Number Sense Review. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Which point is farther right on a number line: 1,000 or 1,250?

  1. Numbers increase as you move right.
  2. 1,250 is greater than 1,000.

Very beginner explanation: On a standard number line, greater numbers are placed farther to the right.

Answer: 1,250

Worked Example 2

Problem: Which point is farther right on a number line: 2,000 or 2,250?

  1. Numbers increase as you move right.
  2. 2,250 is greater than 2,000.

Very beginner explanation: On a standard number line, greater numbers are placed farther to the right.

Answer: 2,250

Worked Example 3

Problem: Which point is farther right on a number line: 3,000 or 3,250?

  1. Numbers increase as you move right.
  2. 3,250 is greater than 3,000.

Very beginner explanation: On a standard number line, greater numbers are placed farther to the right.

Answer: 3,250

Worked Example 4

Problem: Which point is farther right on a number line: 4,000 or 4,250?

  1. Numbers increase as you move right.
  2. 4,250 is greater than 4,000.

Very beginner explanation: On a standard number line, greater numbers are placed farther to the right.

Answer: 4,250

Worked Example 5

Problem: Which point is farther right on a number line: 5,000 or 5,250?

  1. Numbers increase as you move right.
  2. 5,250 is greater than 5,000.

Very beginner explanation: On a standard number line, greater numbers are placed farther to the right.

Answer: 5,250

Worked Example 6

Problem: Which point is farther right on a number line: 6,000 or 6,250?

  1. Numbers increase as you move right.
  2. 6,250 is greater than 6,000.

Very beginner explanation: On a standard number line, greater numbers are placed farther to the right.

Answer: 6,250

Worked Example 7

Problem: Which point is farther right on a number line: 7,000 or 7,250?

  1. Numbers increase as you move right.
  2. 7,250 is greater than 7,000.

Very beginner explanation: On a standard number line, greater numbers are placed farther to the right.

Answer: 7,250

Worked Example 8

Problem: Which point is farther right on a number line: 8,000 or 8,250?

  1. Numbers increase as you move right.
  2. 8,250 is greater than 8,000.

Very beginner explanation: On a standard number line, greater numbers are placed farther to the right.

Answer: 8,250

Worked Example 9

Problem: Which point is farther right on a number line: 9,000 or 9,250?

  1. Numbers increase as you move right.
  2. 9,250 is greater than 9,000.

Very beginner explanation: On a standard number line, greater numbers are placed farther to the right.

Answer: 9,250

Worked Example 10

Problem: Which point is farther right on a number line: 10,000 or 10,250?

  1. Numbers increase as you move right.
  2. 10,250 is greater than 10,000.

Very beginner explanation: On a standard number line, greater numbers are placed farther to the right.

Answer: 10,250

Practice exercise

Create one new Grade 8 problem involving Number-line reasoning. Show the important steps, include units when needed, and explain how you checked the answer.

1.13 Estimation and reasonableness

Estimation and reasonableness is an important Grade 8 concept in Whole Numbers, Integers, and Number Sense Review. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Round 483,726 to the nearest 10.

  1. Find the 10 place.
  2. Look one digit to the right.
  3. 0–4 means keep the digit; 5–9 means increase it by 1.
  4. Replace lower places with zeros.

Very beginner explanation: Rounding replaces a number with a nearby easier number while keeping its size reasonable.

Answer: 483,730

Worked Example 2

Problem: Round 5,904,218 to the nearest 100.

  1. Find the 100 place.
  2. Look one digit to the right.
  3. 0–4 means keep the digit; 5–9 means increase it by 1.
  4. Replace lower places with zeros.

Very beginner explanation: Rounding replaces a number with a nearby easier number while keeping its size reasonable.

Answer: 5,904,200

Worked Example 3

Problem: Round 72,050,601 to the nearest 1,000.

  1. Find the 1,000 place.
  2. Look one digit to the right.
  3. 0–4 means keep the digit; 5–9 means increase it by 1.
  4. Replace lower places with zeros.

Very beginner explanation: Rounding replaces a number with a nearby easier number while keeping its size reasonable.

Answer: 72,051,000

Worked Example 4

Problem: Round 908,004,315 to the nearest 10,000.

  1. Find the 10,000 place.
  2. Look one digit to the right.
  3. 0–4 means keep the digit; 5–9 means increase it by 1.
  4. Replace lower places with zeros.

Very beginner explanation: Rounding replaces a number with a nearby easier number while keeping its size reasonable.

Answer: 908,000,000

Worked Example 5

Problem: Round 1,250,700,049 to the nearest 100,000.

  1. Find the 100,000 place.
  2. Look one digit to the right.
  3. 0–4 means keep the digit; 5–9 means increase it by 1.
  4. Replace lower places with zeros.

Very beginner explanation: Rounding replaces a number with a nearby easier number while keeping its size reasonable.

Answer: 1,250,700,000

Worked Example 6

Problem: Round 64,999 to the nearest 10.

  1. Find the 10 place.
  2. Look one digit to the right.
  3. 0–4 means keep the digit; 5–9 means increase it by 1.
  4. Replace lower places with zeros.

Very beginner explanation: Rounding replaces a number with a nearby easier number while keeping its size reasonable.

Answer: 65,000

Worked Example 7

Problem: Round 7,305,040 to the nearest 100.

  1. Find the 100 place.
  2. Look one digit to the right.
  3. 0–4 means keep the digit; 5–9 means increase it by 1.
  4. Replace lower places with zeros.

Very beginner explanation: Rounding replaces a number with a nearby easier number while keeping its size reasonable.

Answer: 7,305,000

Worked Example 8

Problem: Round 800,080 to the nearest 1,000.

  1. Find the 1,000 place.
  2. Look one digit to the right.
  3. 0–4 means keep the digit; 5–9 means increase it by 1.
  4. Replace lower places with zeros.

Very beginner explanation: Rounding replaces a number with a nearby easier number while keeping its size reasonable.

Answer: 800,000

Worked Example 9

Problem: Round 39,640,125 to the nearest 10,000.

  1. Find the 10,000 place.
  2. Look one digit to the right.
  3. 0–4 means keep the digit; 5–9 means increase it by 1.
  4. Replace lower places with zeros.

Very beginner explanation: Rounding replaces a number with a nearby easier number while keeping its size reasonable.

Answer: 39,640,000

Worked Example 10

Problem: Round 999,999,999 to the nearest 100,000.

  1. Find the 100,000 place.
  2. Look one digit to the right.
  3. 0–4 means keep the digit; 5–9 means increase it by 1.
  4. Replace lower places with zeros.

Very beginner explanation: Rounding replaces a number with a nearby easier number while keeping its size reasonable.

Answer: 1,000,000,000

Practice exercise

Create one new Grade 8 problem involving Estimation and reasonableness. Show the important steps, include units when needed, and explain how you checked the answer.

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Chapter 1 Review Questions and Answers

Q1. What is important to remember about Whole numbers and place value?

Answer: Whole numbers (0, 1, 2, 3, and so on) use place value to show the value of each digit. In this section, the focus is Whole numbers and place value.

Q2. What is important to remember about Reading and writing large numbers?

Answer: Reading and writing large numbers is an important Grade 8 concept in Whole Numbers, Integers, and Number Sense Review. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

Q3. What is important to remember about Expanded form?

Answer: Expanded form is an important Grade 8 concept in Whole Numbers, Integers, and Number Sense Review. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

Q4. What is important to remember about Comparing whole numbers?

Answer: Whole numbers (0, 1, 2, 3, and so on) use place value to show the value of each digit. In this section, the focus is Comparing whole numbers.

Q5. What is important to remember about Ordering whole numbers?

Answer: Whole numbers (0, 1, 2, 3, and so on) use place value to show the value of each digit. In this section, the focus is Ordering whole numbers.

Q6. What is important to remember about Rounding whole numbers?

Answer: Whole numbers (0, 1, 2, 3, and so on) use place value to show the value of each digit. In this section, the focus is Rounding whole numbers.

Q7. What is important to remember about Integer review?

Answer: Integers (negative whole numbers, zero, and positive whole numbers) are useful for temperature, elevation, scores, gains, and losses. In this section, the focus is Integer review.

Q8. What is important to remember about Positive and negative integers?

Answer: Integers (negative whole numbers, zero, and positive whole numbers) are useful for temperature, elevation, scores, gains, and losses. In this section, the focus is Positive and negative integers.

Q9. What is important to remember about Absolute value?

Answer: Absolute value (distance from zero on a number line) is always zero or positive. In this section, the focus is Absolute value.

Q10. What is important to remember about Comparing integers?

Answer: Integers (negative whole numbers, zero, and positive whole numbers) are useful for temperature, elevation, scores, gains, and losses. In this section, the focus is Comparing integers.

Q11. What is important to remember about Ordering integers?

Answer: Integers (negative whole numbers, zero, and positive whole numbers) are useful for temperature, elevation, scores, gains, and losses. In this section, the focus is Ordering integers.

Q12. What is important to remember about Number-line reasoning?

Answer: Number-line reasoning is an important Grade 8 concept in Whole Numbers, Integers, and Number Sense Review. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

Q13. What is important to remember about Estimation and reasonableness?

Answer: Estimation and reasonableness is an important Grade 8 concept in Whole Numbers, Integers, and Number Sense Review. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

Q14. Why should you show your steps?

Answer: Showing steps makes reasoning easier to verify and helps locate mistakes.

Q15. Why is estimation useful?

Answer: Estimation helps you judge whether a calculated answer is reasonable.

Q16. Why do units matter?

Answer: Units tell what a value measures and prevent incompatible quantities from being mixed.

Q17. When should you round?

Answer: Usually round near the end unless the question specifically asks for earlier rounding.

Q18. How can you check an answer?

Answer: Use an inverse operation, substitution, estimation, a graph, or a second method.

Q19. What should you identify first in a word problem?

Answer: Identify what is known, what is unknown, and the mathematical relationship connecting them.

Q20. What makes a final answer complete?

Answer: Give the value, correct units when needed, and a brief interpretation in context.

Q21. What should you do if an answer seems unreasonable?

Answer: Re-read the question and check copied values, signs, units, formulas, and calculations.

Q22. Why are diagrams useful?

Answer: A labelled diagram can reveal relationships among lengths, angles, areas, and coordinates.

Q23. Why are tables useful?

Answer: Tables organize values and help reveal patterns, rates, relationships, and missing information.

Q24. Why should you explain your reasoning?

Answer: Reasoning shows why a method works, not just what answer was obtained.

Q25. How do mistakes help learning?

Answer: Analyzing a mistake identifies the misunderstood rule or step and helps prevent repetition.

Q26. When is a calculator useful?

Answer: A calculator helps with lengthy arithmetic after the mathematical setup is understood and estimated.

Q27. Why compare more than one strategy?

Answer: Different strategies may make a problem easier and provide a way to verify the result.

Q28. What is mathematical communication?

Answer: It is presenting ideas clearly with words, symbols, diagrams, tables, graphs, and justified steps.

Q29. What is mathematical modelling?

Answer: It is representing a real situation with mathematics, testing the model, and revising it if needed.

Q30. Why should assumptions be stated?

Answer: Assumptions show the conditions the model depends on and help readers judge whether it is reasonable.