Grade 6 Math Chapter 2: Place Value and Number Representation

Chapter Overview

This chapter teaches Place Value and Number Representation in original, simple language with step-by-step reasoning, worked examples, practice exercises, common mistakes, and review questions.

Key Technical Terms

Value of each digit (Value of each digit is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.)
Base-ten relationships (Base-ten relationships is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.)
Place-value charts (Place-value charts is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.)
Regrouping numbers (Regrouping numbers is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.)
Equivalent number representations (Equivalent number representations is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.)
Expanded notation (Expanded notation is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.)

How to Learn This Chapter

Read the explanation first, follow each worked example slowly, cover the answer and retry the problem yourself, then use the practice and review questions to check understanding.

2.1 Value of each digit

Value of each digit is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.

Beginner Note

Identify what is given, what must be found, and which rule or representation fits this topic. Work one step at a time and explain why each step is valid.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Use Value of each digit to analyze the values 24 and 11. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Value of each digit is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 2

Problem: Use Value of each digit to analyze the values 6 and 2. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Value of each digit is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 3

Problem: Use Value of each digit to analyze the values 11 and 14. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Value of each digit is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 4

Problem: Use Value of each digit to analyze the values 3 and 13. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Value of each digit is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 5

Problem: Use Value of each digit to analyze the values 30 and 16. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Value of each digit is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 6

Problem: Use Value of each digit to analyze the values 20 and 17. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Value of each digit is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 7

Problem: Use Value of each digit to analyze the values 26 and 3. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Value of each digit is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 8

Problem: Use Value of each digit to analyze the values 29 and 19. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Value of each digit is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 9

Problem: Use Value of each digit to analyze the values 8 and 7. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Value of each digit is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 10

Problem: Use Value of each digit to analyze the values 6 and 10. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Value of each digit is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Practice Exercise

Create one new Grade 6 problem about Value of each digit. Solve it step by step, label any units, and explain how you checked your answer.

Common Mistake

A common mistake is starting a calculation before identifying what is given, what must be found, and which rule fits the problem.

2.2 Base-ten relationships

Base-ten relationships is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.

Beginner Note

Identify what is given, what must be found, and which rule or representation fits this topic. Work one step at a time and explain why each step is valid.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Use Base-ten relationships to analyze the values 22 and 6. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Base-ten relationships is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 2

Problem: Use Base-ten relationships to analyze the values 16 and 18. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Base-ten relationships is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 3

Problem: Use Base-ten relationships to analyze the values 7 and 15. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Base-ten relationships is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 4

Problem: Use Base-ten relationships to analyze the values 9 and 9. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Base-ten relationships is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 5

Problem: Use Base-ten relationships to analyze the values 26 and 15. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Base-ten relationships is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 6

Problem: Use Base-ten relationships to analyze the values 4 and 5. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Base-ten relationships is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 7

Problem: Use Base-ten relationships to analyze the values 8 and 20. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Base-ten relationships is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 8

Problem: Use Base-ten relationships to analyze the values 5 and 3. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Base-ten relationships is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 9

Problem: Use Base-ten relationships to analyze the values 8 and 19. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Base-ten relationships is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 10

Problem: Use Base-ten relationships to analyze the values 27 and 16. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Base-ten relationships is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Practice Exercise

Create one new Grade 6 problem about Base-ten relationships. Solve it step by step, label any units, and explain how you checked your answer.

Common Mistake

A common mistake is starting a calculation before identifying what is given, what must be found, and which rule fits the problem.

2.3 Place-value charts

Place-value charts is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.

Beginner Note

Identify what is given, what must be found, and which rule or representation fits this topic. Work one step at a time and explain why each step is valid.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Use Place-value charts to analyze the values 17 and 14. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Place-value charts is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 2

Problem: Use Place-value charts to analyze the values 15 and 16. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Place-value charts is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 3

Problem: Use Place-value charts to analyze the values 2 and 16. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Place-value charts is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 4

Problem: Use Place-value charts to analyze the values 23 and 9. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Place-value charts is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 5

Problem: Use Place-value charts to analyze the values 11 and 2. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Place-value charts is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 6

Problem: Use Place-value charts to analyze the values 16 and 19. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Place-value charts is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 7

Problem: Use Place-value charts to analyze the values 12 and 3. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Place-value charts is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 8

Problem: Use Place-value charts to analyze the values 29 and 7. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Place-value charts is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 9

Problem: Use Place-value charts to analyze the values 25 and 12. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Place-value charts is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 10

Problem: Use Place-value charts to analyze the values 14 and 2. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Place-value charts is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Practice Exercise

Create one new Grade 6 problem about Place-value charts. Solve it step by step, label any units, and explain how you checked your answer.

Common Mistake

A common mistake is starting a calculation before identifying what is given, what must be found, and which rule fits the problem.

2.4 Regrouping numbers

Regrouping numbers is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.

Beginner Note

Identify what is given, what must be found, and which rule or representation fits this topic. Work one step at a time and explain why each step is valid.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Use Regrouping numbers to analyze the values 7 and 17. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Regrouping numbers is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 2

Problem: Use Regrouping numbers to analyze the values 4 and 11. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Regrouping numbers is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 3

Problem: Use Regrouping numbers to analyze the values 27 and 8. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Regrouping numbers is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 4

Problem: Use Regrouping numbers to analyze the values 4 and 10. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Regrouping numbers is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 5

Problem: Use Regrouping numbers to analyze the values 7 and 4. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Regrouping numbers is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 6

Problem: Use Regrouping numbers to analyze the values 26 and 3. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Regrouping numbers is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 7

Problem: Use Regrouping numbers to analyze the values 23 and 20. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Regrouping numbers is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 8

Problem: Use Regrouping numbers to analyze the values 3 and 5. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Regrouping numbers is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 9

Problem: Use Regrouping numbers to analyze the values 24 and 11. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Regrouping numbers is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 10

Problem: Use Regrouping numbers to analyze the values 24 and 18. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Regrouping numbers is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Practice Exercise

Create one new Grade 6 problem about Regrouping numbers. Solve it step by step, label any units, and explain how you checked your answer.

Common Mistake

A common mistake is starting a calculation before identifying what is given, what must be found, and which rule fits the problem.

2.5 Equivalent number representations

Equivalent number representations is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.

Beginner Note

Identify what is given, what must be found, and which rule or representation fits this topic. Work one step at a time and explain why each step is valid.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Use Equivalent number representations to analyze the values 8 and 13. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Equivalent number representations is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 2

Problem: Use Equivalent number representations to analyze the values 12 and 14. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Equivalent number representations is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 3

Problem: Use Equivalent number representations to analyze the values 15 and 14. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Equivalent number representations is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 4

Problem: Use Equivalent number representations to analyze the values 5 and 20. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Equivalent number representations is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 5

Problem: Use Equivalent number representations to analyze the values 27 and 17. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Equivalent number representations is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 6

Problem: Use Equivalent number representations to analyze the values 8 and 4. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Equivalent number representations is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 7

Problem: Use Equivalent number representations to analyze the values 11 and 17. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Equivalent number representations is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 8

Problem: Use Equivalent number representations to analyze the values 20 and 16. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Equivalent number representations is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 9

Problem: Use Equivalent number representations to analyze the values 27 and 17. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Equivalent number representations is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 10

Problem: Use Equivalent number representations to analyze the values 25 and 19. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Equivalent number representations is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Practice Exercise

Create one new Grade 6 problem about Equivalent number representations. Solve it step by step, label any units, and explain how you checked your answer.

Common Mistake

A common mistake is starting a calculation before identifying what is given, what must be found, and which rule fits the problem.

2.6 Expanded notation

Expanded notation is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.

Beginner Note

Identify what is given, what must be found, and which rule or representation fits this topic. Work one step at a time and explain why each step is valid.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Use Expanded notation to analyze the values 28 and 19. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Expanded notation is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 2

Problem: Use Expanded notation to analyze the values 10 and 4. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Expanded notation is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 3

Problem: Use Expanded notation to analyze the values 18 and 11. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Expanded notation is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 4

Problem: Use Expanded notation to analyze the values 10 and 15. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Expanded notation is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 5

Problem: Use Expanded notation to analyze the values 4 and 6. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Expanded notation is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 6

Problem: Use Expanded notation to analyze the values 19 and 19. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Expanded notation is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 7

Problem: Use Expanded notation to analyze the values 14 and 17. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Expanded notation is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 8

Problem: Use Expanded notation to analyze the values 29 and 2. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Expanded notation is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 9

Problem: Use Expanded notation to analyze the values 10 and 18. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Expanded notation is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 10

Problem: Use Expanded notation to analyze the values 18 and 17. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Expanded notation is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Practice Exercise

Create one new Grade 6 problem about Expanded notation. Solve it step by step, label any units, and explain how you checked your answer.

Common Mistake

A common mistake is starting a calculation before identifying what is given, what must be found, and which rule fits the problem.

2.7 Missing-digit place-value puzzles

Missing-digit place-value puzzles is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.

Beginner Note

Identify what is given, what must be found, and which rule or representation fits this topic. Work one step at a time and explain why each step is valid.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: In 241,864, what is the value of the digit 2 in the 100,000 place?

Locate the requested place.
Multiply the digit by the place value.

Very beginner explanation: A digit can have different values depending on its position.

Answer: 200,000

Worked Example 2

Problem: In 975,016, what is the value of the digit 9 in the 100,000 place?

Locate the requested place.
Multiply the digit by the place value.

Very beginner explanation: A digit can have different values depending on its position.

Answer: 900,000

Worked Example 3

Problem: In 535,498, what is the value of the digit 4 in the 100 place?

Locate the requested place.
Multiply the digit by the place value.

Very beginner explanation: A digit can have different values depending on its position.

Answer: 400

Worked Example 4

Problem: In 191,204, what is the value of the digit 1 in the 100,000 place?

Locate the requested place.
Multiply the digit by the place value.

Very beginner explanation: A digit can have different values depending on its position.

Answer: 100,000

Worked Example 5

Problem: In 694,540, what is the value of the digit 5 in the 100 place?

Locate the requested place.
Multiply the digit by the place value.

Very beginner explanation: A digit can have different values depending on its position.

Answer: 500

Worked Example 6

Problem: In 13,925, what is the value of the digit 3 in the 1,000 place?

Locate the requested place.
Multiply the digit by the place value.

Very beginner explanation: A digit can have different values depending on its position.

Answer: 3,000

Worked Example 7

Problem: In 269,174, what is the value of the digit 1 in the 100 place?

Locate the requested place.
Multiply the digit by the place value.

Very beginner explanation: A digit can have different values depending on its position.

Answer: 100

Worked Example 8

Problem: In 632,819, what is the value of the digit 1 in the 10 place?

Locate the requested place.
Multiply the digit by the place value.

Very beginner explanation: A digit can have different values depending on its position.

Answer: 10

Worked Example 9

Problem: In 940,289, what is the value of the digit 8 in the 10 place?

Locate the requested place.
Multiply the digit by the place value.

Very beginner explanation: A digit can have different values depending on its position.

Answer: 80

Worked Example 10

Problem: In 950,119, what is the value of the digit 5 in the 10,000 place?

Locate the requested place.
Multiply the digit by the place value.

Very beginner explanation: A digit can have different values depending on its position.

Answer: 50,000

Practice Exercise

Create one new Grade 6 problem about Missing-digit place-value puzzles. Solve it step by step, label any units, and explain how you checked your answer.

Common Mistake

A common mistake is starting a calculation before identifying what is given, what must be found, and which rule fits the problem.

2.8 Using benchmark numbers

Using benchmark numbers is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.

Beginner Note

Identify what is given, what must be found, and which rule or representation fits this topic. Work one step at a time and explain why each step is valid.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: In 181,956, what is the value of the digit 1 in the 100,000 place?

Locate the requested place.
Multiply the digit by the place value.

Very beginner explanation: A digit can have different values depending on its position.

Answer: 100,000

Worked Example 2

Problem: In 826,386, what is the value of the digit 8 in the 10 place?

Locate the requested place.
Multiply the digit by the place value.

Very beginner explanation: A digit can have different values depending on its position.

Answer: 80

Worked Example 3

Problem: In 662,312, what is the value of the digit 2 in the 1,000 place?

Locate the requested place.
Multiply the digit by the place value.

Very beginner explanation: A digit can have different values depending on its position.

Answer: 2,000

Worked Example 4

Problem: In 866,737, what is the value of the digit 8 in the 100,000 place?

Locate the requested place.
Multiply the digit by the place value.

Very beginner explanation: A digit can have different values depending on its position.

Answer: 800,000

Worked Example 5

Problem: In 326,017, what is the value of the digit 2 in the 10,000 place?

Locate the requested place.
Multiply the digit by the place value.

Very beginner explanation: A digit can have different values depending on its position.

Answer: 20,000

Worked Example 6

Problem: In 125,281, what is the value of the digit 1 in the 100,000 place?

Locate the requested place.
Multiply the digit by the place value.

Very beginner explanation: A digit can have different values depending on its position.

Answer: 100,000

Worked Example 7

Problem: In 318,431, what is the value of the digit 4 in the 100 place?

Locate the requested place.
Multiply the digit by the place value.

Very beginner explanation: A digit can have different values depending on its position.

Answer: 400

Worked Example 8

Problem: In 480,816, what is the value of the digit 0 in the 1,000 place?

Locate the requested place.
Multiply the digit by the place value.

Very beginner explanation: A digit can have different values depending on its position.

Answer: 0

Worked Example 9

Problem: In 515,149, what is the value of the digit 1 in the 10,000 place?

Locate the requested place.
Multiply the digit by the place value.

Very beginner explanation: A digit can have different values depending on its position.

Answer: 10,000

Worked Example 10

Problem: In 608,517, what is the value of the digit 6 in the 100,000 place?

Locate the requested place.
Multiply the digit by the place value.

Very beginner explanation: A digit can have different values depending on its position.

Answer: 600,000

Practice Exercise

Create one new Grade 6 problem about Using benchmark numbers. Solve it step by step, label any units, and explain how you checked your answer.

Common Mistake

A common mistake is starting a calculation before identifying what is given, what must be found, and which rule fits the problem.

2.9 Comparing different representations

Comparing different representations is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.

Beginner Note

Identify what is given, what must be found, and which rule or representation fits this topic. Work one step at a time and explain why each step is valid.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Use Comparing different representations to analyze the values 25 and 4. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Comparing different representations is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 2

Problem: Use Comparing different representations to analyze the values 4 and 4. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Comparing different representations is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 3

Problem: Use Comparing different representations to analyze the values 18 and 18. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Comparing different representations is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 4

Problem: Use Comparing different representations to analyze the values 19 and 17. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Comparing different representations is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 5

Problem: Use Comparing different representations to analyze the values 26 and 18. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Comparing different representations is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 6

Problem: Use Comparing different representations to analyze the values 18 and 17. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Comparing different representations is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 7

Problem: Use Comparing different representations to analyze the values 21 and 12. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Comparing different representations is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 8

Problem: Use Comparing different representations to analyze the values 11 and 11. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Comparing different representations is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 9

Problem: Use Comparing different representations to analyze the values 15 and 20. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Comparing different representations is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Worked Example 10

Problem: Use Comparing different representations to analyze the values 20 and 4. What should be checked first?

Identify what each value represents.
Choose the rule, representation, or comparison that matches the topic.
Show the reasoning before accepting the result.

Very beginner explanation: Comparing different representations is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Answer: Check the meaning of the values and the rule required by the problem.

Practice Exercise

Create one new Grade 6 problem about Comparing different representations. Solve it step by step, label any units, and explain how you checked your answer.

Common Mistake

A common mistake is starting a calculation before identifying what is given, what must be found, and which rule fits the problem.

2.10 Reading calculator displays

Reading calculator displays is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.

Beginner Note

Identify what is given, what must be found, and which rule or representation fits this topic. Work one step at a time and explain why each step is valid.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: In 918,178, what is the value of the digit 1 in the 100 place?

Locate the requested place.
Multiply the digit by the place value.

Very beginner explanation: A digit can have different values depending on its position.

Answer: 100

Worked Example 2

Problem: In 283,131, what is the value of the digit 8 in the 10,000 place?

Locate the requested place.
Multiply the digit by the place value.

Very beginner explanation: A digit can have different values depending on its position.

Answer: 80,000

Worked Example 3

Problem: In 326,291, what is the value of the digit 6 in the 1,000 place?

Locate the requested place.
Multiply the digit by the place value.

Very beginner explanation: A digit can have different values depending on its position.

Answer: 6,000

Worked Example 4

Problem: In 247,401, what is the value of the digit 4 in the 100 place?

Locate the requested place.
Multiply the digit by the place value.

Very beginner explanation: A digit can have different values depending on its position.

Answer: 400

Worked Example 5

Problem: In 279,392, what is the value of the digit 9 in the 1,000 place?

Locate the requested place.
Multiply the digit by the place value.

Very beginner explanation: A digit can have different values depending on its position.

Answer: 9,000

Worked Example 6

Problem: In 926,172, what is the value of the digit 9 in the 100,000 place?

Locate the requested place.
Multiply the digit by the place value.

Very beginner explanation: A digit can have different values depending on its position.

Answer: 900,000

Worked Example 7

Problem: In 938,612, what is the value of the digit 9 in the 100,000 place?

Locate the requested place.
Multiply the digit by the place value.

Very beginner explanation: A digit can have different values depending on its position.

Answer: 900,000

Worked Example 8

Problem: In 647,472, what is the value of the digit 7 in the 1,000 place?

Locate the requested place.
Multiply the digit by the place value.

Very beginner explanation: A digit can have different values depending on its position.

Answer: 7,000

Worked Example 9

Problem: In 926,340, what is the value of the digit 9 in the 100,000 place?

Locate the requested place.
Multiply the digit by the place value.

Very beginner explanation: A digit can have different values depending on its position.

Answer: 900,000

Worked Example 10

Problem: In 179,457, what is the value of the digit 4 in the 100 place?

Locate the requested place.
Multiply the digit by the place value.

Very beginner explanation: A digit can have different values depending on its position.

Answer: 400

Practice Exercise

Create one new Grade 6 problem about Reading calculator displays. Solve it step by step, label any units, and explain how you checked your answer.

Common Mistake

A common mistake is starting a calculation before identifying what is given, what must be found, and which rule fits the problem.

2.11 Flexible number decomposition

Flexible number decomposition is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.

Beginner Note

Identify what is given, what must be found, and which rule or representation fits this topic. Work one step at a time and explain why each step is valid.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Write 248,067 in expanded form.

Read each digit by place value.
Write the value of every non-zero digit.
Join the values with plus signs.

Very beginner explanation: Expanded form shows the value contributed by each digit.

Answer: 200,000 + 40,000 + 8,000 + 60 + 7

Worked Example 2

Problem: Write 245,450 in expanded form.

Read each digit by place value.
Write the value of every non-zero digit.
Join the values with plus signs.

Very beginner explanation: Expanded form shows the value contributed by each digit.

Answer: 200,000 + 40,000 + 5,000 + 400 + 50

Worked Example 3

Problem: Write 295,597 in expanded form.

Read each digit by place value.
Write the value of every non-zero digit.
Join the values with plus signs.

Very beginner explanation: Expanded form shows the value contributed by each digit.

Answer: 200,000 + 90,000 + 5,000 + 500 + 90 + 7

Worked Example 4

Problem: Write 305,580 in expanded form.

Read each digit by place value.
Write the value of every non-zero digit.
Join the values with plus signs.

Very beginner explanation: Expanded form shows the value contributed by each digit.

Answer: 300,000 + 5,000 + 500 + 80

Worked Example 5

Problem: Write 851,430 in expanded form.

Read each digit by place value.
Write the value of every non-zero digit.
Join the values with plus signs.

Very beginner explanation: Expanded form shows the value contributed by each digit.

Answer: 800,000 + 50,000 + 1,000 + 400 + 30

Worked Example 6

Problem: Write 413,520 in expanded form.

Read each digit by place value.
Write the value of every non-zero digit.
Join the values with plus signs.

Very beginner explanation: Expanded form shows the value contributed by each digit.

Answer: 400,000 + 10,000 + 3,000 + 500 + 20

Worked Example 7

Problem: Write 829,149 in expanded form.

Read each digit by place value.
Write the value of every non-zero digit.
Join the values with plus signs.

Very beginner explanation: Expanded form shows the value contributed by each digit.

Answer: 800,000 + 20,000 + 9,000 + 100 + 40 + 9

Worked Example 8

Problem: Write 737,779 in expanded form.

Read each digit by place value.
Write the value of every non-zero digit.
Join the values with plus signs.

Very beginner explanation: Expanded form shows the value contributed by each digit.

Answer: 700,000 + 30,000 + 7,000 + 700 + 70 + 9

Worked Example 9

Problem: Write 187,332 in expanded form.

Read each digit by place value.
Write the value of every non-zero digit.
Join the values with plus signs.

Very beginner explanation: Expanded form shows the value contributed by each digit.

Answer: 100,000 + 80,000 + 7,000 + 300 + 30 + 2

Worked Example 10

Problem: Write 926,358 in expanded form.

Read each digit by place value.
Write the value of every non-zero digit.
Join the values with plus signs.

Very beginner explanation: Expanded form shows the value contributed by each digit.

Answer: 900,000 + 20,000 + 6,000 + 300 + 50 + 8

Practice Exercise

Create one new Grade 6 problem about Flexible number decomposition. Solve it step by step, label any units, and explain how you checked your answer.

Common Mistake

A common mistake is starting a calculation before identifying what is given, what must be found, and which rule fits the problem.

2.12 Number sense strategies

A rate compares two quantities measured in different units. In this Grade 6 lesson, focus on meaning first, then use a small example, show the steps, and check whether the result fits the question.

Beginner Note

Identify what is given, what must be found, and which rule or representation fits this topic. Work one step at a time and explain why each step is valid.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: 16 units are used in 4 equal groups. Find the unit rate.

Divide the first quantity by the second.
16 ÷ 4 = 4.

Very beginner explanation: A unit rate tells the amount for one unit of the second quantity.

Answer: 4 per group

Worked Example 2

Problem: 12 units are used in 4 equal groups. Find the unit rate.

Divide the first quantity by the second.
12 ÷ 4 = 3.

Very beginner explanation: A unit rate tells the amount for one unit of the second quantity.

Answer: 3 per group

Worked Example 3

Problem: 72 units are used in 12 equal groups. Find the unit rate.

Divide the first quantity by the second.
72 ÷ 12 = 6.

Very beginner explanation: A unit rate tells the amount for one unit of the second quantity.

Answer: 6 per group

Worked Example 4

Problem: 30 units are used in 5 equal groups. Find the unit rate.

Divide the first quantity by the second.
30 ÷ 5 = 6.

Very beginner explanation: A unit rate tells the amount for one unit of the second quantity.

Answer: 6 per group

Worked Example 5

Problem: 140 units are used in 14 equal groups. Find the unit rate.

Divide the first quantity by the second.
140 ÷ 14 = 10.

Very beginner explanation: A unit rate tells the amount for one unit of the second quantity.

Answer: 10 per group

Worked Example 6

Problem: 72 units are used in 9 equal groups. Find the unit rate.

Divide the first quantity by the second.
72 ÷ 9 = 8.

Very beginner explanation: A unit rate tells the amount for one unit of the second quantity.

Answer: 8 per group

Worked Example 7

Problem: 24 units are used in 6 equal groups. Find the unit rate.

Divide the first quantity by the second.
24 ÷ 6 = 4.

Very beginner explanation: A unit rate tells the amount for one unit of the second quantity.

Answer: 4 per group

Worked Example 8

Problem: 165 units are used in 15 equal groups. Find the unit rate.

Divide the first quantity by the second.
165 ÷ 15 = 11.

Very beginner explanation: A unit rate tells the amount for one unit of the second quantity.

Answer: 11 per group

Worked Example 9

Problem: 84 units are used in 14 equal groups. Find the unit rate.

Divide the first quantity by the second.
84 ÷ 14 = 6.

Very beginner explanation: A unit rate tells the amount for one unit of the second quantity.

Answer: 6 per group

Worked Example 10

Problem: 12 units are used in 3 equal groups. Find the unit rate.

Divide the first quantity by the second.
12 ÷ 3 = 4.

Very beginner explanation: A unit rate tells the amount for one unit of the second quantity.

Answer: 4 per group

Practice Exercise

Create one new Grade 6 problem about Number sense strategies. Solve it step by step, label any units, and explain how you checked your answer.

Common Mistake

A common mistake is starting a calculation before identifying what is given, what must be found, and which rule fits the problem.

30 Review Questions and Answers

Q1. What is important to understand about Value of each digit?

Answer: Value of each digit is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Q2. What is important to understand about Base-ten relationships?

Answer: Base-ten relationships is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Q3. What is important to understand about Place-value charts?

Answer: Place-value charts is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Q4. What is important to understand about Regrouping numbers?

Answer: Regrouping numbers is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Q5. What is important to understand about Equivalent number representations?

Answer: Equivalent number representations is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Q6. What is important to understand about Expanded notation?

Answer: Expanded notation is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Q7. What is important to understand about Missing-digit place-value puzzles?

Answer: Missing-digit place-value puzzles is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Q8. What is important to understand about Using benchmark numbers?

Answer: Using benchmark numbers is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Q9. What is important to understand about Comparing different representations?

Answer: Comparing different representations is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Q10. What is important to understand about Reading calculator displays?

Answer: Reading calculator displays is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Q11. What is important to understand about Flexible number decomposition?

Answer: Flexible number decomposition is a Grade 6 mathematics idea in Place Value and Number Representation. The goal is to understand the meaning, choose a suitable strategy, show the steps clearly, and check that the result is reasonable.

Q12. What is important to understand about Number sense strategies?

Answer: A rate compares two quantities measured in different units.

Q13. Why should you show your steps?

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

Q14. Why is estimation useful?

Answer: Estimation helps you judge whether a final answer is reasonable before accepting it.

Q15. Why do units matter?

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

Q16. When should you round?

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

Q17. How can you check an answer?

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

Q18. What should you identify first in a word problem?

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

Q19. What makes a final answer complete?

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

Q20. What should you do if an answer seems unreasonable?

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

Q21. Why are diagrams useful?

Answer: A labelled diagram can reveal relationships that are harder to see in words alone.

Q22. Why are tables useful?

Answer: Tables organize values so patterns, rates, comparisons, and missing information are easier to see.

Q23. Why are graphs useful?

Answer: Graphs make relationships, changes, trends, and comparisons visible.

Q24. Why should you explain your reasoning?

Answer: An explanation shows why a method works instead of only giving a final number.

Q25. Why can more than one strategy be correct?

Answer: Different correct strategies can represent the same mathematics in different but equivalent ways.

Q26. What should you do before using a formula or rule?

Answer: Check what each quantity means and whether the rule matches the situation.

Q27. How does practice help in mathematics?

Answer: Practice builds accuracy and fluency while helping you recognize which strategies fit different problems.

Q28. What is a good way to learn from a mistake?

Answer: Find the exact step where the reasoning changed from correct to incorrect, then solve a similar problem again.

Q29. Why should you compare an exact answer with an estimate?

Answer: The estimate gives a quick reasonableness check for the exact calculation.

Q30. What does representing mathematics mean?

Answer: It means showing an idea with numbers, symbols, words, diagrams, tables, graphs, or models.