Grade 4 · Number & Operations in Base Ten · MAT-04-NBT-002
Explain Place-Value Relationships
Represent and solve explain place-value relationships problems using accurate mathematical models, notation, and reasoning.
Learning objectives
What you should be able to do
- Represent and solve explain place-value relationships problems using accurate mathematical models, notation, and reasoning.
- Explain why a strategy for explain place-value relationships works and verify the result with an independent check.
Prerequisite check
Make sure the foundation is ready.
The new lesson builds directly on this prior concept, so the prerequisite should be usable rather than merely familiar.
Learn
Build the idea from meaning, not memorization.
Meaning and structure
Explain Place-Value Relationships relies on the base-ten system: each place is ten times the value of the place immediately to its right. Algorithms work because they preserve those place-value units while regrouping.
Represent the idea
The Math Box uses base ten blocks as the primary representation. Translate the situation into numbers or geometry first, identify what stays fixed, then change one quantity at a time and observe which relationships remain invariant.
Reason, calculate, and verify
A convincing solution names the quantities, shows the mathematical relationship, computes accurately, and includes a reasonableness check rather than relying on an unsupported answer.
Math Box · Interactive lesson
Touch the math. Change it. See what stays true.
Use the interactive model before and after the worked examples. Change the inputs, make a prediction, then use the model to test whether your reasoning holds.
Decimal Place-Value Lab
Change one place at a time and watch how its value contributes to the complete decimal.
- Build one example of explain place value relationships in the Math Box.
- Change one input and predict the effect before moving the control.
- Explain which mathematical relationship stayed invariant after the change.
Worked examples
Twenty different ways to see the concept work.
Compare the value of the digit 2 in 200 with the same digit value 20.
200 = 10 × 20Moving one place to the left multiplies a digit's value by 10.
Compare the value of the digit 3 in 3,000 with the same digit value 300.
3000 = 10 × 300Moving one place to the left multiplies a digit's value by 10.
Compare the value of the digit 4 in 40,000 with the same digit value 4,000.
40000 = 10 × 4000Moving one place to the left multiplies a digit's value by 10.
Compare the value of the digit 5 in 500 with the same digit value 50.
500 = 10 × 50Moving one place to the left multiplies a digit's value by 10.
Compare the value of the digit 6 in 6,000 with the same digit value 600.
6000 = 10 × 600Moving one place to the left multiplies a digit's value by 10.
Compare the value of the digit 7 in 70,000 with the same digit value 7,000.
70000 = 10 × 7000Moving one place to the left multiplies a digit's value by 10.
Compare the value of the digit 8 in 800 with the same digit value 80.
800 = 10 × 80Moving one place to the left multiplies a digit's value by 10.
Compare the value of the digit 2 in 2,000 with the same digit value 200.
2000 = 10 × 200Moving one place to the left multiplies a digit's value by 10.
Compare the value of the digit 3 in 30,000 with the same digit value 3,000.
30000 = 10 × 3000Moving one place to the left multiplies a digit's value by 10.
Compare the value of the digit 4 in 400 with the same digit value 40.
400 = 10 × 40Moving one place to the left multiplies a digit's value by 10.
Compare the value of the digit 5 in 5,000 with the same digit value 500.
5000 = 10 × 500Moving one place to the left multiplies a digit's value by 10.
Compare the value of the digit 6 in 60,000 with the same digit value 6,000.
60000 = 10 × 6000Moving one place to the left multiplies a digit's value by 10.
Compare the value of the digit 7 in 700 with the same digit value 70.
700 = 10 × 70Moving one place to the left multiplies a digit's value by 10.
Compare the value of the digit 8 in 8,000 with the same digit value 800.
8000 = 10 × 800Moving one place to the left multiplies a digit's value by 10.
Compare the value of the digit 2 in 20,000 with the same digit value 2,000.
20000 = 10 × 2000Moving one place to the left multiplies a digit's value by 10.
Compare the value of the digit 3 in 300 with the same digit value 30.
300 = 10 × 30Moving one place to the left multiplies a digit's value by 10.
Compare the value of the digit 4 in 4,000 with the same digit value 400.
4000 = 10 × 400Moving one place to the left multiplies a digit's value by 10.
Compare the value of the digit 5 in 50,000 with the same digit value 5,000.
50000 = 10 × 5000Moving one place to the left multiplies a digit's value by 10.
Compare the value of the digit 6 in 600 with the same digit value 60.
600 = 10 × 60Moving one place to the left multiplies a digit's value by 10.
Compare the value of the digit 7 in 7,000 with the same digit value 700.
7000 = 10 × 700Moving one place to the left multiplies a digit's value by 10.
Practice
Guided practice with hints
01Compare the value of the digit 8 in 80,000 with the same digit value 8,000.
Hint: Name the quantities and representation first; then choose the operation or comparison that preserves their meaning.
Answer: 80000 = 10 × 8000
02Compare the value of the digit 2 in 200 with the same digit value 20.
Hint: Name the quantities and representation first; then choose the operation or comparison that preserves their meaning.
Answer: 200 = 10 × 20
03Compare the value of the digit 3 in 3,000 with the same digit value 300.
Hint: Name the quantities and representation first; then choose the operation or comparison that preserves their meaning.
Answer: 3000 = 10 × 300
04Compare the value of the digit 4 in 40,000 with the same digit value 4,000.
Hint: Name the quantities and representation first; then choose the operation or comparison that preserves their meaning.
Answer: 40000 = 10 × 4000
05Compare the value of the digit 5 in 500 with the same digit value 50.
Hint: Name the quantities and representation first; then choose the operation or comparison that preserves their meaning.
Answer: 500 = 10 × 50
06Compare the value of the digit 6 in 6,000 with the same digit value 600.
Hint: Name the quantities and representation first; then choose the operation or comparison that preserves their meaning.
Answer: 6000 = 10 × 600
Practice
Independent practice
01Compare the value of the digit 7 in 70,000 with the same digit value 7,000.
Answer: 70000 = 10 × 7000
02Compare the value of the digit 8 in 800 with the same digit value 80.
Answer: 800 = 10 × 80
03Compare the value of the digit 2 in 2,000 with the same digit value 200.
Answer: 2000 = 10 × 200
04Compare the value of the digit 3 in 30,000 with the same digit value 3,000.
Answer: 30000 = 10 × 3000
05Compare the value of the digit 2 in 200 with the same digit value 20.
Answer: 200 = 10 × 20
06Compare the value of the digit 3 in 3,000 with the same digit value 300.
Answer: 3000 = 10 × 300
07Compare the value of the digit 4 in 40,000 with the same digit value 4,000.
Answer: 40000 = 10 × 4000
08Compare the value of the digit 5 in 500 with the same digit value 50.
Answer: 500 = 10 × 50
Common mistakes
Learn to catch the error, not just the answer.
Identify what each number represents and how the quantities are related before selecting an operation.
Estimate, use an inverse operation, or compare with a second representation.
Applications & challenge
Use the idea beyond a single exercise.
Applications
- Use explain place-value relationships when solving multi-step problems in measurement, data, money, design, or everyday quantitative decisions.
- Explain the result with units and a second check so another person can audit the reasoning.
Challenge problems
Compare the value of the digit 6 in 6,000 with the same digit value 600.
6000 = 10 × 600
Compare the value of the digit 7 in 70,000 with the same digit value 7,000.
70000 = 10 × 7000
Compare the value of the digit 8 in 800 with the same digit value 80.
800 = 10 × 80
Compare the value of the digit 2 in 2,000 with the same digit value 200.
2000 = 10 × 200
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 4
Game mode
Grade 4 Math Strategy Sprint
Solve and explain explain place-value relationships problems accurately, then check each result with a second method.
Practice
Mastery check
01Compare the value of the digit 3 in 30,000 with the same digit value 3,000.
Answer: 30000 = 10 × 3000
02Compare the value of the digit 4 in 400 with the same digit value 40.
Answer: 400 = 10 × 40
03Compare the value of the digit 5 in 5,000 with the same digit value 500.
Answer: 5000 = 10 × 500
04Compare the value of the digit 6 in 60,000 with the same digit value 6,000.
Answer: 60000 = 10 × 6000
05Compare the value of the digit 7 in 700 with the same digit value 70.
Answer: 700 = 10 × 70
06Compare the value of the digit 8 in 8,000 with the same digit value 800.
Answer: 8000 = 10 × 800
07Compare the value of the digit 2 in 20,000 with the same digit value 2,000.
Answer: 20000 = 10 × 2000
08Compare the value of the digit 3 in 300 with the same digit value 30.
Answer: 300 = 10 × 30
Terminology
Words to know
- representation
- A diagram, model, equation, table, graph, or other form showing a mathematical idea.
- strategy
- A purposeful method chosen to solve or analyze a problem.
- estimate
- A nearby value used to judge size or reasonableness.
- verify
- To confirm a result using a second calculation, representation, or logical check.
Continue learning
Connected concepts
Curriculum references
Standards and instructional references
California Mathematics Framework (2023)
Grade-level instructional guidance, mathematical practices, and coherent progression.Common Core State Standards for Mathematics — Grades 4–5
Grade-level expectations for operations, number and fraction work, measurement, data, and geometry.Principles to Actions: Ensuring Mathematical Success for All
Evidence-based emphasis on reasoning, representations, discourse, and conceptual understanding.