Standards Alignment
- TEKS.MATH.3.02A primary
compose and decompose numbers up to 100,000 as a sum of so many ten thousands, so many thousands, so many hundreds, so many tens, and so many ones using objects, pictorial models, and numbers, including expanded notation as appropriate; - TEKS.MATH.3.02B primary
describe the mathematical relationships found in the base-10 place value system through the hundred thousands place;
Lesson Overview
This lesson guides Grade 3 students to compose and decompose numbers up to 100,000 by expressing them as sums of ten thousands, thousands, hundreds, tens, and ones. Students will use concrete objects, pictorial models, and numbers, including expanded notation, to understand place value relationships through the hundred thousands place. The lesson emphasizes understanding the base-10 system and applying mathematical process standards to represent and compare whole numbers.
Learning Objectives
- Compose and decompose numbers up to 100,000 as sums of ten thousands, thousands, hundreds, tens, and ones using objects, pictorial models, and numbers, including expanded notation.
- Describe the mathematical relationships found in the base-10 place value system through the hundred thousands place.
Success Criteria
- Students can represent numbers up to 100,000 using concrete objects and pictorial models.
- Students can write numbers as sums of place values (e.g., 3 ten thousands + 4 thousands + 2 hundreds + 5 tens + 7 ones).
- Students can explain how each place value relates to the next in the base-10 system.
- Students can use expanded notation to express numbers up to 100,000.
Prerequisite Knowledge
Students should understand basic place value concepts up to at least 1,000, including ones, tens, and hundreds, and be familiar with counting and grouping objects.
Key Vocabulary
- compose
- decompose
- place value
- ten thousands
- thousands
- hundreds
- tens
- ones
- expanded notation
- base-10 system
Materials and Resources
- Base-10 blocks or place value disks representing ones, tens, hundreds, thousands, and ten thousands
- Place value charts
- Pictorial models (drawings showing place values)
- Whiteboard and markers
- Number cards up to 100,000
Teacher Preparation
- Prepare sets of base-10 blocks for demonstration and student use.
- Create place value charts for classroom display and student copies.
- Prepare pictorial models illustrating numbers decomposed by place value.
- Prepare number cards for guided and independent practice activities.
Detailed Lesson Notes
Understanding Place Value up to 100,000
Place value is the value of a digit depending on its position in a number. In the base-10 system, each place is ten times the value of the place to its right. The places from right to left are ones, tens, hundreds, thousands, ten thousands, and hundred thousands. For example, in the number 43,256, the digit 4 is in the ten thousands place and represents 40,000, while the digit 3 is in the thousands place and represents 3,000.
Composing Numbers Using Place Value
To compose a number means to build it by adding the values of its digits according to their place. For example, the number 52,314 can be composed as 5 ten thousands + 2 thousands + 3 hundreds + 1 ten + 4 ones. Using base-10 blocks, students can physically group and count the blocks to represent each place value and then combine them to form the whole number.
Decomposing Numbers Using Place Value
Decomposing a number means breaking it down into the sum of its place values. For example, 67,809 can be decomposed as 6 ten thousands + 7 thousands + 8 hundreds + 0 tens + 9 ones. Students can use pictorial models or drawings to show this breakdown, helping them visualize the value of each digit.
Expanded Notation
Expanded notation expresses a number as the sum of each digit multiplied by its place value. For example, 34,205 in expanded notation is (3 × 10,000) + (4 × 1,000) + (2 × 100) + (0 × 10) + (5 × 1). This notation helps students understand the value of each digit and the structure of the number.
Mathematical Relationships in the Base-10 System
Each place value is ten times the value of the place to its right. For example, 1 ten thousand is equal to 10 thousands, 1 thousand is equal to 10 hundreds, and so on. Understanding these relationships helps students grasp how numbers grow and how to regroup when adding or subtracting.
Worked Examples
Worked Example 1
Scenario
Decompose the number 54,321 using place value and write it in expanded notation.
Explanation
The digit 5 is in the ten thousands place, so it represents 50,000. The digit 4 is in the thousands place, representing 4,000. The digit 3 is in the hundreds place, representing 300. The digit 2 is in the tens place, representing 20. The digit 1 is in the ones place, representing 1. Adding these together gives the original number.
Answer Guide
54,321 = 50,000 + 4,000 + 300 + 20 + 1 = (5 × 10,000) + (4 × 1,000) + (3 × 100) + (2 × 10) + (1 × 1)
Engage
Teacher Activity
Show students a large number, such as 48,372, written on the board. Ask them what they notice about the number and if they can identify any parts of it.
Student Activity
Students share observations about the digits and their positions in the number, guessing what each digit might represent.
Explanation
This activity activates prior knowledge about place value and introduces the concept of composing and decomposing numbers by place.
Examples
- Students identify digits and their positions.
- Students suggest that digits represent groups of tens, hundreds, thousands, etc.
Explore
Teacher Activity
Provide students with base-10 blocks and place value charts. Ask them to build the number 36,514 using the blocks and then draw a pictorial model showing the number decomposed by place value.
Student Activity
Students use the blocks to represent the number and draw models showing ten thousands, thousands, hundreds, tens, and ones.
Explanation
Hands-on manipulation helps students concretely understand how numbers are composed of different place values.
Examples
- Students correctly use blocks to represent each place value.
- Drawings show the number decomposed into place values.
Explain
Teacher Activity
Demonstrate how to write the number 36,514 as a sum of its place values and in expanded notation on the board. Explain the value of each digit and how the base-10 system works.
Student Activity
Students copy the examples and explain in their own words what each part of the number means.
Explanation
Explicit teaching clarifies the concept of composing and decomposing numbers and the mathematical relationships in the base-10 system.
Examples
- Students write the number as 30,000 + 6,000 + 500 + 10 + 4.
- Students understand the value of each digit.
Elaborate
Teacher Activity
Present a real-world problem where students must compose or decompose a number, such as counting a large number of items (e.g., 42,367 apples). Guide students to represent the number using blocks, pictorial models, and expanded notation.
Student Activity
Students solve the problem by composing and decomposing the number using different representations and explain their reasoning.
Explanation
Applying concepts to real-world contexts deepens understanding and shows the usefulness of place value knowledge.
Examples
- Students correctly represent the number in multiple ways.
- Students explain the usefulness of decomposing numbers.
Evaluate
Teacher Activity
Ask students to independently compose and decompose a given number up to 100,000 using a place value chart and write it in expanded notation. Collect their work for assessment.
Student Activity
Students complete the task individually and explain their answers if asked.
Explanation
This assessment checks students' mastery of composing and decomposing numbers and understanding place value relationships.
Examples
- Students accurately compose and decompose numbers.
- Students write correct expanded notation.
Classroom Activity
Students use the blocks to represent the number and draw models showing ten thousands, thousands, hundreds, tens, and ones.
Guided Practice
Guided Practice 1
Prompt
What place value is the digit 7 in the number 27,489?
Teacher Answer Guide
The digit 7 is in the thousands place.
Guided Practice 2
Prompt
Compose the number 63,205 using ten thousands, thousands, hundreds, tens, and ones.
Teacher Answer Guide
63,205 = 6 ten thousands + 3 thousands + 2 hundreds + 0 tens + 5 ones.
Guided Practice 3
Prompt
Decompose the number 48,316 and write it as a sum of place values.
Teacher Answer Guide
48,316 = 40,000 + 8,000 + 300 + 10 + 6.
Guided Practice 4
Prompt
Explain why the digit 5 in 50,000 is ten times the value of the digit 5 in 5,000.
Teacher Answer Guide
Because each place value to the left is ten times greater than the place to its right, the 5 in the ten thousands place (50,000) is ten times the value of the 5 in the thousands place (5,000).
Guided Practice 5
Prompt
Write the number 72,409 in expanded notation and explain how the base-10 system helps you understand this notation.
Teacher Answer Guide
72,409 = (7 × 10,000) + (2 × 1,000) + (4 × 100) + (0 × 10) + (9 × 1). The base-10 system shows that each digit's value depends on its place, so expanded notation breaks the number into parts that reflect these place values multiplied by their digits.
Independent Practice
- Foundational: What place value is the digit 7 in the number 27,489?
- Developing: Compose the number 63,205 using ten thousands, thousands, hundreds, tens, and ones.
- Application: Decompose the number 48,316 and write it as a sum of place values.
- Analysis: Explain why the digit 5 in 50,000 is ten times the value of the digit 5 in 5,000.
- Challenge: Write the number 72,409 in expanded notation and explain how the base-10 system helps you understand this notation.
Independent Practice Teacher Answer Key
- 1. The digit 7 is in the thousands place.
- 2. 63,205 = 6 ten thousands + 3 thousands + 2 hundreds + 0 tens + 5 ones.
- 3. 48,316 = 40,000 + 8,000 + 300 + 10 + 6.
- 4. Because each place value to the left is ten times greater than the place to its right, the 5 in the ten thousands place (50,000) is ten times the value of the 5 in the thousands place (5,000).
- 5. 72,409 = (7 × 10,000) + (2 × 1,000) + (4 × 100) + (0 × 10) + (9 × 1). The base-10 system shows that each digit's value depends on its place, so expanded notation breaks the number into parts that reflect these place values multiplied by their digits.
Guiding Questions
- What does each digit in a number represent?
- How can you break a number into parts using place value?
- What is expanded notation and why is it useful?
- How do place values relate to each other in the base-10 system?
Common Misconceptions
- Students may think that the digit itself determines value without considering its place.
- Students might confuse the order of place values or think the places are additive rather than multiplicative by powers of ten.
- Students may struggle to understand zero's role in place value and expanded notation.
Differentiation
Support and Intervention
Use more concrete manipulatives and visual aids for students struggling with abstract concepts. Provide step-by-step guided practice with frequent checks for understanding. Use number lines to show place value increments.
English-Language Learner Support
Use visuals and gestures to explain place value terms. Provide vocabulary cards with definitions and pictures. Use partner talk to practice explaining place value concepts in simple language.
Advanced and Extension
Challenge students to compose and decompose numbers beyond 100,000. Introduce problems involving regrouping and place value in addition and subtraction. Have students create their own pictorial models for large numbers.
Assessment
- Observe students using base-10 blocks to represent numbers.
- Ask students to explain their reasoning during guided practice.
- Use exit tickets where students write numbers in expanded notation.
- Write the number 29,473 as a sum of ten thousands, thousands, hundreds, tens, and ones.
- Explain what the digit 9 represents in the number 29,473.
- Compose and decompose a number up to 100,000 using objects and pictorial models.
- Describe the relationship between place values in the base-10 system.
Answer Guide
- What place value is the digit 7 in the number 27,489?
Answer: The digit 7 is in the thousands place. - Compose the number 63,205 using ten thousands, thousands, hundreds, tens, and ones.
Answer: 63,205 = 6 ten thousands + 3 thousands + 2 hundreds + 0 tens + 5 ones. - Decompose the number 48,316 and write it as a sum of place values.
Answer: 48,316 = 40,000 + 8,000 + 300 + 10 + 6. - Explain why the digit 5 in 50,000 is ten times the value of the digit 5 in 5,000.
Answer: Because each place value to the left is ten times greater than the place to its right, the 5 in the ten thousands place (50,000) is ten times the value of the 5 in the thousands place (5,000). - Write the number 72,409 in expanded notation and explain how the base-10 system helps you understand this notation.
Answer: 72,409 = (7 × 10,000) + (2 × 1,000) + (4 × 100) + (0 × 10) + (9 × 1). The base-10 system shows that each digit's value depends on its place, so expanded notation breaks the number into parts that reflect these place values multiplied by their digits.
Real-Life Application
Ask students to find numbers in their home environment (e.g., house numbers, prices) and practice composing and decomposing those numbers using place value charts or drawings. Encourage them to explain what each digit represents to a family member.
Homework or Home Connection
- Ask students to find numbers in their home environment (e.g., house numbers, prices) and practice composing and decomposing those numbers using place value charts or drawings. Encourage them to explain what each digit represents to a family member.
Lesson Summary
In this lesson, students learned to compose and decompose numbers up to 100,000 by breaking them into sums of ten thousands, thousands, hundreds, tens, and ones. They used concrete objects, pictorial models, and expanded notation to understand the value of each digit and the mathematical relationships in the base-10 place value system. This foundational skill supports their ability to work with large numbers and understand number structure deeply.
Teacher Notes
Use the exact standards alignment and retrieved-source provenance stored with this enrichment.