3A · Chapter 1 · Lesson 03
Place Value — Part 2
Compose and decompose a numeral using standard and expanded forms., Rename a number using equivalent groups., Explain regrouping without changing the quantity.
Your starting challenge
Can 4,306 be made using only hundreds and ones? Find two different ways to represent the same amount.
By the end, you can…
- Compose and decompose a numeral using standard and expanded forms.
- Rename a number using equivalent groups.
- Explain regrouping without changing the quantity.
Before starting, you should know…
- Identify digits and place values.
Entry · 2 questions
Quick prerequisite check
Try these two before the lesson. If either is confusing, review the listed prerequisites first.
- E1 In 5,294, what is the value of the digit in the hundreds place?
Show answer and reasoning
Answer: 200
The hundreds digit is 2; each unit has value 100, so its value is 2 × 100 = 200.
If your answer differs — Digit versus value: Circle the requested digit, name its column, then multiply the digit by 1,000, 100, 10, or 1. Review this concept. - E2 In 3,806, what is the value of the digit in the tens place?
Show answer and reasoning
Answer: 0
The tens digit is 0; each unit has value 10, so its value is 0 × 10 = 0.
If your answer differs — Digit versus value: Circle the requested digit, name its column, then multiply the digit by 1,000, 100, 10, or 1. Review this concept.
Vocabulary to know
| expanded form | A number written as the sum of its place values. |
|---|---|
| compose | Build one number from its parts. |
| decompose | Break a number into place-value parts. |
| rename | Express the same amount with different unit groups. |
Learn the concept
Expanded form
4,306 = 4,000 + 300 + 6. The missing tens value is zero, so writing + 0 is optional.
Flexible renaming
Draw 4 labeled thousand-bundles, 3 hundred-bundles, and 6 ones. Trade one thousand-bundle for ten hundred-bundles: now there are 3 thousands, 13 hundreds, and 6 ones. Add the values before (4,000 + 300 + 6) and after (3,000 + 1,300 + 6): both total 4,306. To write the number using hundreds and ones only, change the remaining 3 thousands into 30 more hundreds, producing 43 hundreds and 6 ones.
Counting complete groups
To find the number of complete tens in 4,300, calculate 4,300 ÷ 10 = 430, not 0. The tens *digit* is 0; the number of complete tens is 430.
ONE 1,000 bundle becomes TEN 100 bundles; 4 thousands becomes 3, while 3 hundreds becomes 13. Alternatively: 43 hundreds + 6 ones = 4,306.
Follow the mathematics: Build → Sketch → Explain → Verify
- Build: Represent 4,306 as 4 thousands, 3 hundreds, 0 tens, and 6 ones. Exchange ONE thousand for TEN hundreds.
- Sketch: Before: 4 thousands | 3 hundreds | 0 tens | 6 ones. After: 3 thousands | 13 hundreds | 0 tens | 6 ones.
- Name: Both representations equal 4,306 because 1,000 and 10 × 100 have equal value. A second renaming is 43 hundreds + 6 ones.
- Verify: A digit in the tens place is not the number of complete tens. 4,300 has tens digit 0 but contains 430 complete tens.
No physical blocks? Use identical paper squares or draw labeled bundles. Try the steps before opening the worked examples.
Work through examples
Study the explanation; then try each check before opening its answer.
Worked example 1
Break a number into place values
Write 6,205 in expanded form.
- 6 thousands = 6,000.
- 2 hundreds = 200; 0 tens = 0; 5 ones = 5.
- Combine only the nonzero values.
Result: 6,000 + 200 + 5
Now you try · Understanding check 1
Expand 7,408 into the sum of its nonzero place values.Reveal answer and explanation
7,000 + 400 + 8 — Use thousands, hundreds, tens, ones in order. Zero-value places may be omitted: 7,000 + 400 + 8.
Worked example 2
Rename thousands as hundreds
Write 4,300 entirely in hundreds.
- One thousand contains 10 hundreds.
- Four thousands contain 40 hundreds; add 3 hundreds.
Result: 43 hundreds
Now you try · Understanding check 2
How many complete hundreds are in 5,300?Reveal answer and explanation
53 — Each hundred is worth 100. Divide 5,300 by 100: there are 53 complete hundreds.
Worked example 3
Regroup in a different way
Is 3 thousands + 13 hundreds + 6 ones equal to 4,306?
- 13 hundreds = 1,300.
- 3,000 + 1,300 + 6 = 4,306.
- Both representations have the same total.
Result: Yes; both are 4,306
Now you try · Understanding check 3
Is 5 thousands + 12 hundreds + 4 ones equal to 6,204?Reveal answer and explanation
Yes — 5,000 + 1,200 + 4 = 6,204.
Guided · 4 questions
Guided practice
Try each problem; return to a visual model if you need one.
- G1 Expand 6,082 into the sum of its nonzero place values.
Show answer and reasoning
Answer: 6,000 + 80 + 2
Use thousands, hundreds, tens, ones in order. Zero-value places may be omitted: 6,000 + 80 + 2.
If your answer differs — Lost place value: Use four labeled columns and write the value of every nonzero digit. Add the parts to check. Review this concept. - G2 What number is 5 thousands + 2 hundreds + 0 tens + 7 ones?
Show answer and reasoning
Answer: 5,207
Add 5,000 + 200 + 0 + 7 = 5,207.
If your answer differs — Lost place value: Use four labeled columns and write the value of every nonzero digit. Add the parts to check. Review this concept. - G3 How many complete hundreds are in 3,600?
Show answer and reasoning
Answer: 36
Each hundred is worth 100. Divide 3,600 by 100: there are 36 complete hundreds.
If your answer differs — Group count versus digit: Exchange 1 thousand for 10 hundreds or 1 hundred for 10 tens. Divide by the unit size to count complete groups. Review this concept. - G4 Expand 9,450 into the sum of its nonzero place values.
Show answer and reasoning
Answer: 9,000 + 400 + 50
Use thousands, hundreds, tens, ones in order. Zero-value places may be omitted: 9,000 + 400 + 50.
If your answer differs — Lost place value: Use four labeled columns and write the value of every nonzero digit. Add the parts to check. Review this concept.
Independent · 8 questions
Independent practice
Work without looking at examples first. Show your thinking on paper.
- I1 Expand 7,293 into the sum of its nonzero place values.
Show answer and reasoning
Answer: 7,000 + 200 + 90 + 3
Use thousands, hundreds, tens, ones in order. Zero-value places may be omitted: 7,000 + 200 + 90 + 3.
If your answer differs — Lost place value: Use four labeled columns and write the value of every nonzero digit. Add the parts to check. Review this concept. - I2 What number is 2 thousands + 8 hundreds + 3 tens + 6 ones?
Show answer and reasoning
Answer: 2,836
Add 2,000 + 800 + 30 + 6 = 2,836.
If your answer differs — Lost place value: Use four labeled columns and write the value of every nonzero digit. Add the parts to check. Review this concept. - I3 How many complete hundreds are in 7,200?
Show answer and reasoning
Answer: 72
Each hundred is worth 100. Divide 7,200 by 100: there are 72 complete hundreds.
If your answer differs — Group count versus digit: Exchange 1 thousand for 10 hundreds or 1 hundred for 10 tens. Divide by the unit size to count complete groups. Review this concept. - I4 Expand 4,015 into the sum of its nonzero place values.
Show answer and reasoning
Answer: 4,000 + 10 + 5
Use thousands, hundreds, tens, ones in order. Zero-value places may be omitted: 4,000 + 10 + 5.
If your answer differs — Lost place value: Use four labeled columns and write the value of every nonzero digit. Add the parts to check. Review this concept. - I5 What number is 3 thousands + 6 hundreds + 0 tens + 9 ones?
Show answer and reasoning
Answer: 3,609
Add 3,000 + 600 + 0 + 9 = 3,609.
If your answer differs — Lost place value: Use four labeled columns and write the value of every nonzero digit. Add the parts to check. Review this concept. - I6 How many complete tens are in 8,400?
Show answer and reasoning
Answer: 840
Each ten is worth 10. Divide 8,400 by 10: there are 840 complete tens.
If your answer differs — Group count versus digit: Exchange 1 thousand for 10 hundreds or 1 hundred for 10 tens. Divide by the unit size to count complete groups. Review this concept. - I7 Expand 8,900 into the sum of its nonzero place values.
Show answer and reasoning
Answer: 8,000 + 900
Use thousands, hundreds, tens, ones in order. Zero-value places may be omitted: 8,000 + 900.
If your answer differs — Lost place value: Use four labeled columns and write the value of every nonzero digit. Add the parts to check. Review this concept. - I8 What number is 7 thousands + 0 hundreds + 4 tens + 2 ones?
Show answer and reasoning
Answer: 7,042
Add 7,000 + 0 + 40 + 2 = 7,042.
If your answer differs — Lost place value: Use four labeled columns and write the value of every nonzero digit. Add the parts to check. Review this concept.
Challenge · 2 questions
Challenge and transfer
These nonroutine problems require you to explain or adapt what you learned.
- C1 Represent 5,406 using 4 thousands, ___ hundreds, 0 tens, and 6 ones.
Show answer and reasoning
Answer: 14 hundreds
The missing thousand becomes 10 hundreds; combine with the original 4 hundreds to make 14.
If your answer differs — Group count versus digit: Exchange 1 thousand for 10 hundreds or 1 hundred for 10 tens. Divide by the unit size to count complete groups. Review this concept. - C2 How many complete hundreds are in 7,890, and how many ones are left over?
Show answer and reasoning
Answer: 78 hundreds, 90 ones
7,890 = 78 × 100 + 90. A leftover 90 cannot make another complete hundred.
If your answer differs — Group count versus digit: Exchange 1 thousand for 10 hundreds or 1 hundred for 10 tens. Divide by the unit size to count complete groups. Review this concept.
Exit · 3 questions
Exit check
Use these fresh problems to decide whether to proceed. Answers should be independent.
- E1 Expand 8,054 into the sum of its nonzero place values.
Show answer and reasoning
Answer: 8,000 + 50 + 4
Use thousands, hundreds, tens, ones in order. Zero-value places may be omitted: 8,000 + 50 + 4.
If your answer differs — Lost place value: Use four labeled columns and write the value of every nonzero digit. Add the parts to check. Review this concept. - E2 How many complete hundreds are in 4,700?
Show answer and reasoning
Answer: 47
Each hundred is worth 100. Divide 4,700 by 100: there are 47 complete hundreds.
If your answer differs — Group count versus digit: Exchange 1 thousand for 10 hundreds or 1 hundred for 10 tens. Divide by the unit size to count complete groups. Review this concept. - E3 What number is 3 thousands + 14 hundreds + 0 tens + 8 ones?
Show answer and reasoning
Answer: 4,408
Add 3,000 + 1,400 + 0 + 8 = 4,408.
If your answer differs — Lost place value: Use four labeled columns and write the value of every nonzero digit. Add the parts to check. Review this concept.