A.4 Write numbers in standard and expanded form
What is place value?
Place value is the value of a digit based on its position in a number. In the base-ten number system, each place has a value ten times greater than the place to its right. Understanding place value allows a person to read, write, and compare large numbers accurately.
- In the number 4,326, the digit 4 is in the thousands place. Its value is 4,000.
- In the number 2,587, the digit 5 is in the hundreds place. Its value is 500.
- In the number 9,014, the digit 0 is in the hundreds place. Its value is zero, but it holds the place so that the other digits remain in their correct positions.
A digit receives its value from its location within a number. The digit 7 alone represents seven, but in the number 7,000, it represents seven thousand.
Place value models using base ten blocks
Base ten blocks provide a visual model for place value. A unit cube represents one. A long rod, composed of ten cubes connected, represents ten. A flat square, composed of ten rods, represents one hundred. A large cube, composed of ten flats, represents one thousand. When modeling numbers, one thousand block is used to represent 1,000. Numbers up to 10,000 can be represented by combining multiple thousands blocks.
- To model 3,245: use 3 thousands blocks, 2 hundreds flats, 4 tens rods, and 5 ones cubes.
- To model 8,064: use 8 thousands blocks, 0 hundreds flats, 6 tens rods, and 4 ones cubes.
- To model 10,000: use ten thousands blocks.
When drawing models, a large cube represents a thousand, a square represents a hundred, a line represents a ten, and a dot represents a one. Labeling each group prevents confusion.
Using commas in large numbers
Commas separate digits into groups of three to make large numbers easier to read. Starting from the right, a comma is placed after every three digits. These groups are called periods: ones, thousands, millions, and so on.
- 4,000
- 12,845
- 342,961
- 1,546,258
- 9,000,000
Numbers with four digits may be written with or without a comma (for example, 3,200 or 3200). For five or more digits, commas are necessary to separate the periods.
Place value chart up to ten thousands
A place value chart organizes digits according to their positions. The places include ones, tens, hundreds, thousands, and ten thousands.
| Ten Thousands | Thousands | Hundreds | Tens | Ones |
|---|---|---|---|---|
| 8 | 3 | 7 | 4 |
8 thousands + 3 hundreds + 7 tens + 4 ones = 8,374
When identifying places, begin from the right: ones, tens, hundreds, thousands, ten thousands. Each shift to the left multiplies the value by ten.
Writing numbers in expanded form
Expanded form expresses a number as the sum of the values of each digit. This representation clarifies the contribution of each digit to the total.
- 2,456 = 2,000 + 400 + 50 + 6
- 9,035 = 9,000 + 0 + 30 + 5
- 14,208 = 10,000 + 4,000 + 200 + 0 + 8
- 6,790 = 6,000 + 700 + 90 + 0
- 28,431 = 20,000 + 8,000 + 400 + 30 + 1
When a digit is zero, it may be omitted from the expanded form because it does not add value. Including it, however, serves as a reminder that the place is empty.
Writing numbers in word form
Word form is the number written out using words. Commas separate the periods, and hyphens connect words for numbers between twenty-one and ninety-nine.
- 3,472 = three thousand four hundred seventy-two
- 8,051 = eight thousand fifty-one
- 15,689 = fifteen thousand six hundred eighty-nine
- 940,123 = nine hundred forty thousand one hundred twenty-three
- 2,008 = two thousand eight
The word "and" is not used when writing whole numbers in word form. For example, 4,506 is written as four thousand five hundred six, not four thousand five hundred and six.
Comparing numbers using place value
To compare two numbers, examine the digits from left to right. The first position where the digits differ determines which number is larger or smaller. The symbols > (greater than), < (less than), or = (equal) indicate the relationship.
- Compare 3,842 and 3,921. The thousands digits are both 3. The hundreds digits are 8 and 9. Since 8 < 9, 3,842 < 3,921.
- Compare 12,456 and 12,461. The ten thousands digits are both 1. The thousands digits are both 2. The hundreds digits are both 4. The tens digits are 5 and 6. Since 5 < 6, 12,456 < 12,461.
- Compare 7,235 and 7,235. All digits match, so 7,235 = 7,235.
Align numbers by place value before comparing. Examining only the first digit is insufficient; each place must be considered.
Rounding numbers to the nearest ten, hundred, or thousand
Rounding means finding the closest multiple of ten, hundred, thousand, or another place value. To round, examine the digit immediately to the right of the target place. If that digit is 5 or greater, round up. If it is 4 or less, round down.
- Round 3,672 to the nearest hundred. The hundreds digit is 6. The tens digit is 7. Since 7 ≥ 5, round up: 3,700.
- Round 8,431 to the nearest thousand. The thousands digit is 8. The hundreds digit is 4. Since 4 ≤ 4, round down: 8,000.
- Round 14,589 to the nearest ten. The tens digit is 8. The ones digit is 9. Since 9 ≥ 5, round up: 14,590.
- Round 5,362 to the nearest hundred. The hundreds digit is 3. The tens digit is 6. Since 6 ≥ 5, round up: 5,400.
When rounding up, digits to the left may change. For example, rounding 1,987 to the nearest hundred results in 2,000 because 9 hundreds become 10 hundreds, which adds one thousand.
Understanding the relationship between place values
Each place to the left is ten times greater than the place to its right. Conversely, each place to the right is one-tenth of the place to its left. This pattern is fundamental to the base-ten number system.
- 1 thousand = 10 hundreds = 100 tens = 1,000 ones.
- In 5,000, the 5 is in the thousands place. Its value is ten times the value of 5 hundreds (500). 5,000 ÷ 10 = 500.
- In 6,400, the 4 is in the hundreds place. Its value is ten times 4 tens (40). 400 ÷ 10 = 40.
- In 20,000, the 2 represents 2 ten thousands, which is ten times 2 thousands (2,000).
The "ten times" relationship is essential when multiplying or dividing by 10, 100, or 1,000. Digits shift left or right accordingly.
Modeling numbers up to 10,000 with one thousand block
A single thousand block represents 1,000. To represent numbers greater than 1,000 but less than 10,000, multiple thousands blocks are used alongside hundreds flats, tens rods, and ones cubes. For example, 4,000 is represented by four thousands blocks. Numbers up to ten thousand can be shown with up to ten thousands blocks.
- 2,418: draw 2 large cubes (each representing 1,000), 4 flats (each representing 100), 1 rod (representing 10), and 8 small cubes (each representing 1).
- 5,030: draw 5 large cubes, 0 flats, 3 rods, and 0 ones.
- 9,999: draw 9 large cubes, 9 flats, 9 rods, and 9 ones.
When ten of any block are collected, they are exchanged for the next larger block. Ten ones become a ten rod; ten tens become a hundred flat; ten hundreds become a thousand block.
Real-world application of place value
Place value is used in everyday situations such as reading prices, understanding distances, working with money, and interpreting sports scores. Large numbers appear in populations, measurements, and scores.
- A new car costs $23,450. The 2 represents twenty thousand dollars.
- The population of a town is 8,792 people. The 8 represents eight thousand.
- A video game score is 1,248,000 points. The 1 represents one million.
- The distance from New York to Los Angeles is approximately 2,790 miles. The 2 represents two thousand miles.
When encountering a large number, reading it aloud using correct place value names reinforces understanding.
Common mistakes to avoid with place value
Several errors frequently occur when working with place value. Recognizing them helps prevent mistakes.
- Mistake: Writing four thousand twelve as 400012. Fix: Use a place value chart: 4,012.
- Mistake: Thinking 3,482 is greater than 3,512 because 482 is greater than 512. Fix: Compare thousands (both 3), then hundreds: 4 vs. 5. 3,482 is less.
- Mistake: Omitting commas in large numbers, such as writing 12589 instead of 12,589. Fix: Insert commas every three digits from the right.
- Mistake: Confusing the tens and hundreds places. Fix: Remember that the tens place is second from the right; the hundreds place is third.
Practicing with a partner: one person says a number aloud, and the other writes it with commas and in expanded form, then checks the work.
Summary of key place value ideas
The following concepts are fundamental to understanding place value.
- Place value determines the value of a digit by its position.
- Base ten blocks model numbers: cubes (ones), rods (tens), flats (hundreds), large cubes (thousands). Numbers up to 10,000 use up to ten thousands blocks.
- Commas separate groups of three digits from the right: 1,000; 10,000; 100,000; 1,000,000.
- Expanded form shows a number as the sum of its digit values.
- Word form uses words; "and" is not used for whole numbers.
- Comparing numbers requires examining digits from left to right.
- Rounding uses the digit to the right to determine whether to round up or down.
- Each place is ten times the place to its right.
Consistent practice with numbers encountered in daily life strengthens place value understanding.
Common Core alignment:
CCSS.MATH.CONTENT.4.NBT.A.1 – Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right.
CCSS.MATH.CONTENT.4.NBT.A.2 – Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.
CCSS.MATH.CONTENT.4.NBT.A.3 – Use place value understanding to round multi-digit whole numbers to any place.
Notes for teachers
This lesson is aligned with CCSS.MATH.CONTENT.4.NBT.A.1, 4.NBT.A.2, and 4.NBT.A.3. It may be used for whole-class instruction, independent practice, or homework.
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