Learning Goals
- Read and write whole numbers up to the millions in standard, word, and expanded form.
- Explain the value of a digit based on its place, not just the digit itself.
- Compare and order whole numbers using place value and symbols (
>,<,=). - Round whole numbers to a given place (nearest 10, 100, or 1,000) using clear reasoning.
- Check whether an answer makes sense by estimating with place value.
Intuitive Introduction
Think of place value like seats in a stadium. If one person sits in the “ones” seat, that is just 1. But if one person sits in the “thousands” seat, that same one now represents 1,000. The digit stays the same, but the seat (the place) changes its value.
You can also think of money. One dollar bill and one hundred-dollar bill both show a “1,” but they are not worth the same amount. In numbers, place value works the same way. In the number 4,582, the digit 5 means 500 because it is in the hundreds place. The digit 8 means 80 because it is in the tens place.
A place value chart helps us see this clearly. Imagine columns labeled: thousands, hundreds, tens, ones. When we place digits into those columns, we can read the number correctly and understand how big it is. This matters for almost every topic in math, because adding, subtracting, multiplying, dividing, estimating, and even working with decimals later all depend on strong place value understanding.
Place value also helps us compare numbers quickly. If two numbers have different digits in the largest place, that place decides which number is bigger. For example, 9,245 is greater than 8,999 because 9 thousands is more than 8 thousands, even though 999 looks “big.”
Finally, place value makes rounding useful. When we round, we decide which place is most important right now, then look one place to the right to choose whether to keep or increase the digit. Rounding helps us estimate and check if detailed answers are reasonable.
Formal Definition / Precise Language
A whole number is a number in the set {0, 1, 2, 3, ...}.
In base-10, each place value is 10 times the place to its right:
- ones (
10^0) - tens (
10^1) - hundreds (
10^2) - thousands (
10^3) - ten-thousands (
10^4), etc.
For a numeral written with digits, the value of each digit is:
digit × place value
Example:
7,304 = 7×1,000 + 3×100 + 0×10 + 4×1
To compare two whole numbers:
- Compare digits from the greatest place value to the least.
- At the first place where digits differ, the number with the greater digit is greater.
- If all corresponding digits are equal, the numbers are equal.
To round to a target place:
- Identify the target place digit.
- Look at the digit one place to the right.
- If that digit is 5–9, increase the target digit by 1.
- If that digit is 0–4, keep the target digit the same.
- Replace all digits to the right of the target place with 0.
Worked Examples
Example 1: Expanded Form and Digit Value
Write 63,418 in expanded form, and state the value of the digit 3.
Step 1: Identify each digit’s place.
- 6 is in the ten-thousands place.
- 3 is in the thousands place.
- 4 is in the hundreds place.
- 1 is in the tens place.
- 8 is in the ones place.
Step 2: Write as a sum.
63,418 = 60,000 + 3,000 + 400 + 10 + 8
Step 3: State the value of 3.
- The digit 3 has value 3,000.
Example 2: Comparing Whole Numbers
Which is greater: 507,092 or 570,209?
Step 1: Compare from the largest place.
- Hundred-thousands: both are 5.
- Ten-thousands: first number has 0, second number has 7.
Step 2: Decide at first difference.
- Since
7 > 0, the second number is greater.
Conclusion:
570,209 > 507,092
Example 3: Rounding to the Nearest Thousand
Round 48,650 to the nearest thousand.
Step 1: Target place is the thousands digit (8 in 48,650).
Step 2: Look at the hundreds digit (6).
Step 3: Apply rounding rule.
- Since 6 is 5 or more, increase the thousands digit by 1.
- 48 thousand becomes 49 thousand.
Answer:
48,650 ≈ 49,000(to the nearest thousand)
Common Mistakes & Misconceptions
- Mistake 1: “A bigger digit always means a bigger number.”
Why it is wrong: Place matters more than digit size in smaller places.
Correction: Compare from the leftmost (greatest) place first.
- Mistake 2: Reading 405 as “forty-five.”
Why it is wrong: The 0 is a placeholder showing there are no tens.
Correction: Read it as “four hundred five.”
- Mistake 3: In 3,241, saying the 2 is worth 2.
Why it is wrong: The 2 is in the hundreds place, so it is 200.
Correction: Multiply each digit by its place value.
- Mistake 4: Rounding based on the wrong digit.
Why it is wrong: Students sometimes look two places right instead of one place right.
Correction: Always look exactly one place to the right of the target place.
Quick Checks
- What is the value of the digit 9 in 29,304?
- Write 7,050 in expanded form.
- Compare using
>,<, or=: 64,199 ___ 64,091 - Round 12,489 to the nearest hundred.
- Round 305,001 to the nearest thousand.
Quick Check Answers
- 9,000
7,000 + 50>- 12,500
- 305,000
Graded Exercises
- Easy: Write 56,203 in word form and expanded form.
- Easy: What is the value of the underlined digit in 81,476: _4_?
- Medium: Order these numbers from least to greatest: 209,450; 290,405; 204,950; 294,005.
- Medium: Use
>,<, or=: 700,090 ___ 699,999. - Medium: Round 93,449 to the nearest ten and to the nearest hundred.
- Hard: A school has 18,495 books. About how many books is this to the nearest thousand? Explain your rounding decision.
- Hard: Create a 6-digit number where the digit 7 has value 70,000 and the digit 3 has value 30. Write your number in expanded form.
- Hard: Two numbers both have 5 in the ten-thousands place and 2 in the hundreds place. Make one number greater than the other by changing only the tens and ones digits.
Summary / Key Takeaways
- A digit’s value depends on its place, not only on the digit itself.
- Base-10 place values grow by factors of 10 as you move left.
- Expanded form shows the true value of each digit clearly.
- To compare whole numbers, start at the greatest place and move right.
- To round correctly, look one place to the right of the target place.