Place Value
Ones and Tens
What are ones and tens?
When we write a two-digit number, each digit has a special place. The ones place tells us how many single units we have. The tens place tells us how many groups of ten we have.
Think of a bundle of sticks:
- Ones are single sticks (1, 2, 3, ... up to 9)
- Tens are bundles of 10 sticks tied together
Understanding Tens and Ones:
| Number | Tens | Ones | Meaning |
|---|---|---|---|
| 23 | 2 tens | 3 ones | 20 + 3 = 23 |
| 45 | 4 tens | 5 ones | 40 + 5 = 45 |
| 70 | 7 tens | 0 ones | 70 + 0 = 70 |
| 8 | 0 tens | 8 ones | 0 + 8 = 8 |
Important Rules:
- A group of 10 ones makes 1 ten
- The rightmost digit is always the ones place
- The digit to the left of the ones place is the tens place
- You can have 0 to 9 in the ones place
- You can have 1 to 9 in the tens place (for two-digit numbers)
How many tens and ones are in the number 47?
- Look at the number 47
- The first digit (from the right) is 7 → this is the ones place
- The digit to the left is 4 → this is the tens place
- 4 tens means 40
- 7 ones means 7
- 40 + 7 = 47
Answer: 4 tens and 7 ones
I have 5 bundles of ten pencils and 2 single pencils. What number does this represent?
- Each bundle has 10 pencils
- 5 bundles = 5 × 10 = 50 pencils
- 2 single pencils = 2 pencils
- Add them together: 50 + 2 = 52 pencils
- The number is 52 (5 tens and 2 ones)
Answer: 52
A number has 8 ones and 3 tens. What is the number? Is it greater than 40?
- 3 tens = 30
- 8 ones = 8
- Add: 30 + 8 = 38
- Compare 38 with 40
- 38 is less than 40 (because 38 comes before 40 when counting)
- Answer: The number is 38, which is NOT greater than 40
Answer: 38, no
Key Points
- Tens place shows groups of ten (10, 20, 30...)
- Ones place shows single units (1, 2, 3...)
- 10 ones make 1 ten
- Two-digit numbers have both tens and ones
- The rightmost digit is always the ones place
- Numbers 0-9 have 0 tens and that many ones
Understanding Place Value up to 100 and Expanded Form
What is place value up to 100?
Place value means the value of a digit depends on its position in the number. In numbers up to 100, we have two important places: the tens place and the ones place. The number 100 has a special place called the hundreds place.
What is expanded form?
Expanded form is a way to write a number by showing the value of each digit. We "stretch" the number to show how many tens and ones it has.
Examples of expanded form:
- 45 = 40 + 5 (4 tens + 5 ones)
- 72 = 70 + 2 (7 tens + 2 ones)
- 80 = 80 + 0 (8 tens + 0 ones)
- 100 = 100 + 0 + 0 (1 hundred + 0 tens + 0 ones)
| Number | Standard Form | Expanded Form | Words |
|---|---|---|---|
| 36 | 36 | 30 + 6 | thirty-six |
| 57 | 57 | 50 + 7 | fifty-seven |
| 91 | 91 | 90 + 1 | ninety-one |
| 100 | 100 | 100 + 0 + 0 | one hundred |
Write the number 54 in expanded form.
- Find the tens digit: 5 (means 5 tens = 50)
- Find the ones digit: 4 (means 4 ones = 4)
- Write as addition: 50 + 4
- Check: 50 + 4 = 54
Answer: 50 + 4
A number has 7 tens and 2 ones. Write the number in standard form and expanded form.
- 7 tens = 70
- 2 ones = 2
- Add: 70 + 2 = 72 (standard form)
- Expanded form: 70 + 2
- Check: 70 + 2 = 72 ✓
Answer: Standard form: 72, Expanded form: 70 + 2
Neha saw the expanded form 80 + 7. What is the number? How many tens does it have? If she adds 1 ten, what is the new number?
- 80 + 7 = 87 (the number)
- 87 has 8 tens (80) and 7 ones
- Adding 1 ten means add 10: 87 + 10 = 97
- 97 has 9 tens and 7 ones
- Check: 80 + 7 = 87, 87 + 10 = 97
Answer: 87, 8 tens, 97
Key Points
- Place value means the position of a digit tells its value
- Expanded form breaks a number into the sum of its place values
- Numbers up to 100 have tens and ones places
- The number 100 has 1 hundred, 0 tens, and 0 ones
- Expanded form helps us understand the true value of digits
- Always write expanded form from largest to smallest place value
Comparing Two-Digit Numbers
What is comparing numbers?
Comparing numbers means deciding if one number is greater than, less than, or equal to another number.
The Comparison Symbols:
| Symbol | Name | Meaning | Example |
|---|---|---|---|
| > | Greater than | The number on the left is bigger | 25 > 18 |
| < | Less than | The number on the left is smaller | 9 < 15 |
| = | Equal to | Both numbers are the same | 32 = 32 |
How to compare two-digit numbers:
Step 1: Compare the tens place first
- If tens are different → the number with more tens is bigger
- If tens are the same → go to Step 2
Step 2: Compare the ones place
- The number with more ones is bigger
- If ones are also the same → the numbers are equal
Comparison Examples:
| Compare | Tens Comparison | Ones Comparison | Result |
|---|---|---|---|
| 45 and 52 | 4 tens vs 5 tens → 5 is bigger | Not needed | 45 < 52 |
| 37 and 39 | 3 tens vs 3 tens → same | 7 ones vs 9 ones → 9 is bigger | 37 < 39 |
| 84 and 84 | 8 tens vs 8 tens → same | 4 ones vs 4 ones → same | 84 = 84 |
Compare 63 and 59. Use >, <, or =.
- Look at the tens place: 63 has 6 tens, 59 has 5 tens
- 6 tens is greater than 5 tens
- So 63 is greater than 59
- Correct symbol: 63 > 59
Answer: 63 > 59
Compare 72 and 78. Which is smaller?
- Both numbers have 7 tens (same)
- Compare ones place: 72 has 2 ones, 78 has 8 ones
- 2 ones is less than 8 ones
- So 72 is smaller than 78
- 72 < 78, so 72 is the smaller number
Answer: 72 is smaller
Arrange these numbers in order from smallest to largest: 45, 54, 49, 52
- Compare all numbers by looking at tens place first:
- 45 (4 tens), 54 (5 tens), 49 (4 tens), 52 (5 tens)
- Numbers with 4 tens: 45 and 49
- Compare 45 and 49: 45 < 49 (because 5 ones < 9 ones)
- Numbers with 5 tens: 54 and 52
- Compare 54 and 52: 52 < 54 (because 2 ones < 4 ones)
- Put in order: 45, 49, 52, 54
- Check: Each number is bigger than the previous one
Answer: 45, 49, 52, 54
Key Points
- > means greater than (bigger)
- < means less than (smaller)
- = means equal to (the same)
- Always compare the tens place first
- If tens are the same, compare the ones place
- The alligator mouth eats the bigger number
- On a number line, numbers increase from left to right
Greater Than, Less Than, Equal To and Ordering Numbers
What does ordering numbers mean?
Ordering numbers means arranging them in a sequence either from smallest to largest (ascending order) or from largest to smallest (descending order).
Ascending Order (Smallest to Largest):
- Start with the smallest number
- End with the largest number
- Example: 12, 24, 36, 48
Descending Order (Largest to Smallest):
- Start with the largest number
- End with the smallest number
- Example: 85, 73, 61, 49
Rules for Ordering Numbers:
| Rule | Explanation |
|---|---|
| 1 | Look at all the tens digits first |
| 2 | Numbers with fewer tens are smaller |
| 3 | If tens are same, compare ones |
| 4 | Write numbers in a row to compare easily |
| 5 | Check your order by counting up or down |
Practice Ordering:
| Set of Numbers | Ascending Order | Descending Order |
|---|---|---|
| 25, 18, 32, 27 | 18, 25, 27, 32 | 32, 27, 25, 18 |
| 51, 49, 55, 52 | 49, 51, 52, 55 | 55, 52, 51, 49 |
| 70, 68, 72, 69 | 68, 69, 70, 72 | 72, 70, 69, 68 |
Write these numbers in ascending order (smallest to largest): 31, 27, 35, 29
- Compare tens: 31(3), 27(2), 35(3), 29(2)
- Numbers with 2 tens: 27 and 29 → 27 < 29
- Numbers with 3 tens: 31 and 35 → 31 < 35
- Put all together: 27, 29, 31, 35
- Check: 27 < 29 < 31 < 35 ✓
Answer: 27, 29, 31, 35
Arrange in descending order (largest to smallest): 88, 92, 85, 90
- Compare tens: 88(8), 92(9), 85(8), 90(9)
- Numbers with 9 tens (larger): 92 and 90 → 92 > 90
- Numbers with 8 tens: 88 and 85 → 88 > 85
- Put largest first: 92, 90, 88, 85
- Check: 92 > 90 > 88 > 85 ✓
Answer: 92, 90, 88, 85
Riya has 3 cards with numbers: 47, 52, and 39. Her friend has numbers 45, 50, and 41. Whose largest number is bigger? Arrange all six numbers in ascending order.
- Find Riya's largest: 52 (compare 47, 52, 39 → 52 is biggest)
- Find friend's largest: 50 (compare 45, 50, 41 → 50 is biggest)
- Compare 52 and 50: 52 > 50, so Riya's largest is bigger
- Arrange all six numbers: 39, 41, 45, 47, 50, 52
- Check order: 39<41<45<47<50<52 ✓
Answer: Riya's largest is bigger (52 > 50); Ascending order: 39, 41, 45, 47, 50, 52
Key Points
- Ascending order = smallest to largest (going up)
- Descending order = largest to smallest (going down)
- Always compare tens place first when ordering
- If tens are the same, compare ones place
- Write numbers in a row to compare them easily
- Check your order by making sure each number is bigger or smaller than the next
- Real-life ordering helps us with races, heights, ages, and money