K-12 Math (US): K-2 Number & Operations K-12 Math Place Value Tens and ones

Study Guide: Place Value — Tens and Ones (Grade K-2, Math)

1. The Driving Question

If you have 23 crayons, how do you know you don’t just have "two and three" crayons? Why does the order of the numbers matter, and how can you prove that 23 isn’t the same as 32—even though both use a 2 and a 3? What’s really hiding inside those digits?

2. The Core Idea — Built, Not Listed

Imagine you’re packing snacks for a field trip. You have 2 full lunchboxes, each holding 10 granola bars, and 3 extra granola bars rolling around in your backpack. The number 23 isn’t just "two and three"—it’s 2 tens (the full lunchboxes) and 3 ones (the loose bars). The 2 and the 3 aren’t equal partners; the 2 is like the boss of a group of 10, while the 3 is just a small extra. If you wrote 32 instead, you’d have 3 full lunchboxes (30 bars) and 2 loose ones—a totally different snack situation!

This is how numbers work: every digit has a place that tells you its value. The ones place is for single items, like loose crayons or extra granola bars. The tens place is for groups of 10, like full lunchboxes or stacks of 10 blocks. When you write a number, the left digit is always the tens, and the right digit is always the ones—no exceptions.

Key Vocabulary:
- Digit: A single symbol (0, 1, 2, … 9) used to write numbers.
Example: In the number 47, the digits are 4 and 7—not the number "forty-seven" itself.
- Place value: The value of a digit based on its position in a number.
Example: In 52, the 5 is worth 50 (5 tens) because it’s in the tens place, while the 2 is worth 2 (2 ones).
- Group of ten: A bundle of 10 ones that counts as a single unit in the tens place.
Example: If you trade 10 pennies for 1 dime, you’ve made a group of ten.
- Expanded form: Writing a number to show the value of each digit.
Example: 36 in expanded form is 30 + 6 (3 tens + 6 ones).

3. Assessment Translation (Grades K-2)

How this appears in class:
- Exit tickets: "Draw 24 using tens and ones. How many tens? How many ones?" - Show-your-work problems: "Circle the groups of ten in this picture of 35 stars. How many stars are left over?" - Short constructed response: "Explain why 45 is not the same as 54. Use words or pictures."

What "proficient" looks like vs. "developing":
| Proficient | Developing | |----------------|----------------| | Draws 2 full tens (e.g., 2 stacks of 10 blocks) and 4 single blocks for 24. | Draws 24 individual blocks without grouping. | | Writes 53 as 50 + 3 in expanded form. | Writes 53 as "five and three" or 5 + 3. | | Explains that 62 has 6 tens and 2 ones, while 26 has 2 tens and 6 ones. | Says "62 and 26 are the same because they both have a 6 and a 2." |

Model student response (proficient):
Prompt: "Show the number 37 using tens and ones. How many tens? How many ones?" Response: "I drew 3 groups of 10 circles (??????????) and 7 single circles (???????). There are 3 tens and 7 ones in 37."

4. Mistake Taxonomy

Mistake 1: Ignoring place value in counting
Prompt: "Count these cubes. How many tens? How many ones?" Wrong response: "I see 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. There are 12 ones." Why it loses credit: The student counts each cube individually but doesn’t group them into tens and ones. The question asks for place value, not just total count.
Correct approach: 1. Group the cubes into sets of 10.
2. Count the full groups (e.g., 1 group of 10).
3. Count the leftover cubes (e.g., 2 ones).
4. Write the number: 12 = 1 ten and 2 ones.

Mistake 2: Reversing digits
Prompt: "Write the number that has 4 tens and 6 ones." Wrong response: "64" Why it loses credit: The student writes the digits in the wrong order. The tens digit must come first (left), and the ones digit second (right).
Correct approach: 1. Remember: tens = left, ones = right.
2. Write the tens digit first: 4.
3. Write the ones digit next: 6.
4. The number is 46.

Mistake 3: Misreading expanded form
Prompt: "Which is the same as 20 + 8? Circle the answer: 28, 82, 208" Wrong response: "82" Why it loses credit: The student sees the digits 2 and 8 and picks the option with those digits, ignoring place value. 20 + 8 means 2 tens and 8 ones, which is 28.
Correct approach: 1. Break down the expanded form: 20 = 2 tens, 8 = 8 ones.
2. Combine them: 2 tens + 8 ones = 28.
3. Match to the correct choice.

5. Connection Layer

1. Within math: Place value → adding two-digit numbers
   Why it matters: When you add 23 + 15, you’re really adding 20 + 10 (tens) and 3 + 5 (ones). Understanding place value lets you break big problems into smaller, easier ones.

2. Across subjects: Place value → measuring length in centimeters
   Why it matters: A ruler is like a number line for place value! The number 12 cm means 1 ten (10 cm) and 2 ones (2 cm)—just like 12 is 1 ten and 2 ones.

3. Outside school: Place value → money (dollars and cents)
   Why it matters: A price tag like $4.25 is 4 dollars (4 groups of 100 cents) and 25 cents (2 tens + 5 ones). The decimal point is like a place-value separator for money!

6. The Stretch Question

If you have 100 crayons, how many different ways can you group them into tens and ones? Which way uses the fewest groups?

Pointer toward the answer: - You could have 10 groups of 10 (10 tens + 0 ones = 100).
- Or 9 groups of 10 and 10 ones (9 tens + 10 ones = 100), but that’s the same as 10 tens! - The fewest groups come from making as many tens as possible. What if you tried 5 groups of 20? Does that work? (Hint: Think about what "tens and ones" really means!)