K-12 Math (US): 3-5 Number & Operations K-12 Math Decimals Tenths and hundredths

Study Guide: Decimals — Tenths and Hundredths

Grade Band: 3–5 Subject: K–12 Math (Number & Operations)

1. The Driving Question

If you cut a dollar into 10 equal pieces, each piece is a dime—but how do you write that as a number? And if you cut a dime into 10 even smaller pieces, what’s that called? Why can’t you just use fractions like 1/10 or 1/100 to describe money, time, or measurements, and what’s the secret code that makes decimals work?

2. The Core Idea — Built, Not Listed

Imagine you’re at a lemonade stand with a giant pitcher that holds exactly 1 liter of lemonade. If you pour out 1/10 of it into a cup, you’ve got 0.1 liters—that’s one tenth. Now, if you pour out 1/100 of the pitcher (just a tiny sip), that’s 0.01 liters—one hundredth. Decimals are just another way to write fractions where the denominator is 10, 100, or another power of 10, but instead of writing the denominator, we use a decimal point to show where the "whole" ends and the "parts" begin.

Think of the decimal point like a fence in a backyard. Everything to the left of the fence is whole (like dollars or liters), and everything to the right is parts of that whole (like dimes or milliliters). The first spot after the fence is tenths, the second is hundredths, and so on—just like how a ruler has inches (whole) and then little lines for tenths or hundredths of an inch.

Key Vocabulary:
- Decimal point – The dot that separates whole numbers from parts of a whole.
Example: In 3.25, the decimal point separates the 3 (whole dollars) from the 25 (cents).
- Tenth – One part of something divided into 10 equal pieces.
Example: If you eat 0.3 of a candy bar, you’ve eaten 3 out of the 10 equal pieces it was split into.
- Hundredth – One part of something divided into 100 equal pieces.
Example: A stopwatch shows 7.89 seconds—the 8 is tenths, and the 9 is hundredths (like 9 out of 100 tiny time slices).
- Place value – The value of a digit based on its position in a number.
Example: In 0.47, the 4 is in the tenths place (worth 0.4), and the 7 is in the hundredths place (worth 0.07).

3. Assessment Translation

How this appears in class (Grades 3–5):
- Exit tickets: "Write 4/10 as a decimal." (Proficient: 0.4; Developing: 4.0 or 0.04) - Short constructed response: "Explain why 0.5 is the same as 5/10. Use a drawing or words." (Proficient: "0.5 means 5 parts out of 10, just like 5/10. If you split a pizza into 10 slices, 0.5 is 5 slices."; Developing: "They’re the same because 5 is in both.") - Show-your-work problems: "Liam ran 1.25 miles. How many hundredths of a mile is that?" (Proficient: "1.25 = 125 hundredths because 1 = 100 hundredths, 0.2 = 20 hundredths, and 0.05 = 5 hundredths. 100 + 20 + 5 = 125."; Developing: "125" [no explanation])

What teachers look for:
- Proficient: Correct decimal notation, clear connection to fractions, and explanations that show understanding (not just memorization).
- Developing: Correct answers but weak explanations, or answers that mix up tenths/hundredths (e.g., writing 0.05 as "5 tenths").

Model Proficient Response:
Prompt: "Jada has $2.37. How many hundredths of a dollar is that?" Response: "$2.37 is 237 hundredths of a dollar. Here’s why: - 1 dollar = 100 hundredths, so 2 dollars = 200 hundredths.
- 0.30 (3 dimes) = 30 hundredths.
- 0.07 (7 pennies) = 7 hundredths.
- 200 + 30 + 7 = 237 hundredths."

4. Mistake Taxonomy

Mistake 1: Misplacing the decimal point
Prompt: "Write 6/100 as a decimal." Common wrong answer: 0.6 or 6.00
Why it loses credit: The student confuses tenths (0.6) with hundredths (0.06) or treats the fraction as a whole number.
Correct approach: - 6/100 means 6 parts out of 100, so the 6 goes in the hundredths place.
- The decimal is 0.06 (think: "zero whole dollars, zero dimes, six pennies").

Mistake 2: Adding decimals like whole numbers
Prompt: "Add 0.4 + 0.08." Common wrong answer: 0.12 or 0.48
Why it loses credit: The student ignores place value and adds digits as if they’re all tenths (4 + 8 = 12, so 0.12) or stacks them (0.4 + 0.08 = 0.48).
Correct approach: - Line up the decimal points: 0.40 + 0.08.
- Add hundredths: 0 + 8 = 8.
- Add tenths: 4 + 0 = 4.
- Answer: 0.48.

Mistake 3: Comparing decimals as if they’re whole numbers
Prompt: "Which is bigger: 0.7 or 0.65?" Common wrong answer: 0.65 (because 65 > 7) Why it loses credit: The student compares the numbers after the decimal as if they’re whole numbers, ignoring place value.
Correct approach: - Add a zero to 0.7 to make it 0.70.
- Compare tenths: 7 > 6, so 0.70 > 0.65.
- Think: "7 dimes vs. 6 dimes and 5 pennies—7 dimes is more."

5. Connection Layer

1. Within math: Decimals → Metric measurements — Understanding decimals makes it easier to read a ruler in centimeters (1.5 cm = 1 cm and 5 mm) or measure liquids in liters (0.25 L = 250 mL).
2. Across subjects: Decimals → Science data — Scientists use decimals to record precise measurements, like a plant growing 0.3 cm per day or a chemical reaction taking 1.25 seconds.
3. Outside school: Decimals → Sports stats — In basketball, a player’s free-throw percentage might be 0.875 (87.5%), which is easier to compare than fractions like 7/8.

6. The Stretch Question

If 0.9 is the same as 9/10, and 0.99 is 99/100, what happens if you keep adding 9s after the decimal forever—like 0.999...? Is that number equal to 1, or is it just really, really close?

Pointer toward the answer:
- Try subtracting: 1 – 0.999... = ? If you think it’s 0.000...1, that’s not a real number (you can’t have an "infinitely small" 1 at the end).
- Think about money: If you owe someone $1, and you pay them $0.90 + $0.09 + $0.009 + $0.0009..., you’ll never quite reach $1—but in math, that infinite sum is $1.
- This is how mathematicians define limits—something you’ll see in high school calculus!