Mathematics Grade 4: Word Problems Fractions and Decimals

Grade 4 Mathematics Study Guide: Word Problems – Fractions and Decimals

1. The Driving Question

You’re at a lemonade stand with three friends, and you’ve made $12.50 total. If you split the money exactly so everyone gets the same amount, how do you write that share as a fraction and as a decimal? Why do both numbers describe the same piece of the whole, and how do you know which one to use when the problem asks for it?

2. The Core Idea – Built, Not Listed

Imagine you’re cutting a giant chocolate bar into 10 equal squares. Each square is 1/10 of the bar. If you eat 3 squares, you’ve eaten 0.3 of the bar—because "0.3" is just another way to say "3 out of 10." Fractions and decimals are like two languages for the same idea: parts of a whole. When a word problem asks for a fraction, it’s usually about counting parts (like slices of pizza). When it asks for a decimal, it’s often about measuring (like money or length). The key is to see the whole (the $12.50 or the chocolate bar) and then figure out how many equal parts it’s divided into.

Key Vocabulary: - Fraction – A number that shows parts of a whole, written as numerator/denominator (e.g., 3/4 means 3 parts out of 4). Example: If a soccer game is 60 minutes and you play for 15 minutes, you played 15/60 of the game. - Decimal – A number that shows parts of a whole using place value (e.g., 0.75 means 75 hundredths). Example: A pencil that’s 0.15 meters long is 15 centimeters. - Equivalent – Two numbers that represent the same amount (e.g., 1/2 and 0.5). Example: Half of a 10-piece Lego set is 5 pieces, which is also 0.5 of the set. - Denominator – The bottom number in a fraction, telling how many equal parts the whole is divided into. Example: In 3/8 of a pizza, the denominator (8) means the pizza is cut into 8 slices.

3. Assessment Translation

How this appears in class: - Exit tickets: "Liam has 24 marbles. He gives 1/3 of them to his sister. How many marbles does his sister get? Write your answer as a fraction and a decimal." - Show-your-work problems: "A recipe calls for 0.75 cups of sugar. If you only have a 1/4-cup measure, how many times do you need to fill it to get 0.75 cups?" - Short constructed response: "Explain how 0.4 and 4/10 are the same. Use a real-life example."

Proficient vs. Developing Responses: | Proficient | Developing | |----------------|----------------| | Problem: "A ribbon is 2.5 meters long. If you cut it into 5 equal pieces, how long is each piece?" | | | Response: "2.5 ÷ 5 = 0.5 meters. Each piece is 0.5 meters, which is the same as 1/2 meter." (Shows both forms and explains why they’re equal.) | Response: "0.5 meters." (Missing the fraction or explanation.) | | Response: "I know 2.5 is 25 tenths. 25 ÷ 5 = 5 tenths, which is 0.5 or 1/2." (Uses place value to explain.) | Response: "I divided 2.5 by 5 and got 0.5." (No reasoning or connection to fractions.) |

Model Proficient Response: Prompt: "A garden is 3/4 planted with flowers. What decimal shows how much of the garden has flowers?" Response: "3/4 is the same as 0.75 because 3 divided by 4 is 0.75. I can check by thinking of money: 3 quarters make 75 cents, which is $0.75. So 3/4 of the garden is 0.75 of it."

4. Mistake Taxonomy

Mistake 1: Misreading the Whole Question: "A pizza is cut into 8 slices. You eat 3 slices. What fraction of the pizza did you eat? Write it as a decimal." Common Wrong Answer: "3/10 or 0.3." Why It Loses Credit: The student ignored the denominator (8 slices) and used 10 as the whole because "decimal" made them think of tenths. Correct Approach: The whole is 8 slices, so 3 slices = 3/8. To write 3/8 as a decimal, divide 3 by 8: 3 ÷ 8 = 0.375.

Mistake 2: Confusing Fraction and Decimal Forms Question: "A runner finishes 0.6 of a race. What fraction of the race did they complete?" Common Wrong Answer: "6/100." Why It Loses Credit: The student wrote 0.6 as 6/100 instead of 6/10, forgetting that decimals use place value (tenths, hundredths). Correct Approach: 0.6 means 6 tenths, so the fraction is 6/10. Simplify to 3/5 if needed.

Mistake 3: Ignoring Units in Word Problems Question: "A rope is 1.2 meters long. If you cut it into 4 equal pieces, how long is each piece in meters? Write your answer as a fraction." Common Wrong Answer: "0.3 meters." Why It Loses Credit: The student gave a decimal but didn’t write the fraction (3/10 or 3/25) as asked. Correct Approach: 1.2 ÷ 4 = 0.3 meters. To write 0.3 as a fraction, think of it as 3/10. (Or convert 1.2 to 12/10 first: 12/10 ÷ 4 = 3/10.)

5. Connection Layer

• Within Math: Fractions/decimals-Division — Dividing 3 by 4 gives 0.75, which is the same as 3/4. Understanding this helps when dividing larger numbers (e.g., 7 ÷ 8 = 0.875).
• Across Subjects: Fractions/decimals-Science (Measurement) — Scientists use decimals for precision (e.g., 0.5 grams), but fractions for ratios (e.g., 1/2 of a solution). Knowing both helps you read lab results.
• Outside School: Fractions/decimals-Sports Stats — A basketball player’s free-throw percentage might be 0.875 (87.5%), but announcers say "7 out of 8." Recognizing these as the same number helps you follow games.

6. The Stretch Question

If 0.999... (the 9s go on forever) is equal to 1, does that mean fractions and decimals can represent the same number in more than one way? How would you explain this to a friend who thinks 0.999... is "almost" 1 but not quite?

Pointer Toward the Answer: Think of it like this: If 0.999... is less than 1, there must be a number between them—but there isn’t! You can also show it with fractions: 1/3 = 0.333..., so 3 × 1/3 = 1, and 3 × 0.333... = 0.999... Therefore, 0.999... must equal 1. This isn’t just a trick—it’s how our number system works when we let decimals go on forever.