Mathematics Grade 4: Fractions Addition and Subtraction

Grade 4 Mathematics Study Guide: Fractions – Addition and Subtraction

1. The Driving Question

If you and your friends split a chocolate bar into pieces, and then someone gives you another piece from a different bar, how do you figure out how much chocolate you have total—without melting it all down and starting over? And why can’t you just add the numbers on top like regular numbers?

2. The Core Idea – Built, Not Listed

Imagine you’re sharing a giant Hershey’s bar with three friends. You break it into 8 equal squares—that’s your whole. You take 3 squares (that’s 3/8), and your friend takes 2 squares (that’s 2/8). Now, if you put your pieces together, how much of the whole bar do you have? You don’t have to count the squares again—you can just add the numerators (3 + 2 = 5) while keeping the denominator the same (8). So, 3/8 + 2/8 = 5/8.

But what if your friend’s piece came from a different Hershey’s bar, broken into 4 big chunks instead of 8? Now you have 3/8 and 1/4. You can’t add them yet because the pieces aren’t the same size—it’s like trying to add 3 dimes + 1 quarter without converting to the same unit. You need to find a way to make the denominators match, which we call finding a common denominator. Once both fractions are in eighths (or fourths, or twelfths—whatever works), you can add or subtract them just like the first example.

Key Vocabulary: - Fraction – A number that represents part of a whole, written as numerator/denominator. Example: If you eat 5 slices of a 6-slice pizza, you ate 5/6 of the pizza. - Numerator – The top number in a fraction; tells how many parts you have. Example: In 7/10, the 7 means you have 7 out of 10 equal parts. - Denominator – The bottom number in a fraction; tells how many equal parts the whole is divided into. Example: In 3/5, the 5 means the whole is split into 5 equal pieces. - Common Denominator – A shared denominator that lets you add or subtract fractions. Example: To add 1/3 + 1/6, you can change 1/3 to 2/6 so both have the same denominator.

3. Assessment Translation

How This Appears in Classroom Assessments (Grades K–5): - Exit Tickets: "Solve: 2/5 + 1/5. Show your work." - Short Constructed Response: "Liam ate 3/8 of a pizza, and Mia ate 2/8. What fraction of the pizza did they eat together? Explain how you know." - Show-Your-Work Problems: "Solve: 3/4 – 1/2. Draw a picture to show your answer."

What a Proficient Response Looks Like: - Developing: "2/5 + 1/5 = 3/10" (adds denominators by mistake). - Proficient: "2/5 + 1/5 = 3/5 because I added the numerators (2 + 1 = 3) and kept the denominator the same." - Advanced: "I drew a rectangle split into 5 parts. I shaded 2 parts for 2/5 and 1 more part for 1/5. Now 3 parts are shaded, so it’s 3/5."

Model Proficient Response (Short Answer): Prompt: "Jada has 5/6 of a cup of flour. She uses 1/3 of a cup. How much flour does she have left?" Response: "First, I need to make the denominators the same. 1/3 is the same as 2/6. Then I subtract: 5/6 – 2/6 = 3/6. But 3/6 is the same as 1/2, so Jada has 1/2 cup left."

4. Mistake Taxonomy

Mistake 1: Adding Denominators - Question: "Solve: 1/4 + 1/4" - Common Wrong Answer: "1/4 + 1/4 = 2/8" - Why It Loses Credit: The student adds both numerators and denominators, treating fractions like separate numbers instead of parts of the same whole. - Correct Approach: "The denominators are the same, so I only add the numerators: 1 + 1 = 2. The answer is 2/4, which simplifies to 1/2."

Mistake 2: Forgetting to Find a Common Denominator - Question: "Solve: 2/3 – 1/2" - Common Wrong Answer: "2/3 – 1/2 = 1/1 = 1" (subtracts numerators and denominators separately). - Why It Loses Credit: The student doesn’t realize the pieces aren’t the same size, so they can’t subtract directly. - Correct Approach: "I need a common denominator. 3 and 2 both go into 6, so I change 2/3 to 4/6 and 1/2 to 3/6. Then 4/6 – 3/6 = 1/6."

Mistake 3: Simplifying Incorrectly - Question: "Solve: 4/8 + 2/8. Simplify your answer." - Common Wrong Answer: "4/8 + 2/8 = 6/8 = 3/4" (correct addition but simplifies before adding). - Why It Loses Credit: The student simplifies before adding, which can lead to mistakes. Always add first, then simplify. - Correct Approach: "4/8 + 2/8 = 6/8. Then I simplify 6/8 by dividing numerator and denominator by 2 to get 3/4."

5. Connection Layer

1. Within Math: Fractions-Decimals

2. If you know that 3/4 = 0.75, adding fractions is like adding decimals (e.g., 1/4 + 1/2 = 0.25 + 0.5 = 0.75). Understanding one helps you check the other.

3. Across Subjects: Fractions-Music (Rhythm)

4. In music, a quarter note (1/4) plus another quarter note (1/4) equals a half note (2/4 or 1/2). Adding fractions helps you count beats!

5. Outside School: Fractions-Cooking (Recipe Adjustments)

6. If a recipe calls for 1/2 cup of sugar but you’re doubling it, you add 1/2 + 1/2 = 1 cup. If you only have a 1/3 cup measure, you’ll need to find a common denominator to figure out how many scoops to use.

6. The Stretch Question

If you add 1/2 and 1/3, you get 5/6. But if you add 1/2 and 1/4, you get 3/4. Why does the denominator sometimes stay the same as one of the fractions, and other times it changes to a new number?

Pointer Toward the Answer: The denominator changes when the two fractions don’t already share the same-sized pieces. 1/2 and 1/3 don’t have a common denominator, so you have to find one (6). But 1/2 and 1/4 do share a denominator—4 is a multiple of 2—so you only need to adjust the 1/2 to 2/4. The key is whether one denominator is already a multiple of the other!