A smiling girl in an apron holds a covered baking dish

Break It Apart!

Maya baked a pan of brownies. She wants to share her part over several days. How many different ways can she split it up?

4th Grade Math

Step 1: See the Unit Fractions

Maya's pan is cut into 6 equal parts. Her share is the fraction 5/6. Each shaded piece is one unit fraction: 1/6.

5/6 = 1/6 + 1/6 + 1/6 + 1/6 + ?

5/6 is 5 copies of 1/6. When you decompose a fraction, you break it into a sum of smaller pieces. 1/6 + 1/6 + 1/6 + 1/6 + 1/6 = 5/6.

Step 2: Break It Apart a New Way

Maya eats 2/6 today. She saves the rest for tomorrow. What fraction did she save?

5/6 = 2/6 + ?

5/6 = 2/6 + 3/6. The denominator stays 6 because the size of each piece never changes. Only the top numbers add: 2 + 3 = 5.

Let's Remember

Decompose = Break Apart

          Break a fraction into a sum of fractions with the same
          denominator. The
          numerator
          parts must add up to the original top number.

          5/6 = 1/6 + 4/6

          5/6 = 2/6 + 3/6

          5/6 = 1/6 + 1/6 + 3/6

One fraction, many ways to break it apart. All of them are true!

Prove It with a Model

Use the model to justify your answer. Which equation shows another true way to decompose 5/6?

Check the numerators: 4 + 1 = 5, so 4/6 + 1/6 = 5/6. The other choices fail: 2 + 2 = 4 and 3 + 3 = 6, not 5.

Watch Out!

A Common Mistake

          Some students write 2/6 + 3/6 = 5/12. That is wrong!

          The denominator tells the size of each piece. The pieces are
          still sixths.

          Add only the numerators: 2 + 3 = 5, so 2/6 + 3/6 = 5/6.

Maya says: Adding pieces never changes their size. Six pieces make the pan, so sixths stay sixths!

Solve It!

Which equation shows a correct way to decompose 7/8?

3/8 + 4/8 = 7/8 because 3 + 4 = 7 and the eighths stay eighths. Choice b only makes 6/8, and choice c changes the denominator.

What Goes in the Blank?

4/5 = 2/5 + ?

2/5 + 2/5 = 4/5. Think: 2 + ? = 4, so the missing numerator is 2. The denominator stays 5.

Word Problem

Leo runs 9/10 of a mile in three parts. First he runs 3/10, then 4/10. What fraction is left to run?

9/10 = 3/10 + 4/10 + ?

3 + 4 = 7, and 7 + 2 = 9. So 9/10 = 3/10 + 4/10 + 2/10. Leo has 2/10 of a mile left.

Find the Mistake

The model shows 3/4. Which equation is NOT a correct way to decompose 3/4?

2/4 + 2/4 = 4/4, which is one whole, not 3/4. The other equations both have numerators that add to 3.

Three Parts

8/10 = 3/10 + 2/10 + ?

3 + 2 = 5, and 5 + 3 = 8. So 8/10 = 3/10 + 2/10 + 3/10. Every part is still tenths.

Challenge: Bigger Than a Whole!

The fraction 11/8 is more than one whole. Remember that 8/8 = 1. Which equation decomposes 11/8 into a whole plus a part?

8 + 3 = 11, so 11/8 = 8/8 + 3/8. That also shows the mixed number 1 3/8. Choice a only makes 10/8, and choice c makes 8/8.

Maya found three different ways to break apart 5/6. Why can one fraction be decomposed in so many ways? How could a model prove each way is true?

Great Work!

0/9

correct

decompose fraction numerator denominator justify