Same-Sized Pieces
What does a common denominator help us do?
It makes the fraction pieces the same size. Then we compare the number of pieces we have with the number in one group.
Rewrite 1/2 and 1/4 with a common denominator.
1/2 = 2/4; 1/4 stays 1/4. Common denominator: 4.
Solve: 1/2 ÷ 1/4. Show common-denominator steps.
2/4 ÷ 1/4 → 2 ÷ 1 = 2.
You have 3/4 cup of juice. Each serving is 1/4 cup. How many servings can you make?
3/4 ÷ 1/4 → 3 ÷ 1 = 3 servings.
A student says: 4/9 ÷ 2/9 = 2/9. Fix the answer.
The answer is 2 groups. Divide the piece counts: 4 ÷ 2 = 2. We do not keep the denominator 9 because the quotient counts groups.
Solve: 3/8 ÷ 1/8.
3 ÷ 1 = 3 groups. Three eighths make three groups of one eighth.
Rewrite 2/3 and 1/6 with a common denominator.
2/3 = 4/6; 1/6 stays 1/6. Common denominator: 6.
Solve: 2/3 ÷ 1/4. Show common-denominator steps.
8/12 ÷ 3/12 → 8 ÷ 3 = 8/3 = 2 2/3.
A ribbon is 5/6 yard long. Each piece is 1/3 yard. How many piece-lengths is that, including a fractional piece?
5/6 ÷ 2/6 → 5 ÷ 2 = 2 1/2 piece-lengths. Cut 2 full pieces; 1/6 yard remains, half of another piece.
A student rewrites 1/2 as 1/6 to match 1/3 = 2/6. What should 1/2 become?
1/2 = 3/6. Multiply BOTH numerator and denominator by 3 to keep the value the same.
Solve: 6/10 ÷ 2/10.
6 ÷ 2 = 3 groups. Six tenths make three groups of two tenths.
Rewrite 3/4 and 2/3 with a common denominator.
3/4 = 9/12; 2/3 = 8/12. Common denominator: 12.
Solve: 3/5 ÷ 2/3. Show common-denominator steps.
9/15 ÷ 10/15 → 9 ÷ 10 = 9/10.
You have 3/4 cup of oats. One batch uses 1/2 cup. How many batches can you make, including a fractional batch?
3/4 ÷ 2/4 → 3 ÷ 2 = 1 1/2 batches.
A student says: 2/3 ÷ 1/4 = 2 because 2 ÷ 1 = 2. Fix the work.
First use equal-sized pieces: 2/3 = 8/12 and 1/4 = 3/12. Then 8 ÷ 3 = 2 2/3. The original pieces were different sizes.
Solve: 3/7 ÷ 6/7.
3 ÷ 6 = 1/2. Three sevenths are half of a six-sevenths group.
Rewrite 5/6 and 3/4 with a common denominator.
5/6 = 10/12; 3/4 = 9/12. Common denominator: 12.
Solve: 1 1/4 ÷ 3/8. Show common-denominator steps.
1 1/4 = 5/4 = 10/8. Then 10/8 ÷ 3/8 → 10 ÷ 3 = 10/3 = 3 1/3.
A trail is 1 1/2 miles long. Each walking section is 2/5 mile. How many section-lengths is the trail?
15/10 ÷ 4/10 → 15 ÷ 4 = 15/4 = 3 3/4 section-lengths.
A student says: 1/4 ÷ 1/2 = 2. Is the quotient greater or less than 1? Solve.
Less than 1: the amount is smaller than one group. 1/4 ÷ 2/4 → 1 ÷ 2 = 1/2.
Solve: 7/12 ÷ 2/12. Explain the leftover.
7 ÷ 2 = 7/2 = 3 1/2 groups. The leftover 1/12 is half of a 2/12-size group.
Rewrite 1 1/2 and 2/5 with a common denominator.
1 1/2 = 3/2 = 15/10; 2/5 = 4/10. Common denominator: 10.
Solve: 2 1/3 ÷ 1 1/2. Show common-denominator steps.
2 1/3 = 7/3 = 14/6; 1 1/2 = 3/2 = 9/6. Then 14 ÷ 9 = 14/9 = 1 5/9.
You have 2 1/4 cups of flour. Each full batch needs 2/3 cup. How many FULL batches can you make, and how much flour remains?
9/4 ÷ 2/3 = 27/12 ÷ 8/12 → 27 ÷ 8 = 3 3/8 batch amounts. Make 3 full batches. They use 2 cups, leaving 1/4 cup. The leftover is 3/8 of a batch.
A student says: 5/6 ÷ 1/3 = 2 1/6 because 1/6 is left over. Fix the answer and explain.
5/6 ÷ 2/6 → 5 ÷ 2 = 2 1/2 groups. The leftover 1/6 is HALF of a 1/3-size group.