Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?.
Official wording from the Common Core State Standards for Mathematics (© 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers). View on thecorestandards.org
Dividing by a fraction answers one of two questions: how many groups of this size fit, or how much is in one whole group. For 2/3 ÷ 1/6, students ask how many sixths fit into two thirds; drawing a bar cut into sixths shows there are 4. For 1/2 ÷ 3, they ask how much each person gets if three people share half a pound equally, which is 1/6 of a pound. Sixth grade is the first time students divide a fraction by a fraction, and the standard insists they understand the meaning, not only the shortcut.
The general rule (a/b) ÷ (c/d) = ad/bc follows from the link between multiplication and division. Because 3/4 of 8/9 equals 2/3, it must be true that 2/3 ÷ 3/4 = 8/9. Students are expected to write their own story contexts for a division, draw a visual model, and check the quotient by multiplying it back. Contexts include servings (how many 3/4-cup servings are in 2/3 cup?) and area (the width of a strip with a known area and length).
Students sometimes invert the first fraction instead of the divisor. Writing the check 'quotient × divisor = dividend' exposes the error at once.
6 ÷ 1/2 = 12 surprises students. Ask how many half-pizzas are in 6 pizzas, and the larger answer makes sense.
Dividing numerators and denominators directly works only in special cases. Students who rely on it get stuck on problems like 2/3 ÷ 3/4.
A story for 1/2 ÷ 1/4 that asks 'what is a quarter of a half?' actually models 1/2 × 1/4. The story must ask how many quarters fit into a half.
How many 3/4-cup servings are in 2/3 of a cup of yogurt?
Answer: There are 8/9 of a serving in 2/3 cup.
Begin with whole numbers divided by unit fractions (3 ÷ 1/4) where the 'how many fit' meaning is easy to draw, then move to fractions with common denominators (5/6 ÷ 1/6), and only then to unlike denominators. Asking students to write a story before computing keeps the meaning in view.
Test items commonly ask students to pick the story that matches an expression, compute a quotient, or solve an area or servings problem. The reciprocal rule is fine to use once students can explain why it works.
Original questions written for this standard. Choose an option or type your answer, then press Check. Every question has a worked explanation.
Answer: 6
Count how many eighths fit in 3/4. Since 3/4 = 6/8, there are 6 eighths. Or 3/4 × 8/1 = 24/4 = 6.
Answer: C) 5/6
Multiply by the reciprocal of the divisor: 2/3 × 5/4 = 10/12, which simplifies to 5/6. Flipping the first fraction instead gives 6/5, a common slip.
Answer: 1/6
1/2 ÷ 3 = 1/2 × 1/3 = 1/6. Each friend gets 1/6 of a pound.
Answer: A) 2/3 mile
Width = area ÷ length = 1/2 ÷ 3/4 = 1/2 × 4/3 = 4/6 = 2/3 mile.
Answer: B) How many 2/3-cup scoops are in 4 cups?
Dividing 4 by 2/3 asks how many groups of 2/3 fit into 4. The scoop story asks exactly that (the answer is 6).
Answer: 15
5 ÷ 1/3 = 5 × 3 = 15. Each yard gives 3 pieces, so 5 yards give 15.
A full lesson with slides, activities and an exit ticket on dividing fractions by fractions, pitched to grade 6 and editable in PowerPoint or Google Slides.
Make a lesson →A printable, differentiated worksheet on 6.NS.A.1 with an answer key, ready in about a minute.
Make a worksheet →Turn dividing fractions by fractions into a quiz students answer online that marks itself, with a class summary for you.
Build a test →The standard asks students to interpret and explain fraction division with models and equations. The reciprocal method is the general rule (a/b) ÷ (c/d) = ad/bc, but students should be able to justify it, for example by multiplying back.
Sharing problems (how much per person), measurement problems (how many servings or pieces fit) and missing-side problems with area.