In plain English: 6.NS.A.1 is the Common Core grade 6 math standard that asks students to interpret and compute quotients of fractions and to solve word problems that divide a fraction by a fraction. Students use tape diagrams, number lines, area models and equations, and check each quotient with multiplication. It builds on grade 5 unit fraction division and prepares for rational number operations in grade 7.
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?
Common Core State Standards for Mathematics · Domain: The Number System (NS) · Cluster: Apply and extend previous understandings of multiplication and division to divide fractions by fractions. Also written as 6.NS.1 · Official standard
Students learn what it means to divide a fraction by a fraction and how to compute the answer. The answer to a division problem is called the quotient. The number being divided is the dividend, and the number you divide by is the divisor. In 2/3 ÷ 3/4, the dividend is 2/3 and the divisor is 3/4. Students read a quotient in two ways: "how many groups of the divisor fit in the dividend?" and "how much is in one whole group?"
Students show each problem with a visual fraction model (a drawing that shows fractions as parts of a whole): a tape diagram (a bar split into equal parts), a number line, or an area model (a rectangle split into equal parts). Students also compute quotients with common denominators or by multiplying by the reciprocal of the divisor (the fraction turned upside down, so the reciprocal of 3/4 is 4/3). They write equations, and they check every quotient with multiplication, as the official example does: 2/3 ÷ 3/4 = 8/9 because 3/4 of 8/9 is 2/3. The lesson works through all three official word problems about chocolate, yogurt and a strip of land. All numbers are positive fractions or mixed numbers, as expected in grade 6.
Learning Objectives
By the end of this lesson, students will be able to:
Explain a fraction quotient as "how many groups" or "how much in one group" and tell which meaning a story uses
Show a fraction division with a tape diagram, a number line or an area model
Compute quotients of fractions with common denominators or by multiplying by the reciprocal, and check them with multiplication
Write a story problem for a fraction division expression
Solve word problems that divide a fraction by a fraction and explain what the quotient means in the story
Prior Knowledge Required
Students should already be comfortable with:
The two meanings of whole-number division: equal shares and equal groups 3.OA.A.2
Writing equivalent fractions, such as 2/3 = 8/12 4.NF.A.1
Reading a fraction as division, such as 3/4 = 3 ÷ 4 5.NF.B.3
Multiplying a fraction by a fraction 5.NF.B.4
Dividing a unit fraction (a fraction with numerator 1) by a whole number and a whole number by a unit fraction 5.NF.B.7
Write the prompt on the board. Students answer on mini whiteboards or scrap paper, with a drawing if they like.
Warm-Up Prompt
"You have 3 cups of trail mix. How many 1/2-cup bags can you fill? How many 3/4-cup bags can you fill? Draw a picture for one of your answers."
Collect answers: 6 bags of 1/2 cup and 4 bags of 3/4 cup. Ask how students knew. Many count groups on a drawing: 3 cups hold 6 halves, and 3 cups hold 12 fourths, which make 4 groups of 3 fourths. Write 3 ÷ 1/2 = 6 and 3 ÷ 3/4 = 4, and check with multiplication: 6 × 1/2 = 3 and 4 × 3/4 = 3. Point out that dividing by a number less than 1 gave a quotient bigger than 3. Today the dividend will be a fraction too.
Direct Instruction20-25 minutes
Part 1: Two meanings of a quotient. Division can ask how many groups fit (how many 3/4-cup servings are in 2/3 cup?) or how much is in one group (if 3 people share 1/2 lb equally, how much does one person get?). Division is the inverse of multiplication (it undoes it), so every division has a matching multiplication: if 2/3 ÷ 3/4 = ?, then 3/4 × ? = 2/3. Work the examples below with the class, drawing a model for each one.
How many groups (official example)
How many 3/4-cup servings are in 2/3 of a cup of yogurt? Tell this story for (2/3) ÷ (3/4) and draw a tape diagram in twelfths of a cup (Diagram 1).
Equation: 2/3 = 8/12 and 3/4 = 9/12, so the yogurt fills 8 of the 9 twelfths in a serving: (2/3) ÷ (3/4) = 8/9 serving. Check: 3/4 of 8/9 is 24/36 = 2/3
How much in one group (official example)
How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally?
Equation: (1/2) ÷ 3 = 1/6 lb, because 3 × 1/6 = 1/2. On a tape diagram, split the 1/2 lb bar into 3 equal parts: each part is 1/6 of a whole pound
Missing side length (official example)
How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi? (Diagram 2)
Equation: Area = length × width, so 3/4 × width = 1/2 and width = (1/2) ÷ (3/4) = 2/3 mi. Check: 3/4 × 2/3 = 6/12 = 1/2
How many groups, whole-number quotient
A ribbon is 5/6 yard long. How many 1/12-yard pieces can be cut from it?
Leo pours 3/4 gallon of water into a watering can, and it fills 2/3 of the can. How much does the full can hold?
Equation: 2/3 of the can is 3/4 gallon, so 1/3 of the can is 3/8 gallon and 3/3 is 9/8 gallon: (3/4) ÷ (2/3) = 9/8 = 1 1/8 gallons. Check: 2/3 × 9/8 = 18/24 = 3/4
Part 2: Two ways to compute. Use the examples to name two methods.
Common denominators: write both fractions with the same denominator, then divide the numerators. 2/3 ÷ 3/4 = 8/12 ÷ 9/12 = 8 ÷ 9 = 8/9. This matches the tape diagram: 8 twelfths out of a 9-twelfth serving.
Multiply by the reciprocal: the reciprocal of a fraction swaps its numerator and denominator (the reciprocal of 3/4 is 4/3), and a number times its reciprocal is 1. Dividing by 3/4 gives the same answer as multiplying by 4/3: 2/3 × 4/3 = 8/9. In general, (a/b) ÷ (c/d) = (a/b) × (d/c) = ad/bc.
Check with multiplication: divisor × quotient = dividend. For 2/3 ÷ 3/4 = 8/9, check 3/4 × 8/9 = 24/36 = 2/3. If the check fails, the quotient is wrong.
Part 3: Does the answer make sense? Before computing, ask whether the quotient should be more or less than 1. If the divisor is smaller than the dividend (1/12 is smaller than 5/6), more than one group fits, so the quotient is greater than 1. If the divisor is bigger (3/4 is bigger than 2/3), less than one group fits, so the quotient is less than 1.
Guided Practice15 minutes
Pairs solve four problems. For each one they draw a model, write a division equation and its matching multiplication equation, and say which meaning of division the problem uses. After each problem, one pair shows its drawing.
Guided practice problems and answers
Problem
Answer
1. A recipe needs 7/8 cup of oats, and the only scoop holds 1/4 cup. How many scoops?
7/8 ÷ 1/4 = 7/8 ÷ 2/8 = 7/2 = 3 1/2 scoops (3 full scoops and half a scoop)
2. Compute 1/4 ÷ 2/3 two ways.
3/12 ÷ 8/12 = 3/8, and 1/4 × 3/2 = 3/8
3. Jo has 4/5 hour of free time. Each video lasts 2/15 hour. How many videos can she watch?
4/5 = 12/15, so 12/15 ÷ 2/15 = 6 videos (48 minutes of 8-minute videos)
4. Write the multiplication equation for 3/5 ÷ 1/2, then solve it.
1/2 × ? = 3/5, so ? = 6/5 = 1 1/5
Listen for students who flip the first fraction instead of the second, and for students who read the 1/2 left over in Problem 1 as "1/2 cup". It is half of a scoop, which is 1/8 cup.
Independent Practice10-15 minutes
Students solve five problems on their own and show a model or an equation for each. (1) 9/10 ÷ 3/10 (3). (2) 2/5 ÷ 3/4 (8/15). (3) A painter uses 2/3 gallon of paint to cover 4/5 of a wall. How much paint covers the whole wall? (2/3 ÷ 4/5 = 10/12 = 5/6 gallon.) (4) A rectangle has area 3/8 square foot and length 3/4 foot. How wide is it? (3/8 ÷ 3/4 = 1/2 foot.) (5) Four friends share 2/3 of a pizza equally. What fraction of the whole pizza does each friend get? (2/3 ÷ 4 = 1/6 of the pizza.)
Closure5 minutes
Exit ticket: (1) Compute 5/6 ÷ 2/3 and check your answer with multiplication. (5/6 ÷ 2/3 = 5/6 × 3/2 = 15/12 = 5/4 = 1 1/4; check: 2/3 × 5/4 = 10/12 = 5/6.) (2) Write a short story problem that 3/4 ÷ 1/2 answers, and give the answer with its unit. (For example, "How many 1/2-cup scoops are in 3/4 cup of rice?" The answer is 1 1/2 scoops.)
Differentiation Strategies
For Struggling Students
Give pre-cut fraction strips and have students lay the divisor strip along the dividend strip and count how many fit before writing any equation
Start with "how many groups" problems that have whole-number quotients, such as 3/4 ÷ 1/8, before quotients with a fraction left over
Give a sentence frame for every answer: "___ groups of ___ fit in ___" or "one whole group is ___"
For Advanced Students
Ask students to explain with a common-denominator model why (a/b) ÷ (c/d) = ad/bc always works
Ask for two different fractions whose quotient is exactly 3/2, and a story for one of them
Give a quotient with a mixed number, such as 2 2/3 ÷ 4/9, and ask for a tape diagram and an equation
Assessment Guidance
What to Look For
Check that students can say which meaning of division a story uses and draw a model that fits it. In computation, look for the divisor, not the dividend, being replaced by its reciprocal, and for a multiplication check written next to each answer. Ask students to predict whether a quotient is more or less than 1 before they compute. In word problems, look for a unit with every answer (servings, pieces, gallons, miles) and for a correct reading of any fraction left over: it is a fraction of one group, not of the whole.
02
Classroom Activities
3 Activities
1
Fraction Strip Division
20 minPairs
Pairs cut paper fraction strips (strips of equal length, each one whole, split into halves, thirds, fourths, sixths, eighths or twelfths). For each of 6 cards, they lay the divisor along the dividend, count how many divisors fit, and write the division and the matching multiplication.
Division Cards (6 cards)
Card A: 1/2 ÷ 1/6 (3)
Card B: 2/3 ÷ 1/6 (4)
Card C: 3/4 ÷ 3/8 (2)
Card D: 1/2 ÷ 1/3 (1 1/2)
Card E: 1/3 ÷ 1/2 (2/3)
Card F: 5/6 ÷ 1/4 (3 1/3)
Procedure
Lay the dividend strip on the desk. Lay copies of the divisor piece along it, starting at the same end
Count the whole divisor pieces that fit. If part of a piece is left, find what fraction of one divisor piece it is (use a strip with a common denominator, such as twelfths)
Write the division equation and the matching multiplication, for example 1/6 × 3 = 1/2
Compute the same quotient by multiplying by the reciprocal and compare
Discussion Questions
Which card has a quotient less than 1? (Only Card E.) What is true about its divisor? (It is bigger than the dividend.)
Cards D and E use the same two fractions in the opposite order. How are their quotients related? (3/2 and 2/3 are reciprocals.)
On Card F, 3 fourths fit with 1/12 of the strip left over. Why is the answer 3 1/3 and not 3 1/12? (The 1/12 left over is 1/3 of a fourth, so it is 1/3 of a group.)
Modification for Distance Learning
Share a slide of fraction bars that students can drag and copy. Students post a screenshot of each card's model with the division and multiplication equations typed under it.
2
Story Match and Story Writing
20 minGroups of 3-4
Groups match 4 story cards to 4 expression cards, solve each story, and then write their own stories for a division expression. One expression is a multiplication, so students must decide which stories really call for division.
Expression Cards (4 cards)
3/4 ÷ 1/8
1/8 ÷ 3/4
3/4 × 1/8
3/4 ÷ 3
Story Cards (4 cards)
Story 1: A 3/4-mile trail has a marker every 1/8 mile. How many 1/8-mile sections does the trail have? (3/4 ÷ 1/8 = 6 sections)
Story 2: Zoe has hiked 1/8 mile, which is 3/4 of the way to a lookout. How far is the lookout from the start? (1/8 ÷ 3/4 = 1/6 mile)
Story 3: A pan is 3/4 full of lasagna, and Sam eats 1/8 of what is in the pan. What fraction of a full pan does Sam eat? (3/4 × 1/8 = 3/32 of a pan)
Story 4: Three friends share 3/4 of a pizza equally. What fraction of the whole pizza does each friend get? (3/4 ÷ 3 = 1/4 of the pizza)
Procedure
Match each story to an expression and glue the pair on a sheet
Draw a tape diagram or number line for each division story and solve it
Write your own story for (2/3) ÷ (3/4), different from the yogurt example, and a second story for (3/4) ÷ (2/3). Solve both and say which meaning of division each story uses
Trade stories with another group and check their answers with multiplication
Discussion Questions
Which story is not a division story, and how can you tell? (Story 3: it asks for a part of an amount, which is multiplication)
Which story has a quotient greater than 1? (Only Story 1)
Stories 1 and 2 use the same fractions. Which one asks "how many groups" and which one asks "how much in one whole group"?
Why is the answer to your (2/3) ÷ (3/4) story less than 1, and the answer to your (3/4) ÷ (2/3) story more than 1?
Challenge Variation
Groups write one story for a division of two mixed numbers, such as 2 1/2 ÷ 1 1/4, and trade it with another group, which must draw a model and solve it.
3
Quilt Patch Area Models on Grid Paper
20 minPairs
Pairs draw rectangular quilt patches on grid paper, using 12 grid squares for 1 foot, so each square is 1 inch. Each card gives the area of a patch and one side length in feet. Students find the missing side with an area model and with division, as in the official strip-of-land example.
Patch Cards (4 cards)
Card 1: area 1/2 square foot, length 2/3 foot. Width? (1/2 ÷ 2/3 = 3/4 foot, or 9 inches)
Card 2: area 1/6 square foot, length 1/2 foot. Width? (1/6 ÷ 1/2 = 1/3 foot, or 4 inches)
Card 3: area 5/12 square foot, length 5/6 foot. Width? (5/12 ÷ 5/6 = 1/2 foot, or 6 inches)
Card 4: area 1/4 square foot, length 1/3 foot. Width? (1/4 ÷ 1/3 = 3/4 foot, or 9 inches)
Procedure
Draw a 12-by-12 square for 1 square foot. It holds 144 grid squares
Find how many grid squares the patch area is (1/2 square foot = 72 squares), and mark the given length along the bottom
Shade that many squares in a rectangle with the given length, and measure the width
Write the division that gives the width and check it with length × width = area
Discussion Questions
Which two cards have the same width? (Cards 1 and 4, both 3/4 foot)
Why does dividing the area by the length give the width?
How does counting grid squares match the common-denominator method?
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: A Tape Diagram for 2/3 ÷ 3/4
The official yogurt example drawn to scale in twelfths of a cup. The 2/3 cup of yogurt is 8 twelfths, and one 3/4-cup serving is 9 twelfths. The yogurt fills 8 of the 9 parts of one serving, so 2/3 ÷ 3/4 = 8/9 of a serving, and 3/4 × 8/9 = 2/3 checks it.
Diagram 2: An Area Model for the Strip of Land
A 1 mile by 1 mile square split into 4 columns and 3 rows, so each small rectangle is 1/12 square mile. The strip is 3 columns long (3/4 mile) and has area 1/2 square mile, or 6 small rectangles, so it is 2 rows wide: (1/2) ÷ (3/4) = 2/3 mile.
04
Homework Assignment
~30 min
6.NS.A.1 Homework: Dividing Fractions by Fractions
Directions: Show a model (tape diagram, number line or area model) or an equation for every problem. Check each quotient with multiplication, and write a unit with every answer to a word problem.
Part 1: Compute and Interpret (Problems 1-2)
Compute each quotient and check it with multiplication: (a) 6/7 ÷ 2/7 (b) 5/9 ÷ 2/3 (c) 7/10 ÷ 2/5
(a) Write a "how many" question that 4/5 ÷ 1/10 answers, then answer it. (b) Compute 1/10 ÷ 4/5. (c) Explain why one quotient is greater than 1 and the other is less than 1.
Part 2: Visual Models (Problems 3-4)
Draw a tape diagram or number line for 5/6 ÷ 1/3. How many whole groups of 1/3 fit, and what fraction of a group is left? Write the quotient as a mixed number.
A rectangular piece of poster board has area 7/20 square meter and width 1/2 meter. Draw an area model and find its length. Check with length × width = area.
Part 3: Word Problems (Problems 5-6)
A cook has 2 1/4 pounds of ground beef and makes burgers that each use 3/8 pound. How many burgers can the cook make? Write a division equation and a multiplication check.
Nia walks 2/5 mile in 1/6 hour at a steady pace. (a) How far does she walk in 1 hour? (b) How long does it take her to walk 1 mile? Give (b) in hours and in minutes.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Computation
All quotients correct and in simplest form, each checked with multiplication
One quotient wrong, or checks missing
Two or more quotients wrong
Meaning of the Quotient
Questions and explanations match the division and say what the quotient counts
Question or explanation partly matches
Question or explanation missing or describes multiplication
Visual Models
Models drawn with equal parts and match the equations
Model drawn but parts unequal or not labeled
No model
Word Problems
Correct equation, answer with a unit, and a sensible reading of the answer
Correct answer but missing equation or unit
Wrong operation or answer
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Answer the questions in order and draw a model on scrap paper when it helps. Your score updates after each answer, and Reset quiz clears the page for another try.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
What is 3/5 ÷ 1/4?
Answer: B
Multiply by the reciprocal of the divisor: 3/5 × 4/1 = 12/5 = 2 2/5. Check: 1/4 × 12/5 = 12/20 = 3/5. Choice A multiplies 3/5 × 1/4 without flipping the divisor. Choice C computes 1/4 ÷ 3/5, the division in the wrong order. Choice D flips both fractions: 5/3 × 4/1 = 20/3.
Question 2 of 20 · Multiple Choice
Which expression has the same value as 5/8 ÷ 2/3?
Answer: A
Dividing by 2/3 gives the same result as multiplying by its reciprocal, 3/2, so 5/8 ÷ 2/3 = 5/8 × 3/2 = 15/16. Choice B flips the dividend instead of the divisor. Choice C multiplies without flipping anything. Choice D flips both fractions.
Question 3 of 20 · Multiple Choice
Which question can be answered by 3/4 ÷ 1/6?
Answer: C
3/4 ÷ 1/6 asks how many groups of 1/6 fit in 3/4: 9/12 ÷ 2/12 = 9/2 = 4 1/2 scoops. Choice A describes 1/6 × 3/4, a multiplication. Choice B describes 1/6 ÷ 3/4, the division in the wrong order. Choice D describes the subtraction 3/4 - 1/6.
Question 4 of 20 · Multiple Choice
What is 7/9 ÷ 7/3?
Answer: D
7/9 × 3/7 = 21/63 = 1/3. With common denominators: 7/9 ÷ 21/9 = 7 ÷ 21 = 1/3. The divisor is bigger than the dividend, so the quotient must be less than 1. Choice A computes 7/3 ÷ 7/9, the division in the wrong order. Choice B multiplies 7/9 × 7/3 without flipping. Choice C flips both fractions: 9/7 × 3/7 = 27/49.
Question 5 of 20 · Multiple Choice
Without computing, which quotient is greater than 1?
Answer: C
A quotient is greater than 1 when the divisor is smaller than the dividend, because more than one group fits. Only in choice C is the divisor (1/2) smaller than the dividend (4/7): 4/7 ÷ 1/2 = 8/7. In choices A, B and D the divisor is bigger, so less than one group fits: the quotients are 2/3, 4/7 and 2/5. A student who picks A, B or D may think that dividing by a fraction always gives an answer greater than 1. That happens only when the divisor is smaller than the dividend.
Question 6 of 20 · Multiple Choice
Liam says 3/8 ÷ 1/2 = 3/4. Which multiplication checks his answer?
Answer: B
To check a division, multiply the divisor by the quotient and see whether you get the dividend: 1/2 × 3/4 = 3/8, so Liam is right. Choice A multiplies the dividend by the divisor. Choice C multiplies the dividend by the quotient. Choice D multiplies the quotient by the reciprocal of the divisor, which does not lead back to 3/8.
Question 7 of 20 · Multiple Choice
A number line from 0 to 1 is split into tenths. Kai starts at 0 and makes jumps of 2/5 (4 tenths each) toward 9/10. He makes 2 full jumps and has 1/10 of the line left. What is 9/10 ÷ 2/5?
Answer: D
The piece left over is 1 tenth, and one jump is 4 tenths, so it is 1/4 of a jump: 9/10 ÷ 2/5 = 2 1/4. Check: 2/5 × 9/4 = 18/20 = 9/10. Choice B counts the leftover as 1/10 of the number line instead of as a fraction of one jump. Choice A computes 2/5 ÷ 9/10, the division in the wrong order. Choice C multiplies: 9/10 × 2/5 = 18/50 = 9/25.
Question 8 of 20 · Multiple Choice
A 3/4-pound block of cheese is cut into slices that each weigh 1/16 pound. How many slices are there?
Answer: A
The question asks how many 1/16-pound groups fit in 3/4 pound: 3/4 = 12/16, so 12/16 ÷ 1/16 = 12 slices. Check: 12 × 1/16 = 12/16 = 3/4. Choice B multiplies 3/4 × 1/16. Choice C computes 1/16 ÷ 3/4, the division in the wrong order. Choice D flips both fractions: 4/3 × 16/1 = 64/3 = 21 1/3.
Question 9 of 20 · Multiple Choice
Eli uses 1/2 gallon of paint to cover 2/5 of a fence. At that rate, how much paint does the whole fence need?
Answer: C
2/5 of the fence takes 1/2 gallon, so 1/5 of the fence takes 1/4 gallon, and the whole fence (5/5) takes 5/4 = 1 1/4 gallons. As an equation: 1/2 ÷ 2/5 = 1/2 × 5/2 = 5/4. Choice A multiplies 1/2 × 2/5. Choice B computes 2/5 ÷ 1/2, the division in the wrong order. Choice D subtracts: 1/2 - 2/5 = 1/10.
Question 10 of 20 · Multiple Choice
A rectangular rug has an area of 5/8 square yard and a length of 5/6 yard. How wide is the rug?
Answer: B
Length × width = area, so width = 5/8 ÷ 5/6 = 5/8 × 6/5 = 30/40 = 3/4 yard (27 inches). Check: 5/6 × 3/4 = 15/24 = 5/8. Choice A multiplies the area by the length. Choice C computes 5/6 ÷ 5/8, the division in the wrong order. Choice D flips both fractions: 8/5 × 6/5 = 48/25.
Question 11 of 20 · Multiple Choice
Six hikers share 3/4 pound of trail mix equally. How much trail mix does each hiker get?
Answer: D
Sharing 3/4 pound among 6 hikers is 3/4 ÷ 6 = 3/4 × 1/6 = 3/24 = 1/8 pound each. Check: 6 × 1/8 = 6/8 = 3/4. Choice A multiplies 3/4 × 6. Choice B computes 6 ÷ 3/4, the division in the wrong order. Choice C drops the 3 in the numerator and finds 1/4 ÷ 6 = 1/24.
Question 12 of 20 · Multiple Choice
What number makes this equation true? 3/4 × ? = 3/10
Answer: A
The missing factor is 3/10 ÷ 3/4 = 3/10 × 4/3 = 12/30 = 2/5. Check: 3/4 × 2/5 = 6/20 = 3/10. Choice B multiplies 3/10 × 3/4 instead of dividing. Choice C computes 3/4 ÷ 3/10, the division in the wrong order. Choice D flips both fractions: 10/3 × 4/3 = 40/9.
Question 13 of 20 · Multiple Choice
Mia computes 4/5 ÷ 1/4 and gets 1/5. Why can Mia tell, without redoing the work, that her answer is wrong?
Answer: A
Dividing by 1/4 asks how many fourths fit in 4/5. The divisor is smaller than the dividend, so more than one fourth fits and the quotient is greater than 4/5: in fact 4/5 × 4 = 16/5 = 3 1/5. Mia multiplied 4/5 × 1/4 = 4/20 = 1/5. Choice B uses a whole-number rule that fails for divisors less than 1. Choice C is false: the quotient changes. Choice D is false: 4/5 ÷ 1/4 is greater than 1 even though both fractions are less than 1.
Question 14 of 20 · Multiple Choice
Which shows 3/4 ÷ 5/8 rewritten with a common denominator, and the correct quotient?
Answer: B
3/4 = 6/8, so the problem becomes 6 eighths ÷ 5 eighths, which is 6 ÷ 5 = 6/5 = 1 1/5. Check: 5/8 × 6/5 = 30/40 = 3/4. Choice A divides 5 by 6, the numerators in the wrong order. Choice C changes the denominator of 3/4 without changing its numerator, so 3/8 is not equal to 3/4. Choice D subtracts the numerators instead of dividing.
Question 15 of 20 · Short Answer
Compute 5/12 ÷ 3/4. Then check your answer with multiplication.
5/12 ÷ 3/4 = 5/12 × 4/3 = 20/36 = 5/9. Check: 3/4 × 5/9 = 15/36 = 5/12. The quotient is less than 1 because the divisor, 3/4, is bigger than the dividend, 5/12.
Question 16 of 20 · Short Answer
Write a story problem that 1/2 ÷ 1/8 answers. Then solve it and say what the answer means in your story.
Sample story: "A bag holds 1/2 pound of almonds. How many 1/8-pound snack packs can you fill?" 1/2 = 4/8, so 4/8 ÷ 1/8 = 4 snack packs. The answer counts how many 1/8-pound groups fit in 1/2 pound. Check: 4 × 1/8 = 1/2. Any story that asks how many 1/8-size groups fit in 1/2, or how much is in one whole group when 1/2 fills 1/8 of it, earns credit if it is solved correctly.
Question 17 of 20 · Short Answer
How many 3/4-cup servings are in 1/3 cup of juice? Draw a tape diagram and explain the answer.
Use twelfths of a cup: 1/3 cup = 4/12 and one serving is 3/4 cup = 9/12. The juice fills 4 of the 9 twelfths in one serving, so 1/3 ÷ 3/4 = 4/9 of a serving. Check: 3/4 × 4/9 = 12/36 = 1/3. The tape diagram shows a 9-part serving bar with 4 parts shaded.
Question 18 of 20 · Short Answer
A rectangular park has an area of 3/10 square mile and a width of 1/2 mile. How long is the park? Show an equation and a check.
Rosa rides her bike 2 1/2 miles in 1/4 hour at a steady speed. What is her speed in miles per hour?
Miles per hour asks how far she rides in 1 whole hour. 2 1/2 = 5/2, so 5/2 ÷ 1/4 = 5/2 × 4 = 10 miles per hour. Check: in 1/4 hour at 10 miles per hour she rides 10 × 1/4 = 2 1/2 miles. A common error is to multiply 2 1/2 × 1/4 = 5/8, which gives far less than the distance she already rode.
Question 20 of 20 · Short Answer
A jug holds 5/6 gallon of lemonade. Each cup holds 1/16 gallon. How many full cups can be poured, and how much of a cup is left over?
5/6 ÷ 1/16 = 5/6 × 16 = 80/6 = 40/3 = 13 1/3. So 13 full cups can be poured, and 1/3 of a cup is left over. The 1/3 is a fraction of one cup, not of a gallon: 13 cups use 13/16 gallon, and 5/6 - 13/16 = 1/48 gallon, which is 1/3 of a 1/16-gallon cup.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does 6.NS.A.1 mean?
6.NS.A.1 means students can divide a fraction by a fraction, explain what the answer means, and use it to solve word problems. Students show the division with drawings such as tape diagrams, number lines and area models, and with equations. They also check a quotient with multiplication, for example 2/3 ÷ 3/4 = 8/9 because 3/4 of 8/9 is 2/3.
Is 6.NS.A.1 taught in grade 5 or grade 6?
It is a grade 6 standard. In grade 5 (5.NF.B.7), students divide only a unit fraction by a whole number or a whole number by a unit fraction, such as 1/3 ÷ 4 or 5 ÷ 1/2. Grade 6 extends this to any fraction divided by any fraction, including mixed numbers.
Why do you flip the second fraction and multiply?
Because dividing by a number gives the same result as multiplying by its reciprocal. One way to see it: with a common denominator, 3/5 ÷ 2/5 is 3 fifths ÷ 2 fifths, which is 3 ÷ 2 = 3/2, and 3/5 × 5/2 = 15/10 = 3/2 as well. In general, (a/b) ÷ (c/d) = ad/bc. Students should learn why the rule works, not only the rule, so they flip the divisor and not the dividend.
What is a visual fraction model?
A visual fraction model is a drawing that shows fractions as equal parts of a whole. For division, the common ones are a tape diagram (a bar split into equal parts), a number line with equal jumps, and an area model (a rectangle split into rows and columns). The model shows how many divisor-sized groups fit in the dividend, or how much is in one whole group.
How can dividing make a number bigger?
When the divisor is less than 1, more than one group fits. For example, 1/2 ÷ 1/10 = 5, because five tenths fit in one half. Division by a whole number greater than 1 makes a number smaller, and students sometimes carry that rule over to fractions, so ask them to predict "more or less than 1?" before they compute.
What are common mistakes when dividing fractions?
A common mistake is flipping the dividend instead of the divisor, or flipping both. Another is multiplying the two fractions without flipping at all. In word problems, many students divide in the wrong order, or they read a leftover part as a fraction of the whole instead of a fraction of one group. A multiplication check catches all of these.
How do you know a word problem needs fraction division?
Look for one of three questions. "How many groups of this size fit?" (how many 1/4-cup scoops in 1 1/2 cups). "How much is in one whole group?" (a part of a container is filled, how much fills all of it). "What is the missing side?" (a rectangle's area and one side are known). A story that asks for a part of an amount, such as "1/3 of 3/4 of a pan", is multiplication instead.
What does the remainder mean in fraction division?
A leftover part is a fraction of one group, the divisor. In 7/8 ÷ 1/4, three quarter-cup scoops use 6/8 cup, and 1/8 cup is left. That 1/8 cup is half of a scoop, so the quotient is 3 1/2 scoops, not 3 1/8.
How is 6.NS.A.1 usually assessed?
6.NS.A.1 is usually assessed with three kinds of items: computing a quotient, choosing the equation or model that matches a story, and solving a word problem that divides a fraction by a fraction. Some items ask students to write their own story for an expression or to explain a quotient with a multiplication check, so explanations matter as much as answers.
How does 6.NS.A.1 connect to grade 7 and beyond?
6.NS.A.1 prepares students for dividing all rational numbers, including negative fractions, in grade 7 (7.NS.A.2), and for unit rates found from ratios of fractions, such as miles per hour from 1/2 mile in 1/4 hour (7.RP.A.1). Solving equations such as (3/4)x = 1/2 in Algebra I also relies on dividing by a fraction.
07
Related Standards
6 standards
These standards connect to 6.NS.A.1: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
5.NF.B.7Prerequisite
Divide unit fractions by whole numbers and whole numbers by unit fractions
Lesson coming soon
5.NF.B.4Prerequisite
Multiply a fraction or whole number by a fraction
Lesson coming soon
Alongside
6.EE.B.7Parallel
Solve equations of the form x + p = q and px = q with nonnegative rational numbers
