Dividing fractions
Dividing a fraction by a whole number and by a fraction, and why you multiply by the reciprocal
About four lessons of 45 to 60 minutes
Dividing by a fraction can make the answer bigger
You have half a pizza left and three friends who all want a share. How much of a whole pizza does each one get? Now flip the question: a recipe needs quarter-cup scoops, and you have 2 full cups of flour. How many scoops can you fill? The first question makes the pieces smaller, the second gives a surprisingly big answer, 8 scoops. Both are division of fractions.
That second answer is the shock of this topic: dividing by a number less than 1 makes the answer bigger, not smaller. Today you will see why with pictures, then learn the fast rule (multiply by the reciprocal) and, crucially, understand where it comes from so it never becomes a trick you forget.
- Half a pizza shared among 31/2 divided by 3, each friend gets 1/6 of a whole pizza
- Quarter-cup scoops from 2 cups2 divided by 1/4, that is 8 scoops, the answer got bigger
- How many 1/8 fit in 3/4?3/4 divided by 1/8 is 6, count the eighths that fit
- Dividing by 1/2 is the same as multiplying by 2flip the divisor, then multiply
What students will be able to do
Students will divide a fraction by a whole number and a number by a fraction, model each as either equal sharing or as measuring how many of the divisor fit into the dividend, use the reciprocal to compute a quotient, and explain why dividing by a number less than one gives a larger result.
- I can divide a fraction by a whole number by making the parts smaller.
- I can work out how many of one fraction fit into another.
- I can write dividing by a fraction as multiplying by its reciprocal.
- I can explain why dividing by a number less than 1 makes the answer bigger.
- I can solve a word problem that involves dividing fractions and check the answer is reasonable.
Standards this unit teaches
- 5.NF.B.7Common Core (US)Divide with unit fractions
Divide unit fractions by whole numbers and whole numbers by unit fractions, using visual fraction models and equations to represent and interpret the quotient.
- 6.NS.A.1Common Core (US)Divide fractions by fractions
Interpret and compute quotients of fractions, and solve word problems involving division of a fraction by a fraction using visual models and equations.
- AC9M7N06Australian Curriculum v9 (ACARA)Four operations with positive fractions
Use all four operations with positive rational numbers, including fractions and decimals, to solve problems using efficient strategies. ACARA folds dividing fractions into this Year 7 descriptor rather than naming it earlier.
- AC9M6N06Australian Curriculum v9 (ACARA)Division, remainders as fractions (Year 6 bridge)
Solve division problems using efficient strategies and interpret the remainder in context, expressing results as whole numbers, fractions or decimals. The idea that a quotient can be a fraction sets up this unit.
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
Words to teach and display
- Dividend
- the amount being divided, the number you start with
- Divisor
- the number you divide by, the size of each share or scoop
- Quotient
- the answer to a division
- Reciprocal
- a fraction flipped upside down, the reciprocal of 3/4 is 4/3
- Unit fraction
- a fraction with 1 on top, such as 1/4
- Mixed number
- a whole number and a fraction together, such as 1 1/2
Teach it: concrete, pictorial, abstract
The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.
2. Dividing by a fraction: how many fit?
PictorialThe second meaning of division is measuring: how many of the divisor fit into the dividend? Ask how many quarter-cup scoops you can fill from a supply of flour. On a number line from 0 to 1, mark quarters and count the jumps: four 1/4 scoops fit in 1 whole cup.
So 1 divided by 1/4 is 4. If you have 2 cups, that is twice as far, so 4 and 4 more, which is 8 scoops. Written out: 2 divided by 1/4 is 8.
Here is the surprise made plain: the answer (8) is bigger than the number you started with (2), because each scoop is small, so many of them fit. Dividing by a number smaller than 1 always makes the result larger.
How many quarter-cup scoops can you fill from 2 cups of flour?
- Count how many 1/4 fit in 1 cup: four.
- You have 2 cups, so double it: 4 plus 4.
- That is 8, and 2 divided by 1/4 is 8.
Answer: 8 scoops. The answer is bigger than 2 because each scoop is small.
- How many 1/3 cups fit in 1 cup? In 2 cups?
- Why is the answer to 2 divided by 1/4 bigger than 2?
3. Dividing a fraction by a fraction
PictorialNow both numbers are fractions: how many 1/8 fit into 3/4? The measuring idea still works. Draw 3/4 shaded, then remember that each quarter is 2 eighths, so 3/4 is the same as 6/8. That means six 1/8 pieces fit inside 3/4.
So 3/4 divided by 1/8 is 6. The trick that makes this reliable is to line the two fractions up over the same-size parts (eighths here) and then just count.
When the fit is not a whole number, the quotient is itself a fraction. How many halves fit in 3/4? One half fits, and then a quarter is left, which is half of another half, so the answer is 1 and 1/2.
How many 1/8 fit into 3/4?
- Rewrite 3/4 with eighths: each quarter is 2 eighths, so 3/4 is 6/8.
- Count the eighths: there are 6 of them.
- So 3/4 divided by 1/8 is 6.
Answer: 6 eighths fit into 3/4.
- How many 1/6 fit into 2/3? (Rewrite 2/3 in sixths.)
- How many 1/2 fit into 3/4, and why is the answer not a whole number?
4. The fast rule: multiply by the reciprocal
AbstractEvery picture above hides one rule: to divide by a fraction, multiply by its reciprocal (the fraction flipped over). Dividing by 1/4 gave the same answer as multiplying by 4, and 4 is the reciprocal of 1/4. Dividing by 1/8 matched multiplying by 8. The flip is not magic, it is the shortcut the diagrams keep proving.
For a whole-number divisor, write it as a fraction over 1 first: dividing by 4 is dividing by 4/1, whose reciprocal is 1/4. So 2/3 divided by 4 is 2/3 times 1/4, which is 2/12, and that simplifies to 1/6.
For a fraction divisor, flip it and multiply: 3/4 divided by 1/2 is 3/4 times 2/1, which is 6/4, or 3/2, the mixed number 1 and 1/2. This matches the count from section 3.
Keep, flip, multiply is a handy name for the steps: keep the first fraction, flip the divisor, change divide to multiply. But say why every time, so it stays understanding and not a spell.
Work out 2/3 divided by 4, then 3/4 divided by 1/2.
- 2/3 divided by 4: write 4 as 4/1, flip to 1/4, multiply: 2/3 times 1/4 is 2/12.
- Simplify 2/12 by dividing top and bottom by 2: 1/6.
- 3/4 divided by 1/2: flip 1/2 to 2/1, multiply: 3/4 times 2/1 is 6/4.
- Simplify 6/4 to 3/2, which is the mixed number 1 and 1/2.
Answer: 2/3 divided by 4 is 1/6, and 3/4 divided by 1/2 is 1 and 1/2.
- What is the reciprocal of 5? Of 2/3?
- Rewrite 5/6 divided by 2/3 as a multiplication, then solve it.
Common misconceptions and how to address them
MisconceptionDividing always makes the answer smaller, so 2 divided by 1/4 must be less than 2.
Why it happens: Every division students met before this used whole numbers bigger than 1, where the result does shrink. That experience is now working against them.
How to address it: Return to the scoops: how many 1/4 cups fit in 2 cups? Counting gives 8. Small divisor, many fit, bigger answer. Dividing by a number under 1 always increases the result.
MisconceptionWhen you flip, flip the first fraction (the dividend) instead of the divisor.
Why it happens: Keep, flip, multiply gets remembered as 'flip a fraction' without holding on to which one.
How to address it: Only the divisor flips, because dividing by it is the same as multiplying by its reciprocal. The dividend is the amount you have and it never changes. Point to the second fraction every time.
MisconceptionTo divide fractions, divide the tops and divide the bottoms, like a mirror of multiplying.
Why it happens: Multiplying across works (tops times tops, bottoms times bottoms), so students expect dividing across to work too.
How to address it: Test it on a known case: 3/4 divided by 1/8 is 6 from the picture, but dividing across gives 3 over (a half), which is wrong. Dividing needs keep, flip, multiply, not divide across.
Misconception1/2 divided by 3 is 3/2, because you just put the 3 somewhere.
Why it happens: Dividing by 3 gets muddled with flipping or with multiplying by 3.
How to address it: Sharing half among 3 people must give each person less than a half, so the answer has to be smaller than 1/2. It is 1/6. If a share came out bigger than what you started with, something went wrong.
MisconceptionThe reciprocal of a whole number is just the same whole number, so dividing by 4 means multiplying by 4.
Why it happens: Whole numbers do not look like fractions, so the flip is not obvious.
How to address it: Write the whole number over 1 first: 4 is 4/1, and its reciprocal is 1/4. Dividing by 4 is multiplying by 1/4, which makes the result smaller, as sharing should.
Guided practice (with answers)
1. What is 1/2 divided by 4?
Answer: 1/8. Splitting a half into 4 makes each part 4 times smaller: 1 over (2 times 4).
2. How many 1/3 cups fit into 2 cups?
Answer: 6. Three fit in each cup, and there are 2 cups, so 3 times 2 is 6. (2 divided by 1/3 is 6.)
3. How many 1/8 fit into 3/4?
Answer: 6. Rewrite 3/4 as 6/8 and count the eighths.
4. Rewrite 5/6 divided by 2/3 as a multiplication and solve.
Answer: 5/6 times 3/2 is 15/12, which simplifies to 5/4, the mixed number 1 and 1/4.
5. What is the reciprocal of 2/5, and of 7?
Answer: The reciprocal of 2/5 is 5/2. The reciprocal of 7 is 1/7 (write 7 as 7/1, then flip).
6. True or false: 3 divided by 1/2 is less than 3.
Answer: False. It is 6. Dividing by 1/2 (a number under 1) makes the answer bigger, and 3 times 2 is 6.
Independent practice worksheets
Set the matching ChalkBee worksheets for independent work. The answer keys are computed in code, so they are never wrong. Start with dividing by whole numbers, then move to dividing by fractions.
Differentiation
- Stay with the models: fold paper or draw bars for every problem before touching the reciprocal rule.
- Keep divisors as unit fractions (1/2, 1/3, 1/4) until the how-many-fit idea is secure.
- Give the reciprocal step as a fill-in frame: divided by 2/3 becomes times __ over __.
- Use an estimate first (is the answer bigger or smaller than the dividend?) so a wrong keep-flip-multiply gets caught.
- Divide two mixed numbers by converting to improper fractions first, such as 2 1/2 divided by 3/4.
- Ask for a word problem whose answer is 8 that uses dividing by a fraction, then one whose answer is 1/6.
- Explore what happens as the divisor shrinks toward 0 (the quotient grows without bound) and why you cannot divide by 0.
- Connect to decimals: check 3/4 divided by 1/2 by converting to 0.75 divided by 0.5.
Assessment: exit ticket
A three-question exit ticket for the last five minutes. It samples dividing by a whole number, dividing by a fraction, and the reasoning about size.
1. What is 2/3 divided by 4?
Answer: 1/6 (2/3 times 1/4 is 2/12, which simplifies to 1/6).
2. How many 1/4 fit into 3?
Answer: 12 (3 divided by 1/4 is 3 times 4).
3. Is 5 divided by 1/2 bigger or smaller than 5? Give the answer.
Answer: Bigger. It is 10, because dividing by 1/2 is multiplying by 2.
Teacher notes and timings
- Rough timing across four lessons: Lesson 1 dividing a fraction by a whole number (section 1), Lesson 2 dividing by a fraction as measuring (sections 2 to 3), Lesson 3 the reciprocal rule (section 4), Lesson 4 word problems plus the exit ticket.
- Lead with meaning, not the rule. Let students count how many fit for a full lesson before keep, flip, multiply appears, so the rule lands as a shortcut they have already earned.
- The estimate-first habit (bigger or smaller than the dividend?) is the single best guard against a flipped-the-wrong-fraction error. Insist on it.
- The number line here labels quarters as fractions, not decimals, so a student who has not consolidated decimals can still read it. The step value is 0.25 under the hood, but the ticks show 1/4, 2/4, 3/4.
- US and AU alignment, a genuine divergence: the US introduces dividing with unit fractions in Grade 5 (5.NF.B.7) and full fraction-by-fraction division in Grade 6 (6.NS.A.1). ACARA does not name dividing fractions in Year 6, folding it into Year 7 four-operations work (AC9M7N06), with Year 6 (AC9M6N06) only setting up that a quotient can be a fraction. This unit therefore fits US Grade 6 or an AU Year 6 to 7 bridge.
- Present mode and print both work: use the Print button for a clean teacher copy or a student handout, and project the page to teach straight from the diagrams.