Fractions as division, and multiplying fractions in real problems
Interpreting a fraction as division, sharing word problems, and multiplying fractions and mixed numbers in context
About four lessons of 45 to 60 minutes
3 pizzas, 4 people: how much does each person get?
Three pizzas are shared equally among 4 people. There is no whole pizza each, so what is the fair share, written as one fraction? It turns out that any division you cannot finish evenly, like 3 divided by 4, is exactly the fraction 3/4. A fraction is not just 'parts of a whole' -- it is also a division waiting to be shared out.
That same fraction thinking powers real-world multiplication too: if a recipe needs 2/3 cup of sugar per batch and you are making 4 batches, or if 2/3 of a planted garden grows tomatoes, you are multiplying fractions and mixed numbers to answer real, useful questions.
- 3 pizzas shared by 4 people3 divided by 4 is 3/4 of a pizza each
- 5 sandwiches shared by 3 people5 divided by 3 is 5/3, or 1 and 2/3 sandwiches each
- 2/3 cup of sugar per batch, 4 batches4 x 2/3 = 8/3 = 2 and 2/3 cups total
- 2/3 of a 3/4-planted garden grows tomatoes2/3 x 3/4 = 6/12 = 1/2 of the whole garden
What students will be able to do
Students will interpret a fraction a/b as the division of a by b, use this to solve word problems that share a whole number of items among a whole number of people, and solve real-world problems that multiply a fraction or mixed number by a whole number, a fraction, or another mixed number, using visual models and equations.
- I can explain that a fraction such as 3/4 is the same as 3 divided by 4.
- I can solve a sharing word problem and give the answer as a fraction or mixed number.
- I can multiply a fraction by a fraction, giving the answer in simplest form.
- I can multiply a mixed number by a fraction or another mixed number.
- I can solve a real-world problem that requires multiplying fractions or mixed numbers.
Standards this unit teaches
- 5.NF.B.3Common Core (US)Fractions as division
Interpret a fraction as the division of its numerator by its denominator and solve sharing word problems.
- 5.NF.B.6Common Core (US)Multiply fractions in problems
Solve real world problems that multiply fractions and mixed numbers using models and equations.
- AC9M7N06Australian Curriculum v9 (ACARA)Four operations with rationals (Year 7)
Use the four operations with positive fractions, decimals and percentages, choosing efficient strategies to solve problems. Australia's descriptor for fully combining fraction operations, including multiplying fractions and mixed numbers, sits at Year 7, so this US Grade 5 unit runs about two years ahead of the ACARA placement.
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
- Grade 5 multiplying a fraction by a whole number teaching unitthe repeated-addition meaning of fraction times whole number this unit extends
- Grade 5 fraction of a quantity teaching unitfinding a fraction of an amount, which is exactly what multiplying two fractions models
- Grade 5 long division teaching unitdivision and remainders, which reappear here as a fraction
- How to teach fractionsthe meaning of a fraction as parts of a whole
Words to teach and display
- Fraction as division
- the fact that a/b means exactly the same thing as a divided by b
- Numerator
- the top number of a fraction, the number of parts being counted
- Denominator
- the bottom number of a fraction, the number of equal parts the whole is split into
- Mixed number
- a whole number and a fraction written together, such as 2 and 1/3
- Product
- the answer to a multiplication
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.
1. A fraction is a division
ConcreteShare 3 pizzas equally among 4 people. There is not one whole pizza for each person, so cut every pizza into 4 equal pieces (giving 12 quarter-pieces in total) and give each person 3 of those quarter-pieces: 3/4 of a pizza. That 3/4 is exactly what you get from 3 divided by 4.
This works for any sharing situation: the fraction a/b is always the same amount as a divided by b, where a is the total being shared and b is the number of people sharing it. When the sharing does not come out to a whole number, the leftover naturally becomes the fraction's numerator.
5 sandwiches are shared equally among 3 people. How much does each person get?
- The sharing is 5 divided by 3, so the answer is the fraction 5/3.
- 5/3 is an improper fraction (top bigger than bottom), so change it to a mixed number: 5 divided by 3 is 1 remainder 2, so 5/3 = 1 and 2/3.
Answer: Each person gets 5/3, or 1 and 2/3, sandwiches.
- Why is 3 pizzas shared by 4 people the same as the fraction 3/4?
- What operation is 5/3 secretly recording?
2. Multiplying a fraction by a fraction
PictorialTo find a fraction of a fraction, such as 2/3 of a garden that is already 3/4 planted, multiply the fractions: multiply the numerators together, multiply the denominators together, then simplify. 2/3 x 3/4 = (2 x 3)/(3 x 4) = 6/12, which simplifies to 1/2.
The area model makes this visible: draw a rectangle, shade 3/4 of it one way to show the planted part, then shade 2/3 of that shaded region a different way to show the tomato part. The doubly shaded region is exactly 1/2 of the whole rectangle, matching the equation.
Find 3/5 of 2/3.
- Multiply the numerators: 3 x 2 = 6.
- Multiply the denominators: 5 x 3 = 15.
- So 3/5 x 2/3 = 6/15.
- Simplify: divide top and bottom by their common factor, 3: 6/15 = 2/5.
Answer: 3/5 of 2/3 is 2/5.
- What two numbers do you multiply to get the new numerator?
- Why must an answer like 6/15 be simplified?
3. Multiplying mixed numbers in real problems
AbstractA recipe calls for 1 and 1/2 cups of flour per full batch. To scale the recipe down to 2/3 of a batch, multiply: 1 and 1/2 x 2/3. First change the mixed number to an improper fraction, 1 and 1/2 = 3/2, then multiply straight across: 3/2 x 2/3 = 6/6 = 1.
Word problems are where this standard earns its keep. Read carefully for the two quantities being multiplied, convert any mixed number to an improper fraction, multiply, simplify, and convert back to a mixed number if the result is improper. Always finish by checking the answer makes sense in the story: 1 cup of flour for a scaled-down recipe is a reasonable, checkable amount.
A recipe uses 2 and 1/2 cups of juice per batch. A chef makes 4/5 of a batch. How much juice is needed?
- Change the mixed number to an improper fraction: 2 and 1/2 = 5/2.
- Multiply: 5/2 x 4/5 = (5 x 4)/(2 x 5) = 20/10.
- Simplify: 20/10 = 2.
Answer: The chef needs 2 cups of juice.
- What is the first step before multiplying a mixed number by a fraction?
- Why is checking that an answer makes sense in the story an important last step?
Common misconceptions and how to address them
MisconceptionSharing 3 pizzas among 4 people gives the same amount as sharing 4 pizzas among 3 people, since it is the same two numbers.
Why it happens: Students do not yet see that the order in a/b, which number is being shared and which number is doing the sharing, changes the result.
How to address it: 3 pizzas among 4 people is 3/4 each (less than a whole pizza). 4 pizzas among 3 people is 4/3, or 1 and 1/3, each (more than a whole pizza). Compare both concretely to show the order changes the answer.
MisconceptionMultiplying two fractions always gives a bigger answer, the way multiplying two whole numbers does.
Why it happens: Students over-generalise from whole-number multiplication, where the product is usually bigger than either factor.
How to address it: Multiplying by a fraction less than 1 means finding a part of something, which makes the result smaller. 2/3 x 3/4 = 1/2, and 1/2 is smaller than both 2/3 and 3/4. The area model shows this directly: the doubly shaded region is always smaller than either single shading.
MisconceptionTo multiply mixed numbers, multiply the whole-number parts and the fraction parts separately and add them.
Why it happens: Students apply the addition strategy for mixed numbers (add whole parts, add fraction parts) to multiplication, where it does not work.
How to address it: Always convert a mixed number to an improper fraction first, then multiply straight across. Show the wrong shortcut failing on a checkable example: 1 and 1/2 x 2/3 by the wrong method gives 1 x 2/3 + (1/2 x 2/3) parts muddled incorrectly, while converting to 3/2 x 2/3 correctly gives 1.
Guided practice (with answers)
1. 7 candy bars are shared equally among 2 people. How much does each person get?
Answer: 7/2, or 3 and 1/2, candy bars each, since 7 divided by 2 is 3 remainder 1.
2. Find 2/5 of 15.
Answer: 6, since 2/5 x 15 = 2/5 x 15/1 = 30/5 = 6.
3. Find 3/4 of 2/3 cup.
Answer: 1/2 cup, since 3/4 x 2/3 = 6/12 = 1/2.
4. 4 pizzas are shared equally among 5 friends. How much does each friend get?
Answer: 4/5 of a pizza each, since 4 divided by 5 is 4/5.
5. Find 2 and 1/2 x 4/5.
Answer: 2, since 2 and 1/2 = 5/2, and 5/2 x 4/5 = 20/10 = 2.
Independent practice worksheets
Set the matching ChalkBee fraction and division worksheets for independent work. The answer keys are computed in code, so they are never wrong. Start with sharing word problems, then move to multiplying fractions and mixed numbers.
Differentiation
- Use physical sharing (paper circles cut into equal pieces) for the fraction-as-division section before moving to the abstract a/b notation.
- Keep the area model visible for every fraction-times-fraction problem until the multiply-across shortcut is trusted.
- Provide a written checklist for mixed-number multiplication: convert to improper, multiply across, simplify, convert back.
- Start word problems with the two quantities to multiply already identified, before requiring students to pull them out of the text unaided.
- Multiply three fractions together in one real-world problem, such as scaling a recipe down twice.
- Compare the size of a product to its factors for several examples (both factors less than 1, one factor greater than 1) and generalise a rule for when a product is smaller or bigger than the starting amount.
- Write an original sharing word problem whose answer is an improper fraction, and a multiplication word problem involving two mixed numbers.
- Preview Grade 6 by dividing a fraction by a fraction and comparing the result to multiplying by a fraction.
Assessment: exit ticket
A three-question exit ticket for the last five minutes, sampling fraction-as-division and both kinds of fraction multiplication.
1. 9 cookies are shared equally among 4 kids. How much does each kid get?
Answer: 9/4, or 2 and 1/4, cookies each, since 9 divided by 4 is 2 remainder 1.
2. Find 3/5 of 2/3.
Answer: 2/5, since 3/5 x 2/3 = 6/15, which simplifies to 2/5.
3. A board is 6 feet long. 2/3 of the board is used for shelving. How many feet is that?
Answer: 4 feet, since 2/3 x 6 = 12/3 = 4.
Teacher notes and timings
- Rough timing across four lessons: Lesson 1 fraction as division (section 1), Lesson 2 multiplying fraction by fraction (section 2), Lesson 3 mixed numbers and word problems (section 3), Lesson 4 mixed practice and the exit ticket.
- Language to keep saying: a/b means a divided by b, multiplying by a fraction less than 1 makes something smaller, convert mixed numbers to improper fractions before multiplying. These target the three main misconceptions.
- This unit assumes the repeated-addition meaning of fraction times whole number (from the Grade 4/5 multiply-fraction-by-whole-number unit) and the meaning of a fraction of a quantity are already secure; if either is shaky, revisit those units first.
- Curriculum note: the US formalises fraction-as-division and multiplying fractions and mixed numbers in context at Grade 5. ACARA's matching descriptor for the four operations with fractions, including multiplication, sits at Year 7 (AC9M7N06), so this unit runs about two years ahead of the Australian placement.
- Present mode and print both work: use Present to build the area model live for a fraction-times-fraction example, then print for independent word-problem practice.