Teaching unit · Grade 5 (ages 10 to 11)

Finding a fraction of a quantity

Dividing by the denominator, multiplying by the numerator, and problems like 2/3 of 12

About three lessons of 45 to 60 minutes

Start here · hook

Two thirds of the class chose pizza

The teacher runs a vote for the class party. There are 12 students, and 2/3 of them chose pizza over tacos. Everyone can picture two thirds of the class, but the caterer needs a number: exactly how many pizza orders is that? Saying 2/3 is not enough to place the order. You need 2/3 of 12 as a whole number of students.

You already have both tools you need for this, you just have not combined them yet. Sharing 12 into equal groups is division, which you know. Taking several of those groups is multiplication, which you know. Today you put them together into one reliable routine: to find a fraction of a quantity, divide by the bottom number, then multiply by the top number. By the end you will turn 2/3 of 12 into the number 8 in two quick steps.

2/3 of a class of 12 chose pizza12 shared into 3 groups of 4, take 2 groups, 8 students
3/4 of a bag of 20 sweets are red20 shared into 4 groups of 5, take 3 groups, 15 sweets
5/6 of an 18 minute break already gone18 shared into 6 groups of 3, take 5 groups, 15 minutes
1/4 off a $12 toy12 shared into 4 groups of 3, one group is the $3 discount

Learning objective

What students will be able to do

Students will find a fraction of a whole-number quantity by dividing the quantity by the denominator to find the unit-fraction amount and then multiplying by the numerator, model the process with a bar and with equal groups, and apply it to word problems where the quantity divides evenly by the denominator.

Success criteria

I can find a unit fraction of a quantity by dividing the quantity by the denominator.
I can find a non-unit fraction of a quantity by dividing by the denominator and then multiplying by the numerator.
I can draw a bar model that shows a fraction of a quantity.
I can solve a word problem that asks for a fraction of a group.
I can explain why finding a fraction of a quantity uses both division and multiplication.

Curriculum anchor

Standards this unit teaches

5.NF.B.4Common Core (US)
A fraction of a quantity
Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. Interpret the product a/b times q as a parts of a partition of q into b equal parts, so that finding a/b of a quantity q means dividing q into b equal parts and taking a of them.
4.NF.B.4Common Core (US)
A fraction as a multiple of a unit fraction (foundation)
Understand a fraction a/b as a multiple of the unit fraction 1/b. This is the foundation the unit builds on: once you know 1/3 of a quantity, the amount 2/3 is simply two copies of it.
AC9M5N07Australian Curriculum v9 (ACARA)
Find a fraction of a quantity (Year 5)
Find a familiar fraction, decimal or percentage of a quantity, including in everyday contexts such as discounts, and choose efficient strategies. Finding a fraction of a quantity is the fraction case of this Year 5 descriptor.
AC9M4N02Australian Curriculum v9 (ACARA)
Unit fractions and their multiples (Year 4 foundation)
Recognise and represent unit fractions and their multiples in different ways, combining same-denominator fractions to make a whole. A non-unit fraction of a quantity is built from the unit-fraction amount, so this Year 4 idea is the launch point.

Before you start

Prior knowledge

This unit builds on skills students should already have met. Revisit any that are shaky first.

Key vocabulary

Words to teach and display

Numerator: the top number, how many equal groups you take
Denominator: the bottom number, how many equal groups you share into
Unit fraction: a fraction with 1 on top, one single equal group, such as 1/3
Quantity: the amount you are taking a fraction of, such as 12 students or 20 sweets
Of: in fraction problems, the word 'of' signals finding a fraction of an amount

Teaching sequence

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 unit fraction of a group

Concrete

Start with 12 counters for the 12 students and the vote from the hook. Ask for 1/3 of the group first. The denominator 3 tells you how many equal groups to share the counters into. Share the 12 counters into 3 equal piles and each pile has 4. So 1/3 of 12 is 4. The unit-fraction amount is just the quantity divided by the denominator.

Say the connection out loud: finding 1/3 of a set is the same as dividing the set into 3 equal parts, which is division. 12 divided by 3 is 4. The denominator is the number you divide by.

Try another out loud with counters: 1/4 of 12 shares 12 into 4 groups of 3, so 1/4 of 12 is 3. The bar below shows 12 split into equal thirds.

12 shared into 3 equal groups. Each group of 4 is 1/3 of 12, so 1/3 of 12 is 4.

Check for understanding, ask

To find 1/3 of a group, what do you divide by?
What is 1/4 of 12, and how did you share it?

2. Taking several groups

Pictorial

Now the real question: 2/3 of 12. You already found that 1/3 of 12 is 4, one group. The numerator 2 tells you to take 2 of those equal groups. Two groups of 4 is 8. So 2/3 of 12 is 8, and the caterer orders 8 pizzas.

The two steps in order: divide by the denominator to find one group (12 divided by 3 is 4), then multiply by the numerator to take that many groups (4 times 2 is 8). Divide by the bottom, multiply by the top.

The bar makes the answer visible: three equal parts of 4, with two of them shaded for the two thirds that chose pizza.

2/3 of 12: take 2 of the 3 groups of 4. That is 8 students, so 2/3 of 12 is 8.

The picture of 2/3: two of three equal parts shaded, matching the two groups taken.

Worked example

Find 3/4 of 20.

Divide by the denominator to find 1/4 of 20: 20 divided by 4 is 5.
Multiply by the numerator to take 3 of those groups: 5 times 3 is 15.
So 3/4 of 20 is 15.

20 shared into 4 groups of 5. Take 3 of them for 3/4: that is 15.

Answer: 3/4 of 20 is 15.

Check for understanding, ask

After you find 1/3 of 12 is 4, what do you do to get 2/3 of 12?
Which number tells you how many groups to take, the top or the bottom?

3. The divide then multiply rule

Abstract

Write the routine as a rule you can use without counters: to find a/b of a quantity, divide the quantity by b, then multiply by a. Divide by the bottom, multiply by the top. Because you can also do it in either order, this is the same as multiplying the quantity by the fraction, which connects it to fraction multiplication you meet later.

Worked in symbols: 5/6 of 18. Divide 18 by 6 to get 3 (that is 1/6 of 18). Multiply 3 by 5 to get 15. So 5/6 of 18 is 15.

Choose friendly numbers where the quantity divides exactly by the denominator, so the group size is a whole number. When it divides evenly, the two steps stay simple and the answer is a whole number.

Worked example

Find 5/6 of 18.

Divide by the denominator: 18 divided by 6 is 3, so 1/6 of 18 is 3.
Multiply by the numerator: 3 times 5 is 15.
So 5/6 of 18 is 15.

18 shared into 6 groups of 3. Take 5 of them for 5/6: that is 15.

Answer: 5/6 of 18 is 15.

Check for understanding, ask

State the rule for finding a/b of a quantity in your own words.
Why do we pick a quantity that divides evenly by the denominator?

4. Solving word problems

Abstract

Now use the routine in the kind of problem you meet in real life. The word 'of' after a fraction is your signal to divide by the denominator and multiply by the numerator. Draw the bar, label the whole, and answer the exact question asked.

A bag holds 20 sweets and 3/4 of them are red. How many red sweets? Divide 20 by 4 to get 5, multiply by 3 to get 15. There are 15 red sweets. A follow-up worth asking: how many are not red? The other 1/4 is 5.

Reading tip: check what the question wants. Sometimes it asks for the fraction amount (the red sweets), sometimes for the rest (the sweets that are not red). The bar shows both at once.

3/4 of 20 sweets are red: 15 red and 5 not red.

Worked example

A class of 12 students voted, and 2/3 chose pizza. How many chose pizza, and how many did not?

Find 2/3 of 12: divide 12 by 3 to get 4, then multiply by 2 to get 8. So 8 chose pizza.
The rest is the other 1/3: 12 minus 8 is 4, or 1/3 of 12 is 4.
Check: 8 plus 4 is 12, the whole class.

Answer: 8 students chose pizza and 4 did not.

Check for understanding, ask

In a word problem, what does the word 'of' tell you to do?
If 2/3 of the class chose pizza, what fraction did not, and how many students is that?

Watch for

Common misconceptions and how to address them

MisconceptionTo find 2/3 of 12, divide by the numerator, so 12 divided by 2 is 6.

Why it happens: Students grab the first number in the fraction and divide by it, mixing up the roles of the top and bottom.

How to address it: The denominator names how many equal groups, so it is the number you divide by. For 2/3 of 12, divide by 3 first (12 into 3 groups of 4), then take 2 groups. Point to the bottom number and say 'this is how many groups'.

The denominator 3 sets the number of groups. Divide by 3, not by 2.

MisconceptionTo find 2/3 of 12, multiply 12 by the top number, so the answer is 24.

Why it happens: The word problem contains a multiplication step, and students apply it to the whole quantity without dividing first.

How to address it: You multiply by the numerator only after you have divided by the denominator to find one group. 12 divided by 3 is 4, then 4 times 2 is 8. A fraction of an amount is never bigger than the amount.

MisconceptionThe word 'of' means add, so 2/3 of 12 is 2/3 plus 12.

Why it happens: Students match the small connecting word to an operation from earlier work without reading the fraction context.

How to address it: In fraction problems 'of' signals finding a part of an amount, which is division then multiplication. Read '2/3 of 12' as two thirds of the group of 12, and act it out with counters.

MisconceptionA fraction of a whole number must still be a fraction, so 2/3 of 12 cannot be a whole number like 8.

Why it happens: Students expect a fraction answer whenever a fraction appears in the question.

How to address it: When the quantity divides evenly by the denominator, the answer is a whole number of objects. 2/3 of 12 is 8 whole students, and you cannot order two thirds of a pizza box for a person, you order 8 pizzas.

MisconceptionYou share the quantity into the numerator of groups, so for 3/4 of 20 you make 3 groups.

Why it happens: The numerator is the more prominent number, so students use it to set the number of groups.

How to address it: The denominator sets the number of groups: 3/4 of 20 makes 4 groups of 5, then take 3 of them for 15. Say the routine as divide by the bottom, then multiply by the top.

3/4 of 20 makes 4 groups of 5 (the denominator sets the groups), then takes 3 of them.

Do it together

Guided practice (with answers)

1. What is 1/4 of 20?
Answer: 5. Share 20 into 4 equal groups: each group is 5.
2. What is 2/3 of 12?
2/3 of 12: two of the three groups of 4 is 8.
Answer: 8. Divide 12 by 3 to get 4, then multiply by 2 to get 8.
3. What is 3/5 of 15?
Answer: 9. Divide 15 by 5 to get 3, then multiply by 3 to get 9.
4. A team of 18 players, and 5/6 of them are present. How many are present?
Answer: 15. Divide 18 by 6 to get 3, then multiply by 5 to get 15.
5. What is 3/4 of 20?
Answer: 15. Divide 20 by 4 to get 5, then multiply by 3 to get 15.
6. If 2/3 of the 12 students chose pizza, how many chose tacos?
Answer: 4. The other 1/3 of 12 is 4 students.

On their own

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 fraction practice, then move to word problems once the two-step routine is secure.

Grade 5 fractions worksheet (set A)fraction work that includes a fraction of an amount Grade 5 fractions worksheet (set B)more fraction practice Grade 5 word problems worksheetapply fraction-of-a-quantity in context All Grade 5 fraction worksheetsthe full set, six sets each All word problem worksheetsmixed contexts across grades

Reach every student

Differentiation

Support

Stay with counters: physically share the quantity into equal piles before writing any numbers.
Start with unit fractions only (1/3, 1/4, 1/5 of a set) until the divide step is automatic, then add the multiply step.
Pre-draw the bar with the correct number of parts so the student only fills in the group size.
Choose quantities that divide easily, such as 12, 20 and 18, so the group size is a friendly whole number.

Extension

Find the remaining fraction as well, such as how many are not red, and check the two amounts add to the whole.
Work backward: if 2/3 of a number is 8, what was the number?
Compare two fractions of the same quantity, such as 2/3 of 12 against 3/4 of 12, and say which is more.
Try a quantity that does not divide evenly, such as 2/3 of 10, and discuss why the numbers are chosen to divide cleanly at this stage.

Check it stuck

Assessment: exit ticket

A three-question exit ticket for the last five minutes. It samples a unit fraction of a set, a non-unit fraction of a set, and a word problem.

1. What is 1/3 of 12?
Answer: 4. Share 12 into 3 groups of 4.
2. What is 3/4 of 20?
Answer: 15. Divide 20 by 4 to get 5, then multiply by 3.
3. A class of 12 voted, and 2/3 chose pizza. How many chose pizza?
Answer: 8. Divide 12 by 3 to get 4, then multiply by 2.

For the teacher

Teacher notes and timings

Rough timing across three lessons: Lesson 1 unit fraction of a set with counters and bars (sections 1 to 2), Lesson 2 the divide-then-multiply rule (section 3), Lesson 3 word problems and the exit ticket (section 4 and assessment).
Keep the counters out through the pictorial sections. When a student is stuck on the abstract rule, hand them the counters and let them share the set into groups.
Language to keep saying: divide by the bottom, multiply by the top, and the denominator is how many groups. These phrases pre-empt the divide-by-the-numerator error.
All quantities in this unit divide exactly by their denominators so the group size is a whole number. That is a deliberate teaching choice for the first meeting; quantities that leave a remainder come later once the routine is secure.
US and AU alignment: the US sets this inside multiplying by a fraction at Grade 5 (5.NF.B.4), built on a fraction as a multiple of a unit fraction at Grade 4 (4.NF.B.4). ACARA names finding a fraction, decimal or percentage of a quantity explicitly at Year 5 (AC9M5N07), grown from unit fractions and their multiples at Year 4 (AC9M4N02). The two frameworks align closely for the fraction case at this level.
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.

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