B. Number · Number Sense: Numbers to 100 000, Fractions, Decimals and Percents

Improper Fractions and Mixed Numbers

B1.3About 40 minutes

Learning goals

I can show amounts greater than 1 as improper fractions and as mixed numbers.
I can change an improper fraction into a mixed number.
I can change a mixed number into an improper fraction.
I can find equivalent improper fractions and mixed numbers.

Words to know

improper fraction
A fraction whose numerator is equal to or greater than its denominator, like 7/4 or 3/3.

mixed number
A whole number and a fraction together, like 1 3/4.

proper fraction
A fraction less than 1, where the numerator is less than the denominator, like 3/4.

Remember from Grade 4

In Grade 4 you counted by fractions past 1, like 1/4, 2/4, 3/4, 4/4, 5/4... You wrote amounts greater than 1 in two ways: as a fraction like 5/4 or as a mixed number like 1 1/4. This year you'll change between these two forms quickly, and find equivalent ones.

Let's start with pizza

At a family party, every pizza is cut into 4 equal slices. The kids eat 7 slices. How much pizza is that?

7/4 pizzas (7 fourths), or
1 3/4 pizzas (1 whole pizza and 3 more fourths).

Both names are correct! 7/4 is an improper fraction. 1 3/4 is a mixed number.

Two pizzas cut into fourths. The first is all eaten (4/4 = 1). The second has 3 fourths eaten. 4/4 + 3/4 = 7/4 = 1 3/4.

A number line from 0 to 3 marked in fourths. 7/4 is 3 fourths past 1.

Improper fraction → mixed number

Ask: How many wholes can I make? Each whole needs as many parts as the denominator.

11/4: 4 fourths make 1 whole, 8 fourths make 2 wholes. That leaves 3 fourths. So 11/4 = 2 3/4.

Mixed number → improper fraction

Change each whole into parts, then add the extra parts.

3 2/5: each whole is 5 fifths, so 3 wholes = 15 fifths. Add 2 more fifths: 17/5.

Worked example

Example 1: Improper to mixed

Write 11/4 as a mixed number.

How many groups of 4 fit in 11? 4, 8... that's 2 groups, with 3 left over.
2 wholes and 3 fourths.

11/4 = 2 3/4

Worked example

Example 2: Mixed to improper

Write 3 2/5 as an improper fraction.

Each whole has 5 fifths. 3 wholes = 3 × 5 = 15 fifths.
Add the 2 extra fifths: 15 + 2 = 17 fifths.

3 2/5 = 17/5

Worked example

Example 3: Equivalent mixed numbers

Write 10/6 as a mixed number, then find an equivalent one with bigger pieces.

6 sixths make 1 whole. 10 − 6 = 4 sixths left. So 10/6 = 1 4/6.
4/6 = 2/3 (divide top and bottom by 2). So 1 4/6 = 1 2/3.
As improper fractions: 10/6 = 5/3.

10/6 = 1 4/6 = 1 2/3 = 5/3

Worked example

Example 4: When there's nothing left over

Write 12/3 another way.

3 thirds make 1 whole. 12 thirds make 12 ÷ 3 = 4 wholes.
There are no extra thirds.

12/3 = 4. Some improper fractions are whole numbers!

Improper fraction	Mixed number or whole number	Equivalent
3/2	1 1/2	6/4 = 1 2/4
5/4	1 1/4	10/8 = 1 2/8
7/3	2 1/3	14/6 = 2 2/6
8/4	2	4/2 = 2

Each row shows different names for the same amount.

Quick check

A fraction bar means "divided by". 11/4 is 11 ÷ 4 = 2 with 3 left over. The 2 is the whole number and the 3 left over goes on top of the 4: 2 3/4.

Just for fun

Why do improper fractions wobble when they walk?

Because they're top-heavy: the numerator is bigger than the denominator!

Hands-on activity

Scoop and count

You need: a 1 cup measuring cup, a 1/4 cup and a 1/3 cup measure, a large bowl, and dry rice, oats or water (do this over a tray or sink).

Scoop 1/4 cup at a time into the bowl. Count your scoops out loud in fourths: 1/4, 2/4, 3/4, 4/4...
Stop at 9 scoops. Write the amount as an improper fraction (9/4 cups) and as a mixed number (2 1/4 cups).
Check: pour the bowl back into the 1 cup measure. Can you fill it 2 times with a little left over?
Now use the 1/3 cup scoop. Scoop 7 times. Write the amount both ways. (7/3 = 2 1/3 cups.)
Your grown-up names a mixed number, like 1 2/3 cups. How many 1/3 scoops do you need? (5.)

Your turn

Write 5/4 as a mixed number.

Write 7/2 as a mixed number.

Write 13/5 as a mixed number.

Write 1 2/3 as an improper fraction.

Write 2 1/6 as an improper fraction.

9/3 = ?

Mom has 3 1/2 sandwiches cut into halves. How many halves is that?

More practice: warm-up

Is 6/5 a proper or an improper fraction?

4/4 = ?

Write 3/2 as a mixed number.

Write 1 1/3 as an improper fraction.

2 = ?/5

More practice: on your own

Write 17/6 as a mixed number.

Write 4 3/8 as an improper fraction.

Write 14/4 as a mixed number, then as an equivalent mixed number with halves.

A muffin tray holds 12 muffins. You bake 29 muffins. How many trays is that, as a mixed number?

Is 6/4 equal to 1 1/2? Explain.

Challenge

Write 2 1/2 as an improper fraction three ways: in halves, in fourths and in tenths.

Kai says 2 3/4 = 23/4. What mistake did he make? What is the right answer?

You fill a 1/3 cup measure with oats 8 times. How many cups of oats do you have, as a mixed number?

Check your understanding

Quiz

9/4 = ?

3 1/5 = ?

Which is an improper fraction?

12/4 = ?

Which is equal to 1 1/2?

11/3 is between which two whole numbers?

For parents and teachers

Big idea: B1.3 includes representing improper fractions and mixed numbers and their equivalents. Children should see that 7/4 and 1 3/4 are the same amount, and be able to move between them using wholes made of parts (not just a memorized rule).

Common difficulties:

Pushing digits together (2 3/4 = 23/4). Ask "How many fourths are in 2 wholes?"
Forgetting leftovers (11/4 = 2). Ask "Did you use all 11 fourths?"
Thinking improper means wrong. "Improper" is just a name. 7/4 is a perfectly good number.

How to help: Measuring cups in baking are ideal. "The recipe needs 1 1/2 cups but we only have the 1/2 cup scoop. How many scoops?" (3). Keep the activity over a tray or sink to make cleanup easy.