B. Number · Number Sense: Numbers to 10 000, Fractions and Decimal Tenths

Fractions and Decimals: Same Amount, Two Names

B1.9 B1.4 A1About 40 minutes

Learning goals

I can write a fraction in tenths as a decimal, and a decimal as a fraction.
I can show that 1/2 = 5/10 = 0.5.
I can show that fifths can be written as tenths and as decimals, like 2/5 = 4/10 = 0.4.
I can write mixed numbers in tenths as decimals, like 2 3/10 = 2.3.

Words to know

equivalent
Equal in amount. 1/2 and 0.5 are equivalent.

equivalent fractions
Fractions that name the same amount, like 1/2 and 5/10.

relationship
How two things are connected.

Remember from Grade 3

In Grade 3 you found equivalent fractions in families: halves, fourths and eighths; thirds and sixths; fifths and tenths. You saw that 1/5 and 2/10 cover the same amount. Today you'll connect fractions to decimals.

Let's start with a story

A pancake recipe needs 1/2 L of milk. Liam's measuring jug has 10 equal marks from 0 to 1 L, labelled 0.1, 0.2, 0.3 ... up to 1. Which mark should he fill to?

One half of a whole is the same as 5 tenths. So he fills to the 0.5 L mark.

1/2 = 5/10 = 0.5. Three names, one amount!

Three same-size bars: one half, five tenths and 0.5 cover the same amount.

Tenths: fraction and decimal

Any number of tenths can be written both ways:

1/10 = 0.1
4/10 = 0.4
9/10 = 0.9
10/10 = 1

With wholes, use a mixed number or a decimal:

2 3/10 = 2.3
13/10 = 1 3/10 = 1.3

Fifths and tenths

Each fifth is the same as 2 tenths. Look at the bars: a fifth lines up with two tenths.

Two fifths line up exactly with four tenths. 2/5 = 4/10 = 0.4.

Fraction	In tenths	Decimal
1/5	2/10	0.2
2/5	4/10	0.4
3/5	6/10	0.6
4/5	8/10	0.8
5/5	10/10	1
1/2	5/10	0.5

Fifths and halves written as tenths and as decimals.

Worked example

Example 1: Fraction to decimal

Write 3/5 as a decimal.

Each fifth is 2 tenths.
3 fifths = 3 × 2 = 6 tenths = 6/10.
6 tenths as a decimal is 0.6.

3/5 = 6/10 = 0.6

Worked example

Example 2: Decimal to fraction

Write 0.8 as a fraction in tenths, and in fifths.

0.8 is 8 tenths: 8/10.
Every 2 tenths make 1 fifth. 8 tenths = 4 groups of 2 tenths = 4 fifths.

0.8 = 8/10 = 4/5

Worked example

Example 3: Mixed numbers and decimals

Write 3 1/2 as a decimal. Write 4.2 as a mixed number with fifths.

3 1/2: 3 wholes and 1 half. A half is 5 tenths, so 3 1/2 = 3 and 5 tenths = 3.5.
4.2: 4 wholes and 2 tenths. 2 tenths = 1 fifth, so 4.2 = 4 1/5.

3 1/2 = 3.5 and 4.2 = 4 1/5

What about fourths?

1/4 is half of a half. Half of 5 tenths is 2 and a half tenths, which can't be written with only a tenths digit. You'll learn how to write 1/4 as a decimal in a later grade. For now, you can say 1/4 is between 0.2 and 0.3.

0 to 1 by tenths. 0.5 is the same point as 1/2. 0.8 is the same point as 4/5.

Using what you know

0.5 L of juice is the same as 1/2 L.
A 1.5 km walk is the same as 1 1/2 km.
0.2 m is 2/10 m, or 1/5 m: that is 20 cm.

Make them match

When a question uses fractions and decimals together, change them to the same kind of number first (both tenths, or both decimals). Then compare. It's like translating so everyone speaks the same language!

Just for fun

Riddle: What has three names but is only one amount?

Answer: 1/2, 5/10 and 0.5. It's like a cat called Whiskers, Mr. Whiskers and Sir Fluffington. Still the same cat!

Hands-on activity

Fraction–decimal memory game

You need: 18 small pieces of paper or index cards, a pencil.

Make 9 pairs of cards. On one card of each pair write a fraction, and on the other write the matching decimal:

1/10 and 0.1
3/10 and 0.3
7/10 and 0.7
1/2 and 0.5
1/5 and 0.2
2/5 and 0.4
3/5 and 0.6
4/5 and 0.8
1 1/2 and 1.5

Mix the cards and lay them face down in rows.
Take turns flipping two cards. If they show the same amount, keep them and explain why they match ("3/5 is 6 tenths, so it's 0.6"). If not, flip them back.
The player with the most pairs wins. Be a kind player: help each other remember where cards are if someone is stuck!

Math talk: "Which pairs were easiest? Which were hardest? Why?"

Your turn

Write 7/10 as a decimal.

Write 0.3 as a fraction.

Write 1/2 as a decimal.

Write 1/5 as a decimal.

Write 2 9/10 as a decimal.

Write 6.5 as a mixed number with a half.

Is 0.6 more than, less than, or equal to 3/5?

More practice: warm-up

Write 0.9 as a fraction.

Write 4/10 as a decimal.

How many tenths make 1/2?

Write 10/10 as a whole number.

Write 1.1 as a mixed number.

More practice: on your own

Write 4/5 as a decimal.

Write 0.4 as a fraction in fifths.

Write 15/10 as a decimal.

Which is greater: 0.7 or 1/2? Explain.

A trail is 2 1/2 km. Another trail is 2.3 km. Which is longer?

Challenge

Order from least to greatest: 0.7, 1/2, 3/5, 0.2

Amir says 1/4 = 0.4 because there's a 4 in both. Explain why he is wrong.

Write 3 4/5 as a decimal, and explain your thinking.

Check your understanding

Quiz

Which decimal equals 9/10?

Which is equal to 1/2?

Which is equal to 2/5?

Which mixed number equals 3.7?

Which is greatest?

Which is the same as 1 1/2 km?

For parents and teachers

Big idea: B1.9 asks Grade 4 students to describe relationships and show equivalences among fractions and decimal tenths, in various contexts. Key facts: tenths can be written as fractions or decimals (7/10 = 0.7); 1/2 = 5/10 = 0.5; each fifth is 2 tenths (2/5 = 4/10 = 0.4).

Common difficulties:

Matching digits instead of amounts (thinking 1/4 = 0.4 or 1/5 = 0.5). Always ask "How many tenths is that?"
Forgetting that 10 tenths make a whole (writing 15/10 as 0.15).
Fourths: 1/4 can't be written with tenths alone (it is 0.25, which uses hundredths, a later-grade topic). It's fine to say it's between 0.2 and 0.3.

How to help: Use measuring cups, litre containers and metre sticks marked in tenths. Play the memory game. Strand A: in games, model being a kind and fair player.