B. Number · Number Sense: Numbers to One Million, Integers, Decimals and Fractions

Fractions and Decimals to Thousandths

B1.6About 45 minutes

Learning goals

I can write fractions with denominators of 10, 100 and 1000 as decimals, and back again.
I can use equivalent fractions to change a fraction like 3/8 into a decimal.
I can write a decimal as a fraction in simplest form.
I can explain that a fraction is a division, and that some fractions make repeating decimals.

Words to know

equivalent fractions
Fractions that name the same amount, like 3/8 and 375/1000.

simplest form
A fraction whose numerator and denominator have no common factor except 1, like 3/8.

common factor
A number that divides evenly into two numbers. 5 is a common factor of 15 and 20.

benchmark
A familiar number used to compare or estimate, like 0, 1/2 or 1.

Remember from Grade 5

In Grade 5 you connected fractions and decimals to hundredths: 7/10 = 0.7, 23/100 = 0.23 and 3/4 = 75/100 = 0.75. You also found equivalent fractions by multiplying or dividing the numerator and denominator by the same number. Now we'll go to thousandths and meet fractions that turn into repeating decimals.

A measuring tape mystery

Maya is helping her aunt build a birdhouse. The plan says to cut a board 3/8 m long. But her aunt's measuring tape shows millimetres, and her calculator wants decimals.

How long is 3/8 m in metres as a decimal? In millimetres?

Idea 1: Tenths, hundredths and thousandths are already decimals

If the denominator is 10, 100 or 1000, you can write the decimal right away. The number of zeros in the denominator tells you how many decimal places.

7/10 = 0.7 (1 zero, 1 place)
7/100 = 0.07 (2 zeros, 2 places)
7/1000 = 0.007 (3 zeros, 3 places)
431/1000 = 0.431

Idea 2: Make the denominator 10, 100 or 1000

Find an equivalent fraction. Ask: what can I multiply the denominator by to get 10, 100 or 1000? Multiply the numerator by the same number.

For 3/8: 8 × 125 = 1000. So 3/8 = (3 × 125)/(8 × 125) = 375/1000 = 0.375.

Maya cuts the board at 0.375 m, which is 375 mm.

Eighths from 0 to 1. Each eighth is 0.125 (125 thousandths). 3/8 = 0.375 and 5/8 = 0.625.

Worked example

Example 1: Fraction to decimal

Write 9/20 and 7/40 as decimals.

9/20: 20 × 5 = 100. So 9/20 = 45/100 = 0.45.
7/40: 40 × 25 = 1000. So 7/40 = 175/1000 = 0.175.

9/20 = 0.45 and 7/40 = 0.175

Worked example

Example 2: Decimal to fraction in simplest form

Write 0.625 and 0.08 as fractions in simplest form.

0.625 has three decimal places, so it is 625/1000. 125 is a common factor: 625 ÷ 125 = 5 and 1000 ÷ 125 = 8. So 5/8.
0.08 has two decimal places, so it is 8/100. Divide both by 4: 2/25.

0.625 = 5/8 and 0.08 = 2/25

Idea 3: A fraction is a division

The fraction bar means divide. 3/8 means 3 ÷ 8. Check on a calculator: 3 ÷ 8 = 0.375. It matches!

Some fractions can't be made into tenths, hundredths or thousandths. No whole number times 3 equals 10, 100 or 1000. So for 1/3, divide: 1 ÷ 3 = 0.333…, a repeating decimal. To thousandths, 1/3 ≈ 0.333.

Fraction	Decimal	Thousandths
1/2	0.5	0.500
1/4	0.25	0.250
3/4	0.75	0.750
1/5	0.2	0.200
1/8	0.125	0.125
1/3	0.333…	≈ 0.333
2/3	0.666…	≈ 0.667
1/10	0.1	0.100
1/100	0.01	0.010
1/1000	0.001	0.001

Benchmark fractions and their decimals.

Worked example

Example 3: Mixed numbers

Write 2 3/8 as a decimal and 4.05 as a mixed number in simplest form.

2 3/8: the whole number part stays 2. 3/8 = 0.375. So 2 3/8 = 2.375.
4.05: the whole number part is 4. 0.05 = 5/100 = 1/20 (divide both by 5).

2 3/8 = 2.375 and 4.05 = 4 1/20

Relationships to notice

1/10 = 0.1, 1/100 = 0.01, 1/1000 = 0.001. Each one is 10 times the next one. 0.1 is 100 times 0.001.
The bigger the denominator of a unit fraction, the smaller the piece: 1/1000 < 1/100 < 1/10.
Halving: 1/2 = 0.5, 1/4 = 0.25, 1/8 = 0.125. Each time the fraction is cut in half, so is the decimal.

Each shaded bar is half of the one above: 1/2 = 0.5, 1/4 = 0.25, 1/8 = 0.125. Halving the fraction halves the decimal.

Look for the factor

To change a fraction to thousandths, these are handy: 2 × 500 = 1000, 4 × 250 = 1000, 5 × 200 = 1000, 8 × 125 = 1000, 20 × 50 = 1000, 25 × 40 = 1000, 40 × 25 = 1000, 50 × 20 = 1000, 125 × 8 = 1000, 200 × 5 = 1000, 250 × 4 = 1000.

Just for fun

Fractions walk into a calculator and come out as decimals. 3/8 walks in and comes out as 0.375. Then 1/3 walks in and... it never comes out. It's still in there: 0.333333...

Hands-on activity

Metre strip fractions

You need: a measuring tape or metre stick with millimetres, a long strip of paper at least 1 m long (tape some sheets together end to end), a pencil and a ruler.

Mark 0 at one end of the strip and 1 m at 1000 mm.
Fold the strip in half (matching 0 to 1 m) to find 1/2. Measure it: it should be at 500 mm, so 1/2 m = 0.5 m.
Fold again to find 1/4 and 3/4. Measure and label them in mm and in metres (250 mm = 0.25 m, 750 mm = 0.75 m).
Fold once more to find all the eighths. Label 1/8 = 0.125 m, 3/8 = 0.375 m, 5/8 = 0.625 m and 7/8 = 0.875 m. Check each with the tape.
Bonus: Try to find 1/3 m. Fold the strip into three equal parts. Measure: it should be about 333 mm, so 1/3 m ≈ 0.333 m. Why can't it be exactly a whole number of millimetres?

Folding into thirds takes a few tries. If it's uneven, unfold and try again!

Your turn

Write 59/1000 as a decimal.

Write 0.307 as a fraction.

Write 3/5 as a decimal.

Write 5/8 as a decimal.

Write 0.25 as a fraction in simplest form.

Write 1 7/20 as a decimal.

Write 2/3 as a decimal. Is it terminating or repeating?

More practice: warm-up

Write 3/10 as a decimal.

Write 3/100 as a decimal.

Write 3/1000 as a decimal.

Write 0.5 as a fraction in simplest form.

Write 1/4 as a decimal.

More practice: on your own

Write 11/25 as a decimal.

Write 0.875 as a fraction in simplest form.

Write 0.045 as a fraction in simplest form.

A recipe uses 1/8 kg of cheese. How many grams is that? Write it in kilograms as a decimal.

Which is greater: 1/100 or 0.009? Explain.

Challenge

Write 13/250 as a decimal.

Write 1/6 as a decimal. Round it to the nearest thousandth.

Jordan says 0.3 = 1/3. Is he right? Explain.

Check your understanding

Quiz

What is 3/8 as a decimal?

What is 0.016 as a fraction?

Which fraction is equal to 0.2?

Which fraction makes a repeating decimal?

0.35 in simplest form is...

Which is 10 times greater than 0.001?

For parents and teachers

Big idea: B1.6 asks Grade 6 students to describe relationships and show equivalences among fractions and decimal numbers up to thousandths, using tools and drawings. Converting with equivalent fractions (3/8 = 375/1000) and seeing a fraction as a division are the key ideas.

Common difficulties:

Copying digits. Children may write 3/8 as 0.38 or 3.8. Remind them that the fraction must first have a denominator of 10, 100 or 1000.
Wrong number of places. 59/1000 written as 0.59. The denominator 1000 has three zeros, so the decimal needs three places: 0.059.
Thinking 0.3 = 1/3. Show 1 ÷ 3 on a calculator.

How to help: Use a metric measuring tape or kitchen scale: 1/8 kg = 125 g = 0.125 kg; 3/4 m = 750 mm = 0.75 m. Folding a paper strip is a great hands-on way to see eighths.