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

Comparing and Ordering Decimals to Thousandths

B1.4 B1.3 A1About 35 minutes

Learning goals

I can compare two decimals to thousandths using >, < and =.
I can put decimals in order from least to greatest or greatest to least.
I can place decimals on a number line.
I can explain why more digits does not always mean a greater decimal.

Words to know

equivalent decimals
Decimals with the same value, like 0.3 and 0.300.

ascending order
From least to greatest.

descending order
From greatest to least.

Remember from Grade 5

In Grade 5 you compared decimals to hundredths by lining up the decimal points and comparing place by place, starting at the left. Now we just add one more place.

A close finish

In some sliding sports, like luge, races are so close that times are measured to the thousandth of a second. Here are the times from a made-up race at a winter carnival:

Asha: 47.316 s
Ben: 47.361 s
Chen: 47.306 s
Dana: 47.4 s

In a race, the least time wins. Who won?

Steps to compare decimals

Line up the decimal points.
If needed, add zeros at the end so every number has the same number of decimal places. 47.4 becomes 47.400.
Compare digits from left to right. The first place where they are different decides.

Racer	Tens	Ones	.	Tenths	Hundredths	Thousandths
Asha	4	7	.	3	1	6
Ben	4	7	.	3	6	1
Chen	4	7	.	3	0	6
Dana	4	7	.	4	0	0

Race times lined up by place value. Dana's 47.4 is written as 47.400.

The four times on a number line from 47.3 to 47.4 (each small tick is 0.01). From left: Chen 47.306, Asha 47.316, Ben 47.361, Dana 47.4.

Worked example

Example 1: Who won?

Order the race times from least to greatest: 47.316, 47.361, 47.306, 47.4.

Write Dana's time as 47.400.
Tens and ones are all 47. Tenths: Asha, Ben and Chen have 3; Dana has 4. So Dana's time is the greatest (slowest).
Compare hundredths for the others: Chen 0, Asha 1, Ben 6.

47.306, 47.316, 47.361, 47.4. Chen won! Dana came fourth.

Worked example

Example 2: The longer-number trap

Which is greater: 0.45 or 0.405?

Write both to thousandths: 0.450 and 0.405.
Tenths are the same (4). Hundredths: 5 > 0.

0.45 > 0.405. 0.405 has more digits, but it is smaller: 450 thousandths is more than 405 thousandths.

Drawn to scale in millimetres: 0.45 m (450 mm) is clearly longer than 0.405 m (405 mm), even though 0.405 has more digits.

Worked example

Example 3: Compare masses

Two backpacks have masses of 3.08 kg and 3.075 kg. Which is heavier?

Write 3.08 as 3.080.
Ones and tenths match (3 and 0). Hundredths: 8 > 7.

3.08 kg > 3.075 kg, so the first backpack is heavier (by 0.005 kg, or 5 g).

Check with a second strategy

After you compare with place value, check another way: think of thousandths as whole numbers (450 vs 405), or picture metres and millimetres (0.45 m = 450 mm). Good mathematicians check their answers with a different method.

Just for fun

Why did 0.405 stop bragging about having more digits?

Because 0.45 pointed out that 450 thousandths beats 405 thousandths!

Hands-on activity

Paper airplane contest

You need: a sheet of paper for each player, a measuring tape with millimetres (or a metre stick), masking tape and a pencil.

Each player folds a paper airplane.
Put a piece of masking tape on the floor as the start line. Each player throws their plane three times.
Measure each throw in millimetres from the start line to the nose of the plane. Write it in metres to the thousandth (for example, 3462 mm = 3.462 m).
Write all the distances on a list. Order them from greatest to least. Who threw the farthest?
Find two throws that are close. Which place value decided?

Safety: throw planes away from faces, and only in a clear space. Shake hands at the end: congratulate the winner and cheer everyone's best throw.

Your turn

Use >, < or =: 0.725 ___ 0.752

Use >, < or =: 3.9 ___ 3.900

Use >, < or =: 1.06 ___ 1.059

Order from least to greatest: 0.3, 0.033, 0.303

Order from greatest to least: 5.21, 5.201, 5.012, 5.2

Write a decimal between 0.62 and 0.63.

More practice: warm-up

Use >, < or =: 0.8 ___ 0.08

Use >, < or =: 2.345 ___ 2.346

Write 0.6 with three decimal places.

Which is greater: 4.1 or 4.099?

Order from least to greatest: 0.002, 0.2, 0.02

More practice: on your own

Order from least to greatest: 7.08, 7.8, 7.008, 7.080

Three seedlings are 0.125 m, 0.12 m and 0.152 m tall. Which is tallest? Which is shortest?

Ella says 2.75 < 2.705 because 705 > 75. Explain her mistake.

Write three decimals between 1.4 and 1.41.

Five speed skaters' times (made up) are 36.47 s, 36.074 s, 36.407 s, 36.4 s and 36.704 s. Order them from fastest (least) to slowest.

Challenge

Use the digits 0, 3, 5, 8 once each to make the greatest and least decimals of the form _._ _ _.

Fill in the blank with a digit to make it true. Find all answers. 4.3_7 > 4.36

A number line goes from 0.1 to 0.2. What decimal is exactly halfway? What decimal is halfway between that one and 0.2?

Check your understanding

Quiz

Which is greatest?

Which is true?

Which list is in order from least to greatest?

Which decimal is between 3.45 and 3.46?

In a race, the least time wins. Who wins?

Why can we write 0.4 as 0.400 to compare?

For parents and teachers

Big idea: Comparing and ordering decimals to thousandths is part of B1.4, and it is a big part of B1.3 (comparing and ordering integers, decimals and fractions).

Common difficulties:

"Longer is bigger". Children who think 0.405 > 0.45 are treating the digits after the point like whole numbers. Writing both to the same number of places (0.450 vs 0.405) fixes this.
"Shorter is bigger". Some children over-correct and think fewer digits always means bigger. Check with place value each time.

How to help: Measure things in millimetres and write them in metres to compare. Sports results (skating, sliding sports, swimming) are often written to hundredths or thousandths of a second. Encourage your child to check answers using a second strategy, a key mathematical process in Strand A.