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

Rounding Decimals: Terminating and Repeating

B1.5 A1About 40 minutes

Learning goals

I can round a decimal to the nearest whole number, tenth or hundredth.
I can tell the difference between terminating and repeating decimals.
I can round a repeating decimal.
I can decide how to round a decimal so it makes sense in a real situation.

Words to know

round
Replace a number with a nearby number that is easier to use.

terminating decimal
A decimal that ends, like 0.25 or 3.125.

repeating decimal
A decimal with a digit (or group of digits) that repeats forever, like 0.333…

ellipsis (…)
Three dots that mean "and it keeps going".

Remember from Grade 5

In Grade 5 you rounded decimal hundredths to the nearest tenth. You looked at the hundredths digit: 5 or more, round up; less than 5, keep the tenths digit the same. This year you'll round to the nearest whole number, tenth or hundredth, and round decimals that never end!

Sharing snow-shovelling money

Three friends earn $10 shovelling snow off a neighbour's driveway. They want to share it equally. On a calculator, 10 ÷ 3 = 3.333333333…. The 3s go on forever!

But there are no coins for $0.003. So each friend gets about $3.33. We rounded the decimal to the nearest hundredth (the nearest cent). (There is 1 cent left over. Maybe they can buy a treat later!)

A cleared driveway in front of a snowy house, with three snow shovels leaning on a snowbank. — Three shovels, three friends, and $10 to share. 10 ÷ 3 = 3.333…, a repeating decimal, so each share has to be rounded to the nearest cent. AI-generated image

The calculator says 3.333..., but there is no coin for a fraction of a cent. Rounded to the nearest hundredth, each friend gets $3.33, and 1¢ is left over.

How to round a decimal

Find the rounding place (whole number, tenths or hundredths).
Look at the digit one place to the right.
If that digit is 5 or more, round up: add 1 to the rounding place.
If that digit is less than 5, keep the rounding place the same.
Drop all the digits after the rounding place.

The number line shows why: a number rounds to whichever nearby tenth (or hundredth, or whole) it is closer to.

3.472 is between 3.4 and 3.5. It is closer to 3.5, so 3.472 rounded to the nearest tenth is 3.5.

Round 3.472 to the nearest…	Look at this digit	Answer
whole number	4 (tenths)	3
tenth	7 (hundredths)	3.5
hundredth	2 (thousandths)	3.47

Rounding the same number three ways.

Terminating and repeating decimals

Some decimals end, like 0.5, 0.75 or 3.125. These are terminating decimals.

Other decimals go on forever in a pattern. These are repeating decimals:

1 ÷ 3 = 0.333… (the 3 repeats)
2 ÷ 3 = 0.666… (the 6 repeats)
1 ÷ 6 = 0.1666… (the 6 repeats)
1 ÷ 11 = 0.0909… (the 09 repeats)

The three dots (…) mean "keeps going". Some books draw a short line over the digits that repeat instead.

A calculator screen can only show so many digits, so it rounds the last one. That's why 2 ÷ 3 may show 0.6666666667 instead of all 6s.

Worked example

Example 1: Round to the nearest whole number

A hiking trail is 4.62 km long. Round to the nearest kilometre.

The rounding place is the ones: 4.
Look one place to the right, at the tenths: 6. 6 is 5 or more, so round up.

5 km

Worked example

Example 2: Round a repeating decimal

Round 2/3 = 0.666… to the nearest tenth, hundredth and whole number.

Tenth: look at the hundredths digit, 6. Round up: 0.7.
Hundredth: look at the thousandths digit, 6. Round up: 0.67.
Whole number: look at the tenths digit, 6. Round up: 1.

0.666… ≈ 0.7 (tenth) ≈ 0.67 (hundredth) ≈ 1 (whole). The sign ≈ means "is about equal to".

Worked example

Example 3: When rounding up changes more than one digit

Round 6.97 to the nearest tenth.

The tenths digit is 9. Look at the hundredths: 7, so round up.
9 tenths + 1 tenth = 10 tenths = 1 whole. The ones digit goes from 6 to 7, and the tenths digit becomes 0.

7.0. Keep the zero to show you rounded to the tenths place.

Worked example

Example 4: Round so it makes sense

A gas station sign shows $1.459 per litre. Write the price in dollars and cents.

We pay in cents, so round to the nearest hundredth (cent).
Look at the thousandths digit: 9. Round up: 1.45 → 1.46.

About $1.46 per litre.

Which digit decides?

Only the digit just to the right of the rounding place decides. To round 2.349 to the nearest tenth, look at the 4 (hundredths), not the 9. 2.349 rounds to 2.3.

Just for fun

Why did 3.333... never finish its homework?

It just kept going... and going... and going...

Hands-on activity

Calculator pattern hunt

You need: a calculator (a phone calculator is fine, with a grown-up's OK), paper and a pencil.

Make a table with three columns: Division, Calculator shows, Rounded to the nearest hundredth.
Try these: 1 ÷ 9, 2 ÷ 9, 5 ÷ 9, 1 ÷ 3, 1 ÷ 6, 5 ÷ 6, 1 ÷ 11, 1 ÷ 4, 3 ÷ 8.
For each one, decide: terminating or repeating? Write the repeating ones with three dots, like 0.111…
Round each to the nearest hundredth.
What pattern do you notice with the ninths? Predict 7 ÷ 9 before you check it.

Answers to check: 1 ÷ 9 = 0.111… ≈ 0.11; 2 ÷ 9 = 0.222… ≈ 0.22; 5 ÷ 9 = 0.555… ≈ 0.56; 1 ÷ 3 = 0.333… ≈ 0.33; 1 ÷ 6 = 0.1666… ≈ 0.17; 5 ÷ 6 = 0.8333… ≈ 0.83; 1 ÷ 11 = 0.0909… ≈ 0.09; 1 ÷ 4 = 0.25 (terminating); 3 ÷ 8 = 0.375 ≈ 0.38 (terminating). 7 ÷ 9 = 0.777… ≈ 0.78.

Your turn

Round 8.36 to the nearest tenth.

Round 8.36 to the nearest whole number.

Round 0.784 to the nearest hundredth.

Round 5.555 to the nearest hundredth.

Is 0.444… terminating or repeating? Round it to the nearest tenth.

Is 0.875 terminating or repeating?

Round 12.96 to the nearest tenth.

More practice: warm-up

Round 2.7 to the nearest whole number.

Round 4.21 to the nearest tenth.

Round 0.139 to the nearest hundredth.

Round 9.5 to the nearest whole number.

What does 0.333… mean?

More practice: on your own

Round 1/6 = 0.1666… to the nearest tenth and the nearest hundredth.

Four friends share a $25 prize equally. A calculator shows 6.25. Is the answer terminating or repeating? Does it need rounding to be paid in money?

Round 3.049 to the nearest tenth. Explain which digit you looked at.

A jug holds 1.89 L. Round to the nearest tenth of a litre and to the nearest litre.

Ava rounded 7.248 to the nearest tenth and got 7.3. She said "8 rounds the 4 up to 5, then 5 rounds the 2 up to 3." Is she right?

Challenge

Three friends share $20 equally. How much should each get, rounded to the nearest cent? How much money is left over?

Write a decimal with thousandths that rounds to 4.5 (nearest tenth) and to 4.48 (nearest hundredth).

1 ÷ 7 = 0.142857142857… Which group of digits repeats? Round the decimal to the nearest thousandth.

Check your understanding

Quiz

Round 6.48 to the nearest tenth.

Round 0.333… to the nearest hundredth.

Which is a repeating decimal?

Round 14.5 to the nearest whole number.

Round 2.97 to the nearest tenth.

Which digit decides when you round 5.618 to the nearest hundredth?

For parents and teachers

Big idea: B1.5 asks Grade 6 students to round decimal numbers, both terminating and repeating, to the nearest tenth, hundredth or whole number, as applicable, in various contexts. Choosing the sensible rounding place (cents for money, for example) is part of the expectation.

Common difficulties:

Chain rounding. Rounding 7.248 "from the back" (8 makes the 4 a 5, which makes the 2 a 3) gives a wrong answer. Only the digit just right of the rounding place matters.
Rounding up a 9. 6.97 to the nearest tenth is 7.0, not 6.10. Show it on a number line.
Trusting the calculator display. A calculator showing 0.6666666667 has already rounded; the true decimal repeats forever.

How to help: Divide things at home (a bill split three ways, a bag of 100 g of trail mix shared by 3) and round the answer. Ask, "Does it make sense to round to the cent here, or to the nearest whole?" When your child gets stuck, encourage them to try a number line; persevering through tricky problems is part of Strand A.