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Recurring Decimals: Free Printable Worksheet for Year 9
- ForYear 9
- StageKey Stage 3
- CostFree
- FormatPrintable PDF
Recurring Decimals is a free printable maths worksheet for Year 9 (ages 13 to 14). Students turn fractions into decimals, spot the digits that repeat and write them using dot notation. It takes about 20 minutes, needs only a pencil, and page 2 gives the grown-up notes, help ideas, a stretch task and the answers.
It comes from the Mini Minds Hub, the free library of printables from Jaques of London, the British games and toy company that has been making games for families since 1795. Print both pages, give page 1 to the student and keep page 2 for yourself. Black and white printing works well.
What is on the sheet
Students work through numbered questions with space to write each answer. They start with simple fractions and write them as decimals, then say which digit or digits repeat. Later questions ask them to go the other way, from a repeating decimal back to a fraction, and to give a reason for a decision about a decimal. A calculator may be used for checking.
- Write 1/3 as a decimal. How many digits repeat?
- Write 2/3 as a decimal.
- Write 1/9 as a decimal.
- Write 5/9 as a decimal.
- Write 1/6 as a decimal. Which digit repeats?
- Write 1/11 as a decimal. Which two digits repeat?
- Write 0.777... as a fraction with a bottom of 9.
- Is 0.25 a recurring decimal? Say why.
A different fraction, worked two ways
Write 7/11 as a decimal and show which digits repeat.
This question is not on the sheet. It uses a different fraction, so you can show the method without giving anything away.
Method 1: long division- Divide 7 by 11. 11 does not go into 7, so write 0. and add zeros after the decimal point.
- 70 ÷ 11 = 6, because 6 × 11 = 66. The remainder is 4.
- Bring down a zero. 40 ÷ 11 = 3, because 3 × 11 = 33. The remainder is 7.
- The remainder is 7 again, which is where the sum started. So the digits 6 and 3 will repeat for ever.
- Type 7 ÷ 11 and read the display.
- Look for a block of digits that keeps coming back in the same order.
- Remember the display is cut off, so the true decimal carries on. Write the repeating block with dots.
Why do some decimals never end?
Every fraction is a division. 1/3 means 1 divided by 3. When you divide using long division, you keep getting remainders. If a remainder comes back that you have had before, the same digits must follow, and the decimal repeats. That is why a repeating pattern is not a coincidence. It is a direct result of how division works.
Some decimals end, and these are called terminating decimals. Others repeat in a pattern, and these are called recurring decimals. A fraction in its simplest form gives a decimal that ends only if its bottom number has no prime factors other than 2 and 5. Otherwise the decimal recurs. Dot notation is a short, exact way to write the repeating digits.
Skills and curriculum link
- Fractions to decimals
- Long division
- Pattern spotting
- Dot notation
- Reasoning about number
This sheet supports the area below. It is matched to the published England framework. Please check it against your own school plan.
| Framework | Subject or area | Strand | Year |
|---|---|---|---|
| National Curriculum (England) | Maths | Number | Year 9 |
What children will be able to do
- I can turn a simple fraction into a decimal.
- I can spot a repeating pattern in the digits.
- I can write a recurring decimal using dot notation.
Ideas that help with recurring decimals
- A fraction is a divisionThe line in a fraction means divide. The top number is divided by the bottom number. Writing a fraction as a decimal is just doing that division.
- Remainders tell the storyWatch the remainders in long division. When one repeats, the digits repeat too. This is a good way to know when to stop.
- Dot notationA dot goes above the first and last digit of the repeating block. If only one digit repeats, it gets one dot.
- Equal fractions, equal decimalsA fraction and its simplified form are worth the same, so they give the same decimal. Simplifying first can make the division easier.
Key words
- Recurring decimalA decimal where a digit or a block of digits repeats for ever.
- Terminating decimalA decimal that ends after a fixed number of digits.
- Dot notationA way to write a recurring decimal using dots above the digits that repeat.
- RemainderWhat is left over when one number does not divide exactly into another.
- DenominatorThe bottom number of a fraction. It tells you how many equal parts make a whole.
- NumeratorThe top number of a fraction. It tells you how many parts you have.
How to use it
- Print both pages. Page 1 is for the student. Keep page 2, which has the notes and answers.
- Read the first question together. Ask the student to say what the fraction means as a division.
- Let them choose how to work: long division on paper, or a calculator to check. Ask them to show their working.
- Check against page 2. Ask, "How do you know which digits repeat?" The reasoning matters as much as the answer.
- Come back another day and redo any questions that felt tricky. This sheet takes about 20 minutes, and repeating it builds confidence.
Help, stretch and game ideas
If it feels too hard
Use long division for 1/3 and 1/9 only, and look for the pattern. Write each remainder in a column beside the working. When a remainder repeats, the digits repeat. Once that pattern is clear, try the other questions one at a time.
To stretch confident children
Predict what 1/7 will look like as a decimal, then check the six digit repeat on a calculator. Next, try 2/7 and 3/7. Compare the digits and describe what you notice about the order they appear in.
Make it a game
Challenge a friend to guess the fraction from a recurring decimal. One player writes a fraction secretly and works out its decimal. The other sees only the decimal and guesses. Swap roles and keep score over several rounds.
Mistakes to watch for
Watch forA student rounds the decimal, for example writing the last digit up, instead of showing the repeat.
Try thisExplain that rounding gives an estimate, but dot notation gives the exact value. Ask them to find the repeating block and mark it.
Watch forA student puts the dots over the wrong digits, or over every digit.
Try thisRemind them that dots go over the first and last digit of the repeating block only. One repeating digit gets one dot.
Watch forA student divides the bottom number by the top number.
Try thisSay it aloud as "top divided by bottom". Check with a calculator that the answer is sensible for the fraction.
Talk about it
- What does a fraction bar mean?
- How can you tell that a decimal is going to repeat?
- Why does the same remainder give the same digits again?
- Where do the dots go in dot notation, and why?
- Is a recurring decimal exact or approximate? Explain.
- Which fractions do you think will give decimals that end? Why?
Take it off the page
Remainder detective
Pick a fraction and do the long division on paper. Circle each remainder in a different colour. Stop when a colour repeats and explain what that tells you.
Fraction and decimal cards
Write fractions on one set of paper cards and their decimals on another. Mix them up and match the pairs. Use dot notation on the recurring ones.
Pattern poster
Choose a fraction and make a small poster showing its division, its repeating digits and the dot notation. Add a sentence explaining why the pattern repeats.
Who it is for
Parents
Sit beside your teenager for the first question, then let them lead. You do not need to be a maths expert. Ask them to explain how they know which digits repeat, and use page 2 to check answers.
Teachers
Use it as a starter, a plenary, an early finisher task or homework. It needs no preparation beyond printing. Page 2 gives answers for quick marking. Ask students to explain why remainders repeat, and use the stretch idea for those ready for more.
Home educators
Print the sheet as part of a wider maths plan and let your learner work at their own pace. It pairs well with a calculator for checking and a notebook for long division. Use it again later for revision, or for a brother or sister.
Where this fits in maths
Before this sheet, students need to be confident with fractions, place value in decimals and short and long division. They should know that a fraction can be read as one number divided by another. Earlier work on equivalent fractions also helps.
After this sheet, students can move on to converting recurring decimals into fractions using algebra, and to comparing and ordering fractions and decimals. The idea of exact values also links to rational numbers and later work on surds and irrational numbers.
Check it worked
Ask these when the sheet is finished. Listen for the reasoning, not just the answer.
- How do you show which digits repeat using dot notation?
- What happens when you divide 1 by 3, and why does it never end?
- Which fractions give recurring decimals, and which give ones that stop?
Questions parents and teachers ask
- What age is Recurring Decimals for?
It is written for Year 9 (ages 13 to 14), in Key Stage 3. Students who are confident with fractions and long division will find it most accessible. Younger or less confident students can use the help idea on page 2.
- What will students practise?
Students turn fractions into decimals, spot repeating digits and write them using dot notation. They also work from a repeating decimal back to a fraction and give a reason about a decimal. The sheet builds fluency and reasoning together.
- How long does it take?
It takes about 20 minutes for one student. A calculator may be used for checking. You can make it shorter by choosing fewer questions, or longer by adding the stretch idea from page 2.
- What if it feels too hard?
Go back to basics. Use long division for 1/3 and 1/9 only and look for the pattern in the remainders. Once students see why digits repeat, return to the other questions one at a time.
- Does the sheet include answers?
Yes. Page 2 is for the grown-up and includes the answers, notes on how to use the sheet, help for students who find it hard and a stretch idea. Keep page 2 back until the student has finished.
- Is it suitable for teachers to use in class?
Yes. Teachers can use it as a starter, plenary, early finisher task, homework or table activity. It needs no preparation beyond printing, and page 2 gives support and stretch ideas for mixed ability groups.
- Is it free and screen free?
Yes. It is free to print for home and classroom use, and it needs only paper and a pencil. A calculator is optional, only for checking. Black and white printing on plain A4 paper works well.
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Made by Jaques of London, making games for families since 1795. A printable supports classroom learning and does not replace a teacher.