Terminating and recurring decimals: a week of ready-to-teach maths
Five lessons for Year 8 Number. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet with practice and a stretch section, plus remainder clocks, terminate-or-recur sort cards, a mini-check and every answer with its working.
Start here: five minutes to Monday
- Skim the week at a glance on the next page.
- Print the five days. Each day is two A4 sheets: a plan and a worksheet.
- Print the two cut-out sheets, one copy per pair: the remainder clocks are for Day 1, the sort cards for Day 4 (cut them before then).
- Open the free interactive unit on the board. Every plan says which picture to show and which buttons to press.
- Teach straight from the plan. Timings, talk prompts, misconceptions and worked answers are all on the one page.
The idea in plain words
Dividing the top of a fraction by the bottom gives its decimal. At each step of the division there is a remainder. If a remainder of 0 turns up, the division stops and the decimal terminates, as with 3/8 = 0.375. If not, there are only so many remainders possible, so one must come back, and from then on the same digits repeat for ever: 1/7 = 0.142857142857..., a recurring decimal. Which will happen can be told in advance. Write the fraction in lowest terms and factorise the denominator: only 2s and 5s, the factors of 10, means it terminates; any other prime means it recurs. A calculator rounds its last digit, so its screen is a check, never the whole story.
One day, one lesson
The five lessons fill a week of maths, one lesson of about 60 minutes a day. Run them in order: each day stands on the one before. Every plan marks a split point for two shorter sessions, and every worksheet ends with a stretch section for students who finish early.
How we write recurring decimals
On Days 1 and 2 we write the repeat with three dots after it has shown at least twice, as in 0.41666... From Day 3 we use a bar over the repeating block, 0.416 and 0.142857, and show the other common form, with a dot over the first and the last digit of the block: 0.416˙ and 0.1˙42857˙. Both mean exactly the same number.
The week at a glance
One lesson a day for a week. Each day stands on the day before, so run them in order.
| Day | Lesson | Students learn and do | On screen |
|---|---|---|---|
| 1 | Two ways a fraction turns out | Divide fractions out keeping every remainder, see a remainder of 0 stop the decimal or an old remainder start a repeat, and measure repeating blocks | Two ways a fraction can turn out; remainder clocks |
| 2 | Terminating decimals | Rewrite fractions over a power of ten, turn terminating decimals back into fractions, and see why halves of halves stop | A terminating decimal |
| 3 | Recurring decimals | Write recurring decimals with a bar or dots, expand the notation, compare decimals that look alike, and turn simple ones into fractions | A recurring decimal |
| 4 | The denominator test | Predict from the prime factors of the denominator, simplify first, sort 24 fraction cards and count decimal places | The denominator test; sort cards |
| 5 | Checking with a digital tool | Compare calculator screens with exact decimals, decide when a screen cannot tell, and play the quiz as a class game | Checking with a digital tool; self-check quiz |
How the week builds
Day 1 watches the remainders, which decide everything; Day 2 explains the decimals that stop through powers of ten; Day 3 gives the decimals that repeat a notation of their own; Day 4 turns all of it into a test you can do before dividing; and Day 5 asks what a calculator can and cannot tell you. It builds on Year 7, where students wrote fractions as decimals and met 1/3 = 0.333... It stands beside irrational numbers in Year 8, whose decimals neither stop nor repeat, and prepares the real number system of Year 9 and 10.
Materials for the week (one trip)
- From the classroom: pencils, coloured pencils, scissors, this pack printed.
- Cut-out sheet 1, one per pair: two remainder clocks, for dividing by 7 and by 13 (Day 1).
- Cut-out sheet 2, one per pair: 24 terminate-or-recur sort cards (Day 4).
- Calculators, any kind, for Day 5 (and for checking from Day 2). No other equipment to buy.
Words for the word wall
decimal, terminating decimal, recurring decimal, repeating block, remainder, long division, short division, power of ten, tenths, hundredths, thousandths, lowest terms, denominator, prime factor, factor tree, bar notation, dot notation, rational number, round, digital tool
Dear families
This week in maths, Year 8 is learning why some fractions make decimals that stop, like 3/8 = 0.375, while others repeat for ever, like 1/3 = 0.333... A quick test tells which: in lowest terms, if the bottom number is made only of 2s and 5s (the factors of 10), the decimal stops. We also look closely at what a calculator shows, because it rounds its last digit.
Try this at home
- On a calculator, work out 1 ÷ 7 and 2 ÷ 7. Can you spot the same six digits in both, starting in a different place?
- Ask: does 1/4 stop or repeat? What about 1/6? (Stops at 0.25; repeats, 0.1666...)
- Ask: why does the calculator show 0.6666667 for 2/3 when the 6s never end? (It rounds the last digit it has room for.)
- Ask: is 0.333... the same as 0.33? (No: 0.333... is 1/3, a little more than 0.33.)
My decimals this week
Fill one row a day. Choose a fraction, predict whether it stops or repeats, then check.
| Day | Fraction | Prediction and why | The decimal |
|---|---|---|---|
| Monday | |||
| Tuesday | |||
| Wednesday | |||
| Thursday | |||
| Friday |
Printed from the free seegongsik Terminating and recurring decimals teaching pack · seegongsik.com/au/y8/number/AC9M8N03/pack
Two ways a fraction can turn out
The remainders of a long division decide everything. Students divide fractions out keeping every remainder, see a 0 stop the decimal or an old remainder start a repeat, and measure how long each repeating block is.
We are learning to
- divide a fraction out by short division, writing the remainder at every step,
- say from the remainders whether a decimal terminates or recurs, and where the repeat starts,
- find the length of a repeating block and explain why it cannot be longer than the denominator.
Success criteria
- I can show from the remainders why 1/4 stops but 1/3 repeats.
- I can find the repeating block of a fraction such as 2/7 and say how long it is.
You need
The worksheet; the remainder clocks (cut-out sheet 1), one per pair; coloured pencils. No calculators.
Lesson flow (about 60 minutes)
| 10 min | Stop or repeat? Show the board (see the box). Divide 1/4 and 1/3 out on the board, writing each remainder in a ring: 1/4 leaves 2, 0; 1/3 leaves 1. Ask: “Which remainder ended the first division, and which one keeps the second going?” |
| 15 min | Remainder clocks Pairs take the 7 clock: from 1, draw an arrow to the remainder left after dividing 10 by 7, and so on round. They meet every remainder except 0 before returning to 1: six steps, six digits. Ask: “On the 13 clock, the path from 1 misses some numbers. Where do they go?” |
| 35 min | Divide, decide, measure Parts A to C; early finishers do Part D. Share Part C, lining the blocks up on the board by length. Ask: “Can a fraction with denominator 11 have a block longer than 10 digits? Why not?” |
Two shorter sessions instead? Split after the clocks; start the second with Part A.
Watch for these ideas
- Stopping after a few digits and calling the decimal terminating: only a remainder of 0 ends it.
- Thinking the repeat starts at the first digit: 5/6 = 0.8333333..., where only the 3 repeats.
- Losing track of remainders: writing them down is what shows the repeat.
Answers
- Parts A and B: 3/8 = 0.375, remainders 6, 4, 0, terminates; 2/3 = 0.666666..., remainders 2, recurs; 5/11 = 0.454545..., remainders 6, 5, recurs; 7/20 = 0.35, remainders 10, 0, terminates; 5/6 = 0.8333333..., remainders 2, 2, recurs; 9/16 = 0.5625, remainders 10, 4, 8, 0, terminates.
- Part C: 1/9: block 1, 1 digit; 1/11: block 09, 2 digits; 1/27: block 037, 3 digits; 1/13: block 076923, 6 digits.
- Part D: dividing by d leaves only the remainders 1 to d − 1 (0 would stop it), so within d − 1 steps one comes back and the block has at most d − 1 digits. 1/7 uses all six; 1/13 uses six of twelve: the 13 clock splits into two rings (1, 10, 9, 12, 3, 4 and 2, 7, 5, 11, 6, 8).
Two ways a fraction can turn out
A. Divide it out, keeping every remainder
| Fraction | Remainders in order | Decimal | B. Terminates or recurs? |
|---|---|---|---|
| 3/8 | |||
| 2/3 | |||
| 5/11 | |||
| 7/20 | |||
| 5/6 | |||
| 9/16 |
For each recurring one, ring the remainder that came back.
C. How long is the block?
Divide until a remainder comes back. Write the repeating block and count its digits.
| Fraction | Working | Block | Digits |
|---|---|---|---|
| 1/9 | |||
| 1/11 | |||
| 1/27 | |||
| 1/13 |
D. Stretch: how long can a block be?
When you divide by 7, which remainders are possible? Use that to explain why the block of 1/7 can have at most six digits. The block of 1/13 has only six digits, not twelve: use your 13 clock to say why.
Terminating decimals
A fraction terminates when it can be rewritten over 10, 100, 1000 and so on. Students scale fractions up to a power of ten, turn terminating decimals back into fractions, and see why halves of halves always stop.
We are learning to
- rewrite a fraction over a power of ten and read off its decimal,
- write a terminating decimal as a fraction in lowest terms,
- decide whether a denominator can be scaled up to a power of ten.
Success criteria
- I can show that 3/8 = 375/1000 = 0.375 without long division.
- I can explain why no whole number times 3 gives 10, 100 or 1000.
You need
The worksheet, one per student; calculators for checking Part A only.
Lesson flow (about 60 minutes)
| 10 min | Long division, then a short cut Show the board (see the box). Then rewrite 3/8: 8 × 125 = 1000, so 3/8 = 375/1000 = 0.375, the same digits as the division. Ask: “Why does 8 fit into 1000 exactly? What is 1000 made of?” |
| 15 min | Back to a fraction Model 0.36 = 36/100 = 9/25 on the board, then pairs do Part B and check each answer by dividing. Ask: “How many decimal places tells you which power of ten to write?” |
| 35 min | Scale, convert, decide Parts A to C; early finishers do Part D. Share Part C by writing each denominator as primes. Ask: “12 = 2 × 2 × 3. Which factor stops us reaching a power of ten?” |
Two shorter sessions instead? Split after Part B; start the second with Part C.
Watch for these ideas
- Reading 0.08 as 8/10: two decimal places means hundredths, 8/100 = 2/25.
- Forgetting to simplify: 0.45 = 45/100 = 9/20.
- Thinking every even denominator terminates: 12 and 30 are even but carry a 3.
Answers
- Part A: 7/20 = 35/100 = 0.35; 3/25 = 12/100 = 0.12; 9/40 = 225/1000 = 0.225; 11/16 = 6875/10000 = 0.6875; 13/50 = 26/100 = 0.26.
- Part B: 0.45 = 9/20; 0.125 = 1/8; 2.6 = 13/5; 0.0625 = 1/16; 0.08 = 2/25.
- Part C: 8 = 2 × 2 × 2, yes; 12 = 2 × 2 × 3, no; 25 = 5 × 5, yes; 30 = 2 × 3 × 5, no; 64 = 2 × 2 × 2 × 2 × 2 × 2, yes; 50 = 2 × 5 × 5, yes.
- Part D: 1/2 = 0.5, 1/4 = 0.25, 1/8 = 0.125, 1/16 = 0.0625, 1/32 = 0.03125. Each halving adds one decimal place, because 1/2ⁿ = 5ⁿ/10ⁿ: n places, ending in 5.
Terminating decimals
A. Over a power of ten
Multiply top and bottom so the bottom becomes 10, 100, 1000 or 10 000. Then write the decimal.
| Fraction | Multiply by | Over a power of ten | Decimal |
|---|---|---|---|
| 7/20 | |||
| 3/25 | |||
| 9/40 | |||
| 11/16 | |||
| 13/50 |
B. Back to a fraction
Write each decimal over a power of ten, then simplify to lowest terms.
| 0.45 = | |
| 0.125 = | |
| 2.6 = | |
| 0.0625 = | |
| 0.08 = |
C. Can it reach a power of ten?
Write each denominator as a product of primes. Circle yes if some whole number times it gives 10, 100, 1000...
| 8 | yes · no | |
| 12 | yes · no | |
| 25 | yes · no | |
| 30 | yes · no | |
| 64 | yes · no | |
| 50 | yes · no |
D. Stretch: halves of halves
Write 1/2, 1/4, 1/8, 1/16 and 1/32 as decimals. Count the decimal places each time. Predict 1/64 without dividing, and explain the pattern using 2 × 5 = 10.
Recurring decimals
A recurring decimal repeats a fixed block for ever, so it needs a notation: a bar over the block, or a dot over its first and last digits. Students write, read and compare recurring decimals, and turn simple ones into fractions.
We are learning to
- write a recurring decimal with a bar or dots over exactly the repeating block,
- expand the notation back into digits and compare decimals that look alike,
- explain why 0.3 with a bar is bigger than 0.33.
Success criteria
- I can write 1/6 = 0.16 and say why the bar covers only the 6.
- I can put 0.72, 0.72 and 0.72 in order.
You need
The worksheet, one per student. No calculators: the point is the digits a screen cannot show.
Lesson flow (about 60 minutes)
| 10 min | Where does the repeat start? Show the board (see the box). Write 1/6 = 0.1666666... and 2/11 = 0.181818... and agree where each block starts and ends. Ask: “Three dots say “and so on”. Why is that not quite enough for 2/11?” |
| 15 min | Bar and dots Write 1/6 as 0.16 and as 0.16˙, then 2/11 as 0.18 and 0.1˙8˙. Pairs write Part A. Ask: “The dots mark only two digits of 18. How do we know the block is not longer?” |
| 35 min | Read, compare, convert Parts B and C; early finishers do Part D. Share Part C by writing eight digits of each number under the other. Ask: “Two numbers can agree for many digits. Where do you look to tell them apart?” |
Two shorter sessions instead? Split after Part A; start the second with Part B.
Watch for these ideas
- Putting the bar over every digit: 0.1666... is 0.1 with a bar over the 6 only.
- Thinking a bar over 27 means 0.2777...: it means 0.272727...
- Thinking 0.33 is 1/3: it stops; 0.3 with a bar is a little bigger.
Answers
- Part A: 2/3 = 0.6; 5/9 = 0.5; 1/12 = 0.083; 4/11 = 0.36; 3/7 = 0.428571; 7/30 = 0.23.
- Part B: 0.8 = 0.88888888...; 0.27 = 0.27777777...; 0.27 = 0.27272727...; 1.405 = 1.40540540....
- Part C: 0.3 > 0.33; 0.16 > 0.16; 0.45 < 0.45.
- Part D: 0.4 = 4/9 (check: 4/9 = 0.444444...); 0.27 = 3/11 (check: 3/11 = 0.272727...). A block of one digit sits over 9, a block of two over 99.
Recurring decimals
A. Write it with a bar
Divide each fraction out until the block repeats. Write the decimal with a bar over the block only.
| Fraction | Working | With a bar |
|---|---|---|
| 2/3 | ||
| 5/9 | ||
| 1/12 | ||
| 4/11 | ||
| 3/7 | ||
| 7/30 |
B. Read the notation
Write each decimal to eight decimal places.
| 0.8 = | |
| 0.27 = | |
| 0.27 = | |
| 1.405 = |
C. Which is bigger?
Write both to eight decimal places, one under the other. Then put > or < between them.
| Pair | Eight decimal places each |
|---|---|
| 0.3 ___ 0.33 | |
| 0.16 ___ 0.16 | |
| 0.45 ___ 0.45 |
D. Stretch: back to a fraction
Work out 1/9, 2/9 and 1/99, 2/99 as decimals. Use the pattern to write 0.4 and 0.27 as fractions in lowest terms, then check by dividing.
The denominator test
In lowest terms, a denominator made only of 2s and 5s gives a terminating decimal; any other prime makes it recur. Students predict before dividing, learn to simplify first, sort 24 cards and count decimal places.
We are learning to
- write a denominator as a product of primes and predict whether the decimal terminates or recurs,
- simplify a fraction before testing, because a shared factor can hide the true denominator,
- predict how many decimal places a terminating decimal will have.
Success criteria
- I can say whether 1/20 and 5/12 terminate or recur, and why, without dividing.
- I can explain why 3/12 terminates although 12 has a factor of 3.
You need
The worksheet; the sort cards (cut-out sheet 2), one set per pair; calculators for checking Part A.
Lesson flow (about 60 minutes)
| 10 min | Predict, then check Show the board (see the box). Then test 1/20 (20 = 2 × 2 × 5: terminates) and 5/12 (12 = 2 × 2 × 3: recurs) on the board. Ask: “Why do 2 and 5 behave differently from 3 or 7?” |
| 20 min | Sort the cards Pairs sort the 24 cards into terminates and recurs, simplifying first and writing the true denominator on the back. Pairs compare; argue out 21/28, 6/24 and 18/48 as a class. Ask: “Which cards changed pile once you simplified them?” |
| 30 min | Test and count Parts A to C; early finishers do Part D. Share Part B, writing the fraction before and after simplifying. Ask: “How can you tell from 40 = 2 × 2 × 2 × 5 that 7/40 has three decimal places?” |
Two shorter sessions instead? Split after the sort; start the second with Part A.
Watch for these ideas
- Testing before simplifying: 21/35 = 3/5 terminates, though 35 has a 7.
- Testing the numerator: only the denominator decides.
- Thinking a big denominator means recurring: 7/125 = 0.056 terminates.
Answers
- Part A: 3/16: 2 × 2 × 2 × 2, terminates, 0.1875; 5/24: 2 × 2 × 2 × 3, recurs, 0.208333333...; 4/15: 3 × 5, recurs, 0.2666666...; 7/125: 5 × 5 × 5, terminates, 0.056; 11/32: 2 × 2 × 2 × 2 × 2, terminates, 0.34375; 9/14: 2 × 7, recurs, 0.6428571428571....
- Part B: 3/12 = 1/4 = 0.25; 21/35 = 3/5 = 0.6; 6/15 = 2/5 = 0.4; 14/28 = 1/2 = 0.5; 9/12 = 3/4 = 0.75: all terminate.
- Part C (sort): terminates 3/20, 21/28, 7/8, 3/15, 9/25, 6/24, 11/40, 3/30, 17/50, 13/32, 18/48, 3/75, 5/16; recurs 1/6, 2/9, 5/7, 4/21, 1/3, 2/11, 7/12, 8/9, 1/18, 5/14, 7/15.
- Part D: 7/40: 2 × 2 × 2 × 5, 3 places, 0.175; 3/64: 2 × 2 × 2 × 2 × 2 × 2, 6 places, 0.046875; 1/80: 2 × 2 × 2 × 2 × 5, 4 places, 0.0125. The places are the larger count of 2s or 5s.
The denominator test
A. Predict, then check
Each fraction is in lowest terms. Factorise the denominator, predict, then divide (or use a calculator) to check.
| Fraction | Denominator as primes | Prediction | Decimal |
|---|---|---|---|
| 3/16 | |||
| 5/24 | |||
| 4/15 | |||
| 7/125 | |||
| 11/32 | |||
| 9/14 |
B. Simplify first
Each denominator has a prime other than 2 or 5. Simplify, test again, and write the decimal.
| 3/12 | |
| 21/35 | |
| 6/15 | |
| 14/28 | |
| 9/12 |
C. Card sort record
After sorting your cards, list three that changed pile when you simplified them, with their true denominators.
| 1. | |
| 2. | |
| 3. |
D. Stretch: how many places?
Without dividing, predict how many decimal places 7/40, 3/64, 1/80 will have. Then check. What rule links the number of 2s and 5s to the number of places?
Checking with a digital tool
A calculator shows only so many digits and rounds the last one. Students compare screens with exact decimals, settle what a screen cannot show with the denominator test, then play the quiz.
We are learning to
- read a calculator screen as a rounded value, not always the whole decimal,
- decide when a screen cannot show whether a decimal terminates or recurs,
- use the denominator test to settle what the screen cannot.
Success criteria
- I can explain why a screen shows 0.1428571 for 1/7, and what it hides.
- I can name a screen that cannot show a repeat, and check it another way.
You need
The worksheet, one per student; a calculator each (8-digit screens show 7 decimal places; note if yours shows more).
Lesson flow (about 60 minutes)
| 10 min | What the screen shows Show the board (see the box). Then work 1/7 by hand to eight places, 0.142857142857..., beside the screen’s 0.1428571. Ask: “The screen’s last digit is a 1, but the next true digit is a 4. What has the screen done?” |
| 30 min | Screens on trial Parts A to C; early finishers do Part D. Share Part B: the screens look alike, the denominators do not. Ask: “Which tells you more about 1/81: the screen or the denominator?” |
| 20 min | Self-check quiz as a class game Teams of four hold up one, two or three fingers for their option, then reveal and read the explanation aloud. A team that was wrong says what fooled it. Ask: “Before we reveal it: what would the denominator test say?” |
Two shorter sessions instead? Split after Part B; start the second with Part C and the quiz.
Watch for these ideas
- Trusting the last digit: 2/3 shows 0.6666667, but no 7 ever appears in 2/3.
- Thinking a screen that fills up means recurring: 1/1024 fills the screen and terminates.
- Thinking a screen that stops early means terminating: check the denominator.
Answers
- Part A: 2/3: screen 0.6666667, exact 0.666666...; 5/8: screen 0.625, exact 0.625 (the same); 4/7: screen 0.5714286, exact 0.571428571428...; 1/32: screen 0.03125, exact 0.03125 (the same); 5/6: screen 0.8333333, exact 0.8333333....
- Part B: 1/1024: screen 0.0009766; 1024 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2, so it terminates (0.0009765625); 1/81: screen 0.0123457; 81 = 3 × 3 × 3 × 3, so it recurs (0.012345679012345679...); 3/256: screen 0.0117188; 256 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2, so it terminates (0.01171875).
- Part C: the screen alone settles none of Part B; the denominator settles all of them.
- Part D: 1/17 = 0.0588235294117647 repeating, a 16-digit block, longer than two screens: divide by hand until a remainder comes back.
Checking with a digital tool
A. Screen against exact
Write what your calculator shows, then the exact decimal (with a bar if it recurs). Ring the screens that rounded.
| Fraction | Screen shows | Exact decimal |
|---|---|---|
| 2/3 | ||
| 5/8 | ||
| 4/7 | ||
| 1/32 | ||
| 5/6 |
B. Screens that cannot tell
Each of these fills the screen. From the screen alone, guess: terminates or recurs? Then use the denominator test.
| Fraction | Screen | Guess | Denominator test |
|---|---|---|---|
| 1/1024 | |||
| 1/81 | |||
| 3/256 |
C. Who settles it?
For Part B, which told you more, the screen or the denominator? Write two sentences a younger student could follow.
D. Stretch: a block longer than the screen
1/17 repeats, but its block does not fit on a screen. Divide 1 by 17 by hand until a remainder comes back. How long is the block? Could a calculator ever show you all of it at once?
Remainder clocks
For Day 1, one pair of clocks per pair
Start at 1. Divide 10 by the clock’s number: the remainder is where your first arrow goes. From there, multiply by 10 and divide again, drawing an arrow each time, until you come back to a number you have visited. Each arrow is one digit of the decimal.
Teacher note: from 1 the 7 clock runs 1 → 3 → 2 → 6 → 4 → 5 → 1 (all six, so 1/7 has a six-digit block); the 13 clock runs 1 → 10 → 9 → 12 → 3 → 4 → 1 and never meets 2, 7, 5, 11, 6, 8, which form a second ring (the remainders of 2/13).
Terminate or recur? Sort cards
For Day 4, one set per pair: 24 cards
Simplify each fraction first and write its true denominator on the back. Then sort: terminates or recurs.
Teacher note: 13 terminate and 11 recur. The full lists are in the Day 4 answers.
What we know: Terminating and recurring decimals
Work on your own. Use a calculator only for Question 7. Show your working.
- Divide 5/8 out, writing each remainder. Does it terminate or recur? ______________________
- Divide 4/9 out. Which remainder comes back, and what is the decimal? ______________________
- Write 0.35 as a fraction in lowest terms. ______________________
- Without dividing, say whether each terminates or recurs, and why: 7/16, 2/15, 9/30. ______________________
- Write 5/12 as a decimal with a bar over the repeating block. ______________________
- Write 0.18 to six decimal places. Is it bigger or smaller than 0.18? ______________________
- A calculator shows 0.4285714 for 3/7. Is that the exact decimal? Explain. ______________________
- Ali says 7/28 must recur because 28 = 2 × 2 × 7. Is Ali right? ______________________
How did this week feel? Circle one: I can teach it · I am getting there · I need more help
Answers and marking guide
Answers with working
- Remainders 2, 4, 0: it reaches 0, so it terminates: 5/8 = 0.625.
- The remainder 4 comes back every step: 4/9 = 0.444444..., a recurring decimal.
- 0.35 = 35/100 = 7/20.
- 7/16: 2 × 2 × 2 × 2, terminates; 2/15: 3 × 5, recurs; 9/30 = 3/10: 2 × 5, terminates.
- 5/12 = 0.41666666... = 0.416, with the bar over the 6 only.
- 0.188888..., which is bigger than 0.18 from the third decimal place.
- No: 3/7 = 0.428571428571... recurs for ever; the screen rounds after seven decimal places (7 has no 2s or 5s).
- No: 7/28 = 1/4 in lowest terms, so it terminates: 0.25. Simplify before testing.
A quick three-level guide
| Idea | Working towards | At standard | Beyond |
|---|---|---|---|
| Dividing out (Q1, Q2) | stops after a few digits | divides to a remainder of 0 or a repeated remainder | explains the repeat from the remainders |
| Notation (Q5, Q6) | bars every digit, or misreads a bar | bars exactly the block and expands bar notation | compares decimals that agree for many digits |
| The denominator test (Q4, Q8) | tests the numerator, or skips simplifying | simplifies, factorises and predicts correctly | predicts the number of decimal places |
| Terminating decimals and screens (Q3, Q7) | trusts the last digit on a screen | converts terminating decimals to fractions and reads screens as rounded | settles any screen with the denominator test |
Eight questions, four ideas. A student at standard answers most questions with working and always simplifies before testing a denominator.
Weekly class record
Jot a tick as you move around the room; the mini-check fills any gaps. A tick a day is plenty.
| Name | Remainders | Terminating | Notation | Denominator test | Digital tools |
|---|---|---|---|---|---|
The five columns are the five days: remainders, terminating decimals, notation, the denominator test and digital tools.