Write the fraction in simplest form and break the denominator into primes. If its only prime factors are 2 and 5, the decimal terminates; any other prime factor makes it recur. It works because 10 = 2 × 5.
Without dividing anything, you can tell that 7/40 stops and 7/30 goes on forever. The secret is in the denominator's prime factors.
A terminating decimal is a fraction over 10, 100, 1000…: 0.175=1000175. Since 10=2×5, those denominators are made only of 2s and 5s. So:
Write the fraction in simplest form.
Break the denominator into prime factors.
Only 2s and 5s: it terminates. Any other prime (3, 7, 11…): it recurs.
407: 40=2×2×2×5, so it terminates (0.175). 307: 30=2×3×5 has a 3, so it recurs (0.2333…).
Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.
Make your prediction first. Help opens once you've locked it in.
PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Eleven fiftieths
Break 50 into primes.
TryTry it on the board and compare the result with your prediction.
Test the rule
Try it yourselfYour edits are saved in this browser.
Set 11/50 and check your prediction. Then try 11/30.
Use the board. This ticks when you get there.
InvestigateWork out why it happens, one step at a time.
Why 2s and 5s
Read Max's working. What is 3/12 in simplest form?
Max decides whether 3/12 terminates. Click the line where his reasoning goes wrong.
Terminate or recur?
1/16
3/20
1/12
2/15
7/25
6/156/15 = 2/5: simplify first.
ModifyChange the problem on purpose and see what happens.
Find one that stops
Need a hint?
15 = 3 × 5.
Another hint
Choose a numerator that cancels the 3.
Need a hint?
24 = 2 × 2 × 2 × 3: cancel the 3.
Try it yourselfYour edits are saved in this browser.
Find a fraction with denominator 15 that terminates.
Use the board. This ticks when you get there.
Find a fraction with denominator 24 that terminates.
Use the board. This ticks when you get there.
PractiseA few on your own, to check you have it.
Which stop
1/16 terminates. How many decimal places does it have?
Review: how many digits are in the repeating block of 1/9?
MakeMake your own problem using what you have learned. Expect a few tries; that is normal.
Sixtieths
Then explain, using primes, why your fraction stops and 1/60 does not.
Try it yourselfYour edits are saved in this browser.
Need a hint?
60 = 2 × 2 × 3 × 5.
Another hint
The numerator must cancel the 3.
Find a fraction with denominator 60 that terminates.
Use the board. This ticks when you get there.
For teachers
Students predict whether a fraction's decimal terminates from the prime factors of its simplified denominator, and check on the long-division board (AC9M8N03, node 2). About 35 minutes.
I can predict whether a fraction terminates without dividing.
I can explain the rule using 10 = 2 × 5.
I can simplify a fraction before using the rule.
Before class
Do the Run board yourself first. Students need prime factors (Number Foundations) and the previous lesson.
The one idea: in simplest form, a denominator made only of 2s and 5s gives a terminating decimal, because terminating decimals are fractions over powers of 10 = 2 × 5. The common mistake is skipping the simplifying (Max's mistake).
The board shows the simplest form and the prime factors with the long division.
Wider: in base 12 (12 = 2 × 2 × 3), thirds would terminate and fifths would recur.
Running the lesson
Predict: ask what 40 is made of.
Run: ask for a denominator under 30 that is all 2s and 5s.
Investigate: ask Max to simplify.
Modify: ask which numerators work for denominator 15, and why.
Make: ask for the reason in primes.
The hard part (on purpose)
Simplifying first. Ask students to write the fraction in simplest form before they say anything about its denominator.
When students get stuck
A student judges by the denominator's size or digits. Ask: What are its prime factors? Fix: Only the primes matter: 2s and 5s stop.
A student does not simplify first. Ask: Is the fraction in simplest form? Fix: Cancel common factors, then look at the denominator.
A student thinks 8 = 2 × 4 counts as other primes. Ask: Is 4 a prime? Fix: Break it down to primes: 8 = 2 × 2 × 2.
Stretch and support
Stretch: In base 12, which unit fractions from 1/2 to 1/12 would terminate? Why?
Support: A list of primes to 20 and factor trees for the denominators.
Where it leads
This completes AC9M8N03.
Where it leads: irrational numbers (next unit), whose decimals never stop and never repeat.
What each step is for
eleven fiftieths: Committing to whether 11/50 stops, without dividing, shows whether they look at the denominator's factors or its size.
test the rule: Trying many denominators with their prime factors shown connects 2s and 5s to decimals that stop.
why 2s and 5s: Terminating decimals are fractions over powers of 10, made of 2s and 5s; simplify first (Max does not); sorting fractions applies the rule.
find one that stops: Choosing numerators that cancel the bad primes shows simplifying decides the answer.
which stop: The rule with simplifying, a count of decimal places, and a review of block lengths, mixed; not simplifying first is the planted slip.
sixtieths: Finding a terminating fraction over 60 combines simplifying and the prime rule.
Now you can
Predict whether a fraction terminates without dividing.