Decimals and fractions (age 11+)

Activities to try

1. 1

   Scale to a power of ten — does it land?

   Why it works: A fraction in lowest terms terminates exactly when its denominator is built only from the primes 2 and/or 5 — because those are the primes of ten, so the denominator can be scaled to a power of ten. Any other prime factor forces a recurring decimal.

   1. Take 1/8 and try to turn it into tenths, then hundredths, then thousandths: find the multiplier (×125) that makes the denominator 1000, giving 125/1000 = 0.125; confirm it 'lands' exactly.
   2. Now try 1/3 the same way: aim for 10, 100, 1000… and find that no power of ten works; carry out the division 1 ÷ 3 and watch the 3s continue (0.333…), a recurring decimal that never stops.
   3. Write each denominator as a product of primes side by side: 8 = 2×2×2; 6 = 2×3; 3 = 3. Mark which denominators are built only from 2s and 5s.
   4. State the rule you discover in your own words, then test it on 1/20, 1/40 (should terminate) and 1/6, 1/7 (should recur), factorising each denominator first.
2. 2

   Read a decimal by its last place value

   Why it works: A terminating decimal IS a fraction with a power-of-ten denominator; reading the last place value gives that denominator, and simplifying gives lowest terms. Fraction→decimal is division (numerator ÷ denominator), equivalently scaling to a power of ten.

   1. Name the place value of the LAST digit of 0.375 (thousandths), so the starting denominator is 1000: write 0.375 = 375/1000.
   2. Simplify stepwise to lowest terms: 375/1000 = 3/8 (divide both by 125). Check the simplified fraction against the original decimal.
   3. Repeat with other decimals (0.2 = 2/10 = 1/5; 0.25 = 25/100 = 1/4; 0.35 = 35/100 = 7/20), each time naming the last place value first.
   4. Reverse the direction: convert 3/8 to a decimal two ways — divide 3 ÷ 8, and scale 3/8 to 375/1000 — and confirm both give 0.375.
3. 3

   Sort, match and fill the gaps (consolidation)

   Why it works: Bidirectional conversion to simplest form becomes fluent, and the terminate/recur judgement is made from the denominator's prime factors rather than by trial division.

   1. Match terminating decimals to their simplest fractions: 0.375 ↔ 3/8, 0.2 ↔ 1/5, 0.25 ↔ 1/4.
   2. Sort a set of fractions into two columns, 'terminates' and 'recurs', by inspecting each denominator's prime factors (not by dividing): e.g. 7/40 and 1/20 terminate; 5/12 and 1/6 recur.
   3. Fill in the chain 0.375 = __/1000 = __/8, then explain each step (last place value → denominator, then the ÷125 simplification).
   4. Transfer to unfamiliar fractions: predict whether 9/25, 4/15 and 6/15 terminate, justify from the prime factors of each denominator in lowest terms (6/15 = 2/5), then check by converting (9/25 = 0.36; 4/15 = 0.2666…; 6/15 = 0.4).