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Number line

Where does 3/7 sit among the ticks 0, 0.25, 0.5, 0.75, 1? Is 5/9 or 4/7 nearer to ½? Place fractions and decimals, with denominators that keep growing.

  • Number sense
  • Timed
  • Endless levels

How it plays

A fraction or a decimal, and a line with labelled ticks: pick the two ticks it falls between. The ticks may be fractions when the number is a decimal, or decimals when it is a fraction — the conversion is the point.

Or two fractions: say which is nearer to one half. No common denominator is needed: the distance of a/b from ½ is |2a − b| / 2b.

Levels raise the denominators without end, bring in finer and stranger ticks (fifths, eighths, thirds, a line from 0 to 3), and shorten the time. The working puts everything over one denominator.

Keys

Space starts and skips ahead · 1–9 pick · Enter submits · Esc stops. On a phone, tap.

What it trains

This trains placing fractions and decimals by size. It is practice at this, and claims nothing wider.

Learn

How to get better at number line: the method, the techniques, what each new level asks of you, and the traps the wrong answers are made of.

Your approach to each question

  1. Read the ticks first and count how many steps make one whole: 0 · 0.25 · 0.5 · 0.75 · 1 is four to a unit; 0 · 1/12 · 1/6 … 1 is twelve.
  2. Multiply the number by that count. The whole part of the result is how many steps along it sits.
  3. Count that many steps from 0 along the ticks: the number lies between that tick and the next one.
  4. For “nearer to ½”, skip the line altogether: work out each fraction’s distance from ½ and compare the two distances.

Techniques and tricks

  • Multiply by the ticks per unit

    If the ticks are steps of 1/m, multiply the number by m. Between k and k + 1 means between the k-th tick and the next, counting from 0.

    e.g. 3/7 on quarters: 3/7 × 4 = 12/7 = 1.71, so it is between tick 1 and tick 2 — 0.25 to 0.5

  • Decimals the same way

    A decimal is multiplied just like a fraction, and the ticks can be fractions when the number is a decimal.

    e.g. 0.62 on eighths: 0.62 × 8 = 4.96, so between 4/8 and 5/8 — written 1/2 to 5/8

  • Distance from ½ is |2a − b| over 2b

    a/b − 1/2 = (2a − b)/2b. Double the top, take away the bottom, ignore the sign; that over twice the bottom is the distance.

    e.g. 5/9: |10 − 9| = 1, so 1/18. 4/7: |8 − 7| = 1, so 1/14. 1/18 is smaller, so 5/9 is nearer.

  • Same gap, bigger bottom wins

    When two fractions give the same |2a − b|, the one with the bigger bottom is nearer to ½. Only when the gaps differ do you need to cross-multiply.

    e.g. 6/11 or 4/7: both have a gap of 1, and 11 is bigger than 7, so 6/11 is nearer

  • Benchmarks for the odd ticks

    Know the awkward ticks as decimals: 1/3 ≈ 0.333, 2/3 ≈ 0.667, 1/6 ≈ 0.167, 5/6 ≈ 0.833, 1/8 = 0.125, 3/8 = 0.375, 1/12 ≈ 0.083.

    e.g. 0.35 on thirds: 0.35 > 0.333, so it is between 1/3 and 2/3

  • Whole numbers first on longer lines

    On a line to 2 or 3, split off the whole part, place it, then place the rest within that unit.

    e.g. 7/3 = 2⅓, on halves to 3: past 2, and ⅓ is under ½, so between 2 and 5/2

As the levels climb

  • Three things rotate as you climb: the largest denominator grows (9, 12, 16, 22 and on without end), a new pair of tick sets arrives, and the clock shrinks. Expect a dip each time new ticks appear.
  • The tick sets arrive in pairs: halves and quarters as decimals; then tenths, and quarters as fractions; then fifths as decimals and thirds; then eighths, and a line from 0 to 2; then sixths, and a line from 0 to 3; twelfths last. The newest set is asked about half the time while it is new.
  • Decimals gain places as the denominators grow, up to eight. Multiply the first two or three places by the ticks per unit, and look at further places only when the result lands right beside a whole number.
  • The clock closes to 62% of its start: 5.5 seconds plus 0.9 for each option shown when placing a number, 7 seconds for the ½ question. With twelve ticks to a unit there is no time to count along the line; multiply.

Traps to avoid

  • One segment out. The neighbouring segments are the first wrong options, and a common cause is counting the ticks from 1 instead of from 0. Tick 0 is the start; the k-th step ends at tick k.
  • The mirror segment. 1/8 to 1/4 offered for 3/4 to 7/8 is the line read from the wrong end, or a/b confused with 1 − a/b.
  • A decimal taken for the fraction it resembles. 0.3 is not 1/3; it is just below it. 0.6 is below 2/3.
  • Picking the bigger or the smaller fraction for “nearer to ½”. Distance is all that counts: 2/3 is nearer than 21/22, though 21/22 is bigger.

How to practise

  • Use “Questions” to practise “Which segment it is in” on its own until multiplying by the ticks is a reflex, then add “Which is nearer to ½”.
  • Raise “Options shown” towards 8 to make every neighbouring segment an option. Then only the exact segment will do, and the habit of counting from 0 gets tested every time.
  • When you miss, read the working: it puts the number and both ticks over one denominator. Do that once yourself, then go back to multiplying for speed.

Levels

Each question has a clock that comes from the work it asks for. There is no last level: at least one dial turns without end. Your level is judged on your last ten answers, every ten: 90% moves you up, below 80% moves you down, which holds most people near four in five right.

All of it is in the settings beside Start — how many answers a judgement looks at and how often it recalibrates, the thresholds, the time, which questions, how many options, and whether you see each verdict at once or review them together at the judgement. You can also jump to any level.

More number sense

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