Generalising with repeated reasoning · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Finding the rule that always works

Mathematics · Mathematical Thinking · ages 10-11
Name ______________________   Date ____________

  1. A sequence starts 5, 9, 13, 17. What is the next term?

    Answer: ______________

  2. Triangle angles total 180, quadrilaterals 360, pentagons 540. What comes next for a hexagon?

    • 600
    • 660
    • 720
    • 540

  3. A triangle's angles sum to 180° and a quadrilateral's sum to 360°. Which rule fits the pattern for any polygon with n sides?

    3→180°4→360°5→540°
    • n × 180°, one set per side
    • (n − 2) × 180°
    • n + 180°, plus a single triangle

  4. Dividing any whole number by 5 always gives a fraction with denominator 5, like 3 ÷ 5 = 3/5.

    Circle one:   True   False

  5. A linear sequence has rule 4n + 1. What is its 20th term?

    Answer: ______________

  6. Zoe divides 9 by 4 and writes the answer as a fraction. What is the denominator?

    • 4
    • 9
    • 36

  7. What is the interior angle sum of a hexagon in degrees?

    • 540
    • 360
    • 900
    • 720

  8. Pentagon angles total 540. Ana predicts the hexagon total is 720, since each extra side adds 180. Is Ana right?

    Circle one:   True   False

  9. Tom's sequence is 2, 5, 8, 11. He says the rule is 'multiply position by 3', but that gives 3 for term 1, not 2. What went wrong?

    • Fix it to 3n − 1 and check it on two terms
    • The rule is fine; the sequence 2, 5, 8, 11 is wrong
    • Nothing is wrong here at all

  10. Dividing 1/7 by 3 gives 1/21. Dividing 1/7 by 5 gives 1/35. What does dividing 1/7 by 4 give?

    • 4/7
    • 1/28
    • 1/11
    • 7/4
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Answer key

For grown-ups. Fold this page away before handing over the rest.

Finding the rule that always works W1-mt_iyovKgZC1q-s1

  1. 21 · The next term is 21 because each step adds 4: 17 + 4 = 21.
  2. 720 · 720 is next because each extra side adds 180: 540 + 180 = 720.
  3. (n − 2) × 180° · Triangles (3 sides) sum to 180°, quadrilaterals (4 sides) to 360°. Each extra side adds another 180°, so the rule is (n − 2) × 180°.
  4. True · Dividing by 5 always puts 5 on the bottom of the fraction, whatever whole number you start with.
  5. 81 · The 20th term is 81 because 4 times 20 plus 1 equals 81.
  6. 4 · Dividing by 4 always makes 4 the denominator, so 9 ÷ 4 = 9/4.
  7. 720 · 720 degrees is correct because (6 - 2) times 180 = 720.
  8. True · Ana is right. The steady step of 180 carries the pattern from 540 to 720.
  9. Fix it to 3n − 1 and check it on two terms · A rule has to work for every term you check, including the first one. 3n − 1 gives 2, 5, 8, 11, which matches the sequence.
  10. 1/28 · 1/28 is correct because the divisor multiplies the bottom: 7 times 4 = 28.
Worksheet · LightMySky