From Examples to a General Argument · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

From a few cases to every case

Mathematics · Mathematical Thinking · ages 12-13
Name ______________________   Date ____________

  1. A pattern suggests the nth figure uses 3n + 1 sticks. How many sticks does the 10th figure use?

    Answer: ______________

  2. Which sentence claims something about every even number?

    • 3 + 5 = 8, so that one single sum is even
    • Every even is 2 times an integer
    • 6 is even, checked this once
    • 2 + 4 = 6, checked this once

  3. A pattern suggests the nth figure uses 3n + 1 sticks. How many sticks does the 10th figure use?

    Answer: ______________

  4. Why does testing five cases NOT settle a claim about all numbers?

    • Five cases leave out every other number
    • Five cases take too long to check
    • Five cases always contain an error

  5. Spotting that 3 + 5 = 8 and 7 + 9 = 16, which general claim is worth testing?

    • The sum of two numbers is always even
    • Odd numbers are even
    • 3 + 5 always equals 8
    • The sum of two odd numbers is even

  6. Priya turns a pattern into the rule 2n and checks n = 6 gives 12, matching the figure. This one check proves the rule.

    Circle one:   True   False

  7. The claim is that the first n odd numbers total n squared. Check n = 5: what is 1 + 3 + 5 + 7 + 9?

    Answer: ______________

  8. Spotting that 3 + 5 = 8 and 7 + 9 = 16, which general claim is worth testing?

    • The sum of two numbers is always even
    • 3 + 5 always equals 8
    • The sum of two odd numbers is even

  9. Which is a correct short argument that the sum of two even numbers is even?

    • Even numbers are lucky, so their sum is even
    • 2a + 2b = 2(a + b), which is even
    • Examples like 2 + 4 = 6 settle every case

  10. The nth triangular number is n(n + 1) / 2. What is the 8th triangular number?

    Answer: ______________

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Answer key

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

From a few cases to every case W1-mt_vm7cigeRtg-s1

  1. 31 · Substitute n = 10: 3 times 10 + 1 is 31.
  2. Every even is 2 times an integer · Only the 2-times sentence covers every even number; the rest check one sum only.
  3. 31 · Substitute n = 10: 3 x 10 + 1 = 31.
  4. Five cases leave out every other number · Five examples cover five numbers, while the claim covers infinitely many.
  5. The sum of two odd numbers is even · Both examples add two odds to make an even, so the odd-pair claim fits.
  6. False · One fresh case supports the rule but leaves the rest untested.
  7. 25 · Adding gives 25, which equals 5 squared, so the fresh case matches.
  8. The sum of two odd numbers is even · Both examples add two odds to make an even, so the odd-pair claim fits.
  9. 2a + 2b = 2(a + b), which is even · Writing evens as 2a and 2b shows the sum is 2 times something, hence even.
  10. 36 · Substitute n = 8: 8 x 9 / 2 = 36.
Worksheet · LightMySky