Constructing mathematical arguments · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Building a mathematical argument

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

  1. What is 6 x 0.5?

    Answer: ______________

  2. Which example disproves the claim that multiplying always makes things bigger?

    • 10 x 0.5 = 5
    • 3 x 4 = 12
    • 7 x 2 = 14
    • 9 x 100 = 900

  3. A triangle has two angles of 60 degrees each. What must the third angle be?

    • 50°
    • 60°
    • 70°

  4. One example that works is enough to prove a maths rule is always true.

    Circle one:   True   False

  5. What is 180 - 90 - 30?

    Answer: ______________

  6. Which example disproves the claim 'multiplying two numbers always makes a bigger result'?

    • 10 × 10 = 100
    • 5 × 0.5 = 2.5
    • 6 × 2 = 12

  7. A pupil writes (1/2) divided by 2 = 2/2. What is the error?

    • They simplified wrongly
    • Doubled the top, not the bottom
    • They added the halves
    • The answer 2/2 is actually right

  8. Omar claims multiplying by 2 always makes a number bigger, citing 5 x 2 = 10. Priya counters with (-4) x 2 = -8, which is smaller than -4. Does Priya's counter-example disprove Omar?

    Circle one:   True   False

  9. A student says an angle must be 60° 'because the triangle looks equilateral in the picture.' What is the problem with this argument?

    • Pictures are not proof; angle facts must show it is 60°
    • The student is correct because equilateral triangles always look equal
    • The student needs a protractor, not angle facts, to prove it

  10. Which chain proves the third angle is 60 degrees when the other two are 90 and 30?

    • 90 + 30 = 120, then 360 - 120 = 240
    • 90 x 30 = 2700, then 2700 - 2640 = 60
    • 90 + 30 + 60 = 180, so it is 60
    • 180 - 90 = 90, then 90 - 30 = 60
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Answer key

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

Building a mathematical argument W1-mt_j5YqQnN6xe-s1

  1. 3 · The answer is 3, which is smaller than 6, showing multiplication can shrink.
  2. 10 x 0.5 = 5 · 10 x 0.5 = 5 disproves it because 5 is smaller than the starting 10.
  3. 60° · Angles in a triangle always add to 180°. 180° minus 60° minus 60° leaves 60°.
  4. False · You would need to check every case, or use a general argument. One counter-example, though, is enough to prove a rule is false.
  5. 60 · The answer is 60: 180 - 90 = 90, then 90 - 30 = 60.
  6. 5 × 0.5 = 2.5 · 5 × 0.5 = 2.5 is smaller than 5, so this one case shows the claim is not always true.
  7. Doubled the top, not the bottom · They doubled the top instead of the bottom: the correct working is 1/2 times 1/2 = 1/4.
  8. True · Yes. One genuine counter-example is enough: -8 is smaller than -4, so always is false.
  9. Pictures are not proof; angle facts must show it is 60° · A drawing can be inaccurate. A real proof uses angle facts, like the triangle angle sum, that stay true no matter how the picture looks.
  10. 180 - 90 = 90, then 90 - 30 = 60 · 180 - 90 = 90, then 90 - 30 = 60 proves it: it starts from the 180 total and subtracts exactly the known angles.
Worksheet · LightMySky