Sure, or just sure-looking?

The idea

Observing a property by measuring some figures only gives a conjecture, a statement we are highly confident about but not sure always holds; only a deduction or proof shows it always holds.

Meena measures

Meena draws three squares and measures their diagonals. Each time the two diagonals are equal. She says:

“If all four sides of a quadrilateral are equal, its diagonals are equal.”

Is Meena’s statement certain to be true for every such quadrilateral?

• Yes, she checked three
• We cannot be sure yet
• No, it is surely false

Measure more shapes

Meena’s statement is a conjecture: something we feel very sure about. Tap each shape to measure its diagonals.

A conjecture is not a proof

Observing a property by measuring some figures only gives a conjecture, a statement we are highly confident about but not sure always holds; only a deduction or proof shows it always holds.

Three squares agreed, but the leaning shape with four equal sides broke the conjecture.

A proof, like the SAS argument for rectangles, covers every rectangle. Even the 1000th one.

Notes

Measuring some figures only gives a conjecture. Only a deduction or proof shows that a property always holds.

Check yourself

Ravi draws 10 triangles and measures their angles. Each time the angles add up to about 180°. By measuring alone, what has Ravi got?

Which one proves that the diagonals of a rectangle are equal?

Meena finds one quadrilateral with four equal sides whose diagonals are not equal. What does this show?

Rohan tries n² + n + 41 for n = 1, 2, 3, … 39 and gets 43, 47, 53, 61, …, all prime. He says: “It is always prime.” Test n = 40: what is 40² + 40 + 41?

Answer: 1681

1600 + 40 + 41 = 1681 = 41 × 41, which is not prime. The pattern held 39 times and then failed: measuring or testing many cases is only a conjecture.

• A proof that it is always 180°. Ten triangles are only ten. Measuring cannot show that it holds for every triangle.
• A conjecture — correct. Yes! He can feel very sure, but only a proof can show it always holds.
• A wrong answer. His measurements may well be fine. But they give only a conjecture, not a proof.
• Measuring the diagonals of 100 rectangles. A hundred rectangles are still only a hundred. The 101st could be different, unless we prove it.
• The SAS argument with ∆ADC and ∆DAB — correct. Yes! The argument works for every rectangle, so it is a proof.
• Drawing a very big rectangle and measuring carefully. One rectangle, however big or careful, shows nothing about all rectangles.
• Her measurements must be wrong. The measurements are right. The leaning shape really has unequal diagonals.
• Her statement is still true for most shapes, so keep it. A statement that “always holds” fails even if just one shape breaks it.
• Her statement is false — correct. Yes! One shape that breaks it is enough to show that it does not always hold.