Proving or disproving a statement | Maths | Quizalize & Oak National Academy

Starter quiz

Which of these statements are always true?

An odd number subtract an odd number is an odd number.

An even number subtract an even number is an even number. ✓

An even number multiplied by an integer is an even number. ✓

An integer squared is an even number.
Which of these is a general form for any odd number (where $n$ is an integer)?

$n$

$n + 1$

$2n$

$2n+1$ ✓

$4n + 1$
The diagram shows a right-angled triangle. Which of these is an expression for the area?

$ab$

$abc$

$\frac{bc}{a}$

$\frac{ab}{2}$ ✓

$a + b + c$
The diagram shows a right-angled triangle. Which of these formula show the correct relationship between the lengths of the sides?

$a + b = c$

$ab=c$

$\frac{ab}{2}= c$

$2a + 2b = c$

$a^2 + b^2 = c^2$ ✓
Which of these shows the first 4 numbers of the form $9^n$ where $n$ is a positive integer?

1, 9, 17, 26

1, 9, 81, 729

9, 81, 109, 181

9, 81, 512, 4608

9, 81, 729, 6561 ✓
Andeep has written out the first 4 numbers of the form $9^n$ where $n$ is a positive integer. Which of these conjectures hold true for these 4 values?

Numbers of this form end in either 1 or 9 ✓

Numbers of the form $9^{n + 1}$ always end in 1

Numbers of the form $9^{2n}$ always end in 9

Numbers of the form $9^{2n-1}$ always end in 9 ✓

Numbers of the form $9^{5n}$ always end in 9

To __________ is to formulate a statement or rule that applies correctly to all relevant cases.

conjecture

estimate

generalise ✓

prove
Why is testing a conjecture often not a good way to prove it is true?

A counterexample may be found.

It is often easy to make a mistake when substituting values or measuring.

It is often impossible to test all relevant cases. ✓

Proofs are more convincing if they have diagrams or practical demonstrations.
Laura wants to prove Pythagoras' theorem. Her first few steps are shown. Which of these is the correct next step of her proof?

$a^2 + b^2 = \frac{4ab}{2} + c^2$

$a^2 + b^2 = c^2$

$a^2 + ab + ab + b^2 = 2ab + c^2$ ✓

$2a + 2b = 2ab + c^2$

where $a, b, c$ are all integers.
If $p$ is a prime number and $n$ is a positive integer, which is a counterexample to the conjecture " $p^n$ is always odd"?

$p = 2$ and $n = 3$ ✓

$p = 3$ and $n = 4$

$p = 5$ and $n = -2$

$p = 10$ and $n = 1$

$p = 13$ and $n = 5$
Which of these is a counterexample to the conjecture "The digit sum of any number of the form $9n$ where $n$ is a positive integer is always 9"?

$n = 0$

$n = 2$

$n = 8$

$n = 15$

$n = 21$ ✓
Aisha writes the conjecture "For all integer values of $x, (x-2)^2 > x-2$ ". There is a counterexample to this when $x=$ ______.

'2' ✓

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A demonstration is a proof.

A demonstration shows that it works for that one specific case. A proof allows us to know all cases that are true.

Conjecture - A conjecture is a (mathematical) statement that is thought to be true but has not been proved yet.
Generalise - To generalise is to formulate a statement or rule that applies correctly to all relevant cases.