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      Proving or disproving a statement

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      Lesson details

      Learning outcome

      I can appreciate what constitutes the proof of a statement and what is required to disprove.

      Key learning points

      1. Substituting values can reveal whether the conjecture is wrong.
      2. This is referred to as disproving.
      3. A proof requires all possible cases to be considered and accounted for.

      Keywords

      • 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.

      Common misconception

      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.

      Teacher tip

      Give pupils the opportunity to discuss how they remember discovering Pythagoras' theorem, what is the advantage of proving the theorem over demonstrating it?

      Licence

      This content is © Oak National Academy Limited (2025), licensed on Open Government Licence version 3.0
       except where otherwise stated. See Oak's terms & conditions
       (Collection 2).

      Lesson video

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      Prior knowledge starter quiz

      Download starter quiz questions (PDF)

      6 Questions

      Q1.
      Which of these statements are always true?

      An odd number subtract an odd number is an odd number.
      Correct answer: An even number subtract an even number is an even number.
      Correct answer: An even number multiplied by an integer is an even number.
      An integer squared is an even number.

      Q2.
      Which of these is a general form for any odd number (where $$n$$ is an integer)?

      $$n$$
      $$n + 1$$
      $$2n$$
      Correct answer: $$2n+1$$
      $$4n + 1$$

      Q3.
      The diagram shows a right-angled triangle. Which of these is an expression for the area?

      An image in a quiz
      $$ab$$
      $$abc$$
      $$\frac{bc}{a}$$
      Correct answer: $$\frac{ab}{2}$$
      $$a + b + c$$

      Q4.
      The diagram shows a right-angled triangle. Which of these formula show the correct relationship between the lengths of the sides?

      An image in a quiz
      $$a + b = c$$
      $$ab=c$$
      $$\frac{ab}{2}= c$$
      $$2a + 2b = c$$
      Correct answer: $$a^2 + b^2 = c^2$$

      Q5.
      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
      Correct answer: 9, 81, 729, 6561

      Q6.
      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?

      Correct answer: 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
      Correct answer: Numbers of the form $$9^{2n-1}$$ always end in 9
      Numbers of the form $$9^{5n}$$ always end in 9

      Assessment exit quiz

      Download exit quiz questions (PDF)

      6 Questions

      Q1.
      To __________ is to formulate a statement or rule that applies correctly to all relevant cases.

      conjecture
      estimate
      Correct answer: generalise
      prove

      Q2.
      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.
      Correct answer: It is often impossible to test all relevant cases.
      Proofs are more convincing if they have diagrams or practical demonstrations.

      Q3.
      Laura wants to prove Pythagoras' theorem. Her first few steps are shown. Which of these is the correct next step of her proof?

      An image in a quiz
      $$a^2 + b^2 = \frac{4ab}{2} + c^2$$
      $$a^2 + b^2 = c^2$$
      Correct answer: $$a^2 + ab + ab + b^2 = 2ab + c^2$$
      $$2a + 2b = 2ab + c^2$$
      where $$a, b, c$$ are all integers.

      Q4.
      If $$p$$ is a prime number and $$n$$ is a positive integer, which is a counterexample to the conjecture "$$p^n$$ is always odd"?

      Correct answer: $$ p = 2$$ and $$n = 3$$
      $$ p = 3$$ and $$n = 4$$
      $$ p = 5$$ and $$n = -2$$
      $$ p = 10$$ and $$n = 1$$
      $$ p = 13$$ and $$n = 5$$

      Q5.
      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"?

      $$n = 0$$
      $$n = 2$$
      $$n = 8$$
      $$n = 15$$
      Correct answer: $$n = 21$$

      Q6.
      Aisha writes the conjecture "For all integer values of $$x, (x-2)^2 > x-2$$". There is a counterexample to this when $$x=$$ .

      Correct Answer: 2, 3, two, three

      To help you plan your 11 maths lesson on: Proving or disproving a statement, download all teaching resources for free and adapt to suit your pupils' needs...