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20.6 Proof

GCSE Maths Foundation · Edexcel · Chapter 20: More Algebra

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: identities, "show that" questions, odd, even and consecutive numbers, and counter-examples. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

  • Know the difference between an equation and an identity, and use the symbol ≡
  • Show that two expressions are identical by expanding and simplifying
  • Use 2n, 2n + 1 and n, n + 1, n + 2 to prove facts about numbers
  • Disprove a statement with a counter-example

Key words

  • Equation: True only for certain values of the letter. 2x + 1 = 9 is only true when x = 4.
  • Identity: True for every value of the letter. 3(x + 2) ≡ 3x + 6. The symbol ≡ means "is identical to".
  • Proof: An argument that shows a statement is always true. Checking a few examples is not a proof.
  • Counter-example: One example that shows a statement is false.

Equations and identities

An equation can be solved to find the value of the letter. An identity is true whatever the letter is, because both sides simplify to the same expression.

Worked example: equation or identity?

(a) 4(x − 1) = 4x − 4: expanding the left gives the right, so it is an identity.
(b) 4(x − 1) = 12: only true when x = 4, so it is an equation.

Answer: (a) identity (b) equation

Showing that expressions are identical

Start with one side (usually the more complicated one). Expand and simplify step by step until you reach the other side. Do not work on both sides at once.

Worked example: show that 5(x + 2) − 3(x − 4) ≡ 2(x + 11)

Left side: 5x + 10 − 3x + 12 = 2x + 22
Right side: 2(x + 11) = 2x + 22
Both sides are equal for every x.

Answer: shown

Worked example: show that (n + 4)² − (n + 2)² ≡ 4n + 12

(n + 4)² = n² + 8n + 16 and (n + 2)² = n² + 4n + 4
Subtract: n² + 8n + 16 − n² − 4n − 4 = 4n + 12

Answer: shown

Odd, even and consecutive numbers

Use algebra to stand for any number of a certain type, where n is an integer:

  • any even number; Write it as 2n
  • any odd number; Write it as 2n + 1
  • three consecutive integers; Write it as n, n + 1, n + 2
  • a multiple of 3; Write it as 3 × (an integer), e.g. 3(n + 1)

Worked example: prove that the sum of three consecutive integers is a multiple of 3

Let the integers be n, n + 1 and n + 2.
Sum = n + n + 1 + n + 2 = 3n + 3 = 3(n + 1)
3(n + 1) is 3 times an integer, so it is a multiple of 3.

Answer: proved for every integer n

Worked example: prove that the sum of two odd numbers is even

Use different letters, as the odd numbers may be different: 2a + 1 and 2b + 1.
Sum = 2a + 2b + 2 = 2(a + b + 1), which is 2 times an integer, so it is even.

Answer: proved

Counter-examples

To show a statement is false, you only need one example where it does not work.

Worked example: "Every number squared is bigger than the number"

Try 0.5: 0.5² = 0.25, which is smaller than 0.5.

Answer: False: 0.5 is a counter-example (1 and 0 also work)

Key fact: Examples can show a statement is false, but they can never prove it is always true. For that you need algebra.

Common mistakes

  • Checking a few numbers and calling it a proof.
  • Using the same letter for two different odd numbers, e.g. writing (2n + 1) + (2n + 1), which only covers equal numbers.
  • Not finishing the argument. Say clearly why your final expression is even, odd or a multiple.
  • Working on both sides of a "show that" at the same time without starting from one side.

Revision questions

Try these, then check your answers below each one.

Question 1. Which of these is an identity (true for every value of x)? (Choose from: 2(x + 5) = 2x + 10, 2x + 5 = 11, x² = 9, 3x = 12.)

Answer: 2(x + 5) = 2x + 10

Working: Expanding 2(x + 5) always gives 2x + 10. The others are only true for certain values.

Question 2. True or false? Checking that a statement works for five different numbers proves it is always true.

Answer: False

Working: Examples cannot prove that something is always true. You need an algebraic argument.

Question 3. n is an integer. Which expression is always odd? (Choose from: 2n + 1, 2n, n + 1, 3n.)

Answer: 2n + 1

Working: 2n is even, so 2n + 1 is always odd. The others can be even or odd.

Related lessons

Continue in the full course

This lesson is section 20.6 (EGCMF2006) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.

 

 

 

 

 

 

 

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