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Proof Positive

The grade-9 skill examiners love: prove a statement is ALWAYS true using algebra, not lucky examples. Represent any number, manipulate, and land the concluding sentence.

⏱️ 12 min 🎯 12 activities Teachers Not yet rated Students Not yet rated

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Proof Positive ✅

"It works for 3 and 4, so it's always true" — is **not** a proof. Checking examples can never cover *every* number. A real proof uses **algebra** to stand for **any** number, so one argument covers them all. This is a classic grade-9 question, and it follows a reliable recipe you can learn.

Let algebra stand for any number 🔤

The whole trick is choosing the right algebra: • **Any integer** → *n*. The next ones are *n + 1*, *n + 2* (**consecutive**). • **Any even number** → *2n* (2 times something). • **Any odd number** → *2n + 1* (one more than even). And when two quantities are **independent**, give them **different letters** — *2n* and *2m* — so you do not accidentally force them to be equal.

Match the phrase to its algebra

  • Any even number
  • Any odd number
  • Two consecutive integers
  • Any multiple of 3
  • 2n
  • 2n + 1
  • n and n + 1
  • 3n

Represent any even number

Which expression represents ANY even number, for every whole-number value of n?

  • 2n
  • n + 2
  • 2n + 1
  • n²

The three-step recipe 🧭

Every algebraic proof follows the same shape: 1. **Represent** the numbers with algebra (using the toolkit above). 2. **Manipulate** — add, expand or factorise to a useful form. 3. **Conclude** with a sentence that links the result back to the claim. Miss step 3 and you drop the final mark — you must *say why* your algebra proves the statement.

Worked proof ✍️

**Claim:** the sum of two consecutive integers is always odd. 1. **Represent:** let the integers be *n* and *n + 1*. 2. **Manipulate:** *n* + (*n* + 1) = **2n + 1**. 3. **Conclude:** 2n is even, so 2n + 1 is one more than an even number — therefore the sum is always **odd**. ∎

Order the proof

An interactive activity.

What makes it a proof?

A full algebraic proof needs which TWO of these? (Pick two.)

  • Use algebra so the argument covers every possible number
  • End with a sentence linking the result back to the claim
  • Check that it works for three different numbers
  • Draw a graph of the numbers

Simplify the sum

For a proof about three consecutive integers n, n + 1 and n + 2, what does their sum simplify to?

  • 3n + 3
  • 3n
  • n + 3
  • 3n + 6

Finish the argument ➕

You showed n + (n + 1) + (n + 2) = **3n + 3**. Now **factorise** to reveal the structure: 3n + 3 = **3(n + 1)**. This is 3 times a whole number, so **the sum of three consecutive integers is always a multiple of 3**. That final sentence is what turns working into a proof.

Complete the proof

Claim: the sum of two even numbers is even. Write the numbers as 2n and _____, using a different letter because they are independent. Their sum is 2n + 2m = _____. This has a factor of _____, so the sum is always even.

2m 2(n + m) 2 2n n + m

Why a different letter?

In that proof, why write the two even numbers as 2n and 2m rather than 2n and 2n?

  • The two numbers can be different, so they need different letters; 2n and 2n would force them to be equal
  • It makes the algebra look more advanced
  • There is no real difference; 2n and 2n would be fine
  • Because 2m is odd, which is needed here