Proof Positive

Proof Positive

"It works for 3 and 4, so it is 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 before you write a single line.

Match the phrase to its algebra

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 into 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 have to 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

Put the lines of the proof "the sum of two consecutive integers is odd" in order.

• Let the two consecutive integers be n and n + 1
• Add them: n + (n + 1) = 2n + 1
• 2n is even, so 2n + 1 is one more than an even number
• Therefore the sum of two consecutive integers is always odd

What makes it a proof?

Which TWO of these does a full algebraic proof need?

• Algebra chosen so the argument covers every possible number
• A closing sentence linking the result back to the claim
• A check that the statement works for three different numbers
• A graph of the numbers involved
• A single counter-example showing the statement can fail
• A calculator display confirming the arithmetic

Find the broken line

This "proof" that the sum of two even numbers is even has ONE broken line. Tap it.

• Claim: the sum of two even numbers is even.
• Let the two numbers be 2n and 2n
• .
• Adding them gives 2n + 2n = 4n
• .
• 4n has a factor of 2, so the sum is even
• .

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 have shown n + (n + 1) + (n + 2) = _____. Factorising gives _____, which is 3 times a whole number. Therefore the sum of three consecutive integers is always a _____.

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.

Build a proof

Prove that the sum of an even number and an odd number is always odd. Make the decision at each stage.

• First, represent the two numbers. Which pair covers EVERY even number and EVERY odd number?
• Now add them together. What do you get?
• To show that result must be odd, what is the useful next move?
• Finally, which sentence completes the proof?

In the exam

Proof mastered. Grade-9 habits: • Represent first: 2n for any even, 2n + 1 for any odd, n and n + 1 for consecutive, and a different letter for an independent quantity. • Manipulate into a form that shows the structure. Factorising is usually what reveals it. • Conclude in a sentence that names the claim. Working alone is not a proof, and examples are never a proof.

Write the proof

Prove algebraically that the sum of the SQUARES of two consecutive integers is always odd.

• Step 1, represent: let the two consecutive integers be n and n + 1.
• Step 2, manipulate: square each one and add them, expanding carefully.
• Collect the like terms, then take a factor of 2 out of everything you can.
• Step 3, conclude: say why what you are left with has to be odd.
• The concluding sentence is a mark on its own. Do not stop at the algebra.