Algebra
Grade 6-9Algebraic Proof
An algebraic proof shows that a statement is true for every possible value, not just a few examples, by representing the numbers involved as algebraic expressions and simplifying to reach the required result. This lesson covers representing consecutive integers, odd numbers and even numbers algebraically, proving a statement by simplifying an expression, proving a statement about the digits of a number, and disproving a statement with a counter-example.
What you need to know
- Use 2n for any even number and 2n plus 1 for any odd number.
- Consecutive integers are n, n plus 1, n plus 2.
- Show the general case with algebra; giving a few examples is not a proof.
- Finish by explaining why your expression must be even, odd, or a multiple.
Written by Asad, co-founder of Teachably
Video walkthrough coming soon
The written lesson below covers everything you need in the meantime.
Representing numbers algebraically
Consecutive integers can be written as n, n + 1, n + 2, and so on. Any even number can be written as 2n, and any odd number as 2n + 1, since these forms only ever produce even or odd values.
Three consecutive integers can be written as n, n + 1 and n + 2. Three consecutive even numbers can be written as 2n, 2n + 2 and 2n + 4.
Proving a statement by simplifying an expression
Write the numbers involved algebraically, form the expression described in the statement, then expand and simplify it until it clearly matches what needs to be shown.
To prove the sum of two consecutive integers is always odd, let the integers be n and n + 1. Their sum is n + (n + 1) = 2n + 1, which is always one more than a multiple of 2, so the sum is always odd.
Proving a statement about the digits of a number
Write a two-digit number with tens digit a and units digit b as 10a + b. Reversing the digits gives 10b + a. Form and simplify the expression described in the statement using these two forms.
To prove the sum of a two-digit number and its reverse is a multiple of 11, add (10a + b) + (10b + a) = 11a + 11b = 11(a + b), which is a multiple of 11 for any digits a and b.
Disproving a statement with a counter-example
To disprove a statement that claims something is always true, it is enough to find just one specific value for which the statement fails. This is called a counter-example.
To disprove that n² + n + 1 is always prime, try n = 4: 4² + 4 + 1 = 21 = 3 × 7, which is not prime, so the statement is false.
Worked examples
Three exam-style questions, fully solved
Prove that the sum of two consecutive integers is always odd.
Easy- 1.Let the two consecutive integers be n and n + 1
- 2.Find their sum: n + (n + 1) = 2n + 1
Answer: 2n + 1 is one more than a multiple of 2, so the sum is always odd
A two-digit number has tens digit a and units digit b, so the number is 10a + b. Prove that the sum of a two-digit number and the number formed by reversing its digits is always a multiple of 11.
Medium- 1.Write the reversed number: 10b + a
- 2.Add the original number and the reversed number: (10a + b) + (10b + a) = 11a + 11b
Answer: 11a + 11b = 11(a + b), which is a multiple of 11 for any digits a and b
A student claims that n² + n + 1 is a prime number for every positive integer value of n. Show that the student is wrong.
Hard- 1.Choose a value of n to test, such as n = 4
- 2.Substitute and evaluate: 4² + 4 + 1 = 21
- 3.Check whether 21 is prime: 21 = 3 × 7
Answer: 21 is not prime, so the statement is false
Practice
6 questions, marked instantly
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Practice
Now try these yourself.
Type your answer and check it. The worked solution appears once you have had a go.
n is an integer. Write an expression for the next integer after n.
Write an expression for the next even number after 2n.
Expand and simplify (n + 1)² − n².
Expand and simplify (n + 3)² − (n − 3)².
n, n + 1 and n + 2 are three consecutive integers. Write their sum as simply as possible.
Expand and simplify (2n + 1)(2n − 1).
Avoid these
The mistakes students make most often
Using specific numbers to "prove" a statement, which only shows it works for those examples, not that it is true for every possible value.
Choosing the wrong algebraic form for the numbers involved, such as using n and n + 1 for two consecutive even numbers instead of 2n and 2n + 2.
Making an expanding or simplifying error partway through the proof, especially when squaring a bracket or collecting like terms.
Reaching the correct simplified expression but not explicitly stating why it proves the claim, such as forgetting to say "which is a multiple of 3" after reaching 3(n + 1).
When asked to disprove a statement, giving a vague explanation instead of a specific counter-example that clearly shows the statement fails.
FAQ
Questions students and parents ask
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