IB Maths AA SL · Unit 1: Number and Algebra
IB Maths AA SL Proof Questions
Exam-style IB Maths AA SL proof questions with worked solutions. Start with the 3 fully worked examples below — each is solved step by step the way an IB examiner expects — then try the practice questions and check your working against the mark scheme.
- 20 questions
- Paper 1: 16
- Paper 2: 4
- 6 easy
- 9 medium
- 4 hard
- 1 starter
- 3 worked examples
Practise Proof questions →
AA SL formula booklet
What's examined in AA SL proof
The question bank covers these proof question types (number of questions in brackets):
- Number Properties Proofs (10)
- Algebraic Identities (7)
- Formal Proof Techniques (3)
Proof worked examples
Worked example 1: Deductive proof of identities · easy
Prove the algebraic identity $(a - b)^2 - (a + b)^2 \equiv -4ab$.
1. Expand the first binomial squared on the left-hand side (LHS): $(a - b)^2 = a^2 - 2ab + b^2$.
2. Expand the second binomial squared: $(a + b)^2 = a^2 + 2ab + b^2$.
3. Substitute these back into the LHS expression, using brackets to protect signs: $(a^2 - 2ab + b^2) - (a^2 + 2ab + b^2)$.
4. Distribute the negative sign through the second bracket: $a^2 - 2ab + b^2 - a^2 - 2ab - b^2$.
5. Cancel the $a^2$ and $b^2$ terms.
6. Simplify to yield the final expression: $-2ab - 2ab =$ $-4ab$, which equals the RHS.
Examiner tip: When subtracting an expanded bracket, failing to wrap the entire expansion in parentheses before distributing the negative sign is the single most common cause of sign errors in deductive proofs.
Worked example 2: Proving multiples · medium
Prove that the sum of any three consecutive integers is always a multiple of $3$.
1. Define three consecutive integers algebraically. Let the integers be $n$, $n+1$, and $n+2$, where $n \in \mathbb{Z}$.
2. Set up the sum of these integers: $\text{Sum} = n + (n+1) + (n+2)$.
3. Collect like terms together: $\text{Sum} = 3n + 3$.
4. Factorise the expression by factoring out the common divisor: $\text{Sum} = 3(n + 1)$.
5. State the logical conclusion: Since $n$ is an integer, $(n+1)$ is also an integer.
6. Conclude the proof: Because the sum is $3$ times an integer, it is always a multiple of 3.
Examiner tip: A proof requires general algebraic terms. Showing that $4 + 5 + 6 = 15$ (which is $3 \times 5$) is an example, not a proof, and will score zero marks.
Worked example 3: Proving properties of odd/even numbers · hard
Prove that the product of any two distinct odd integers is always an odd integer.
1. Define two distinct odd integers. Let them be $2k+1$ and $2m+1$, where $k, m \in \mathbb{Z}$.
2. Set up the product of these two integers: $(2k+1)(2m+1)$.
3. Expand the brackets using FOIL: $4km + 2k + 2m + 1$.
4. Factorise a $2$ out of the first three terms to isolate the even component: $2(2km + k + m) + 1$.
5. State that since $k$ and $m$ are integers, the expression $(2km + k + m)$ must also evaluate to an integer. Let this be integer $P$.
6. Conclude the proof: The product can be written in the form $\mathbf{2P + 1}$, which is the mathematical definition of an odd number.
Examiner tip: You must use two different variables (e.g., $k$ and $m$) to represent "any two" odd integers. Using $(2k+1)(2k+1)$ only proves that the *square* of an odd number is odd!
FAQ
How many IB Maths AA SL proof questions are there?
There are 20 exam-style proof questions in the AA SL question bank (Paper 1: 16 · Paper 2: 4), graded 6 easy, 9 medium, 4 hard, 1 starter. Every question has a full IB-style mark scheme (M, A and R marks).
Is proof on Paper 1 or Paper 2?
Both. In the bank, Paper 1: 16 · Paper 2: 4. Paper 1 is non-calculator, so practise the exact-value algebra as well as the GDC methods.
Where can I get the mark schemes?
Open the AA SL Unit 1 practice page: every question has a step-by-step IB-style mark scheme, and you can photograph your working for instant AI marking. The worked examples on this page are free.
Proof in other IB Maths courses
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