Algebraic Transformation: Proving Identities

1. What "Show That" Really Means in an Exam

First, let's be crystal clear about what the examiners are looking for. The command word "Show that" has a very specific meaning. It means the answer is already given to you, and your task is to provide the detailed, step-by-step working that proves how to get from the starting point to that given answer.

You are being tested on your logical reasoning and communication. Omitting steps, even ones that seem obvious to you, will cost you marks.

To see exactly what this means in practice, let's analyze an official Cambridge example.

The key takeaway from that guide is crucial: the candidate showed that $A{B}^2=140625$ and then wrote $AB=375$. They lost a mark because they failed to write the intermediate step: $AB=\sqrt{140625}$. In a "show that" question, you must assume the examiner is skeptical and needs to see every link in your logical chain.

The structure of your argument is as important as the calculation itself.

2. The Systematic Approach to Algebraic Proof

For any "show that" question, whether it's proving a geometric result or an algebraic identity, you can follow a consistent three-step mental process.

As the resource outlines, the process is:

1. Start Correctly: Write down the initial expression(s) based on the problem. This might be using the left-hand side of an identity, or it might be setting up algebraic terms for "two consecutive odd numbers."
2. Manipulate Logically: Expand, simplify, or factorise the expression. Each line of your working should be a direct and verifiable consequence of the line before it. Your goal is to methodically transform the expression.
3. Finish Clearly: Your final line should be the exact target expression given in the question. For proofs about properties (e.g., "show it is even"), you must end with an expression that demonstrates that property, such as $2\times (\text{something})$.

Let's apply this framework to the common types of algebraic proofs you'll encounter.

3. Key Techniques for "Show That" Questions

The resource you just read contains excellent worked examples that demonstrate the core techniques. We will work through them, focusing on the logical flow.

Let's break down the strategies from those examples:

• Proving an Identity (Example 1):

  • Strategy: Start with the more complicated side of the identity (here, the Left-Hand Side).
  • Execution: Expand the brackets carefully. Remember that a minus sign in front of a bracket flips the sign of every term inside it. Collect like terms.
  • Goal: Manipulate your expression until it is identical to the Right-Hand Side. In this specific example, the working is a bit more complex, showing how one side can be rearranged to match the structure of the other. The key skill is precise expansion and simplification.

• Proving a Multiple (Example 2 & 4):

  • Strategy: Start with the expression given.
  • Execution: Expand and simplify it completely.
  • Goal: The final, crucial step is to factorise. To prove something is a multiple of 3, you must show you can write it as $3\times (\text{some integer expression})$. To prove it's a multiple of 8, you must factor out an 8. This final factorisation is the "show me" step that gets you the marks.

• Proving a Property (Example 3):

  • Strategy: Use the standard algebraic definitions (e.g., consecutive integers are $n$ and $n+1$).
  • Execution: Set up the expression as described (e.g., difference of squares: $(n+1{)}^2-n^2$). Expand and simplify.
  • Goal: Arrive at a general form that proves the property. To prove a number is odd, you must show it can be written as $2n+1$. Simply stating "it's odd" is not enough; your final line of algebra must be the proof.

4. Avoiding Common Errors

You've mentioned that small arithmetic errors are a recurring issue. In "show that" questions, these errors can completely derail your argument. Being aware of the common traps is the first step to avoiding them.

Let's reinforce those points:

• Bracket Expansion: Always remember that $(a+b{)}^2=a^2+2ab+b^2$. A very common mistake is to write $a^2+b^2$.
• Sign Errors: When subtracting an entire bracketed expression, like in $(...)-(n^2-6n+9)$, every sign inside the bracket must be flipped. This becomes $-n^2+6n-9$. Use your finger to trace the distribution of the negative sign to each term.

Conclusion

Today's lesson was laser-focused on the logic and presentation required for "show that" questions. This is a skill that separates good students from top-scoring students because it demonstrates a deep understanding of the mathematical process.

Key Takeaways:

• No Skipped Steps: "Show that" is a command to display your complete, unbroken chain of reasoning. Even "obvious" steps like writing the square root symbol must be included.
• Start -> Manipulate -> Finish: Follow the structure of setting up the correct initial expression, carefully manipulating it with correct algebra (especially with brackets and signs), and arriving at the exact target form.
• The Goal is the Proof: Your final step must be the proof itself.
  • For an identity, the LHS must equal the RHS.
  • For a multiple of k, you must factor out $k$, showing the form $k\times (\text{expression})$.
  • For an odd/even number, you must show the form $2n+1$ or $2n$.

By treating these questions as a communication exercise, you will not only secure more marks but also build the logical habits that prevent errors in all other types of problems.

Preview of the Next Lesson:
We have now completed the module on Advanced Algebra. In the next lesson, we will move on to a new topic: Functions and Sequences. We will start by learning how to find and use the nth term formula for linear (arithmetic) and quadratic sequences, which will draw upon the pattern-recognition skills you've been honing in algebra.