1-ALGEBRAIC EXPRESSIONS Gr10 In the following expressions, TWENTY possible solutions are provided for the instruction: Pactorise full or simpliti. FIVE of the fiventy solutions are algebraically correct and the rest are neolred. 4. Identify and CIRCLE OR CROSS the LETTER that represent solutions with correct (15) answers. 2 From the FIFTEEN incorrect solutions, identify and underline the mistakes made in EACH. 3. Correct the mistakes made in EACH incorrect solution.

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Below is one example of how to approach the problem step by step. (Note that without the explicit list of the twenty provided expressions and their proposed factorizations/simplifications, we cannot mark exactly which letter is correct and which are wrong. Instead, we describe a method for analysing each solution.)

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1. **Set Up the Task**

We are given twenty candidate answers for the instruction “factorise fully or simplify” from a Grade 10 algebra exercise. Only five of these twenty are algebraically correct. Our steps are:

- Identify and mark (e.g. circle or cross) the letters that represent solutions with correct final answers.
   - For the fifteen incorrect solutions, underline the point(s) in each where the error occurs.
   - Correct each error by providing the proper factorisation or simplification.

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2. **Verification of Each Candidate Solution**

For each of the twenty answers, proceed as follows:

a. **Review the Original Expression**

- Write down the starting algebraic expression (for example, suppose one expression is  
        $$
        x^2 - 9.
        $$
      ).

b. **Re-factorise or Simplify the Expression**

- Check whether the candidate has factored the expression correctly. For our example, the proper factorisation is  
        $$
        x^2 - 9 = (x-3)(x+3).
        $$
      
      - Make sure every factor is correctly derived according to the algebraic formulas (e.g. difference of squares, common factor extraction, etc.).

c. **Compare the Candidate’s Answer with the Correct Answer**

- If they match, then mark the letter corresponding to that candidate as correct (e.g. circle the letter “C” if candidate C’s answer is the correct factorisation above).

- If they do not match, underline the step or symbol in the candidate’s answer that is in error.

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3. **Common Mistakes to Look For**

While reviewing the candidate solutions, some typical mistakes may include:

- **Incorrect application of algebraic identities:**  
     For example, confusing difference of squares with sum of squares.  
     A wrong step might show  
     $$
     x^2 - 9 = (x-9)(x+1),
     $$
     where the factors do not satisfy  
     $$
     (x-9)(x+1)= x^2 - 8x - 9,
     $$
     which is not equal to $$x^2 -9$$.

- **Sign errors:**  
     A candidate might factor  
     $$
     x^2 - 9 = (x+3)(x+3),
     $$
     mistakenly carrying a wrong sign.

- **Leaving Common Factors Unfactored:**  
     If an original expression is  
     $$
     2x^2 - 4x,
     $$
     the proper factorisation is  
     $$
     2x(x-2).
     $$
     An incorrect solution might stop at $$x(x-2)$$ or merely rewrite the expression without removing the common factor.

- **Simplification mishaps:**  
     If the task was to simplify a rational expression and the candidate made cancellation errors (such as cancelling terms that are not common factors).

---

4. **Correction of an Incorrect Example**

Suppose one candidate gave the following answer for an expression:  
   $$
   x^2 - 9 = (x-9)(x+1).
   $$
   
   **Step-by-Step Correction:**

- **Step 1:** Recognise that $$x^2-9$$ is a difference of two squares, since  
     $$
     9 = 3^2.
     $$
     
   - **Step 2:** Write the generic formula for a difference of squares:  
     $$
     a^2 - b^2 = (a-b)(a+b).
     $$
     
   - **Step 3:** Identify $$a=x$$ and $$b=3$$, so the correct factorisation is  
     $$
     x^2-9=(x-3)(x+3).
     $$
     
   - **Step 4:** Underline the error in the candidate’s answer where they incorrectly set the factors as $$(x-9)$$ and $$(x+1)$$.

- **Step 5:** Write the correct factorisation $$ \boxed{(x-3)(x+3)} $$.

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5. **Summary of the Strategy**

$$
   \textbf{For each candidate:}
   $$
   
   - Write down the original expression.
   - Re-derive the factorisation or simplification using proper algebraic identities.
   - Compare the result with the candidate’s answer:
     - If they are equivalent, mark that candidate with a circle (or appropriate marker).
     - If they are not equivalent, underline the point(s) where the mistake occurred.
   - Provide the corrected solution.

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6. **Example Checklist**

For every candidate answer, you might prepare a checklist like:

- [ ] Does the factorisation match the algebraic identity?
   - [ ] Are signs handled correctly?
   - [ ] Are common factors properly taken out?
   - [ ] Is the expression fully factorised or simplified?

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Using this method, you or the students can systematically identify, mark, and correct each of the provided solutions. Without the explicit twenty expressions, this outline provides the general steps needed to tackle the problem.
To identify correct factorizations, circle or cross the letters with accurate answers. For incorrect solutions, underline the mistakes and correct them accordingly.

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