1. **Stating the problem:** We want to understand factorisation and algebraic manipulation from the basics, including identifying common factors and factoring trinomials, in a way that is easy and clear for a 9th standard student. 2. **What is factorisation?** Factorisation means breaking down an expression into simpler parts (factors) that multiply to give the original expression. 3. **Identifying common factors:** - Look for numbers or variables that appear in every term. - For example, in $6x + 9$, both terms have a common factor of 3. - So, $6x + 9 = 3(2x + 3)$. 4. **Formula for factoring out common factors:** $$a b + a c = a(b + c)$$ where $a$ is the common factor. 5. **Factoring trinomials:** - A trinomial is an expression with three terms, like $ax^2 + bx + c$. - We want to write it as $(mx + n)(px + q)$. 6. **Steps to factor a trinomial $x^2 + bx + c$ when $a=1$:** - Find two numbers that multiply to $c$ and add to $b$. - For example, $x^2 + 5x + 6$: - Numbers that multiply to 6 and add to 5 are 2 and 3. - So, $x^2 + 5x + 6 = (x + 2)(x + 3)$. 7. **Visualizing with a mind map:** - Start with "Factorisation" in the center. - Branch 1: "Common Factors" with examples. - Branch 2: "Trinomials" with steps and examples. 8. **Pictorial representation:** - Use area models to show how $(x + 2)(x + 3)$ expands to $x^2 + 5x + 6$. - Draw a rectangle divided into parts representing $x^2$, $2x$, $3x$, and $6$. 9. **Summary:** - Always look for common factors first. - For trinomials, find two numbers that multiply to $c$ and add to $b$. - Use diagrams and mind maps to connect concepts visually. This approach helps build a strong foundation in factorisation and algebra manipulation for a 9th standard student.