The “Invisible” Mark Killers#
These aren’t big concepts — they’re small habits and skills that silently steal marks across every section. Fix them and you’ll see immediate improvement.
1. Negative Signs & Brackets#
The single most common algebraic error is mishandling negative signs, especially when subtracting brackets.
The Rule#
When you subtract a bracket, the sign of every term inside flips:
$$ -(3x^2 - 5x + 2) = -3x^2 + 5x - 2 $$Where this bites you#
- First Principles: $f(x+h) - f(x)$ requires subtracting the entire $f(x)$. Forgetting brackets means half the terms keep the wrong sign.
- Trig Identities: Working both sides of a proof, you often subtract complex expressions.
- Completing the Square: $-2(x^2 - 3x + \frac{9}{4}) + \frac{9}{2}$ — the negative outside the bracket affects every term.
Practice#
Expand: $5 - 2(3x - 4)$
$= 5 - 6x + 8 = 13 - 6x$ ✓
NOT $5 - 6x - 8 = -3 - 6x$ ✗ (forgot to flip the $-4$ to $+8$)
2. Substitution#
Substitution appears everywhere: plugging values into formulas, finding $y$-coordinates of turning points, evaluating $f(a)$.
The Golden Rule: Use Brackets#
When substituting a value, always wrap it in brackets:
If $f(x) = 2x^2 - 3x + 1$, find $f(-2)$:
$f(-2) = 2(-2)^2 - 3(-2) + 1 = 2(4) + 6 + 1 = 15$ ✓
Without brackets: $f(-2) = 2 \times -2^2 - 3 \times -2 + 1$ — your calculator may interpret $-2^2$ as $-(2^2) = -4$ instead of $(-2)^2 = 4$.
Where this bites you#
- Cubic functions: Finding $f(3)$ to check if $x = 3$ is a root (Factor Theorem)
- Finance: Substituting $i = \frac{0.12}{12}$ into $(1 + i)^n$
- Trig: Evaluating $\sin(180° - \theta)$ — the $180° - \theta$ must stay together
3. Solving Equations — The Zero Product Rule#
To solve an equation, you MUST get one side equal to zero, then factor.
$$ \text{If } ab = 0, \text{ then } a = 0 \text{ or } b = 0 $$The Trap: Dividing by a Variable#
NEVER divide both sides by $x$ (or $\sin\theta$, or any expression that could be zero) — you’ll lose a solution.
Wrong approach:
$x^2 = 5x$
$\frac{x^2}{x} = \frac{5x}{x}$ → $x = 5$ (lost the solution $x = 0$!)
Correct approach:
$x^2 - 5x = 0$
$x(x - 5) = 0$
$x = 0$ or $x = 5$ ✓
Same trap in Trig#
$2\sin\theta\cos\theta = \sin\theta$
Wrong: Divide by $\sin\theta$ → $2\cos\theta = 1$ → $\theta = 60°$ (lost solutions where $\sin\theta = 0$!)
Correct: $2\sin\theta\cos\theta - \sin\theta = 0$ → $\sin\theta(2\cos\theta - 1) = 0$
$\sin\theta = 0$ or $\cos\theta = \frac{1}{2}$ ✓
4. Inequalities#
When solving inequalities, remember:
- Multiplying or dividing by a negative number FLIPS the inequality sign.
- Use a number line or sign diagram for quadratic inequalities.
Example#
$-2x > 6$
Divide by $-2$ (flip the sign): $x < -3$
Quadratic Inequality#
$x^2 - 5x + 6 < 0$
Factor: $(x-2)(x-3) < 0$
Critical values: $x = 2$ and $x = 3$
Test intervals: The product is negative between the roots.
Answer: $2 < x < 3$
5. Calculator Skills#
Finance Calculations#
- Store values: Use the
STOandRCLbuttons. Never round $i$ mid-calculation. - Brackets are essential: For $(1.00875)^{-240}$, enter
(1.00875)^(-240). Without brackets around -240, the calculator computes $(1.00875)^{240}$ and then negates it. - ANS button: Use it to chain calculations without retyping.
Trig Calculations#
- Check MODE: Ensure you’re in DEGREES mode (not radians). A common disaster.
- Negative angles: $\sin(-30°) = -\sin(30°) = -0.5$. Your calculator handles this, but you need to interpret it correctly.
- Inverse trig: $\sin^{-1}(0.5) = 30°$ gives you the reference angle. You must then find ALL solutions in the required interval using the CAST diagram.
General Tips#
- Close all brackets: Count your opening and closing brackets before pressing
=. - Estimation: Before pressing
=, estimate the answer in your head. If you expect ~R7 000 and get R700 000, something went wrong.
6. Reading Exam Questions#
Marks are regularly lost because students answer the wrong thing.
Key phrases to watch for#
| Phrase | What it means |
|---|---|
| “Determine the value of $x$” | Find $x$ — the number |
| “Hence determine the maximum value” | Use your previous answer, then find the maximum VALUE (not the $x$-value) |
| “Show that…” | You must PROVE it — don’t just write the answer |
| “For which values of $x$…” | Give an inequality or interval, not a single value |
| “Correct to two decimal places” | Round only at the VERY END |
| “Use first principles” | You MUST use the limit definition — the power rule gets zero marks |
| “Hence or otherwise” | “Hence” = use the previous part; “otherwise” = you may use a different method |
7. The Distributive Law (Expanding Brackets)#
Students still make errors with:
FOIL (Two binomials)#
$(x + 3)(x - 2) = x^2 - 2x + 3x - 6 = x^2 + x - 6$
Squaring a Binomial#
$(x + 3)^2 = x^2 + 6x + 9$
NOT $x^2 + 9$! The middle term $2(x)(3) = 6x$ is always forgotten.
$(a - b)^2 = a^2 - 2ab + b^2$ — note the signs.
Cubing a Binomial (for First Principles)#
$(x + h)^3 = x^3 + 3x^2h + 3xh^2 + h^3$
Use Pascal’s triangle: coefficients are 1, 3, 3, 1.
Build on Your Lower-Grade Foundations#
If these habits still feel weak, revisit the source lessons:
- Grade 10 Fundamentals: Basic Algebra
- Grade 10 Fundamentals: Integers & Number Sense
- Grade 10 Equations & Inequalities
- Grade 11 Fundamentals: Equation Solving
- Grade 11 Equations & Inequalities
🚨 Summary: The Top 10 Silent Mark Killers#
- Dropping the negative sign when subtracting brackets
- Dividing by a variable (losing solutions)
- Not using brackets when substituting negative values
- Forgetting the middle term when squaring binomials
- Calculator in wrong mode (radians vs degrees)
- Rounding mid-calculation in finance
- Answering the wrong part of the question (“find $x$” vs “find the maximum value”)
- Not writing $\lim_{h \to 0}$ on every line of first principles
- Splitting the denominator of a fraction ($\frac{a}{b+c} \neq \frac{a}{b} + \frac{a}{c}$)
- Cancelling terms instead of factors ($\frac{x+3}{x} \neq 3$)
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