Class 8 & 9 Mathematics Foundations: Algebra, Geometry, Probability & Proportional Reasoning: quick revision, 15 bites

Coordinates: Every Point Has an Address

Two numbers can pinpoint any point.

The Cartesian plane uses **two perpendicular number lines** — the x-axis and y-axis — to locate any point as an ordered pair $(x, y)$.

Takeaway. Order matters: $(3, 5)$ and $(5, 3)$ are different points.

What Makes a Polynomial Linear?

Degree 1 = straight line, always.

Takeaway. A linear polynomial has exactly one zero and graphs as a straight line.

The Three Algebraic Identities You Must Know

Three formulas unlock dozens of problems.

Takeaway. Recognising the pattern saves you from multiplying term by term every time.

The Classic $(a+b)^2$ Trap

Squaring a sum is NOT squaring each term.

Takeaway. The middle term $2ab$ is the most-forgotten part of the identity.

Proportional Reasoning: The Unitary Method

Find one unit, then scale up.

Takeaway. Direct proportion: both quantities increase together; inverse: one rises as the other falls.

Direct vs Inverse Proportion

Same direction or opposite?

Takeaway. Check whether the product or the ratio stays fixed — that tells you which proportion it is.

Quadrilaterals: A Family Tree

Every quadrilateral has a special property.

Takeaway. A square is the most special quadrilateral — it satisfies every property of rectangle and rhombus.

Perimeter and Area: The Core Formulas

Boundary vs surface — two different measures.

Takeaway. Perimeter measures the boundary (1-D); area measures the enclosed surface (2-D).

Don't Confuse Perimeter With Area

Units give the answer away.

Takeaway. The unit of your answer tells you whether you computed length or area.

Probability: Measuring Uncertainty

Chance has a number between 0 and 1.

Takeaway. Probability is a fraction — it can never be negative or greater than 1.

Probability in Action: Rolling a Die

Six faces, one target.

Takeaway. Listing the sample space first prevents counting errors.

Remember the Difference of Squares

Same terms, opposite signs.

For $a^2 - b^2$, think: **'Same two numbers, one plus, one minus.'** So $a^2 - b^2 = (a+b)(a-b)$.

Takeaway. The minus sign in the middle is your cue to factor into a sum × difference.

Where Proportional Reasoning Shows Up

Recipes, maps, and speed — all proportions.

Takeaway. Spotting proportions lets you solve everyday problems without heavy algebra.

Show the Identity Before Substituting

Write the formula first — marks follow.

In algebra exams, **state the identity you're using** (e.g. $(a+b)^2 = a^2 + 2ab + b^2$) before plugging in values. Examiners award method marks even if arithmetic slips.

Takeaway. Method marks are free marks — always show the identity line.

Sign Errors Sink Algebra Answers

One minus sign, whole answer wrong.

Takeaway. The middle term carries the sign — treat it as the most error-prone part.