Foundations for Geometry

Negatives show up everywhere in geometry — coordinates, slopes, directed lengths. Get fast and sure at adding and subtracting them on a number line.

Area, scaling, unit rates, splitting a length into equal parts — geometry leans on multiplication and division constantly. Make the facts automatic and the multi-digit work reliable.

One expression, one answer — but only if everyone evaluates it in the same order. Lock in PEMDAS so multi-step formulas in geometry come out right every time.

Round to estimate, to report a sensible answer, and to decide "how many do we need." Master place value rounding and the real-world cases where you must round up.

The same amount can be written as 1/2, 2/4, or 3/6. Seeing why — and building equivalents on demand — is what makes comparing, adding, and scaling fractions possible.

You can only add fractions when the pieces are the same size. Master finding a common denominator, then combining — the move behind every fraction sum in geometry.

Multiplying fractions is "a part of a part"; dividing asks how many fit. Two clean procedures — multiply across, or flip and multiply — used all over geometry.

A proportion is two equal ratios. Cross-multiply to solve for the missing value — the engine behind unit rates, scale drawings, and similar figures in geometry.

Percent means "per hundred" — a ratio in disguise. Set up part over whole equal to percent over 100, and discounts, tax, tips, and percent change all fall out.

Plug numbers into a formula and evaluate — carefully. This single skill gates two-thirds of the Regents — area, perimeter, the Pythagorean theorem, and more all start here.

Combine like terms and distribute to write an expression in its simplest form — the cleanup step that makes solving equations and using formulas manageable.

Isolate the variable using inverse operations, keeping the equation balanced. The other half of the Regents spine — used to find missing sides, angles, and unknowns everywhere.

An exponent is repeated multiplication — squares, cubes, and beyond. Evaluate powers correctly (especially with negative bases) for area, volume, and the Pythagorean theorem.

A square root undoes a square. Find exact roots of perfect squares and estimate the rest between whole numbers — the key to the Pythagorean theorem and the distance formula.

Same-base powers combine by simple rules — add exponents to multiply, subtract to divide, multiply to raise a power to a power. Learn when each applies and why.

Every point on the plane is named by an ordered pair, and the axes split it into four quadrants. Reading coordinates and naming quadrants is the language of all coordinate geometry.

The distance formula is just the Pythagorean theorem on the grid. Find how far apart two points are — directly when they line up, and through a right triangle when they don't.

The midpoint of a segment is the average of its endpoints. Halve the run and the rise to land dead center — the basis for bisectors, centers, and symmetry.

Slope measures how steep a line is and which way it leans — rise over run, the change in y divided by the change in x between any two points.