Mathematics 7

Fact families link multiplication and division together: if 7 × 8 = 56, then 56 ÷ 8 = 7 and 56 ÷ 7 = 8 — knowing one fact unlocks three more.

On a number line, adding a positive moves right; adding a negative moves left. Subtracting an integer is the same as adding its opposite: a − b = a + (−b).

Line up the decimal points (and thus the place values) before adding or subtracting, padding shorter decimals with trailing zeros if needed.

Percent means 'out of 100' — 75% = 75/100 = 0.75, and as a ratio, 75:100 (or simplified, 3:4).

Strategies like doubling (6 × 8 = double of 3 × 8), using a nearby known fact (9 × 7 = 10 × 7 − 7), and skip-counting fill gaps faster than memorizing in isolation.

Substitute each whole-number input into the expression, following order of operations, to get its matching output.

Undo operations in REVERSE order: for ax + b = c, subtract b from both sides first, then divide by a.

'Discrete' means the inputs are separate whole-number steps (like 1, 2, 3 items) rather than every possible value in between.

Every move must be done to BOTH sides of the equation to keep it balanced (preservation of equality).

Each row of a table becomes one ordered pair (input, output) to plot on the Cartesian plane.

Circumference = π × diameter (C = πd), or equivalently C = 2πr since diameter is twice the radius.

Volume = length × width × height, which is the same as (base area) × height, where the base area is length × width.

If given the radius, either double it first to get the diameter, or use the 2πr form directly.

All three dimensions must be in the SAME unit before multiplying, or the answer's units will be wrong.

Area of a circle = π × radius² (A = πr²) — note it's the RADIUS squared, not the diameter.

The sign pattern of an ordered pair (x, y) tells you the quadrant: (+,+) is Quadrant I, (−,+) is II, (−,−) is III, (+,−) is IV.

Apply the transformations ONE AT A TIME, in the stated order — the image after the first transformation becomes the starting shape for the second.

Always move horizontally (x) first, then vertically (y), starting from the origin (0, 0).

The order matters: reflecting then rotating a shape can land it somewhere different than rotating then reflecting the same shape.

Compare the original shape to the final image one step at a time: look for an overall shift (translation), a mirrored orientation (reflection), or a turned orientation (rotation).

Convert each category's share to a fraction of the whole, then to a sector angle: angle = (category ÷ total) × 360°.

Two events are independent when the outcome of one doesn't affect the outcome of the other (like a coin flip and a die roll).

Label each sector with its category and value or percent so the graph is readable at a glance.

Record EVERY trial's combined outcome (like 'heads, 4') in an organized list or tally so nothing is lost or miscounted.

A larger sector directly shows a larger share of the whole — compare sector sizes to compare categories at a glance.

Sales tax and discounts both start with 'percent of' the original price: tax ADDS a percent, a discount SUBTRACTS a percent.

Simple interest on savings is also a percent of an amount, usually calculated per year: interest = amount × rate × time.