Pre-Algebra

In arithmetic, every number is known. In algebra, unknowns are represented by variables — symbols, usually letters like x, y, or n, that stand for values not yet known or that can change. Variables let you write general rules that work for ALL numbers. Instead of memorizing hundreds of specific number facts, you write a single rule with variables. The statement three x plus two x equals five x is true whether x is ten, three point five, or negative seven. This is the power of algebra: expressing relationships in their most general form. The number in front of a variable is a coefficient. Like terms have the same variable raised to the same power and can be combined by adding their coefficients.

A fraction represents part of a whole or a ratio between two quantities. The top number is the numerator — it counts how many equal parts you have. The bottom number is the denominator — it tells you how many equal parts make one whole unit. To add or subtract fractions, they must share the same denominator. Find a common denominator, which is any multiple of both. The least common denominator is the LCM of the denominators. To multiply fractions, multiply numerators and multiply denominators. To divide by a fraction, multiply by its reciprocal (flip it upside down). A fraction bar is itself a division symbol.

A ratio compares two quantities. It can be written with a colon as three to four, or as the fraction three over four. A proportion is an equation stating that two ratios are equal. Proportions solve problems involving scaling, maps, recipes, and similar figures. The key technique is cross multiplication: multiply the numerator of each fraction by the denominator of the other and set the cross products equal. The double arrow symbol means "if and only if" — the two statements are logically equivalent. A rate is a ratio comparing quantities measured in different units, like miles per hour.

The word percent means "per hundred." The percent symbol divides any number by one hundred. To convert a percent to a decimal, move the decimal point two places left. To find a percentage of a number, convert to a decimal and multiply. A percentage increase is the change divided by the original value, expressed as a percent. A percentage decrease works the same way. For example, if a price rises from eighty dollars to one hundred dollars, the increase is twenty dollars, and twenty divided by eighty gives twenty-five percent.

The integers extend the number line in both directions from zero. The absolute value of a number, written with vertical bars, measures its distance from zero on the number line — distance is always non-negative. The absolute value of positive three is three. The absolute value of negative three is also three. Both numbers are three units from zero. Subtracting a negative number is equivalent to adding: five minus negative three equals five plus three, which equals eight. Think of subtraction as "add the opposite": the opposite of negative three is positive three.

See how changing the slope and intercept transforms a straight line. This is the geometry behind every proportion.