GA Milestones Math Grade 8 flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Irrational number

A real number that cannot be written as a ratio of two integers; its decimal form is non-terminating and non-repeating, such as √2, π, or 0.010010001.

Rational number

Any number expressible as a fraction a/b where a and b are integers and b is not zero; its decimal either terminates or repeats, as with 3/8 = 0.375.

Perfect square

The product of an integer multiplied by itself, such as 49 = 7² or 144 = 12²; its square root is an integer, so it is rational.

Perfect cube

The product of an integer used three times as a factor, such as 27 = 3³ or 125 = 5³; its cube root is an integer, such as ∛64 = 4.

Estimating square roots

Locate the two consecutive perfect squares on either side of the radicand; √40 lies between √36 = 6 and √49 = 7, closer to 6 because 40 is nearer 36.

Product rule for radicals

For nonnegative factors, √(ab) = √a · √b; this lets you simplify √72 as √36 · √2 = 6√2, which is useful for comparing and combining radicals.

Quotient rule for radicals

For a nonnegative a and positive b, √(a/b) = √a / √b; for example √(9/16) equals 3/4, and √50/√2 simplifies to √25 = 5.

Zero exponent

For any nonzero base a, a⁰ = 1; the pattern a³, a², a¹ divided repeatedly by a gives a⁰ = 1, and 0⁰ is undefined in this context.

Negative exponent

For a nonzero base a, a^(−n) = 1/a^n; so 2^(−3) = 1/8 and 5^(−1) = 1/5. A negative exponent never makes the value negative.

Product of powers property

When multiplying powers with the same base, add the exponents: a^m · a^n = a^(m+n). For example, 3⁴ · 3² = 3⁶ = 729.

Quotient of powers property

When dividing powers with the same base, subtract the exponents: a^m ÷ a^n = a^(m−n). For example, 7⁵ ÷ 7² = 7³ = 343.

Power of a power property

To raise a power to a power, multiply the exponents: (a^m)^n = a^(mn). For example, (2³)⁴ = 2¹² = 4096.

Scientific notation

A number written as a × 10^n where 1 ≤ |a| < 10 and n is an integer; 3,400,000 = 3.4 × 10⁶ and 0.00071 = 7.1 × 10^(−4).

Multiplying in scientific notation

Multiply the coefficients and add the exponents, then adjust if the new coefficient is not between 1 and 10; (2 × 10³)(4 × 10⁵) = 8 × 10⁸.

Slope formula

For two points (x₁, y₁) and (x₂, y₂), slope m = (y₂ − y₁)/(x₂ − x₁). A horizontal line has slope 0; a vertical line has undefined slope.

Slope-intercept form

The equation y = mx + b, where m is the slope and b is the y-intercept; the graph crosses the y-axis at (0, b) and changes by m for each unit right.

Proportional relationship graph

A straight line passing through the origin, described by y = kx; the constant of proportionality k equals the unit rate y/x and is also the line's slope.

Unit rate as slope

In a proportional graph, the slope equals the amount of change in y per one unit of x; a line through (0,0) and (4,10) has unit rate 2.5.

Solving linear equations

Undo operations in reverse order: distribute and combine like terms first, then add or subtract, then multiply or divide. Whatever is done to one side must be done to the other.

Equations with variables on both sides

Collect variable terms on one side and constants on the other; 5x + 3 = 2x + 18 becomes 3x = 15, so x = 5. Check by substitution.

System of linear equations

Two or more linear equations considered together; the solution is the ordered pair satisfying all equations, found by graphing, substitution, or elimination.

Substitution method

Solve one equation for a variable, substitute that expression into the other equation, solve for the remaining variable, then back-substitute to find the other value.

Elimination method

Add or subtract the equations, after multiplying as needed, so one variable cancels; solve the resulting single-variable equation, then find the other variable.

No solution system

A system whose equations describe parallel lines with the same slope but different intercepts; elimination produces a false statement such as 0 = 5.

Infinitely many solutions

A system whose equations describe the same line; after elimination a true statement such as 0 = 0 remains, and every point on the line satisfies both equations.

Function definition

A rule assigning exactly one output to each input; a set of ordered pairs is a function when no two pairs share the same x-value with different y-values.

Vertical line test

A graph represents a function if and only if no vertical line crosses the graph more than once; a circle or sideways parabola therefore is not a function.

Linear function

A function whose graph is a nonvertical straight line, with constant rate of change; it can be written as y = mx + b or f(x) = mx + b.

Rate of change

The change in the output divided by the corresponding change in the input between two points; for a linear function it is constant and equals the slope.

Initial value

The output of a function when the input is zero; on a graph it is the y-intercept, and in y = mx + b it is b, representing the starting amount.

Transformation of a linear function

Changing y = mx + b: adding to b shifts the line up or down, while changing m rotates the line steeper or flatter about the y-intercept.

Checking a solution

Substitute the ordered pair into every original equation; a true statement in each confirms the solution, while a false one shows an error or no solution.