MTLE Mathematics (Grades 5-12) 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 expansion is non-terminating and non-repeating, as with the square root of 2 or pi.

Absolute value

The distance of a number from zero on the number line. It is never negative, and the absolute value of a minus b gives the distance between a and b.

Negative exponent rule

A base raised to a negative exponent equals the reciprocal of that base raised to the positive exponent: a to the power negative n equals one over a to the n, for nonzero a.

Percent change formula

Percent change equals the difference between new and old values divided by the old value, times one hundred. A positive result is an increase; a negative result is a decrease.

Proportional relationship

A relationship in which the ratio of two quantities stays constant, written y equals k times x. Its graph is a straight line passing through the origin.

Divisibility by 3

A whole number is divisible by 3 exactly when the sum of its digits is divisible by 3. The same digit-sum test works for divisibility by 9.

Slope formula

For two points, slope equals the change in y divided by the change in x, or (y2 minus y1) divided by (x2 minus x1). Vertical lines have undefined slope.

Quadratic formula

For ax squared plus bx plus c equals zero with a nonzero, x equals negative b plus or minus the square root of b squared minus 4ac, all divided by 2a.

Discriminant

The expression b squared minus 4ac from a quadratic equation. Positive gives two real roots, zero gives one repeated real root, and negative gives no real roots.

Perpendicular slopes

Two nonvertical lines are perpendicular exactly when their slopes are negative reciprocals, so the product of the two slopes equals negative one.

Vertical line test

A graph represents a function if and only if no vertical line intersects it more than once. A second intersection means one input has two outputs.

Exponential decay model

A function of the form y equals a times b to the x, where a is the starting value and b is between zero and one, giving repeated proportional decrease.

Inverse functions

Two functions are inverses when composing them in either order returns the input. Their graphs are mirror images across the line y equals x.

Parabola vertex x-coordinate

For a quadratic written as ax squared plus bx plus c, the vertex occurs at x equals negative b divided by 2a, giving the maximum or minimum value.

Pythagorean theorem

In any right triangle, the sum of the squares of the two legs equals the square of the hypotenuse: a squared plus b squared equals c squared.

Trapezoid area formula

The area of a trapezoid equals one half the height times the sum of the two parallel bases: A equals one half times (b1 plus b2) times h.

Cone volume formula

The volume of a cone is one third of the base area times the height: V equals one third times pi times r squared times h.

Interior angle sum

The interior angles of a polygon with n sides sum to (n minus 2) times 180 degrees. A quadrilateral therefore totals 360 degrees.

Similar figure area ratio

When two figures are similar with scale factor k, the ratio of their areas is k squared and the ratio of their volumes is k cubed.

Inscribed angle theorem

An angle inscribed in a circle measures half the intercepted arc. An angle inscribed in a semicircle is therefore a right angle of 90 degrees.

Dilation properties

A dilation with center and scale factor produces a similar image. Scale factors greater than one enlarge, between zero and one reduce, and the shape is preserved.

Median versus mean

The median is the middle ordered value and resists outliers; the mean is the arithmetic average and is pulled toward extreme values, so skewed data favors the median.

Standard deviation meaning

A measure of typical spread around the mean. A larger standard deviation means data points are more dispersed; a smaller one means they cluster near the mean.

Complement rule

The probability that an event does not occur equals one minus the probability that it does occur, since an event and its complement cover all outcomes.

Independent events multiplication rule

When two events are independent, the probability both occur equals the product of their individual probabilities, because neither outcome affects the other.

Conditional probability formula

The probability of A given B equals the probability of both A and B divided by the probability of B, provided B has nonzero probability.

Combination versus permutation

A combination counts unordered selections, while a permutation counts ordered arrangements. Permutations exceed combinations whenever more than one item is chosen.

Correlation and causation

A strong correlation between two variables shows a linear association only. It does not prove that changes in one variable cause changes in the other.

Counterexample role

A single case that satisfies the hypothesis but fails the conclusion disproves a general statement, which is why inductive conjectures need testing.

Contrapositive equivalence

The statement if not Q then not P is logically equivalent to if P then Q. Proving one proves the other, unlike the converse.

Polya problem-solving phases

Four stages: understand the problem, devise a plan, carry out the plan, and look back to verify the answer and consider generalizations or alternative methods.

Concrete-representational-abstract sequence

An instructional progression moving from physical manipulatives to pictorial representations to symbolic procedures, building understanding before fluency with algorithms.