GTLE Mathematics Specialism flashcards

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

Rational number definition

A number expressible as a quotient of two integers with a nonzero denominator. Its decimal form either terminates or repeats, distinguishing it from irrational numbers.

Scientific notation form

A product of a coefficient between 1 and 10 and a power of ten. Multiplying numbers adds exponents; dividing subtracts exponents after dividing coefficients.

Absolute value as distance

The expression |x - c| gives the distance between x and c on a number line. The inequality |x - c| < d describes all points within d units of c.

Successive percent change

Percent changes applied in sequence multiply rather than add. A 20% increase followed by a 20% decrease produces a net 4% decrease from the original amount.

Quadratic formula and discriminant

The roots of ax^2 + bx + c = 0 are x = (-b plus or minus the square root of b^2 - 4ac) divided by 2a. The discriminant's sign gives two, one, or no real roots.

Slope of a line

Slope equals the change in y divided by the change in x between two points. It appears as m in y = mx + b and as the constant rate in linear models.

Function definition

A relation that assigns exactly one output to each input. A graph represents a function only if no vertical line crosses it more than once.

Exponential versus linear growth

Linear models change by a constant difference per step; exponential models change by a constant ratio. Exponential functions take the form y = a(1 + r)^t.

Inverse function property

Functions f and g are inverses when f(g(x)) = x and g(f(x)) = x. Their graphs are reflections of each other across the line y = x.

Logarithm product and power rules

The log of a product equals the sum of logs, and the log of a power equals the exponent times the log. These rules let you solve exponential equations.

Pythagorean theorem

In a right triangle, the squares of the two legs sum to the square of the hypotenuse. Common integer triples include 3-4-5 and 5-12-13.

Similar figure ratios

Corresponding angles are equal and corresponding sides are proportional. The ratio of areas equals the square of the scale factor, and the ratio of volumes equals its cube.

Interior angle sum of polygon

A polygon with n sides has interior angles summing to (n - 2) times 180 degrees. Each exterior angle of a regular n-gon measures 360 divided by n degrees.

Distance and midpoint formulas

Distance between two points is the square root of the sum of squared coordinate differences. The midpoint averages the x-coordinates and averages the y-coordinates.

Volume of cone and sphere

A cone's volume is one third of the base area times height. A sphere's volume is four thirds of pi times the cube of the radius.

Mean versus median

The mean divides the sum of values by the count; the median is the middle ordered value. Outliers shift the mean far more than the median.

Standard deviation meaning

A measure of typical distance of data values from the mean. In a normal distribution about 68% of values fall within one standard deviation.

Independent events rule

Two events are independent when the occurrence of one does not change the probability of the other. Then the probability of both equals the product of their probabilities.

Conditional probability formula

The probability of A given B equals the probability of both divided by the probability of B. Events are independent exactly when this equals the probability of A.

Interquartile range and outliers

The interquartile range is the third quartile minus the first. Values beyond 1.5 interquartile ranges above the third or below the first quartile are flagged as outliers.

Correlation versus causation

Correlation measures the strength and direction of a linear association between two variables. A strong correlation alone never establishes that one variable causes the other.

Contrapositive equivalence

The contrapositive of 'if P then Q' is 'if not Q then not P', and it is logically equivalent to the original. Converse and inverse are not equivalent to it.

Counterexample role in proof

One example that satisfies the hypothesis but fails the conclusion disproves a universal statement. No number of confirming examples can prove one.

Mathematical induction steps

Prove the statement for a base case, assume it holds for an arbitrary case k, then show it must hold for k + 1. This establishes it for all integers beyond the base.

Negating universal quantifier

The negation of 'for all x, P(x)' is 'there exists an x such that not P(x)'. The negation of an existential statement becomes a universal statement.

Polya problem-solving phases

Understand the problem, devise a plan, carry out the plan, and look back. The final phase checks the result and considers whether the method generalizes.

Estimation as answer check

Rounding each quantity to one significant figure before computing gives a rough expected result. Comparing it with the exact answer reveals order-of-magnitude setup errors.

Classifying student errors

Errors are computational, procedural, or conceptual. Conceptual errors reflect a flawed underlying idea and require re-teaching with models rather than repetitive practice.

Concrete to abstract sequence

Instruction moves from physical manipulatives to pictorial representations to symbolic notation. Each stage anchors the next so procedures connect to meaning.

Formative versus summative assessment

Formative assessment happens during instruction to adjust teaching, such as exit tickets. Summative assessment measures achievement after instruction, such as a unit test.

Productive struggle in mathematics

Students work on a challenging task before direct explanation, building persistence and sense-making. The teacher scaffolds with questions rather than supplying the solution.

Law of sines and cosines

In any triangle, side-to-sine ratios are equal. The law of cosines, c squared equals a squared plus b squared minus 2ab cosine C, extends Pythagoras to all triangles.