Praxis Core Academic Skills for Educators: Mathematics (5733) flashcards

The distance along the curved edge of a sector of a circle, proportional to the central angle.

• Calculated as a fraction of the circle's full circumference.
• Formula: L = (θ/360°) × 2πr (for θ in degrees).
• Formula: L = θr (for θ in radians).

Memory trick: Arc length: a 'Fraction' of the 'Circumference'!

The amount of two-dimensional space enclosed by a trapezoid, calculated using its parallel bases and height.

• A trapezoid has at least one pair of parallel sides (bases).
• The height is the perpendicular distance between the parallel bases.
• Formula: A = (1/2) × (b1 + b2) × h.

Memory trick: Trapezoid area: 'Average' the 'Bases', then 'Height' for the 'Area'!

In a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

• Applies only to right-angled triangles.
• Formula: a² + b² = c², where c is the hypotenuse.
• Can be used to find an unknown side length if two sides are known.

Memory trick: A squared plus B squared equals C squared, for right triangles, it's always declared!

The total area of a two-dimensional shape made up of two or more simpler geometric shapes.

• Break down the complex figure into simpler, recognizable shapes (e.g., rectangles, triangles, circles).
• Calculate the area of each individual simpler shape.
• Add or subtract the areas as appropriate to find the total area of the composite figure.

Memory trick: Divide and conquer: Break it down, calculate parts, then add it up.

A line segment connecting two non-adjacent vertices of a rectangle, dividing it into two congruent right-angled triangles.

• The diagonals of a rectangle are equal in length.
• The diagonal forms the hypotenuse of a right triangle with the sides of the rectangle.
• Its length can be calculated using the Pythagorean theorem.

Memory trick: Remember, a rectangle's diagonal 'cuts' it into two 'right' triangles!

The total distance around the outside of a rectangle.

• Calculated by adding all four side lengths.
• Formula: P = 2 * (length + width) or P = 2L + 2W.
• Units are linear (e.g., feet, meters).

Memory trick: Around the rectangle, path you make, two lengths and two widths for goodness sake!

A branch of mathematics dealing with the relationships between the sides and angles of triangles, particularly right-angled triangles.

• SOH: Sine = Opposite / Hypotenuse.
• CAH: Cosine = Adjacent / Hypotenuse.
• TOA: Tangent = Opposite / Adjacent.

Memory trick: SOH CAH TOA – 'Some Old Hippie, Caught Another Hippie, Tripping On Acid'!

The amount of two-dimensional space enclosed within the boundary of a circle.

• Formula: A = πr², where r is the radius.
• The radius is half the diameter (r = d/2).
• Units are square units (e.g., cm², m²).

Memory trick: Pi R squared, for the area's quest, around the circle, it's the best!

The midpoint formula states that the coordinates of the midpoint M of a line segment joining two points (x1, y1) and (x2, y2) are M = ((x1 + x2)/2, (y1 + y2)/2). This can be rearranged to find an unknown endpoint.

• The midpoint is exactly halfway between two endpoints.
• To find an unknown endpoint, double the midpoint coordinates and subtract the known endpoint coordinates.
• Applies in 2D and 3D coordinate systems.

Memory trick: Midpoint is 'Middle', Endpoints are 'Edges'.

The amount of three-dimensional space occupied by a cylinder, calculated by the area of its base times its height.

• Formula: V = πr²h.
• The base is a circle, so its area is πr².
• Diameter (d) is twice the radius (r), so r = d/2.

Memory trick: Cylinder volume: 'Pi' 'R' 'Squared' times 'Height'!

The total measure of all interior angles within any four-sided polygon (quadrilateral).

• The sum of interior angles in any quadrilateral is always 360 degrees.
• This property applies to squares, rectangles, parallelograms, trapezoids, etc.
• Can be used to find an unknown angle if the others are known.

Memory trick: N minus two, times one-eighty, for any polygon's angle sum, it's weighty!

When a transversal intersects two parallel lines, corresponding angles are located in the same relative position at each intersection and are always congruent (equal in measure).

• Form an 'F' shape visually.
• One angle is interior, the other is exterior.
• They are on the same side of the transversal.
• Their measures are equal.

Memory trick: Parallel lines are 'Partners', Transversal 'Travels' through them.

The area of a portion of a circle enclosed by two radii and an arc.

• Formula: A = (θ/360°) * πr² (where θ is in degrees).
• Alternatively: A = (1/2)r²θ (where θ is in radians).
• It's a fraction of the total area of the circle.

Memory trick: Sector is a 'Slice' of the whole pie.

The total area of the outer surface of a cylinder, including its two circular bases and its lateral (curved) surface.

• Formula: SA = 2πrh + 2πr².
• 2πr² accounts for the area of the two circular bases.
• 2πrh accounts for the area of the lateral (curved) surface.

Memory trick: Two 'Circles' for top/bottom, one 'Rectangle' for the curved side.

The amount of three-dimensional space occupied by a sphere.

• Calculated using the formula V = (4/3)πr³.
• r represents the radius of the sphere.
• Units are cubic (e.g., cm³, m³).

Memory trick: Volume is 'Vast' inside, Surface Area is 'Skin' on the outside.

A formula used to find the distance between two points in a Cartesian coordinate system.

• Derived from the Pythagorean theorem.
• Formula: d = √((x2 - x1)² + (y2 - y1)²).
• The order of (x1,y1) and (x2,y2) does not affect the final distance.

Memory trick: X difference squared, Y difference squared, add them, root them, distance is shared!

The amount of three-dimensional space occupied by a cone.

• Formula: V = (1/3) * π * r² * h, where r is the radius and h is the height.
• The height must be perpendicular to the base.
• Units are cubic units (e.g., cubic feet, cubic meters).

Memory trick: Base area times height, then for pointy shapes, a third is just right!

The measure of the two-dimensional space enclosed by a triangle whose vertices are given as coordinates on a plane.

• If two vertices share a common x or y coordinate, that side can be used as the base.
• The height is the perpendicular distance from the third vertex to the line containing the base.
• Can also use the Shoelace Formula for any triangle: A = (1/2) |(x1y2 + x2y3 + x3y1) - (y1x2 + y2x3 + y3x1)|.

Memory trick: Find a 'Flat' base, then measure 'Up' for the height.

The area of a polygon with all sides and all angles equal, calculated using its perimeter and apothem.

• Formula: A = (1/2) * P * a, where P is the perimeter and a is the apothem.
• The apothem is the distance from the center to the midpoint of any side, perpendicular to that side.
• Perimeter P = n * s, where n is the number of sides and s is the side length.

Memory trick: Perimeter times Apothem, then 'Half' it for the area.

A formula used to find the coordinates of the point exactly halfway between two given points in a coordinate plane.

• For two points (x1, y1) and (x2, y2).
• Midpoint M = ((x1 + x2)/2, (y1 + y2)/2).
• Essentially averages the x-coordinates and the y-coordinates separately.

Memory trick: Midpoint: 'Average' the 'X's, 'Average' the 'Y's, 'Comma' in between!

The amount of two-dimensional space enclosed by a rhombus, calculated using its diagonals.

• A rhombus is a quadrilateral with all four sides equal in length.
• Its diagonals bisect each other at right angles.
• Area = (d1 × d2) / 2, where d1 and d2 are the lengths of the diagonals.

Memory trick: Rhombus area: 'D'iagonals 'D'ivide by '2' to get the 'Area'!

In a right-angled triangle, the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (legs, a and b): a² + b² = c².

• Only applies to right-angled triangles.
• Used to find an unknown side length when two sides are known.
• Hypotenuse is always the longest side, opposite the right angle.

Memory trick: Hypotenuse is 'High', the legs 'Lie' on the ground.

The amount of three-dimensional space occupied by a pyramid, calculated as one-third the product of its base area and height.

• Formula: V = (1/3) × A_base × h.
• A_base depends on the shape of the base (e.g., square, triangle).
• The height (h) is the perpendicular distance from the apex to the base.

Memory trick: Pyramid volume: 'Base Area' times 'Height', then 'Divide by 3'!

The angle formed by the horizontal line of sight and the upward line of sight to an object.

• Often involves a right-angled triangle.
• Related to trigonometric ratios: sine, cosine, tangent.
• Use inverse trigonometric functions (arcsin, arccos, arctan) to find the angle.

Memory trick: SOH CAH TOA: Say it often, helps you know, which ratio for which side to go!

An interior angle is an angle inside a polygon, formed by two adjacent sides. An exterior angle is formed by one side of a polygon and the extension of an adjacent side.

• An interior angle and its adjacent exterior angle are supplementary (add to 180°).
• The sum of all exterior angles of any convex polygon is 360°.
• For a regular n-sided polygon, each interior angle = (n-2) * 180 / n.

Memory trick: Interior is 'In', Exterior is 'Out', Together they make a 'Straight' line.

The total area of all the faces (surfaces) of a three-dimensional rectangular prism.

• A rectangular prism has 6 faces (3 pairs of identical rectangular faces).
• Formula: SA = 2(lw + lh + wh).
• Where l=length, w=width, h=height.

Memory trick: Surface Area: 'Two' of 'Length-Width', 'Length-Height', and 'Width-Height'!

The amount of three-dimensional space occupied by a prism whose bases are triangles.

• Formula: V = B × h, where B is the area of the triangular base and h is the height of the prism.
• Area of triangular base (B) = (1/2) × base_triangle × height_triangle.
• The height of the prism is the perpendicular distance between its two triangular bases.

Memory trick: Base Area first, then 'Stack' it up with height.

A transformation that changes the size of a figure by a specific scale factor, with the origin as the center of dilation.

• Each coordinate (x, y) is multiplied by the scale factor 'k'.
• Transformed point becomes (kx, ky).
• If k > 1, the figure gets larger; if 0 < k < 1, it gets smaller.

Memory trick: Dilation from Origin: 'Scale' up 'X' and 'Y' equally!

Using the standard equation of a circle (x-h)² + (y-k)² = r² to solve for unknown dimensions within a circular context.

• For a circle centered at the origin (0,0), the equation simplifies to x² + y² = r².
• This is directly related to the Pythagorean theorem.
• Can be used to find a coordinate (height or width) given another coordinate and the radius.

Memory trick: X squared, plus Y squared, equals R squared, for a circle at the center, it's declared!

The total two-dimensional space enclosed within the boundaries of a regular six-sided polygon.

• A regular hexagon can be decomposed into 6 congruent equilateral triangles.
• If 's' is the side length, the radius of the circumcircle is also 's'.
• Formula: Area = (3√3/2)s², or 6 * (√3/4)s².

Memory trick: Divide and conquer, into triangles you'll see, for symmetric shapes, area is key!

The total area of all the faces (surfaces) that enclose a rectangular prism (box).

• A rectangular prism has 6 faces (3 pairs of identical faces).
• Formula: SA = 2(LW + LH + WH), where L=length, W=width, H=height.
• Units are square units (e.g., cm², m²).

Memory trick: Unfold the box, count all the sides, sum their areas, where the surface hides!

The distance around the edge of a circle.

• Calculated using C = πd or C = 2πr.
• π (pi) is approximately 3.14159.
• Units are linear (e.g., feet, meters).

Memory trick: Circumference is the 'circle' around, Area is the 'apple' inside.

The amount of two-dimensional space enclosed within the boundaries of a triangle.

• Formula: A = (1/2) * base * height.
• The height must be perpendicular to the base.
• Units are square units (e.g., square inches, square meters).

Memory trick: Half of base times height, for a triangle's area, is just right!

The angle formed by two adjacent sides inside a regular polygon, where all interior angles are equal.

• Sum of interior angles: (n-2) × 180°.
• For a regular polygon, all interior angles are equal.
• One interior angle = (Sum of interior angles) / n.

Memory trick: Count the 'n' sides, subtract '2' for triangles, then '180' for degrees!

The middle value in a dataset when the values are arranged in ascending or descending order.

• Not affected by outliers.
• If n is odd, median is the (n+1)/2-th value.
• If n is even, median is the average of the two middle values.

Memory trick: Central tendency: mean is sum, median is middle, mode is most.

The range of a data set is the difference between the maximum and minimum values.

• It is a measure of statistical dispersion.
• Easy to calculate but sensitive to outliers.
• Provides a quick sense of the spread of data.

Memory trick: Range is the edge-to-edge span.

The likelihood of a specific outcome occurring when there are multiple independent events.

• List all possible outcomes (sample space).
• Count favorable outcomes.
• Probability = (Favorable Outcomes) / (Total Outcomes).

Memory trick: Multiple events: list all paths, count the good ones.

Probability is a measure of the likelihood that an event will occur, expressed as a number between 0 and 1.

• Calculated as (Number of favorable outcomes) / (Total number of possible outcomes).
• A probability of 0 means the event is impossible, 1 means it's certain.
• Applies to situations with random outcomes.

Memory trick: Favorable outcomes over Total possibilities.

A permutation is an arrangement of items from a set where the order of arrangement matters.

• Formula: P(n, k) = n! / (n-k)!, where n is total items and k is items to arrange.
• If all items are arranged (k=n), the formula simplifies to n!.
• Used when determining sequences, rankings, or ordered selections.

Memory trick: Permutation: 'Position' always counts.

Extracting information and drawing conclusions from data represented visually in a bar graph.

• Each bar represents a category's value.
• Height or length of bar indicates quantity.
• Useful for comparing discrete categories.

Memory trick: Bars show counts, percentages need total.

The sum of all values in a dataset divided by the number of values in the dataset.

• Most common measure of central tendency.
• Sensitive to outliers.
• Represents the 'balancing point' of the data.

Memory trick: Central tendency: mean is sum, median is middle, mode is most.

The probability of two or more independent events occurring, where the outcome of one event does not affect the outcome of the others.

• Events are 'with replacement' or naturally separate.
• Sample space remains constant for each event.
• Calculated by multiplying individual probabilities.

Memory trick: Independent: Each event stands alone, multiply their chances.

The average outcome of a random variable over a large number of trials, calculated as the sum of (each outcome * its probability).

• For n trials with probability p, E(X) = n * p.
• Represents the long-run average.
• Does not guarantee that outcome will occur in a single trial.

Memory trick: Expected value: what you 'expect' on average, probability times trials.

A measure of the amount of variation or dispersion of a set of values, indicating how spread out the numbers are from the mean.

• Square root of the variance.
• Expressed in the same units as the data.
• A low standard deviation means values are close to the mean; high means spread out.

Memory trick: Dispersion: spread out, how far from the mean.

The value(s) that appear most frequently in a dataset.

• A dataset can have one mode (unimodal), multiple modes (multimodal), or no mode.
• Useful for categorical data.
• Not affected by outliers.

Memory trick: Central tendency: mean is sum, median is middle, mode is most.

A combination is a selection of items from a set where the order of selection does not matter.

• Formula: C(n, k) = n! / (k! * (n-k)!).
• Used when forming groups, committees, or selecting items where arrangement is irrelevant.
• Always results in a smaller number than permutations for the same n and k (if k > 1).

Memory trick: Combination: 'Choose' (order doesn't matter). Permutation: 'Position' (order matters).

For a normal distribution, approximately 68% of data falls within 1 standard deviation, 95% within 2 standard deviations, and 99.7% within 3 standard deviations of the mean.

• Applies specifically to data that follows a normal (bell-shaped) distribution.
• Standard deviation (σ) measures the spread of data around the mean (μ).
• Used to estimate probabilities or proportions within certain ranges without complex calculations.

Memory trick: Normal bell curve, 68-95-99.7 for the steps.

A pie chart is a circular statistical graphic divided into slices to illustrate numerical proportion.

• Each slice represents a category's proportion of the whole.
• The sum of all slice percentages must equal 100%.
• The size of each slice is proportional to the quantity it represents.

Memory trick: Whole pie is 100, slices are parts.

The probability of two or more events occurring where the outcome of one event affects the outcome of the subsequent events.

• Events are 'without replacement'.
• The sample space changes after each event.
• Calculated by multiplying conditional probabilities.

Memory trick: Dependent means the next choice depends on the last.

For two independent events A and B, the probability that both A and B occur is the product of their individual probabilities: P(A and B) = P(A) * P(B).

• Events are independent if the outcome of one does not influence the other.
• Common examples include coin flips, dice rolls, or drawing with replacement.
• The keyword 'AND' often suggests multiplying probabilities for independent events.

Memory trick: Separate chances, multiply for both.

The probability of an event not occurring is 1 minus the probability that it does occur.

• P(not A) = 1 - P(A).
• The sum of the probability of an event and its complement is always 1.
• Useful for calculating 'at least one' type problems.

Memory trick: What you want, or everything else.

The number of ways to choose items from multiple distinct groups where the order of selection within each group does not matter.

• Use the combination formula C(n, k) for each group.
• Multiply the results of each group's combination calculation.
• Order of selection does not matter.

Memory trick: Separate groups, combine choices with multiplication.

Two events are independent if the outcome of one does not affect the outcome of the other.

• P(A and B) = P(A) * P(B).
• Common in 'with replacement' scenarios or events with no causal link.
• The probability of the second event is the same regardless of the first.

Memory trick: No link, just multiply.

The number of distinct ways to arrange a set of 'n' unique items in a specific order.

• Order of arrangement matters.
• Calculated using factorial notation (n!).
• Used when all items are selected and arranged.

Memory trick: Permutation: Position matters; Combination: Committee, order doesn't.

To find a fraction of a whole number, multiply the fraction by the whole number. To find the remainder, subtract this result from the whole.

• Represents a part of a total.
• Numerator is the part, denominator is the whole.
• Can be used to calculate specific quantities within a larger group.

Memory trick: Slice the whole, then count what's left.

A proportion is an equation stating that two ratios are equal. To solve for an unknown in a proportion, use cross-multiplication: multiply the numerator of one ratio by the denominator of the other, and set the products equal.

• Set up two equal ratios.
• Cross-multiply the terms.
• Solve the resulting linear equation for the unknown variable.

Memory trick: Ratios equal, cross-multiply, then solve for the unknown guy!

Sequential fractional changes involve calculating a fraction of a total, then calculating a fraction of the *remaining* amount. Each step alters the base for the next calculation.

• Each fraction applies to the current remaining quantity, not the original total.
• Requires careful step-by-step calculation.
• Common in resource allocation or consumption problems.

Memory trick: First fraction, then what's left, then another fraction of *that* amount, then what's truly left.

To convert a temperature from Celsius to Fahrenheit, use the formula F = (9/5)C + 32. Multiply the Celsius temperature by 9/5, then add 32.

• C represents Celsius temperature, F represents Fahrenheit temperature.
• The fraction 9/5 can also be written as 1.8.
• Always perform multiplication before addition according to order of operations.

Memory trick: Celsius goes to Fahrenheit, multiply then add.

When dealing with quantities defined by ranges (minimum and maximum values), calculations for combined minimums or maximums involve adding the respective bounds.

• To find maximum combined value, add individual maximums.
• To find minimum combined value, add individual minimums.
• This principle applies to addition and subtraction of estimated values.

Memory trick: For max, add the max; for min, add the min. Don't mix 'em in!

Exponential growth occurs when a quantity increases by a constant factor over equal intervals of time. It is often modeled by A = P * r^(t/k).

• P = initial amount
• r = growth factor (e.g., 2 for doubling, 3 for tripling)
• t = total time
• k = time for one growth period

Memory trick: Exponent: Power up, grow fast, like a rocket launching.