Your FREE Praxis 5003 Flashcards 2026 – 200+ Cards

The value a digit has because of its position in a number. In $4{,}873$ the digit $8$ means $8 \times 100 = 800$.

A place-value system where each place is $10$ times the place to its right, so digits use powers of $10$: ones, tens, hundreds, thousands, …

Writing a number as the sum of each digit times its place value, e.g. $3{,}506 = 3000 + 500 + 0 + 6$, or with powers: $3\times10^3 + 5\times10^2 + 6\times10^0$.

$10^1=10$, $10^2=100$, $10^3=1000$. The exponent equals the number of zeros and tells how many places the value shifts left.

Multiplying by $10^n$ shifts every digit $n$ places left (appends $n$ zeros to a whole number): $42 \times 10^3 = 42{,}000$.

Dividing by $10^n$ shifts every digit $n$ places right, moving the decimal point left: $350 \div 10^2 = 3.5$.

Breaking a number into parts by place value (or recombining them), e.g. $47 = 40 + 7$ or $47 = 30 + 17$, to make computation easier.

Replacing a number with a nearby value to a chosen place. Look at the next digit: $5$ or more rounds up, $4$ or less rounds down. $4{,}873 \to 4{,}900$ to the nearest hundred.

The counting numbers together with zero: $0, 1, 2, 3, \dots$ — no fractions or negatives.

The whole numbers and their opposites: $\dots, -2, -1, 0, 1, 2, \dots$ — no fractions or decimals.

A number that can be written as a ratio of two integers $\dfrac{a}{b}$ with $b \neq 0$. Every terminating or repeating decimal is rational.

A real number that cannot be written as a ratio of integers; its decimal never ends and never repeats, e.g. $\sqrt{2}$ and $\pi$.

A line on which each point represents a number, increasing left to right. Used to compare, order, and model operations with integers and rationals.

A number's distance from zero on the number line, always non-negative: $|-7| = 7$ and $|7| = 7$.

The number the same distance from $0$ on the other side; their sum is $0$. The opposite of $5$ is $-5$, since $5 + (-5) = 0$.

Arranging numbers by size using $<, >, =$. On a number line, a number farther right is greater: $-3 < -1 < 2$.

A number $\dfrac{a}{b}$ showing $a$ parts out of $b$ equal parts; $a$ is the numerator and $b$ the denominator.

The top number of a fraction; it counts how many equal parts are taken. In $\dfrac{3}{4}$ the numerator is $3$.

The bottom number of a fraction; it tells how many equal parts make one whole. In $\dfrac{3}{4}$ the denominator is $4$.

Fractions that name the same value: $\dfrac{1}{2} = \dfrac{2}{4} = \dfrac{3}{6}$. Multiply or divide numerator and denominator by the same nonzero number.

Writing a fraction in lowest terms by dividing numerator and denominator by their GCF: $\dfrac{8}{12} = \dfrac{8 \div 4}{12 \div 4} = \dfrac{2}{3}$.

A fraction whose numerator is less than its denominator, so its value is between $0$ and $1$, e.g. $\dfrac{3}{5}$.

A fraction whose numerator is greater than or equal to its denominator, so its value is $\geq 1$, e.g. $\dfrac{7}{4}$.

A whole number plus a proper fraction, e.g. $1\tfrac{3}{4}$. It equals the improper fraction $\dfrac{7}{4}$.

Rewrite with a common denominator, then add the numerators: $\dfrac{1}{3} + \dfrac{1}{6} = \dfrac{2}{6} + \dfrac{1}{6} = \dfrac{3}{6} = \dfrac{1}{2}$.

Rewrite with a common denominator, then subtract numerators: $\dfrac{3}{4} - \dfrac{1}{6} = \dfrac{9}{12} - \dfrac{2}{12} = \dfrac{7}{12}$.

Multiply numerators and multiply denominators, then simplify: $\dfrac{2}{3} \times \dfrac{3}{4} = \dfrac{6}{12} = \dfrac{1}{2}$.

Multiply by the reciprocal of the divisor: $\dfrac{2}{3} \div \dfrac{4}{5} = \dfrac{2}{3} \times \dfrac{5}{4} = \dfrac{10}{12} = \dfrac{5}{6}$.

The number you multiply by to get $1$; flip the fraction. The reciprocal of $\dfrac{4}{5}$ is $\dfrac{5}{4}$, and $\dfrac{4}{5} \times \dfrac{5}{4} = 1$.

A shared multiple of two fractions' denominators used to add or subtract them. The least common denominator (LCD) is their LCM.

A base-ten number with digits to the right of a decimal point representing tenths, hundredths, thousandths, … e.g. $0.25 = \dfrac{25}{100}$.

After the point: tenths $(10^{-1})$, hundredths $(10^{-2})$, thousandths $(10^{-3})$. In $0.347$, the $4$ means $4 \times \dfrac{1}{100}$.

Divide the numerator by the denominator: $\dfrac{3}{8} = 3 \div 8 = 0.375$.

Write the digits over the matching power of ten, then simplify: $0.6 = \dfrac{6}{10} = \dfrac{3}{5}$.

A decimal that ends, such as $0.75$. A fraction terminates when, in lowest terms, its denominator's only prime factors are $2$ and $5$.

A decimal with a digit or block that repeats forever, written with a bar: $\dfrac{1}{3} = 0.\overline{3}$. Repeating decimals are rational.

Line up the decimal points (align place values), then add or subtract as with whole numbers: $2.30 + 0.45 = 2.75$.

Multiply ignoring the points, then place the point so the product has as many decimal places as the factors combined: $0.3 \times 0.4 = 0.12$.

Shift the divisor's point to make it a whole number, shift the dividend's point the same amount, then divide: $1.2 \div 0.4 = 12 \div 4 = 3$.

A ratio out of $100$; $\%$ means "per hundred." So $25\% = \dfrac{25}{100} = 0.25$.

Divide by $100$ (move the decimal point two places left): $7\% = 0.07$ and $150\% = 1.5$.

Multiply by $100$ (move the point two places right) and add $\%$: $0.32 = 32\%$.

Multiply the number by the percent as a decimal: $20\%$ of $80 = 0.20 \times 80 = 16$.

The rise as a percent of the original: $\dfrac{\text{increase}}{\text{original}} \times 100\%$. From $50$ to $60$: $\dfrac{10}{50} = 20\%$.

The drop as a percent of the original: $\dfrac{\text{decrease}}{\text{original}} \times 100\%$. From $80$ to $60$: $\dfrac{20}{80} = 25\%$.

A comparison of two quantities by division, written $a:b$ or $\dfrac{a}{b}$. "$3$ cats to $2$ dogs" is $3:2$.

A ratio comparing quantities with different units, such as $60$ miles per $1$ hour. A unit rate has a denominator of $1$.

A rate per one unit, found by dividing: $\$12$ for $3$ lb is $\dfrac{12}{3} = \$4$ per lb.

An equation stating two ratios are equal: $\dfrac{a}{b} = \dfrac{c}{d}$. Solve by cross multiplying: $ad = bc$.

To solve $\dfrac{a}{b} = \dfrac{c}{d}$, set $a \times d = b \times c$. From $\dfrac{3}{4} = \dfrac{x}{8}$: $4x = 24$, so $x = 6$.

Scaling quantities so a ratio stays constant. If $2$ pens cost $\$3$, then $6$ pens cost $\$9$ (each tripled).

A whole number that divides another with no remainder. The factors of $12$ are $1, 2, 3, 4, 6, 12$.

The product of a number with a whole number. Multiples of $4$ are $4, 8, 12, 16, \dots$

A whole number greater than $1$ with exactly two factors, $1$ and itself, e.g. $2, 3, 5, 7, 11$.

A whole number greater than $1$ with more than two factors, e.g. $4, 6, 8, 9$. $12 = 2 \times 2 \times 3$.

Writing a number as a product of primes: $60 = 2^2 \times 3 \times 5$. Every composite has one unique prime factorization.

The largest factor shared by two numbers. For $12$ and $18$, the GCF is $6$. Used to simplify fractions.

The smallest positive multiple shared by two numbers. For $4$ and $6$, the LCM is $12$. Used to find common denominators.

Same signs: add and keep the sign. Different signs: subtract and take the sign of the larger absolute value: $-8 + 3 = -5$.

Add the opposite: $a - b = a + (-b)$. So $4 - (-6) = 4 + 6 = 10$.

Same signs give a positive product; different signs give a negative product: $(-3)(-4) = 12$ and $(-3)(4) = -12$.

Same signs give a positive quotient; different signs give a negative quotient: $\dfrac{-20}{4} = -5$.

PEMDAS: Parentheses, Exponents, Multiplication/Division (left to right), then Addition/Subtraction (left to right): $2 + 3 \times 4 = 14$.

A small raised number telling how many times to multiply the base by itself: $2^3 = 2 \times 2 \times 2 = 8$.

A number multiplied by itself, written with exponent $2$: $6^2 = 36$. The result is a perfect square.

A value that, squared, gives the number: $\sqrt{49} = 7$ because $7^2 = 49$.

A whole number equal to an integer squared: $1, 4, 9, 16, 25, 36, \dots$

Order does not change the result for addition or multiplication: $a + b = b + a$ and $a \times b = b \times a$.

Multiplication distributes over addition: $a(b+c) = ab + ac$. So $3(4+5) = 12 + 15 = 27$.

Operations that undo each other: addition and subtraction; multiplication and division. They are used to check work and solve equations.

The amount left over after whole-number division. $17 \div 5 = 3$ remainder $2$, since $5 \times 3 + 2 = 17$.

Use a common denominator or cross multiply. $\dfrac{3}{5}$ vs $\dfrac{2}{3}$: $3\times3 = 9$ vs $2\times5 = 10$, so $\dfrac{3}{5} < \dfrac{2}{3}$.

Three ways to write the same value: $\dfrac{1}{4} = 0.25 = 25\%$. Convert freely among them.

Look at the digit just past the target place: $5$ or more rounds up. $3.146$ to the nearest hundredth is $3.15$.

Multiply whole by denominator, add the numerator, keep the denominator: $2\tfrac{1}{3} = \dfrac{2\times3+1}{3} = \dfrac{7}{3}$.

Divide numerator by denominator; the quotient is the whole, the remainder is the new numerator: $\dfrac{7}{3} = 2\tfrac{1}{3}$.

$\dfrac{\text{new} - \text{old}}{\text{old}} \times 100\%$; positive means increase, negative means decrease.

Identify the question, list known and unknown quantities, choose operations in order, solve, then check the answer for reasonableness.

Relationship $\text{part} = \text{percent} \times \text{whole}$. If $15$ is $25\%$ of a number, the whole is $15 \div 0.25 = 60$.

Ratios that simplify to the same value: $2:3 = 4:6 = 6:9$. A ratio table lists equivalent ratios for proportional reasoning.

A letter or symbol standing for an unknown or changing number, such as $x$ in $x + 3 = 7$.

A combination of numbers, variables, and operations with no equals sign, e.g. $3x + 5$. It can be evaluated, not solved.

A statement that two expressions are equal, using $=$, e.g. $2x + 1 = 9$. Solving finds the value(s) of the variable that make it true.

The number multiplied by a variable in a term. In $7x$, the coefficient is $7$.

A term with a fixed value and no variable, such as the $5$ in $3x + 5$.

A single number, variable, or product separated by $+$ or $-$ in an expression. $4x + 2y - 7$ has three terms.

Terms with the same variable and exponent, which can be combined: $3x + 5x = 8x$. $3x$ and $3x^2$ are not like terms.

Adding or subtracting the coefficients of like terms to simplify: $2x + 7 + 4x - 3 = 6x + 4$.

Substituting given values for the variables and computing. For $3x + 2$ at $x = 4$: $3(4) + 2 = 14$.

Multiply the outside factor by each term inside: $4(x + 3) = 4x + 12$.

Apply the inverse operation to both sides. For $x + 5 = 12$, subtract $5$: $x = 7$.

Undo addition/subtraction first, then multiplication/division. For $2x + 3 = 11$: subtract $3$ to get $2x = 8$, then divide to get $x = 4$.

Collect variable terms on one side and constants on the other. For $5x - 2 = 3x + 6$: $2x = 8$, so $x = 4$.

Use the operation that undoes another to isolate the variable: subtract to undo addition, divide to undo multiplication.

Doing the same operation to both sides keeps an equation balanced: if $a = b$, then $a + c = b + c$ and $ac = bc$.

A statement comparing expressions with $<, >, \leq, \geq$, e.g. $x + 2 > 5$. Its solution is a range of values.

Solve like an equation, but reverse the inequality sign when multiplying or dividing by a negative: $-2x < 6$ gives $x > -3$.

An open circle for $<$ or $>$, a closed circle for $\leq$ or $\geq$, with shading toward the solution direction.

Translate phrases into symbols: "$5$ more than twice a number" is $2n + 5$; "the quotient of $x$ and $3$" is $\dfrac{x}{3}$.

A predictable sequence of numbers or figures formed by a rule, e.g. add $4$ each time: $2, 6, 10, 14, \dots$

A sequence with a constant difference between terms. $3, 7, 11, 15, \dots$ adds $4$ each time (common difference $4$).

The fixed amount added to get from one term to the next in an arithmetic sequence; in $5, 9, 13, \dots$ it is $4$.

A sequence with a constant ratio between terms. $2, 6, 18, 54, \dots$ multiplies by $3$ each time (common ratio $3$).

The fixed factor multiplied to get the next term in a geometric sequence; in $3, 6, 12, \dots$ it is $2$.

A growing pattern made of shapes or dots whose count follows a rule, often modeled by an expression like $2n + 1$.

A formula giving any term from its position. For $5, 8, 11, \dots$ the rule is $3n + 2$, so the $10\text{th}$ term is $32$.

A relationship that assigns exactly one output to each input. $y = 2x$ gives one $y$ for every $x$.

The value put into a function and the value it produces. For the rule "multiply by $3$," input $4$ gives output $12$.

A table pairing inputs with outputs to reveal the rule. If $1\!\to\!4$, $2\!\to\!7$, $3\!\to\!10$, the rule is $3x + 1$.

The input you choose, usually $x$, plotted on the horizontal axis.

The output that depends on the input, usually $y$, plotted on the vertical axis.

A relationship that graphs as a straight line and changes by a constant rate. Its equation has the form $y = mx + b$.

The steepness of a line, equal to rise over run: $m = \dfrac{y_2 - y_1}{x_2 - x_1}$. It is the constant rate of change.

Where a line crosses the $y$-axis, the value of $y$ when $x = 0$. In $y = 2x + 3$, the $y$-intercept is $3$.

The line equation $y = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept.

A quantity that increases or decreases by the same amount per step, which produces a linear graph; the slope measures it.

A linear relationship through the origin, $y = kx$. The constant $k$ is the unit rate (constant of proportionality).

The fixed ratio $k = \dfrac{y}{x}$ in $y = kx$. If $3$ items cost $\$12$, then $k = 4$ dollars per item.

A positive slope rises left to right; a negative slope falls. A horizontal line has slope $0$; a vertical line's slope is undefined.

Use inverse operations to isolate the wanted variable. From $A = \ell w$, solve for $w$: $w = \dfrac{A}{\ell}$.

Substitute the value back into the original equation; both sides should be equal. For $x = 4$ in $2x + 1 = 9$: $9 = 9$. \checkmark

Define a variable, translate the relationships into an equation, then solve. "$3$ less than $4$ times $n$ is $17$": $4n - 3 = 17$.

Use the distributive property and combine like terms: $2(x + 3) + 4x = 2x + 6 + 4x = 6x + 6$.

Find the rule from the given terms, then apply it. For $1, 4, 9, 16, \dots$ (perfect squares), the next term is $25$.

A rule that defines each term from the previous one, e.g. "start at $2$, add $5$" gives $2, 7, 12, 17, \dots$

A rule giving a term directly from its position $n$, without earlier terms, e.g. $a_n = 5n - 3$.

Plot the $y$-intercept, then use the slope (rise over run) to find more points, and connect them with a straight line.

Expressions that have the same value for every input, such as $2(x + 3)$ and $2x + 6$.

$<$ less than, $>$ greater than, $\leq$ less than or equal to, $\geq$ greater than or equal to.

Linear data changes by a constant amount (straight-line graph); nonlinear changes by varying amounts, like $y = x^2$.

Evaluate inside grouping symbols and exponents before multiplying/dividing, then adding/subtracting, even with variables present.

"At least" means $\geq$, "at most" means $\leq$, "more than" means $>$, and "fewer than" means $<$.

Distribute, combine like terms, then isolate the variable. For $3(x - 2) = 12$: $3x - 6 = 12$, $3x = 18$, $x = 6$.

Find how the output changes per unit input (the slope) and the value at $x = 0$ (intercept) to write $y = mx + b$.

Use $y = mx + b$, where $m$ is the per-unit rate and $b$ is a starting amount: a $\$5$ fee plus $\$2$/hour is $y = 2x + 5$.

Replacing a variable with a value or another expression, e.g. putting $x = 3$ into $x^2 + 1$ to get $10$.

An ordered pair $(x, y)$ that satisfies the line's equation. $(2, 7)$ lies on $y = 2x + 3$ because $7 = 2(2)+3$.

An expression has no equals sign and is simplified or evaluated; an equation sets two expressions equal and is solved.

Describing a pattern with a variable expression: $n$ tables seat $4n$ people, or $4n + 2$ if the ends seat extra.

$\dfrac{\text{change in } y}{\text{change in } x}$. From $(1, 5)$ to $(3, 11)$: $\dfrac{11 - 5}{3 - 1} = 3$.

A location with no size, named with a capital letter such as point $A$; the basic building block of geometry.

A straight path extending forever in both directions, with no thickness, named by two points, e.g. $\overleftrightarrow{AB}$.

Part of a line between two endpoints, with a measurable length, written $\overline{AB}$.

A figure formed by two rays sharing an endpoint (vertex), measured in degrees from $0^\circ$ to $360^\circ$.

An angle measuring less than $90^\circ$.

An angle measuring exactly $90^\circ$, often marked with a small square.

An angle measuring more than $90^\circ$ but less than $180^\circ$.

Two angles whose measures add to $90^\circ$. If one is $35^\circ$, its complement is $55^\circ$.

Two angles whose measures add to $180^\circ$. If one is $110^\circ$, its supplement is $70^\circ$.

The interior angles of any triangle add to $180^\circ$. If two are $60^\circ$ and $70^\circ$, the third is $50^\circ$.

A closed plane figure made of straight line segments, such as a triangle, quadrilateral, pentagon, or hexagon.

A four-sided polygon; its interior angles sum to $360^\circ$. Examples include squares, rectangles, and trapezoids.

A quadrilateral with both pairs of opposite sides parallel and equal; opposite angles are equal.

A parallelogram with four right angles. Its area is $A = \ell \times w$.

A rectangle with all four sides equal. Its area is $A = s^2$ and perimeter $P = 4s$.

Equilateral (all sides equal), isosceles (two equal), scalene (none equal).

Acute (all angles $< 90^\circ$), right (one $90^\circ$ angle), obtuse (one angle $> 90^\circ$).

The total distance around a polygon, found by adding all side lengths. A $4$ by $6$ rectangle has perimeter $2(4+6) = 20$.

The amount of surface a flat figure covers, measured in square units. A $4$ by $6$ rectangle has area $24$ square units.

Half the base times the height: $A = \dfrac{1}{2} b h$. A triangle with base $8$ and height $5$ has area $20$.

The set of all points a fixed distance (the radius) from a center. Its key parts are the radius, diameter, and circumference.

The radius $r$ is center to edge; the diameter $d$ crosses through the center: $d = 2r$.

The distance around a circle: $C = 2\pi r = \pi d$. With $r = 3$, $C = 6\pi \approx 18.8$.

$A = \pi r^2$. With radius $5$, the area is $25\pi \approx 78.5$ square units.

The constant ratio of a circle's circumference to its diameter, $\pi \approx 3.14159$, an irrational number.

The space a solid occupies, measured in cubic units. For a rectangular prism, $V = \ell \times w \times h$.

Length times width times height: $V = \ell w h$. A $2 \times 3 \times 4$ box holds $24$ cubic units.

In a right triangle, $a^2 + b^2 = c^2$, where $c$ is the hypotenuse. Legs $3$ and $4$ give $c = 5$.

Inches, feet, yards, miles: $12$ in $= 1$ ft, $3$ ft $= 1$ yd, $5280$ ft $= 1$ mi.

Millimeter, centimeter, meter, kilometer, related by powers of $10$: $100$ cm $= 1$ m, $1000$ m $= 1$ km.

Multiply or divide by the conversion factor: $5$ ft $= 5 \times 12 = 60$ in; $250$ cm $= 250 \div 100 = 2.5$ m.

A transformation that flips a figure over a line (the line of reflection), producing a mirror image of the same size.

A transformation that slides a figure a fixed distance in a direction without turning or flipping it.

A transformation that turns a figure about a fixed point by a given angle; size and shape are unchanged.

A transformation that resizes a figure by a scale factor from a center point; the shape stays similar but changes size.

Figures with the same size and shape; one maps onto the other by reflection, translation, or rotation. Symbol: $\cong$.

Figures with the same shape but possibly different sizes; corresponding angles are equal and sides are proportional. Symbol: $\sim$.

A grid formed by a horizontal $x$-axis and vertical $y$-axis meeting at the origin $(0, 0)$, splitting it into four quadrants.

Two numbers $(x, y)$ giving a point's horizontal then vertical position. $(3, -2)$ is $3$ right and $2$ down.

The four regions of the coordinate plane: I $(+,+)$, II $(-,+)$, III $(-,-)$, IV $(+,-)$, numbered counterclockwise.

A visual representation of data such as a bar graph, line plot, pictograph, line graph, or circle graph.

A display using rectangular bars whose lengths show the values of categories; good for comparing groups.

A number line with an X or dot stacked for each data value; it shows frequency and clusters in a small data set.

A display connecting data points with line segments to show change over time, such as temperature across a week.

A circle divided into sectors showing each category's share of the whole; the sectors total $100\%$.

How many times a value or category occurs in a data set; shown by bar height, dot count, or a frequency table.

The sum of the values divided by how many there are. For $4, 6, 11$: $\dfrac{4+6+11}{3} = 7$.

The middle value of an ordered data set (the average of the two middle values if the count is even): for $3, 5, 9$ it is $5$.

The value that appears most often in a data set; a set can have one mode, several, or none. In $2, 4, 4, 7$ the mode is $4$.

The spread of a data set: the greatest value minus the least. For $3, 8, 15$: $15 - 3 = 12$.

Single values that summarize the middle of data: the mean, median, and mode.

The median resists outliers, so it better describes a skewed set; one very large value inflates the mean but not the median.

A number from $0$ to $1$ measuring how likely an event is: $\dfrac{\text{favorable outcomes}}{\text{total outcomes}}$.

Favorable outcomes over total equally likely outcomes. Rolling a $4$ on a die is $\dfrac{1}{6}$.

A certain event has probability $1$; an impossible event has probability $0$. All probabilities fall between.

The chance an event does not happen: $P(\text{not } A) = 1 - P(A)$. If $P(\text{rain}) = 0.3$, then $P(\text{no rain}) = 0.7$.

Events where one outcome does not affect the other; multiply their probabilities: two heads in two flips is $\dfrac{1}{2} \times \dfrac{1}{2} = \dfrac{1}{4}$.

A population is the whole group studied; a sample is a smaller part of it used to make inferences about the whole.

A face is a flat surface, an edge is where two faces meet, a vertex is a corner. A cube has $6$ faces, $12$ edges, $8$ vertices.

Parallel lines never meet and stay the same distance apart; perpendicular lines cross at a right angle $(90^\circ)$.