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# Glossary

Module by: Interactive Mathematics Program. E-mail the author

Summary: This is the glossary for IMP Year 1.

This is the glossary for all five units of IMP Year 1. This glossary may be useful when you encounter a term in bold text that is new or unfamiliar, or it can be used to confirm or clarify your understanding of a term.

Absolute value The distance a number is from 0 on the number line. The symbol size 12{ lline rline } {} stands for the absolute value of that number.

Examples: 2 = 2 ; 7 = 7 ; 0 = 0 2 = 2 ; 7 = 7 ; 0 = 0 size 12{ lline -2 rline =2;" " lline 7 rline =7;" " lline 0 rline =0} {}

Acute angle An angle that measures more than 0° and less than 90°.

Acute triangle A triangle with angles that are all acute.

Adjacent angles Two angles with the same vertex, formed using a shared ray.

Example: Angles A and B are adjacent angles.

Adjacent side (for an acute angle of a right triangle) The side of the right triangle that together with the hypotenuse forms the given angle.

Example: In the right triangle ABC, side BC¯BC¯ size 12{ {overline { ital "BC"}} } {} is adjacent to C¯C¯ size 12{ {overline {C}} } {}, and side AB¯AB¯ size 12{ {overline { ital "AB"}} } {} is adjacent to ÐAÐA size 12{ÐA} {}.

Alternate interior angles If two lines are intersected by a transversal, then the inside angles with different vertices that are on opposite sides of the transversal are alternate interior angles.

Example: Angles K and L are one pair of alternate interior angles, and angles M and N are another pair.

Amplitude (for a pendulum) The angle of a pendulum’s swing, measured from the vertical to the most outward position of the pendulum during its swing.

Example: The pendulum in the diagram has an amplitude of 20°.

Angle Informally, an amount of turn, usually measured in degrees. Formally, the geometric figure formed by two rays with a common point, called the vertex of the angle.

Angle of elevation The angle at which an object appears above the horizontal, as measured from a chosen point.

Example: The diagram shows the angle of elevation to the top of the tree from point A.

Area Informally, the amount of space inside a two-dimensional figure, usually measured in square units.

Area model For probability, a diagram showing the possible outcomes of a particular event. Each portion of the model represents an outcome, and the ratio of the area of that portion to the area of the whole model is the probability of that outcome.

Average Often refers to the mean, but more generally refers to some sort of measure of the center of a data set.

Axis (plural: axes) See coordinate system.

Box-and-whiskers plot (or box plot) A display that shows the minimum, lowerquartile, median, upper quartile, and maximum of a data set.

Closed formula A method to figure the Out value of a table by using its associated In value. Contrast with recursive formula.

Coefficient Usually, a number used to multiply a variable or power of a variable in an

algebraic expression.

Example: In the expression 3x+4x23x+4x2 size 12{3x+4x rSup { size 8{2} } } {}, 3 and 4 are coefficients.

Complementary angles A pair of angles with measures that add to 90°. If two complementary angles are adjacent, together they form a right angle.

Conclusion Informally, any statement arrived at by reasoning or through examples. See also “If . . . , then . . .” statement.

Conditional probability The probability that an event will occur based on the assumption that some other event has already occurred.

Congruent Informally, having the same shape and size. Formally, two polygons are congruent if their corresponding angles have equal measure and their corresponding sides have equal length. The symbol @@ size 12{@} {} means “is congruent to.”

Conjecture A theory or an idea about how something works, usually based on examples.

Consecutive sum A way to write a number as the sum of a sequence of two or more whole numbers, where each number of the sequence is one more than the previous number.

Example: 4 + 5 + 6 is a consecutive sum. This consecutive sum shows a way to represent the number 15.

Constraint Informally, a limitation or restriction.

Continuous graph Informally, a graph that can be drawn without lifting the pencil, in contrast to a discrete graph.

Coordinate system A way to represent points in the plane with ordered pairs of numbers called coordinates. The system is based on two perpendicular number lines, one horizontal and one vertical, called coordinate axes. The point where the lines intersect is called the origin. Traditionally, the axes are labeled with the variables x and y, as shown below. The horizontal axis is often called the x-axis, and the vertical axis is often called the y-axis.

Example: Point A has coordinates (3, –2).

Corresponding angles (for a transversal) If two lines are intersected by a transversal, then two angles are corresponding angles if they occupy the same position relative to the transversal and the other lines that form them.

Example: Angles A and D are a pair of corresponding angles, and angles B and E are another pair of corresponding angles.

Corresponding parts (for similar or congruent polygons) Sides or angles of polygons that have the same relative position.

Example: Side a in the small triangle and side b in the large triangle are corresponding parts.

Cosine The ratio of the length of the leg adjacent to one non-right angle of a right triangle to the length of the hypotenuse. The cosine of ÐAÐA size 12{ÐA} {} is abbreviated as cos A.

cos A = length of leg adjacent to ÐA length of hypotenuse cos A = length of leg adjacent to ÐA length of hypotenuse size 12{"cos"A= { {"length""of""leg""adjacent""to"ÐA} over {"length""of""hypotenuse"} } } {}
(1)

Counterexample An example that demonstrates a conjecture is not true.

Counting number See natural number.

Degree The measurement unit for an angle defined by having a complete turn equal to 360 degrees. The symbol º represents degrees.

Dependent variable Informally, the changing quantities represented by the Out values in an In-Out table. Also the quantity that changes as a result of changes in the independent variable. On a graph, the name of the dependent variable is labeled on the y-axis.

Diagonal In a polygon, a line segment that connects two vertices and is not a side of the polygon.

Discrete graph A graph consisting of isolated or unconnected points, in contrast to a continuous graph.

Divisor A factor of an integer.

Example: 1, 2, 3, 4, 6, and 12 are the positive divisors of 12.

Domain The set of values that can be used as inputs for a given function.

Edge The line segment where two faces of a three-dimensional shape intersect.

Example: When a die is rolled, it never lands teetering on one of its edges.

Equation A statement, using an equal sign, of two equivalent expressions. See also

formula, function.

Equilateral triangle A triangle with all sides the same length.

Equivalent equations Two or more equations that have the same solution (or solutions).

Example: The equation 3x + 5 = 17 is equivalent to the equations 3x = 12 and x = 4. When the solution, 4, is substituted for x, each equation is true (17 = 17, 12 = 12, or 4 = 4).

Example: The equations y = 5 + 2x and 2y = 4x + 10 are equivalent equations. Any coordinate pair (x, y) that makes the first equation true will also be true in the second equation, such as (1, 7), (2, 9), or (3, 11),

Equivalent expressions Two or more expressions that have the same value when any number is substituted for the variable.

Example: The expression 4 + 6x is equivalent to 2(2 + 3x). When any number (for example, 5) is substituted for x, both expressions evaluate to the same number (in the example, 34).

Event The specific set of outcomes from performing an experiment several times, like ways a pair of dice could show a total value of 5. See also outcome.

Expected value In a game or other probability situation, the average amount gained or lost per turn in the long run.

Experimental probability See observed probability.

Exponent A number written as a superscript to another number (the base), to indicate how many times the base is used as a factor of multiplication.

Example:2525 size 12{2 rSup { size 8{5} } } {} = 2 · 2 · 2 · 2 · 2. The 5 is the exponent, with 2 as its base. 2 is used as a factor 5 times.

Expression The written combination of variables and numbers that often represents some situation.

Exterior angle An angle formed outside a polygon by extending one of its sides.

Example: The diagram shows ÐBAX,ÐBAX, size 12{Ð ital "BAX",} {} an exterior angle for polygon ABCDE.

Face A flat surface of a three-dimensional shape.

Example: The face of a die is where the numbers are printed.

Factorial The product of all the whole numbers from a particular number down to 1. The symbol ! stands for factorial.

Example: 5! (read “five factorial”) means 5 · 4 · 3 · 2 · 1.

Fair game A game in which both players are expected to come out equally well in the long run.

Formula A mathematical statement describing a relationship among variables or indicating how to calculate for some unknown. See also equation, function.

Frequency bar graph A bar graph showing how often each result occurs.

Example: This frequency bar graph shows, for instance, that 11 times in 80 rolls, the sum of two dice was 6.

Function Informally, a process or rule for determining the numerical value of one variable in terms of another. A function is often represented as a set of number pairs in which the second number is determined by the first, according to the function rule.

Graph A mathematical diagram for displaying information.

Hexagon A polygon with six sides.

Horizontal Extending side to side, like the horizon.

Hypotenuse The longest side in a right triangle, or the length of this side. The hypotenuse is located opposite the right angle.

Example: In right triangle ABC, the hypotenuse is AC¯AC¯ size 12{ {overline { ital "AC"}} } {}.

Hypothesis Informally, a theory about a situation or about how a certain set of data is behaving. Also, a set of assumptions used to analyze or understand a situation. See also “If . . . , then . . .” statement.

“If . . . , then . . .” statement A specific form of mathematical statement, saying that if one condition is true, then another condition must also be true.

Example: If two angles of a triangle have equal measure, then the sides opposite these angles have equal length.

The condition “two angles of a triangle have equal measure” is the hypothesis. The condition “the sides opposite these angles have equal length” is the conclusion.

Independent events Two (or more) events are independent if the result of one does not influence the result of the other.

Independent variable Informally, the changing quantities represented by the In values in an In-Out table. Also the quantity that, when changed, causes changes in the Out values, or the dependent variable. On a graph, the name of the independent variable is labeled on the x-axis.

Example: In a situation involving time and distance traveled, time is usually the independent variable.

Integer Any number that is either a counting number, 0, or the opposite of a counting number. The integers can be represented using set notation as

{ . . . –3, –2, –1, 0, 1, 2, 3, . . . }

Examples: –4, 0, and 10 are integers.

Interior angle An angle inside a figure, especially within a polygon, formed by sides of the figure.

Example: Angle BAE is an interior angle of the polygon ABCDE.

Isosceles triangle A triangle with at least two sides of equal length.

Justify To make an argument for; to give a reason or explanation.

Leg Either of the two shorter sides in a right triangle. The two legs of a right triangle form the right angle of the triangle.

Linear function A function whose graph is a line. A common form for writing a linear function is f(x) = ax + b, where a is the rate of change and b is the y-coordinate of the starting point.

Line of best fit Informally, the line that comes closest to fitting a given set of points on a discrete graph.

Line segment The portion of a straight line between two given points.

Lower quartile The median of the values below the median of a data set.

Mathematical model A mathematical description or structure used to represent how a real-life situation works.

Mean The numerical average of a data set, found by adding the data items and dividing by the number of items in the set.

Example: For the data set 8, 12, 12, 13, and 17, the sum of the data items is 62. There are 5 items in the data set, so the mean is 62 ÷ 5, or 12.4.

Measurement variation The situation of taking several measurements of the same thing and getting different results.

Median (of a set of data) The “middle number” in a set of data that has been arranged from smallest to largest. If the data set has an even number of values, the median is the mean of the two “middle numbers.”

Example: For the data set 4, 17, 22, 56, and 100, the median is 22, because it is the number in the middle of the ordered list.

Mode (of a set of data) The number that occurs most often in a set of data. A data set may have more than one mode.

Example: For the data set 3, 4, 7, 16, 18, 18, and 23, the mode is 18.

Natural number Any of the numbers used for counting, such as 1, 2, 3, 4, and so on, but not including zero. Also called counting number.

Normal distribution A certain, precisely defined set of probabilities that can often be used to approximate real-life events. Sometimes used to refer to any data set whose graph is approximately “bell-shaped.”

Observed probability The likelihood of a certain event happening based on observed results, as distinct from theoretical probability. Also called experimental probability.

Obtuse angle An angle that measures more than 90° and less than 180°.

Obtuse triangle A triangle with an obtuse angle.

Octagon An eight-sided polygon.

Opposite side The side of a triangle across from a given angle.

Ordered pair Two numbers paired together using the format (x, y), often used to locate a point in the coordinate system.

Order of operations A set of conventions that mathematicians have agreed to use whenever a calculation involves more than one operation.

Example: 2 + 3 · 4 is 14, not 20, because according to the conventions for order of operations, multiplication occurs before addition.

Origin See coordinate system.

Outcome The result of performing one trial of an experiment, like flipping a coin or drawing a card from a deck. See also event.

Parallel lines Two lines in a plane that do not intersect.

Pentagon A five-sided polygon.

Period The length of time for a cyclical event to complete one full cycle.

Perpendicular lines A pair of lines that intersect to form right angles.

Polygon A closed, two-dimensional shape formed by three or more line segments. The line segments that form a polygon are called its sides. The endpoints of these segments are called vertices (singular: vertex).

Examples: All of these figures are polygons.

Prime number A whole number greater than 1 that has only two whole-number divisors, 1 and itself.

Example: 7 is a prime number, because its only whole-number divisors are 1 and 7.

Probability The likelihood of a certain event happening. For a situation involving equally likely outcomes, the probability that the outcome of an event will be an outcome within a given set is defined by a ratio:

Probability = number of outcomes in the set total numbers of possible outcomes Probability = number of outcomes in the set total numbers of possible outcomes size 12{"Probability"= { {"number""of""outcomes""in""the""set"} over {"total""numbers""of""possible""outcomes"} } } {}
(2)

Example: If a cube has 2 red faces and 4 green faces, the probability of rolling the cube and getting a green face is

number of green faces total number of faces = 4 6 number of green faces total number of faces = 4 6 size 12{ { {"number""of""green""faces"} over {"total""number of faces"} } = { {4} over {6} } } {}
(3)

Proof An absolutely convincing argument.

Proportion A statement that two ratios are equal.

Proportional Having the same ratio.

Example: Corresponding sides of triangles ABC and DEF are proportional, because the ratios 4646 size 12{ { {4} over {6} } } {}, 812812 size 12{ { {8} over {"12"} } } {}, and 10151015 size 12{ { {"10"} over {"15"} } } {} are equal.

Quadrant One of the four regions created in a coordinate system by using the x-axis and the y-axis as boundaries. The quadrants have standard numbering, as shown.

Random Used in probability to indicate that any of several events is equally likely. In situations where events are not equally likely to occur, random is used to mean selection from a set of events according to a precisely described distribution.

Range (of a set of data) The difference between the largest and smallest numbers in the set.

Example: For the data set 7, 12, 18, 18, and 29, the range is 29 – 7, or 22.

Rate (or rate of change) A number describing change. It is calculated by computing a ratio of two quantities.

Examples: miles per day, heartbeats per minute.

Ratio A comparison of two numbers.

Ray The part of a line from a single point, called the vertex, through another point on the line and continuing infinitely in that direction.

Rectangle A four-sided polygon with angles that are all right angles.

Recursive formula A method to determine the Out value of a table by using the previous Out value in the table. This method requires the In values to be listed in order, going up by 1 at each step. Contrast with closed formula.

Regular polygon A polygon with sides that all have equal length and angles that all have equal measure.

Rhombus A four-sided polygon with sides that all have the same length.

Right angle An angle that measures 90°.

Right triangle A triangle with a right angle.

Sample standard deviation The calculation on a set of data taken from a larger population of data, used to estimate the standard deviation of the larger population.

Sequence An ordered list of numbers, expressions, or pictures, usually following a pattern or rule.

Example: 1, 3, 5, 7, 9, . . . is the sequence of positive odd numbers.

Sigma notation See summation notation.

Similar Informally, having the same shape. Formally, two polygons are similar if their corresponding angles have equal measure and their corresponding sides are proportional in length. The symbol ~ means “is similar to.”

Simulation An experiment or set of experiments using a model of an event that is based on the same probabilities as the real event. Simulations allow people to estimate the likelihood of an event when it is impractical to experiment with the real event.

Sine The ratio of the length of the leg opposite one non-right angle of a right triangle to the length of the hypotenuse. The sine of ÐAÐA size 12{ÐA} {} is abbreviated as sin A.

sin A = length of leg opposite ÐA length of hypotenuse sin A = length of leg opposite ÐA length of hypotenuse size 12{"sin"A= { {"length""of""leg""opposite"ÐA} over {"length""of""hypotenuse"} } } {}
(4)

Slope Informally, the steepness of a line.

Solution A value that, when substituted for a variable in an equation, makes the equation a true statement.

Example: The value x = 3 is a solution to the equation 2x = 6 because 2 · 3 = 6.

Square A four-sided polygon with all sides of equal length and with four right angles.

Square (of a number) The number multiplied by itself; in other words, the number to the exponent 2.

Square root A number whose square is a given number. The symbol size 12{ sqrt {} } {} is used to denote the nonnegative square root of a number.

Example: Both 6 and –6 are square roots of 36, because 6262 size 12{6 rSup { size 8{2} } } {} = 36 and 6262 size 12{ left (-6 right ) rSup { size 8{2} } } {} = 36; 36=636=6 size 12{ sqrt {"36"} =6} {}.

Standard deviation A specific measurement of how spread out a set of data is, usually represented by the lowercase Greek letter sigma (Σ).

Starting point An informal reference to the amount of something at the beginning of a situation. The use of the word point emphasizes that this starting amount coincides with the y-coordinate associated with the y-intercept.

Example: If a wagon train begins its westward journey with 40 pounds of salt, 40 is referred to as the starting amount, or starting point. The y-intercept of a graph of the salt usage would be (0, 40).

Stem-and-leaf plot (or stem plot) A graphical display with “stems” showing the leftmost digit of the values separated from “leaves” showing the next digit or set of digits.

Straight angle An angle that measures 180°. The rays forming a straight angle together make up a straight line.

Strategy A complete plan about how to proceed in a game or problem situation. A strategy for a game should tell a person exactly what to do under any situation that can arise in the game.

Subscript A symbol written below and to the right of another symbol.

Example: For the variable PBPB size 12{P rSub { size 8{B} } } {}, the letter B is written as a subscript.

Substitute To replace a variable in an expression with a numerical value. Usually followed by evaluation, that is, to compute the numerical value of the resulting expression.

Summary phrase A concise phrase to describe the quantity represented by an expression.

Summation notation A useful technique for writing the sum of a sequence of numbers. The uppercase Greek letter sigma (Σ) stands for summation.

Example: The consecutive sum 3 + 4 + 5 + 6 + 7 is r=37r.r=37r. size 12{ Sum cSub { size 8{r=3} } cSup { size 8{7} } {r "." } } {}

Example: In the expression t=58(4t2+3)t=58(4t2+3) size 12{ Sum cSub { size 8{t=5} } cSup { size 8{8} } { $$4t rSup { size 8{2} } +3$$ } } {}, the number 5 is called the lower limit, the number 8 is called the upper limit, and the expression 4t2+34t2+3 size 12{4t rSup { size 8{2} } +3} {}is called the summand. The variable t is referred to as an index variable, or dummy variable.

Superscript  A symbol written above and to the right of another symbol, such as an exponent.

Supplementary angles A pair of angles with measures that add to 180°. If two supplementary angles are adjacent, together they form a straight angle.

Tangent The ratio of the length of the leg opposite one non-right angle of a right triangle to the length of the leg adjacent to the same angle. The tangent of ÐAÐA size 12{ÐA} {} is abbreviated as tan A.

tan A = length of leg opposite ÐA length of leg adjacent to ÐA tan A = length of leg opposite ÐA length of leg adjacent to ÐA size 12{"tan"A= { {"length""of""leg""opposite"ÐA} over {"length""of""leg""adjacent""to"ÐA} } } {}
(5)

Term (of an algebraic expression) A part of an algebraic expression, combined with other terms using addition or subtraction.

Example: The expression 2x22x2 size 12{2x rSup { size 8{2} } } {} + 3x – 12 has three terms: 2x22x2 size 12{2x rSup { size 8{2} } } {} + 3x,and 12.

Term (of a sequence) One of the items listed in a sequence.

Example: In the sequence 3, 5, 7, . . . , the number 3 is the first term, 5 is the second term, and so on.

Theoretical probability The likelihood of an event occurring, as explained by a theory or model, as contrasted with observed probability.

Transversal A line that intersects two or more other lines.

Example: The line l is a transversal that intersects the lines m and n.

Trapezoid A four-sided polygon with exactly one pair of parallel sides.

Example: Quadrilateral PQRS is a trapezoid, because QR¯QR¯ size 12{ {overline { ital "QR"}} } {} and PS¯PS¯ size 12{ {overline { ital "PS"}} } {}are parallel and PQ¯PQ¯ size 12{ {overline { ital "PQ"}} } {} and SR¯SR¯ size 12{ {overline { ital "SR"}} } {}are not parallel.

Triangle A polygon with three sides.

Triangle inequality principle The principle that the lengths of any two sides of a triangle must add up to more than the length of the third side.

Trigonometric function Any of six functions defined for acute angles in terms of ratios of sides of a right triangle.

Unique A word used to mean one and only one. Often used in reference to the only possible solution.

Upper quartile The median of the values above the median of a data set.

Variable A symbol used to represent a quantity that can have different values.

Variance Like standard deviation, a specific measurement of how spread out a set of data is. The variance is the square of the standard deviation.

Vertex (plural: vertices) A common endpoint of two segments or rays. See how this term is used in angle, polygon, and ray.

Vertical Extending straight up and down, like a flagpole.

Vertical angles A pair of opposite angles formed by a pair of intersecting lines.

Example: Angles F and G are vertical angles.

Whole number A number that is either 0 or a counting number.

x-intercept A place on a graph where a line or curve crosses the x-axis.

y-intercept A place on a graph where a line or curve crosses the y-axis.

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