Mathematics Glossary and Tables
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Addition and subtraction within 5, 10, 20, 100, or 1000. Addition or subtraction of two whole numbers with whole number answers, and with sum or minuend in the range 05, 010, 020, or 0100, respectively. Example: 8 + 2 = 10 is an addition within 10, 14 – 5 = 9 is a subtraction within 20, and 55 – 18 = 37 is a subtraction within 100.
Additive inverses. Two numbers whose sum is 0 are additive inverses of one another. Example: 3/4 and – 3/4 are additive inverses of one another because 3/4 + (– 3/4) = (– 3/4) + 3/4 = 0.
Associative property of addition. See Table 3 in this Glossary.
Associative property of multiplication. See Table 3 in this Glossary.
Bivariate data. Pairs of linked numerical observations. Example: a list of heights and weights for each player on a football team. Box plot. A method of visually displaying a distribution of data values by using the median, quartiles, and extremes of the data set. A box shows the middle 50% of the data.^{1}
Commutative property. See Table 3 in this Glossary.
Complex fraction. A fraction A/B where A and/or B are fractions (B nonzero).
Computation algorithm. A set of predefined steps applicable to a class of problems that gives the correct result in every case when the steps are carried out correctly. See also: computation strategy.
Computation strategy. Purposeful manipulations that may be chosen for specific problems, may not have a fixed order, and may be aimed at converting one problem into another. See also: computation algorithm.
Congruent. Two plane or solid figures are congruent if one can be obtained from the other by rigid motion (a sequence of rotations, reflections, and translations).
Counting on. A strategy for finding the number of objects in a group without having to count every member of the group. For example, if a stack of books is known to have 8 books and 3 more books are added to the top, it is not necessary to count the stack all over again. One can find the total by counting on—pointing to the top book and saying “eight,” following this with “nine, ten, eleven. There are eleven books now.”
Dot plot. See: line plot.
Dilation. A transformation that moves each point along the ray through the point emanating from a fixed center, and multiplies distances from the center by a common scale factor.
Expanded form. A multidigit number is expressed in expanded form when it is written as a sum of singledigit multiples of powers of ten. For example, 643 = 600 + 40 + 3.
Expected value. For a random variable, the weighted average of its possible values, with weights given by their respective probabilities.
First quartile. For a data set with median M, the first quartile is the median of the data values less than M. Example: For the data set {1, 3, 6, 7, 10, 12, 14, 15, 22, 120}, the first quartile is 6.^{2}See also: median, third quartile, interquartile range.
Identity property of 0. See Table 3 in this Glossary.
Independently combined probability models. Two probability models are said to be combined independently if the probability of each ordered pair in the combined model equals the product of the original probabilities of the two individual outcomes in the ordered pair.
Integer. A number expressible in the form a or –a for some whole number a.
Interquartile Range. A measure of variation in a set of numerical data, the interquartile range is the distance between the first and third quartiles of the data set. Example: For the data set {1, 3, 6, 7, 10, 12, 14, 15, 22, 120}, the interquartile range is 15 – 6 = 9. See also: first quartile, third quartile.
Line plot. A method of visually displaying a distribution of data values where each data value is shown as a dot or mark above a number line. Also known as a dot plot.^{3}
Mathematics of Information Processing and the Internet (IA). The Internet is everywhere in modern life. To be informed consumers and citizens in the informationdense modern world permeated by the Internet, students should have a basic mathematical understanding of some of the issues of information processing on the Internet. For example, when making an online purchase, mathematics is used to help you find what you want, encrypt your credit card number so that you can safely buy it, send your order accurately to the vendor, and, if your order is immediately downloaded, as when purchasing software, music, or video, ensure that your download occurs quickly and errorfree. Essential topics related to these aspects of information processing are basic set theory, logic, and modular arithmetic. These topics are not only fundamental to information processing on the Internet, but they are also important mathematical topics in their own right with applications in many other areas.
Mathematics of Voting (IA). The instantrunoff voting (IRV), the Borda method (assigning points for preferences), and the Condorcet method (in which each pair of candidates is run off head to head) are all forms of preferential voting (rank according to your preferences, rather than just voting for your single favorite candidate).
Mean. A measure of center in a set of numerical data, computed by adding the values in a list and then dividing by the number of values in the list.^{4} Example: For the data set {1, 3, 6, 7, 10, 12, 14, 15, 22, 120}, the mean is 21.
Mean absolute deviation. A measure of variation in a set of numerical data, computed by adding the distances between each data value and the mean, then dividing by the number of data values. Example: For the data set {2, 3, 6, 7, 10, 12, 14, 15, 22, 120}, the mean absolute deviation is 20.
Median. A measure of center in a set of numerical data. The median of a list of values is the value appearing at the center of a sorted version of the list—or the mean of the two central values, if the list contains an even number of values. Example: For the data set {2, 3, 6, 7, 10, 12, 14, 15, 22, 90}, the median is 11.
Midline. In the graph of a trigonometric function, the horizontal line halfway between its maximum and minimum values. Multiplication and division within 100. Multiplication or division of two whole numbers with whole number answers, and with product or dividend in the range 0100. Example: 72 ÷ 8 = 9.
Percent rate of change. A rate of change expressed as a percent. Example: if a population grows from 50 to 55 in a year, it grows by 5/50 = 10% per year.
Probability distribution. The set of possible values of a random variable with a probability assigned to each.
Properties of operations. See Table 3 in this Glossary.
Properties of equality. See Table 4 in this Glossary.
Properties of inequality. See Table 5 in this Glossary.
Properties of operations. See Table 3 in this Glossary.
Probability. A number between 0 and 1 used to quantify likelihood for processes that have uncertain outcomes (such as tossing a coin, selecting a person at random from a group of people, tossing a ball at a target, or testing for a medical condition).
Probability model. A probability model is used to assign probabilities to outcomes of a chance process by examining the nature of the process. The set of all outcomes is called the sample space, and their probabilities sum to 1. See also: uniform probability model.
Random variable. An assignment of a numerical value to each outcome in a sample space. Rational expression. A quotient of two polynomials with a nonzero denominator.
Rational number. A number expressible in the form a/b or – a/b for some fraction a/b. The rational numbers include the integers.
Rectilinear figure. A polygon all angles of which are right angles.
Rigid motion. A transformation of points in space consisting of a sequence of one or more translations, reflections, and/or rotations. Rigid motions are here assumed to preserve distances and angle measures.
Repeating decimal. The decimal form of a rational number. See also: terminating decimal.
Sample space. In a probability model for a random process, a list of the individual outcomes that are to be considered.
Scatter plot. A graph in the coordinate plane representing a set of bivariate data. For example, the heights and weights of a group of people could be displayed on a scatter plot.^{5 }
Similarity transformation. A rigid motion followed by a dilation.
Tape diagram. A drawing that looks like a segment of tape, used to illustrate number relationships. Also known as a strip diagram, bar model, fraction strip, or length model.
Terminating decimal. A decimal is called terminating if its repeating digit is 0.
Third quartile. For a data set with median M, the third quartile is the median of the data values greater than M. Example: For the data set {2, 3, 6, 7, 10, 12, 14, 15, 22, 120}, the third quartile is 15. See also: median, first quartile, interquartile range.
Transitivity principle for indirect measurement. If the length of object A is greater than the length of object B, and the length of object B is greater than the length of object C, then the length of object A is greater than the length of object C. This principle applies to measurement of other quantities as well.
Uniform probability model. A probability model which assigns equal probability to all outcomes. See also: probability model.
Vector. A quantity with magnitude and direction in the plane or in space, defined by an ordered pair or triple of real numbers.
VertexEdge Graphs (IA). Vertexedge graphs are diagrams consisting of vertices (points) and edges (line segments or arcs) connecting some of the vertices. Vertexedge graphs are also sometimes called networks, discrete graphs, or finite graphs. A vertexedge graph shows relationships and connections among objects, such as in a road network, a telecommunications network, or a family tree. Within the context of school geometry, which is fundamentally the study of shape, vertexedge graphs represent, in a sense, the situation of no shape. That is, vertexedge graphs are geometric models consisting of vertices and edges in which shape is not essential, only the connections among vertices are essential. These graphs are widely used in business and industry to solve problems about networks, paths, and relationships among a finite number of objects – such as, analyzing a computer network; optimizing the route used for snowplowing, collecting garbage, or visiting business clients; scheduling committee meetings to avoid conflicts; or planning a large construction project to finish on time.
Visual fraction model. A tape diagram, number line diagram, or area model.
Whole numbers. The numbers 0, 1, 2, 3,....
Tables
Table 1. Common addition and subtraction situations. ^{6}
Result Unknown  Change Unknown  Start Unknown  

Add to  Two bunnies sat on the grass. Three more bunnies hopped there. How many bunnies are on the grass now? 2 + 3 = ? 
Two bunnies were sitting on the grass. Some more bunnies hopped there. Then there were five bunnies. How many bunnies hopped over to the first two? 2 + ? = 5 
Some bunnies were sitting on the grass. Three more bunnies hopped there. Then there were five bunnies. How many bunnies were on the grass before? ? + 3 = 5 
Take from  Five apples were on the table. I ate two apples. How many apples are on the table now? 5 – 2 = ? 
Five apples were on the table. I ate some apples. Then there were three apples. How many apples did I eat? 5 – ? = 3 
Some apples were on the table. I ate two apples. Then there were three apples. How many apples were on the table before? ? – 2 = 3 
Total Unknown  Addend Unknown  Both Addends Unknown^{1}  
Put Together/ Take Apart^{2} 
Three red apples and two green apples are on the table. How many apples are on the table? 3 + 2 = ? 
Five apples are on the table. Three are red and the rest are green. How many apples are green? 3 + ? = 5, 5 – 3 = ? 
Grandma has five flowers. How many can she put in her red vase and how many in her blue vase? 5 = 0 + 5, 5 = 5 + 0 5 = 1 + 4, 5 = 4 + 1 5 = 2 + 3, 5 = 3 + 2 
Difference Unknown  Bigger Unknown  Smaller Unknown  
Compare^{3}  (“How many more?” version): Lucy has two apples. Julie has five apples. How many more apples does Julie have than Lucy?
(“How many fewer?” version): Lucy has two apples. Julie has five apples. How many fewer apples does Lucy have han Julie? 
(Version with “more”): Julie has three more apples than Lucy. Lucy has two apples. How many apples does Julie have?
(Version with “fewer”): Lucy has 3 fewer apples than Julie. Lucy has two apples. How many apples does Julie have? 
(Version with “more”): Julie has three more apples than Lucy. Julie has five apples. How many apples does Lucy have?
(Version with “fewer”): Lucy has 3 fewer apples than Julie. Julie has five apples. How many apples does Lucy have? 
^{1}These take apart situations can be used to show all the decompositions of a given number. The associated equations, which have the total on the left of the equal sign, help children understand that the = sign does not always mean makes or results in but always does mean is the same number as. ^{2}Either addend can be unknown, so there are three variations of these problem situations. Both Addends Unknown is a productive extension of this basic situation, especially for small numbers less than or equal to 10. ^{3}For the Bigger Unknown or Smaller Unknown situations, one version directs the correct operation (the version using more for the bigger unknown and using less for the smaller unknown). The other versions are more difficult. 
Table 2. Common multiplication and division situations.^{7}
Unknown Product  Group Size Unknown (How many in each group? Division) 
Number of Groups Unknown (How many groups? Division) 


3 x ? = 18, and 18 ÷ 3 = ?  ? x 6 = 18, and 18 ÷ 6 = ?  
Equal Groups  There are 3 bags with 6 plums in each bag. How many plums are there in all?
Measurement example. You need 3 lengths of string, each 6 inches long. How much string will you need altogether? 
If 18 plums are shared equally into 3 bags, then how many plums will be in each bag?
Measurement example. You have 18 inches of string, which you will cut into 3 equal pieces. How long will each piece of string be? 
If 18 plums are to be packed 6 to a bag, then how many bags are needed?
Measurement example. You have 18 inches of string, which you will cut into pieces that are 6 inches long. How many pieces of string will you have? 
Arrays,^{4} Area^{5} 
There are 3 rows of apples with 6 apples in each row. How many apples are there?
Area example. What is the area of a 3 cm by 6 cm rectangle? 
If 18 apples are arranged into 3 equal rows, how many apples will be in each row?
Area example. A rectangle has area 18 square centimeters. If one side is 3 cm long, how long is a side next to it? 
If 18 apples are arranged into equal rows of 6 apples, how many rows will there be?
Area example. A rectangle has area 18 square centimeters. If one side is 6 cm long, how long is a side next to it? 
Compare  A blue hat costs $6. A red hat costs 3 times as much as the blue hat. How much does the red hat cost?
Measurement example. A rubber band is 6 cm long. How long will the rubber band be when it is stretched to be 3 times as long? 
A red hat costs $18 and that is 3 times as much as a blue hat costs. How much does a blue hat cost?
Measurement example. A rubber band is stretched to be 18 cm long and that is 3 times as long as it was at first. How long was the rubber band at first? 
A red hat costs $18 and a blue hat costs $6. How many times as much does the red hat cost as the blue at?
Measurement example. A rubber band was 6 cm long at first. Now it is stretched to be 18 cm long. How many times as long is the rubber band now as it was at first? 
General  a × b = ?  a × ? = p, and p ÷ a = ?  ? × b = p, and p ÷ b = ? 
^{4}The language in the array examples shows the easiest form of array problems. A harder form is to use the terms rows and columns: The apples in the grocery window are in 3 rows and 6 columns. How many apples are in there? Both forms are valuable. ^{5}Area involves arrays of squares that have been pushed together so that there are no gaps or overlaps, so array problems include these especially important measurement situations. 
Table 3. The properties of operations. Here a, b and c stand for arbitrary numbers in a given number system. The properties of operations apply to the rational number system, the real number system, and the complex number system.
Associative property of addition  (a + b) + c = a + (b + c) 
Commutative property of addition  a + b = b + a 
Additive identity property of 0  a + 0 = 0 + a = a 
Existence of additive inverses  For every a there exists –a so that a + (–a) = (–a) + a = 0. 
Associative property of multiplication  (a × b) × c = a × (b × c) 
Commutative property of multiplication  a × b = b × a 
Multiplicative identity property of 1  a × 1 = 1 × a = a 
Existence of multiplicative inverses  For every a ≠ 0 there exists 1/a so that a × 1/a = 1/a × a = 1. 
Distributive property of multiplication over addition  a × (b + c) = a × b + a × c 
Table 4. The properties of equality. Here a, b and c stand for arbitrary numbers in the rational, real, or complex number systems.
Reflexive property of equality  a = a 
Symmetric property of equality  If a = b, then b = a. 
Transitive property of equality  If a = b and b = c, then a = c. 
Addition property of equality  If a = b, then a + c = b + c. 
Subtraction property of equality  If a = b, then a – c = b – c. 
Multiplication property of equality  If a = b, then a × c = b × c. 
Division property of equality  If a = b and c ≠ 0, then a ÷ c = b ÷ c. 
Substitution property of equality  If a = b, then b may be substituted for a in any expression containing a. 
Table 5. The properties of inequality. Here a, b and c stand for arbitrary numbers in the rational or real number systems.
Exactly one of the following is true: a < b, a = b, a > b. 
If a > b and b > c then a > c. 
If a > b, then b < a. 
If a > b, then –a < –b. 
If a > b, then a ± c > b ± c. 
If a > b and c > 0, then a × c > b × c. 
If a > b and c < 0, then a × c < b × c. 
If a > b and c > 0, then a ÷ c > b ÷ c. 
If a > b and c < 0, then a ÷ c < b ÷ c. 
^{1}Adapted from Wisconsin Department of Public Instruction, http://dpi.wi.gov/ standards/mathglos.html, accessed March 2, 2010.
^{2}Many different methods for computing quartiles are in use. The method defined here is sometimes called the Moore and McCabe method. See Langford, E., “Quartiles in Elementary Statistics,” Journal of Statistics Education Volume 14, Number 3 (2006).
^{3}Adapted from Wisconsin Department of Public Instruction, op. cit.
^{4}To be more precise, this defines the arithmetic mean.
^{5}Adapted from Wisconsin Department of Public Instruction, op. cit.
^{6}Adapted from Box 24 of Mathematics Learning in Early Childhood, National Research Council (2009, pp. 32, 33).
^{7}7The first examples in each cell are examples of discrete things. These are easier for students and should be given before the measurement examples.