# Glossary

The K–12 SMP Glossary has two sections: The first includes terms used in the module, and the second is the CCSS for Mathematics Glossary from the CCSS Initiative Web site.

## Section 1: Glossary of terms from the module

Abstract: Considered apart from concrete existence; summary; having internal logic.

Argument: A coherent series of statements leading from a premise to a conclusion; a reason or set of reasons given with the aim of persuading others that an action is right or wrong.

Argumentation: The action or process of reasoning systematically in support of an idea, action, or theory.

Assumptions: A thing that is accepted as true or as certain to happen, without proof; accepted cause and effect relationship.

Cases: An instance of a particular situation; an example of something occurring.

Claim: An arguable statement; an assertion that something is true or not true.

College and career ready: Possessing the academic skills necessary for successful enrollment in postsecondary education as well as the ability to apply these skills to concrete situations in order to function in the workplace and in routine daily activities; em­ployability skills (such as critical thinking and responsibility) that are essential in any career area; and technical, job-specific skills related to a specific career pathway.

Community of Reasoners: A community of students who use mathematical reasoning as an instrument of inquiry for discovering and exploring new ideas and/or in justifying or proving mathematical claims.

Composite number: An integer that is divisible without remainder by at least one positive integer other than itself and one.

Conjecture: A statement that is believed to be true but not yet proved.

Consecutive positive integers: Positive integers that differ by 1.

Contextualize: To place in a context (the circumstances that form the setting).

Counterexample: An exception to a proposed general rule that illustrates that the rule does not hold in all situations.

Decontextualize: To remove from a context (the circumstances that form the setting).

Discourse: A verbal interchange of ideas; a formal and orderly and usually extended expression of thought on a subject.

Disequilibrium: When new information cannot fit into existing schemas which drives the learning process.

Divisor: A number by which another number is to be divided.

Even: An integer that can be divided exactly by 2.

Evenly: In equal amounts or shares; in a balanced or impartial way; “a class is evenly divided between girls and boys.”

Explanation: A statement or account that makes something clear; a reason or justification. A set of statements constructed to describe a set of facts.

Five (5) C’s for California’s 21st Century Goals: Creativity, critical thinking, communication, collaboration, and construction of new knowledge.

1. Creativity:
• Think creatively using a wide range of techniques, refining and analyzing ideas to improve and maximize efforts.
• Work creatively with others in developing, implementing, and communicating new ideas while being responsive to new and different perspectives and demonstrating originality and inventiveness; understanding real-world limits and viewing failure as an opportunity to learn.
• Implement innovations to make a tangible and useful contributions to the field in which the innovations will occur.

1. Critical Thinking:
• Reason effectively using deductive, inductive, and systems thinking.
• Make judgments and decisions after analyzing and evaluating information.
• Solve non-familiar problems in both conventional and innovative ways; and identify and ask significant questions that clarify various points of view and lead to better solutions.

1. Communication:
• Articulate thoughts and ideas effectively using oral, written, and nonverbal communication skills in a variety of forms and contexts.
• Listen effectively to decipher meaning, including knowledge, values, attitudes, and intentions.
• Use communication for a range of purposes.
• Utilize multiple media and technologies, and know how to judge their effectiveness a priority as well as assess their impact.
• Communicate effectively in diverse environments (including multi-lingual).

1. Collaboration:
• Demonstrate ability to work effectively and respectfully with diverse teams.
• Exercise flexibility and willingness to be helpful in making necessary compromises to accomplish a common goal.
• Assume shared responsibility for collaborative work and value the individual contributions made by each team member.

1. Construction of New Knowledge:
• Construction and exploration of new understandings of knowledge through the integration of content from one subject area to another.
• Provide pupils with multiple modes of demonstrating innovative learning.

Flaw: A defect; impairs soundness; a gap or error that makes something faulty.

Gesture: A movement by a limb to help express thought or to emphasize speech.

Habits of Mind: Dispositions that are skillfully and mindfully employed by characteristically successful people when confronted with problems (the solutions to which are not immediately apparent)

Incenter: The center of the inscribed circle in a given triangle.

Justify: To prove to show right or reasonable; to show a claim; statement to be right.

Logical: Capable of reasoning in an orderly and cogent fashion.

Mathematizing: Describing a situation in terms of mathematics, solving problems, posing problems, playing with patterns and relationships, and proving one’s thinking.

Mathematical language: Symbols, terms, notations, definitions, and representations.

Metacognitive Journal:  A journal in which one writes about their thinking on various issues, such as how to use particular strategies for learning or problem solving.

Misconception: A preconceived notion that is inaccurate; a conceptual misunderstanding.

Necessary and Sufficient: A condition that must be true for the statement to be true; if true, ensures the truth of the statement (e.g., A person must be at least 35 years old to be President of the United States. This is a necessary but not sufficient condition since there are many people who are at least 35 years old who are not President, but every President must be at least 35. Another example: An integer that is divisible by 4 is even. This is a sufficient condition since all numbers divisible by 4 are even, but it is not necessary since there are even numbers that are not divisible by 4).

Non-example: The opposite of an example; a non-example helps to say what the example is not.

Odd: Any integer that cannot be divided exactly by 2.

Premises: A proposition upon which an argument is based or from which a conclusion is drawn.

Prime: A positive integer that is not divisible without remainder by any integer except itself and 1. The integers 2, 3, 5, and 7 are prime numbers.

Proficient: Well advanced or competent; skilled.

Proof: Evidence or argument establishing or helping to establish a fact or the truth of a statement; anything serving as evidence (what proof do you have?); evidence or argument that compels the mind to accept an assertion as true; the process or an instance of establishing the validity of a statement.

Public knowledge: The mathematical ideas, procedures, methods and terms that have already been defined and established within a given community.

Quantitative: Relating to, being measured by, or pertaining to the quantity.

Reasoning: Think, understand, and form judgments by a process of logic; find an answer to a problem by considering various possible solutions.

Referents: The object or idea to which a word or phrase refers; one that refers or is referred to.

Representation: Refers to both process and production (i.e., to the act of capturing a mathematical concept or relationship in some form and to the form itself); processes and products that are observable externally as well as those that occur internally in the minds of people doing mathematics.

Reverse engineering: The process of discovering the technological principles of a device, object, or system through analysis of its structure, function, and operation.

Revision: The process of changing, improving, or making additions to something such as a plan, piece of writing, or law.

Revoicing: When a teacher repeats some or all of a student's mathematical statements during a discussion for the purpose of providing opportunities for students to revise their statements or add clarity and precision; also used to repeat a student's mathematical statements to involve other students in the discussion.

Self-efficacy: A person's belief in his/her own competence, such as the belief that one is capable of performing in a certain manner to attain a certain set of goals.

Sierpinski Triangle: A geometric object first described by the Polish mathematician Waclaw Sierpinski in 1915. It is formed recursively by dividing a triangle into four smaller congruent triangles and removing innermost triangle.

Standards for Mathematical Practice: Practices defined by the CCSS for Mathematics that describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education.

Strategy: A plan of action.

Symbolic representation: An expression, equation, or formula using mathematical symbols and numbers to represent a relationship.

Unpacking: Many mathematical statements contain previously developed concepts, terminology and symbolism. Unpacking refers to taking these compressed statements and describing or discussing the components of the statement. For example a statement like, "An integer that is divisible by 4 is even" could be unpacked to discuss: What does "divisible" mean? What is an integer? What is the logic of the statement?

Valid: Actually supporting the intended point or claim; acceptable as cogent; sound; a valid reason.

Viability: Capable of normal growth and development.

Visualization: Use of a drawing, picture, or model to help conceptualize mathematical concepts and relationships.

## Section II: CCSS Mathematics Glossary

(Adapted from the Common Core State Standards for Mathematics: http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf)

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 0–5, 0–10, 0–20, or 0–100, respectively (e.g., 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 (e.g., 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: For arbitrary numbers a, b, and c in a given number system: (a + b) + c = a + (b + c).

Associative property of multiplication: For arbitrary numbers a, b, and c in a given number system: (a × b) × c = a × (b × c).

Bivariate data: Pairs of linked numerical observations (e.g., 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 of addition: For arbitrary numbers a, b, and c in a given number system: a + b = b + a.

Commutative property of multiplication: For arbitrary numbers a, b, and c in a given number system: a × b = b × a.

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 (e.g., 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,” followed by “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 multi-digit number is expressed in expanded form when it is written as a sum of single-digit multiples of powers of ten (e.g., 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 (e.g., for the data set {1, 3, 6, 7, 10, 12, 14, 15, 22, 120}, the first quartile is 6². See also: median, third quartile, interquartile range).

Fraction: A number expressible in the form a/b where a is a whole number and b is a positive whole number (e.g., the word “fraction” in these standards always refers to a non-negative number). See also: Rational number.

Identity property of 0: For an arbitrary number a in a given number system: a + 0 = 0 + a = a.

Identity property of 1: For an arbitrary number a in a given number system: a × 1 = 1 × a = a.

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 (e.g., 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

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 list4 (e.g., 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 (e.g., 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 (e.g., 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 0­–100 (e.g., 72 ÷ 8 = 9).

Multiplicative inverses: Two numbers whose product is 1 (e.g., 3/4 and 4/3 are multiplicative inverses of one another because 3/4 × 4/3 = 4/3 × 3/4 = 1).

Number line diagram: A diagram of the number line used to represent numbers and support reasoning about them. In a number line diagram for measurement quantities, the interval from 0 to 1 on the diagram represents the unit of measure for the quantity.

Percent rate of change: A rate of change expressed as a percent (e.g., 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 equality: For arbitrary numbers a, b, and c in a given number system:

• 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

Properties of inequality: For arbitrary numbers a, b, and c in a given number system:

• 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

Properties of operations: For arbitrary numbers a, b, and c in a given 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
a × (b + c) = a × b + a × c over addition

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 non-zero denominator.

Rational number: A number expressible in the form a/b, where a and b are whole numbers and b ≠ 0. 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 (e.g., the heights and weights of a group of people could be displayed on a scatter plot5).

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 (e.g., 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.

Visual fraction model: A tape diagram, number line diagram, or area model.

Whole numbers: The numbers 0, 1, 2, 3...

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.