Mathematics Codex & Vocabulary Hub

The non-negative magnitude of a real number on the continuous number line, representing its geometric distance from origin zero ($0$) irrespective of direction.

If a = b, then a + c = b + c for any real number c.

The algorithmic process of composing a single higher base-ten unit whenever the sum of digits in an existing column equals or exceeds the base value ($10$).

Core mathematical concept established in Module 4: Polynomial and Quadratic Expressions, Equations, and Functions, Quadratic Expressions, Equations, and Functions.

An equality relation A = B such that A and B evaluate to the same value for all possible values of their variables.

A mathematical representation using variables and operations to simulate real-world behaviors and make predictions.

Core mathematical concept established in Module 5: A Synthesis of Modeling with Equations and Functions, Elements of Modeling.

Core mathematical concept established in Module 5: A Synthesis of Modeling with Equations and Functions, Elements of Modeling.

Core mathematical concept established in Module 2: Descriptive Statistics, Numerical Data on Two Variables.

The total two-dimensional surface enclosed within a closed boundary, quantified as the aggregate count of square unit tiles ($1 \times 1$) that tile the planar region without overlaps.

A sequence is called arithmetic if there is a real number d such that each term in the sequence is the sum of the previous term and d.

How can a graph be used to make a prediction? Estimate the value from the line or curve.

The property stating that the grouping of numbers being added or multiplied does not change the result: (a + b) + c = a + (b + c) and (ab)c = a(bc).

The change in the dependent variable (elevation) divided by the change in the independent variable (time) over a given interval. Graphically, this is the slope of the line segment.

Geometric postulates establishing that two triangles possess identical size and shape if a specific subset of corresponding sides and angles are proven equal.

The constant factor by which the function value is multiplied each time $x$ increases by 1 unit. When $b > 1$, the function models growth; when $0 < b < 1$, it models exponential decay.

The value of a digit determined strictly by its position within a number, where each unit in the tens column equals an aggregate group of ten individual single units.

A polynomial with exactly two terms, such as x + 4 or 3x² - 2.

The line dividing the coordinate plane into two half-planes in an inequality graph.

A continuous interval of real numbers with finite endpoints on both sides, e.g., (a, b] or [a, b].

A graph that provides a picture of the data ordered and divided into four intervals that each contains approximately 25% of the data.

Geometric relationships for any Euclidean circle, where Circumference ($C$) is the boundary perimeter and Area ($A$) is the internal planar surface, both governed by the transcendental constant $\pi \approx 3.14159$.

When grouping students, consider their reading skill.

A solid filled dot on a number line representing an inclusive boundary that is included in the solution set (≤ or ≥).

A set has closure under an operation if performing that operation on members of the set always produces a member of the same set. Polynomials are closed under addition and subtraction.

I am thinking of a plus 2 pattern, so it continues 10, 12, 14, 16, ….

The numerical multiplicative factor preceding a variable in an algebraic term (e.g., 5 in 5x²).

The total operational output rate achieved when multiple independent agents or machines work concurrently.

Students may subtract 16 outside of the parentheses to balance the +16 inside in Opening Exercise 2.

The property stating that the order in which two real numbers are added or multiplied does not change the result: a + b = b + a and ab = ba.

Core mathematical concept established in Module 2: Descriptive Statistics, Calculating and Interpreting Measures of Center and Variability.

Core mathematical concept established in Module 4: Polynomial and Quadratic Expressions, Equations, and Functions, Function Transformations and Modeling.

Core mathematical concept established in Module 4: Polynomial and Quadratic Expressions, Equations, and Functions, Using Different Forms for Quadratic Functions.

An extension of the one-dimensional real number system into a two-dimensional complex plane ($\mathbb{C}$), defined by the imaginary unit $i$ whose square equals $-1$.

An inequality formed by joining two individual inequalities with the connective word 'and' or 'or'.

Core mathematical concept established in Module 2: Descriptive Statistics, Categorical Data on Two Variables.

A compound sentence that is true only if both of its component statements are simultaneously true. Corresponding to set intersection (∩).

A condition or restriction that must be satisfied by the solutions of a mathematical model.

Core mathematical concept established in Module 4: Polynomial and Quadratic Expressions, Equations, and Functions, Quadratic Expressions, Equations, and Functions.

The total aggregated quantity of water used up to a specified point in time.

Core Math Tools, http://nctm.

A logical process in which a conclusion is based on the concordance of multiple premises that are assumed or proven to be true.

The highest exponent of the variable in any term of the polynomial (when in one variable).

For non-zero polynomials P and Q, deg(P · Q) = deg(P) + deg(Q).

The output value of a function, which depends on the input value (usually elevation/height, graphed on the vertical axis).

The instantaneous rate of change of a differentiable continuous function with respect to its independent variable, geometrically representing the exact slope of the tangent line at any point $x$.

Core mathematical concept established in Module 4: Polynomial and Quadratic Expressions, Equations, and Functions, Using Different Forms for Quadratic Functions.

Core mathematical concept established in Module 2: Descriptive Statistics, Shapes and Centers of Distributions.

Two sets that have no elements in common (their intersection is empty).

A compound sentence that is true if at least one of its component statements is true. Corresponding to set union (∪).

The mathematical axiom asserting that multiplying a sum by a number gives the same result as multiplying each addend separately and adding the products: a(b + c) = ab + ac.

A plot of each data value on a scale or number line.

The actual percentage of total income paid in taxes: (Total Tax Paid / Total Income) * 100%.

A graph where vertical position represents physical altitude or height above ground.

An algebraic technique where equations are added or subtracted to cancel out one of the variables.

This graph of a quadratic function opens up.

Given a quadratic function in the form f(x)=ax^{2}+bx+c (or f(x)=a(x-h)^{2}+k), the quadratic function is said to open up if a>0 and open down if a<0.

A mathematical statement asserting that two algebraic expressions are equal in value.

Two algebraic expressions that yield the exact same numerical result whenever the same values are substituted for their variables.

Two systems of equations that have the exact same solution set.

Core mathematical concept established in Module 2: Descriptive Statistics, Shapes and Centers of Distributions.

An iterative place-value algorithm that determines how many times a positive integer divisor ($b$) partitions into a dividend ($a$), yielding an integer quotient ($q$) and an exact remainder ($r$).

A formula allowing direct calculation of the nth term of a sequence using only the index n, without computing any preceding terms.

How would the relative frequencies look if males and females had the same opinions about their favorite superpowers? This question is a prelude to Lesson 11.

Core mathematical concept established in Module 3: Linear and Exponential Functions, Linear and Exponential Sequences.

A function in which an independent variable appears as an exponent: $f(x) = a \cdot b^x$. The rate of change increases directly in proportion to the magnitude of the function value itself.

A non-linear mathematical model where the rate of change is proportional to the current amount, resulting in geometric compounding over continuous or discrete time intervals $t$.

A solution that emerges from the process of solving an equation but is not a valid solution to the original equation.

An expression expressed as a product of irreducible linear or polynomial factors.

The intersection of all half-planes defined by a system of inequalities representing all viable solutions.

A mathematical relationship or rule expressed with algebraic symbols (e.g., V = lwh).

During this phase of the lesson, students should work in small groups.

I first looked for the function type, which narrowed my search considerably.

I looked at the overall shape first to identify what type of graph it was (quadratic, linear, exponential, piecewise, square root, or cube root).

Core mathematical concept established in Module 3: Linear and Exponential Functions, Transformations of Functions.

The method of renaming fractions into equivalent representations sharing a common unit measure (the Least Common Denominator) prior to combining numerators.

An algebraic law establishing that dividing by a rational fraction is mathematically equivalent to multiplying by its multiplicative inverse (reciprocal).

A function is a correspondence between two sets, X and Y, in which each element of X is matched to one and only one element of Y.

The unifying theorem of mathematical analysis demonstrating that differentiation and integration are inverse operations: the definite integral (net accumulated area) can be calculated via anti-derivatives.

A geometric sequence goes from one term to the next by multiplying (or dividing) by the same value.

Core mathematical concept established in Module 4: Polynomial and Quadratic Expressions, Equations, and Functions, Function Transformations and Modeling.

Core mathematical concept established in Module 4: Polynomial and Quadratic Expressions, Equations, and Functions, Quadratic Expressions, Equations, and Functions.

The set of all points on the coordinate plane whose coordinates (x, y) satisfy the equation.

Core mathematical concept established in Module 3: Linear and Exponential Functions, Transformations of Functions.

Graphs Data are often summarized by graphs; the graphs are the first indicator of variability in the data.

The largest natural number that evenly divides two or more integers without leaving a remainder, forming the basis of simplifying fractions.

The region of a coordinate plane on one side of an infinite boundary line.

A graph of data that groups the data based on intervals and represents the data in each interval by a bar.

A horizontal line ($y = 0$ for basic exponential curves) that the curve approaches closer and closer as $x \to -\infty$, but never intersects or passes.

A segment where slope m = 0, indicating zero flow rate or usage.

An equation that evaluates to True for all possible values of its variables (e.g., 2(x + 1) = 2x + 2).

R→R is the function such that x↦x^{2}, then the range of f is the set of all nonnegative real numbers.

Notice that in the next table in your packet (Brand A), the second row says "Deviation from the Mean.

Sequences can be indexed by starting with any integer.

The complete family of antiderivative functions whose instantaneous derivative equals the integrand $f(x)$, unified by the arbitrary constant of integration $C$.

The input value of a function, which changes independently (usually time, graphed on the horizontal axis).

A mathematical statement comparing two expressions using relation symbols <, ≤, >, or ≥.

If a < b and c < 0, then ac > bc. Multiplying or dividing both sides of an inequality by a negative number reverses the direction of the inequality symbol.

The starting value a_0 (or a_1) required to initiate the evaluation of a recursive process.

The value of the function when $x = 0$, representing the $y$-intercept. Because any nonzero base raised to the power of 0 equals 1 ($b^0 = 1$), $f(0) = a \cdot 1 = a$.

Core mathematical concept established in Module 2: Descriptive Statistics, Numerical Data on Two Variables.

Core mathematical concept established in Module 4: Polynomial and Quadratic Expressions, Equations, and Functions, Quadratic Expressions, Equations, and Functions.

Core mathematical concept established in Module 2: Descriptive Statistics, Calculating and Interpreting Measures of Center and Variability.

The set of elements belonging to both sets A and B (A ∩ B).

The least common multiple of the denominators of a set of fractions.

R→R be the function such that x↦x^{2}.

Core mathematical concept established in Module 3: Linear and Exponential Functions, Functions and Their Graphs.

ExponentialC: Quadratic I looked at the difference in each output.

An expression constructed from a set of terms by multiplying each term by a constant and adding the results.

The canonical algebraic representation of a two-dimensional straight line, where $m$ signifies the constant rate of change (slope) and $b$ represents the coordinate value where the line intercepts the vertical y-axis ($(0, b)$).

An equation consisting predominantly of letters and variables representing known or unknown quantities.

The inverse mathematical function of exponentiation, answering the question: "To what exponent must the base $b$ be raised to produce the value $x$?"

What do you notice about the three sets of data? Can you identify the type of function they represent? Students may observe some of the examples listed below.

A valid solution that is accidentally eliminated by an invalid operation (such as dividing both sides by a variable expression that equals zero).

The tax rate applied to the last dollar of income earned within a specific bracket.

The process of choosing and using appropriate mathematics and statistics to analyze empirical situations, to understand them better, and to improve decisions.

Statistical metrics summarizing an entire data distribution into a single representative central value: the arithmetic average (Mean), the 50th percentile midpoint (Median), and the most frequent value (Mode).

Core mathematical concept established in Module 2: Descriptive Statistics, Calculating and Interpreting Measures of Center and Variability.

Core mathematical concept established in Module 2: Descriptive Statistics, Calculating and Interpreting Measures of Center and Variability.

Core mathematical concept established in Module 5: A Synthesis of Modeling with Equations and Functions, Completing the Modeling Cycle.

Core mathematical concept established in Module 5: A Synthesis of Modeling with Equations and Functions, Completing the Modeling Cycle.

Core mathematical concept established in Module 2: Descriptive Statistics, Numerical Data on Two Variables.

Core mathematical concept established in Module 2: Descriptive Statistics, Numerical Data on Two Variables.

If a = b, then ac = bc for any real number c.

Core mathematical concept established in Module 4: Polynomial and Quadratic Expressions, Equations, and Functions, Quadratic Expressions, Equations, and Functions.

Core mathematical concept established in Module 3: Linear and Exponential Functions, Using Functions and Graphs to Solve Problems.

Any real number that causes a denominator in an expression to equal zero, making the expression undefined.

Just like the last lesson, use the fact that the examples are very similar to reduce the amount of writing on the board.

Every new example or exercise in this lesson usually inserts just one more line of pseudocode into the previous exercise or example.

Now can you see the trends more clearly? Yes, it is obvious that the first is linear.

Any object that is free falling or projected into the air without a power source is under the influence of gravity.

A hollow dot on a number line representing an exclusive boundary that is not included in the solution set (< or >).

If students are struggling with the type of function, have them plot the points.

A pair of numbers (x, y) written in a specific order representing a unique point on a coordinate plane.

The universal algebraic precedence convention dictating the precise sequence in which arithmetic operations must be evaluated to ensure unambiguous solutions.

Core mathematical concept established in Module 3: Linear and Exponential Functions, Using Functions and Graphs to Solve Problems.

Core mathematical concept established in Module 3: Linear and Exponential Functions, Transformations of Functions.

A function defined by multiple linear segments, each covering a specific interval of time or domain. The overall graph looks like joined straight lines with corners.

A single point on the coordinate plane where two lines cross, representing the simultaneous solution.

An expression consisting of variables and coefficients, constructed using only addition, subtraction, multiplication, and non-negative integer exponentiation.

Why does the first scatter plot result in an arch shape in the residual plot? The points in the scatter plot are not in a straight line.

An operational shortcut derived from the binomial theorem and difference quotient that computes the instantaneous rate of change of any monomial power function in a single algebraic step.

A natural number greater than $1$ is Prime if its only positive divisors are $1$ and itself. A number with more than two distinct positive factors is Composite.

Specifies the domain of the variable of x to be the set of integers.

A fundamental theorem of Euclidean geometry stating that in any right triangle, the area of the square erected upon the hypotenuse ($c$) equals the combined sum of the areas of the squares erected upon legs $a$ and $b$.

A polynomial function of degree 2 ($f(x) = ax^2 + bx + c$, with $a \neq 0$). Its graph forms a U-shaped parabola.

An equation containing at least one fraction whose denominator contains a variable expression.

A continuous portion of the coordinate line extending indefinitely in one direction from an endpoint.

What does the graph of a quadratic equation look like?Remind students that when calculating squares in their calculator, they need to watch out for a common error: (-1)^{2}≠-1^{2}.

A geometric configuration of discrete objects aligned into identical horizontal rows and vertical columns, serving as the concrete foundation of multiplication.

An equation that expresses each element of a sequence as a function of the preceding elements (e.g., a_n = 2a_{n-1} + 5).

Core mathematical concept established in Module 3: Linear and Exponential Functions, Linear and Exponential Sequences.

A sequence of numbers where each term after the first is defined as a function of the preceding term(s).

Core mathematical concept established in Module 2: Descriptive Statistics, Numerical Data on Two Variables.

An algebraic operation that preserves the exact solution set, such that applying its inverse returns the original equation.

Fundamental trigonometric ratios relating the acute reference angle ($\theta$) of a right triangle to dimensionless quotients of its side lengths.

The input values that cause an expression or function to evaluate to zero.

We can tell that this problem involves a geometric sequence because we are multiplying each term by either 1.

The differences between consecutive first differences. In any quadratic sequence with equal intervals, the second differences are constant!

Let a_{n+1}=2a_{n},a_{0}=1 for 0≤n≤4 where n is an integer.

A mathematical shorthand notation describing a set by stating the properties that its members must satisfy, e.g., {x | x > -9}.

The set of all values from the domain that make an open sentence or equation true.

Core mathematical concept established in Module 4: Polynomial and Quadratic Expressions, Equations, and Functions, Quadratic Expressions, Equations, and Functions.

A graph where vertical position represents instantaneous rate of motion.

A polynomial written such that its terms are placed in descending order of degree from left to right.

Core mathematical concept established in Module 4: Polynomial and Quadratic Expressions, Equations, and Functions, Function Transformations and Modeling.

The variable that is isolated by itself on one side of an equation with a coefficient of 1.

An algebraic technique where one variable is isolated and substituted into the other equation.

Core mathematical concept established in Module 2: Descriptive Statistics, Categorical Data on Two Variables.

I am thinking of the units digits in the multiples of two, so it continues 2, 4, 6, 8, 0, 2, 4, 6, 8, ….

A collection of two or more linear equations involving the same set of variables.

A geometric organizer where terms of one polynomial form rows and terms of the other form columns, computing all partial products systematically.

Setting an explicit sequence formula equal or unequal to a desired threshold target and solving backwards for the initial term.

The amount of gross income used to calculate tax liability, after subtracting allowable exemptions and deductions.

A single mathematical number, variable, or product of numbers and variables separated by addition or subtraction operators.

A coordinate pair (not on the boundary line) substituted into an inequality to determine which half-plane contains solutions.

Callie: It looks like the U.

X→Y is used to name the function and describes both X and Y.

Core mathematical concept established in Module 3: Linear and Exponential Functions, Linear and Exponential Sequences.

Core mathematical concept established in Module 4: Polynomial and Quadratic Expressions, Equations, and Functions, Function Transformations and Modeling.

Core mathematical concept established in Module 4: Polynomial and Quadratic Expressions, Equations, and Functions, Function Transformations and Modeling.

A polynomial with exactly three terms, such as ax² + bx + c.

The property of a mathematical sentence indicating whether it is True or False under a specific substitution.

The relative frequency table would be found by dividing each of the above cell values by 450.

The set of elements belonging to set A, set B, or both (A ∪ B).

A rational quantity formed by partitioning a single whole into $b$ congruent, equal-sized pieces, where the fraction represents exactly one of those partitioned segments.

A rate in which the second quantity in the comparison is one unit (e.g., 25 pages per minute).

Core mathematical concept established in Module 4: Polynomial and Quadratic Expressions, Equations, and Functions, Using Different Forms for Quadratic Functions.

The speed of an object in a given direction, corresponding to signed slope on a displacement graph.

The point where the graph of a quadratic function and its axis of symmetry intersect is called the vertex.

The total measure of three-dimensional space enclosed within a solid boundary, quantified as the aggregate number of unit cubes ($1 \times 1 \times 1$) that pack the interior without gaps.

Core mathematical concept established in Module 3: Linear and Exponential Functions, Linear and Exponential Sequences.

Core mathematical concept established in Module 3: Linear and Exponential Functions, Functions and Their Graphs.

X→Y, then x is matched to an element of Y called f(x).

If the product of two real numbers is zero, then at least one of the numbers must be zero: ab = 0 implies a = 0 or b = 0.