Definitions and worked examples in the MVP packets

Solution of an equation

Explains that a solution makes an equation true and demonstrates substitution.

Source note: The example initially prints a right side of −21, but checks and solves using −19. The definition is clear; the numerical example has a typo.

Ordered pairs and two-variable solutions

Explains that a two-variable solution includes both coordinates, with x first.

Function notation and evaluating a function

Substitutes a specified input into a linear rule, then explains function notation.

Arithmetic sequence: recursive and explicit rules

A completed table example explains the next-term rule, starting term, explicit rule, and arithmetic classification.

Slope and y-intercept in slope-intercept form

A short reminder identifies m as slope or rate of change and b as the y-intercept.

Arithmetic means

Names the missing terms between supplied arithmetic-sequence terms.

One-step equations: addition and multiplication

Two completed solutions show undoing addition and multiplying by a reciprocal.

Fill missing arithmetic and geometric terms

A completed pair of sequences shows repeated addition and multiplication between the same endpoints.

Equivalent linear rules from a growing pattern

Two students explain completed rules for the same quilt pattern.

Source note: Their reasoning is supplied; the student is still asked to justify equivalence.

Write a line rule from two points

Student dialogue supplies a slope, table, and point-based rules for the displayed line.

Source note: This is supplied reasoning for discussion, not a finished proof of equivalence.

Simplify equivalent expressions with justifications

A completed simplification distributes, combines like terms, and names a property.

Source note: The heading calls them equivalent equations, but the worked comparison is an identity between expressions.

Linear and exponential equation forms

A table names slope-intercept, point-slope, explicit exponential, and recursive forms and explains their parameters.

Additive and multiplicative properties of equality

States the properties and gives two completed one-step solutions with justifications.

Average rate of change

Defines average rate of change as change in output divided by change in input.

Find an intersection algebraically

Sets two linear rules equal, solves for x, and finds the shared y-coordinate.

Solve a two-step equation with justifications

A completed example uses addition and division properties of equality.

Compare expressions using a known variable value

Substitutes a given value into two expressions and writes the resulting inequality.

Substitution in a system with a known variable

Uses a fixed y-value in the second equation to calculate x.

Translate “greater than” into an inequality

One completed word-to-symbol example precedes the inequality-language exercises.

Substitute an expression to solve a linear system

Explains substitution, distributing a negative sign, solving for x, and finding y.

Translate and solve “six less than a number”

A completed dialogue corrects the translation and obtains the solution inequality.

Feasible region

Names the first-quadrant combinations that satisfy all constraints in the pet-boarding situation.

Pythagorean theorem and hypotenuse

Identifies the sides in the theorem and checks the 5–12–13 triangle.

Transformation, pre-image, and image

Introduces transformations as moving figures, with slide, flip, and turn illustrations.

Source note: Pre-image and image label the before-and-after pictures rather than receiving separate formal definitions.

Row reduction of a matrix

Supplies a step-by-step row-reduction procedure for solving a system, making diagonal entries one and other entries zero.

Translations, reflections, and rotations

Introduces translations as slides, reflections as flips, and rotations as turns.

Source note: A brief vocabulary introduction followed by unsolved transformation tasks.

Horizontal and vertical lines are perpendicular

States the relationship and supplies a horizontal-line/vertical-line example.

Line symmetry and rotational symmetry

Defines reflecting or rotating a figure onto itself.

Quadrilateral and its diagonals

Defines a four-sided polygon and a segment joining opposite vertices.

Rectangle, parallelogram, rhombus, square, and trapezoid

Prints the defining side and angle properties before the symmetry investigation.

Polygon diagonal and symmetry reminders

Defines a diagonal using nonconsecutive vertices and repeats line and rotational symmetry definitions.

Quadrilateral definitions used to justify properties

Restates definitions of rectangle, parallelogram, rhombus, and square before asking students to derive further properties.

Compass-and-straightedge construction and arcs

Explains the roles of an unmarked straightedge and a compass, and names part of a circle an arc.

Source note: The construction tasks that follow are student work, not completed examples.

Congruent triangles and ASA criteria

Describes corresponding congruent sides and angles, and names the two-angle/included-side information ASA.

Source note: The justification of ASA is left to the student.

Rectangle, rhombus, and square reminders

Restates the quadrilateral definitions before congruent-triangle investigations.

Slope triangles, rise/run, and the slope formula

A labeled triangle connects coordinate differences to rise and run and gives the slope formula.

Source note: The distance formula is to be developed by students; this is not a worked distance calculation.

What a proof must establish

Explains why a general argument using variables is needed rather than a few numerical examples.

Source note: The perpendicular-slope proof itself is a student task.

Vectors and graphical vector addition

Defines magnitude and direction, explains end-to-end and parallelogram addition, and introduces horizontal and vertical components.

Determinant of a 2 × 2 matrix

Defines the determinant, supplies its formula, and relates a nonzero determinant to the existence of a matrix inverse.

Vector notation and reflections with matrices

Explains column-matrix vector notation and supplies an original vector and its reflected coordinates.

Distribution shape, skew, outliers, variability, and center

A reference list describes uniform/unimodal/bimodal/multimodal shapes, skew, normal distributions, outliers, and variability.

Source note: The normal-distribution and outlier descriptions are classroom shorthand, not a complete statistical definition.

Marginal and joint frequencies

Defines the outer row/column totals and the interior two-way-table counts using the supplied table.

Relative and conditional frequencies

Defines frequency-table language and supplies completed column and whole-table percentage tables.

Source note: The packet groups whole-table relative frequencies with conditional frequencies; the denominator matters. The unfinished row-table exercise is not part of this entry.

Residuals and their signs

Explains observed-versus-predicted differences and the sign for points above or below a regression line.

Source note: The reference to segment lengths is informal: residuals are signed vertical differences. Students calculate the actual residual values.

Distribute and combine like terms

A completed algebraic expansion shows two uses of the distributive property and collection of like terms.

Multiply two binomials

The same expansion is rewritten as a product of two binomials and simplified.

Standard form of a quadratic and its coefficients

Defines standard form and gives a completed identification of a, b, and c.

Trinomial

Names the three-term quadratic expression used in the quilt-block task.

Source note: The wording describes the quadratic trinomials in this task; trinomials in general need not be quadratic.

Multiply binomials with a two-way table

A filled multiplication table supplies the partial products and simplified quadratic.

Square roots, higher roots, and simplifying radicals

Explains roots and supplies two completed radical simplifications.

Solve a cubic equation using roots or rational exponents

Two complete solutions of the same cubic equation compare radical and rational-exponent notation.

Polynomial functions and the Fundamental Theorem of Algebra

Supplies a general polynomial form and states the root-counting theorem.

Source note: The constant term has exponent zero, despite the phrase “positive integers.” Roots are counted over the complex numbers, with multiplicity.

Real numbers and the imaginary unit

Big Idea #1 identifies real numbers; Part 2 introduces i to represent square roots of negative numbers.

Imaginary and complex numbers

Big Idea #2 names imaginary and complex numbers; completed examples rewrite negative square roots using i.

Source note: Real numbers are also complex numbers; the next pages explicitly include a + 0i.

The complex plane and vector representations

Explains the real and imaginary axes and represents a + bi by the point (a, b) or a position vector.

Modulus of a complex number

Defines modulus as distance from the origin in the complex plane.

Multiply complex numbers

Supplies the expansion of (a + bi)(c + di), while asking students to justify it geometrically.

Complex conjugate

Defines the conjugate of a + bi and introduces its notation.

Divide complex numbers using a conjugate

Explains division by a real scalar and introduces multiplying numerator and denominator by a conjugate; the numerical quotient is an exercise.

Distance and average of complex numbers

Defines distance using the modulus of a difference and average using the midpoint in the complex plane.

Piecewise-defined function

Defines a function with different rules on different parts of its domain.

Absolute value written as a piecewise function

A supplied graph and two equivalent formulas illustrate rewriting absolute value piecewise; students explain the connection.

Straight angle and diameter

A circle-folding illustration explains opposite radii and a straight angle.

Linear pair

Defines a linear pair and supplies angle diagrams and real-life illustrations.

Source note: The intended angles are adjacent: sharing a vertex alone is not enough.

Write an argument as a two-column proof

Travis’s completed statements-and-reasons table establishes that a segment is a median of an equilateral triangle.

Organize a triangle-congruence argument

Tehani supplies a complete narrative congruence argument using ASA; students convert it to two-column form.

Source note: The prose uses ED bisecting BC; the given line above the blank proof mistakenly says AC.

Postulates and theorems

Distinguishes statements accepted without proof from statements supported by proof.

Vertical, exterior, and remote interior angles

Defines the angle names with labeled diagrams; conjectures about their measures remain exercises.

Transversals and angle pairs

Defines transversal and labels corresponding, alternate interior, and same-side interior angles.

Congruence and similarity through transformations

Defines congruence using rigid motions and similarity using rigid motions and dilations.

Similar polygons

Gives the corresponding-angle and proportional-side characterization of similarity.

Right Triangle Altitude Theorems

States two geometric-mean relationships for an altitude to the hypotenuse; students supply the proofs.

Sine, cosine, and tangent in a right triangle

Supplies the opposite/hypotenuse, adjacent/hypotenuse, and opposite/adjacent ratios beside a labeled triangle; evaluating them is student work.

Angles of elevation and depression

Explains each angle relative to a horizontal line of sight.

Rotational symmetry

Explains a rotation that makes a polygon coincide with itself.

Cyclic polygon and inscribed angles

Defines a polygon whose vertices lie on one circle and identifies its inscribed angles.

Degree measure of an arc

Defines arc measure using the corresponding central angle; the inscribed-angle theorem is left to investigate.

Circumscribed angle

Names the angle formed by two lines tangent to a circle.

Find arc length from a central angle

A completed example uses the fraction of a full circumference to compute an arc length.

Source note: One sentence calls the result an “area,” but the calculation and units are for length.

Radians and familiar turns

Defines one radian and a full turn, then supplies reasoning for half and quarter turns.

Oblique solids and volume formulas

Names oblique prisms, pyramids, cylinders, and cones and supplies volume statements in terms of base or cross-sectional area and height.

Parabola, focus, and directrix

Defines a parabola as points equally distant from a focus and a directrix.

Sketch a parabola from its focus and directrix

A completed distance calculation, reflected point, vertex, and graph illustrate the construction.

Standard form of a circle

Supplies the circle equation with center (h, k) and radius r.

Ellipse and foci

Defines an ellipse using a constant sum of distances from two foci.

Standard form of an ellipse

Supplies standard-form equations and describes horizontal and vertical semiaxes and the distance from the center to a focus.

Hyperbola sketch and standard forms

Explains a specific asymptote-rectangle construction with its supplied diagram, then gives both standard forms.

Conditional probability and “given” notation

Introduces conditional probability through a supplied tree diagram and the vertical-bar notation.

Independence and conditional probability

States independence using conditional probabilities and recalls the conditional-probability formula.

Undo a function with its inverse

A completed function-machine example adds eight and then subtracts eight, returning to the starting input.

Composition notation

Explains the equivalence between nested function notation and the composition symbol.

Translate powers into logarithms

Completed base-two and base-ten examples show the inverse relationship between exponential and logarithmic expressions.

Logarithm

Defines a logarithm by its equivalent exponential equation and states restrictions on the base and argument.

Product, quotient, and power rules for logarithms

Supplies the three logarithm rules; students justify their generality.

Evaluate a logarithm using the product rule

Mary and Abe compute a base-two logarithm from supplied values using the product rule.

Logarithm rules beside practice problems

Repeats the product, quotient, and power rules next to practice.

Source note: Stephen’s calculation at the top is intentionally incorrect: the product rule does not apply to a sum.

Solve exponential and logarithmic equations

Two completed examples rewrite expressions using equal bases or equal logarithms to solve for an input.

Compound-interest formula

Supplies the compound-interest model and explains principal, annual rate, compounding frequency, and time.

The number e and continuous compounding

Introduces e and supplies the continuous-compounding formula with principal, rate, and time defined.

Natural logarithms and continuous growth or decay

Names the base-e logarithm and gives continuous growth and decay formulas with their parameters.

Integer division with a remainder

A supplied long-division result illustrates a whole-number quotient and remainder.

Cubic functions and polynomial degree

Names the cubic function and connects degree with the highest exponent, contrasting linear and quadratic functions.

Subtract polynomials by changing signs

A completed subtraction is displayed both horizontally and vertically.

Standard form and a polynomial area model

Defines descending-power standard form and supplies a filled multiplication box with its simplified result.

Polynomial long division

A six-step completed division compares polynomial division with integer long division.

Source note: The concluding sentence mistakenly says 121 is divisible by 12; the displayed division is 1452 ÷ 12 = 121.

Polynomial function and degree

Gives a formal definition with real coefficients and nonnegative integer exponents, alongside examples and non-examples.

Closure and counterexamples

Explains closure under an operation and how a counterexample refutes a universal claim.

Conjugate pairs

Defines conjugate pairs and supplies factoring and conjugate examples involving radicals and imaginary numbers.

Fundamental Theorem of Algebra

States the root-counting theorem before students investigate quadratic and higher-degree polynomials.

Source note: Roots are counted over the complex numbers, with multiplicity.

Even function

Gives the algebraic definition of an even function and a supplied example/non-example graph pair.

Odd function

Gives the algebraic definition of an odd function after its examples and non-examples.

Rational function

Defines a rational function as a quotient of polynomials whose denominator is not the zero polynomial.

Pythagorean theorem

Recalls the squared-side relationship and identifies the legs and hypotenuse.

Law of Sines and Law of Cosines

Supplies both triangle-solving laws, including alternate forms.

Rotational symmetry

Explains rotational symmetry using a hubcap and a quarter-turn example.

Triangle area from two sides and their included angle

Supplies three equivalent sine-based area formulas, with the corresponding side and angle labels.

Triangle area formulas and Heron’s formula

Supplies an area formula derived from the Law of Sines and Heron’s formula with semiperimeter.

Angular speed and linear speed

Defines angular speed and contrasts it with linear speed, using clock hands and rotating records.

Positive angles of rotation

States the counterclockwise convention for positive rotation.

Cosine from circle coordinates

Recalls cosine as the horizontal coordinate divided by the radius.

Convert degrees to radians

A completed conversion of 72 degrees demonstrates multiplying by the appropriate unit ratio.

Coterminal angles

Defines coterminal angles and supplies a labeled positive-and-negative-angle example.

Right-triangle tangent ratio

Recalls tangent as opposite side length divided by adjacent side length.

Even and odd functions

Supplies the two algebraic definitions before students classify trigonometric functions.

Secant, cosecant, and cotangent

Supplies the three reciprocal right-triangle ratios and their side-length formulas.

Fundamental trigonometric identities

Lists sine and cosine symmetry, the Pythagorean identity, and the tangent quotient identity; students provide justifications.

Sine and cosine sum and difference identities

Supplies all four sum and difference formulas and explains their use for exact values.

Inverse sine and its principal interval

Supplies a calculator evaluation and explicitly defines inverse sine on its restricted output interval.

Multiply and divide complex numbers in polar form

States the multiply-moduli/add-angles rule and the divide-moduli/subtract-angles rule. Proof exercises are not indexed.

Roots of a complex number in polar form

Supplies the root formula, root count, and equal angular spacing.

Complex modulus and coordinate representations

Explains real and imaginary axes, points and vectors, and supplies the modulus formula.

Polar coordinate conversion and trigonometric form

Supplies polar-to-rectangular coordinate relationships and the trigonometric form of a complex number.

Identify inside and outside operations

A completed substitution separates cubing an input from multiplying the result by five.

Frequency polygon and frequency curve

Explains connecting histogram midpoints and smoothing them into a frequency curve, with a supplied sketch.

Parametric curve and parameter

Defines a parametric curve and explains parameter inputs with ordered-pair outputs.

Standard normal distribution and z-scores

Defines the standardized mean and standard deviation and interprets a negative z-score using an area-table result.

Z-score formula

Supplies the formula for standardizing a data value by subtracting the mean and dividing by the standard deviation.

Population, sample, and parameter of interest

Defines three core terms used to describe statistical studies.

Surveys, observational studies, and experiments

Defines three study methods and illustrates each with a diet study, including control and experimental groups.