Standards are the floor. Mastery and understanding are the goal.

Explore how each subject becomes a coherent sequence of knowledge, reasoning, practice, and transfer—not a disconnected checklist.

Grade 12 Math

69 curriculum outcomes with explicit teaching progressions

Select a record to inspect its progression

Vectors Matrices · CCSS.Math.Content.HS.N-VM.A.1

Represent vector quantities

Independently recognize vector quantities as having both magnitude and direction. represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes. across representations and contexts.

Component progression

  1. 1I can distinguish quantities that have both magnitude and direction (vectors) from quantities that have magnitude only (scalars).
  2. 2I can draw a vector as a directed line segment with correct length representing magnitude and correct direction.
  3. 3I can use appropriate symbols to denote a vector and to denote its magnitude.
Vectors Matrices · CCSS.Math.Content.HS.N-VM.A.2

Find vector components from coordinates

Independently find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point. across representations and contexts.

Component progression

  1. 1I can identify the initial point and terminal point of a vector drawn between two coordinate points.
  2. 2I can compute a vector's components by subtracting the initial point's coordinates from the terminal point's coordinates.
  3. 3I can use a vector's components to redraw it in standard position starting at the origin.
Vectors Matrices · CCSS.Math.Content.HS.N-VM.A.3

Solve problems using vector quantities

Independently solve problems involving velocity and other quantities that can be represented by vectors. across representations and contexts.

Component progression

  1. 1I can represent a real-world quantity, such as a velocity or force, as a vector with appropriate magnitude and direction.
  2. 2I can solve a problem requiring the combination of two or more vector quantities, such as a velocity affected by wind or current.
  3. 3I can interpret the magnitude and direction of a resulting vector in terms of the original real-world situation.
Vectors Matrices · CCSS.Math.Content.HS.N-VM.B.4

Add and subtract vectors

Independently add and subtract vectors: add vectors end-to-end, component-wise, and by the parallelogram rule, understanding that the magnitude of a sum of two vectors is typically not the sum of the magnitudes; given two vectors in magnitude and direction form, determine the magnitude and direction of their sum; and subtract a vector from another by adding its additive inverse, representing the subtraction graphically and performing it component-wise. across representations and contexts.

Component progression

  1. 1I can add two vectors graphically using the tip-to-tail (triangle) or parallelogram method, and compare the magnitude of the sum to the sum of the two original magnitudes.
  2. 2I can given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
  3. 3I can add and subtract vectors by adding or subtracting their corresponding components, representing subtraction as adding the additive inverse.
Vectors Matrices · CCSS.Math.Content.HS.N-VM.B.5

Multiply a vector by a scalar

Independently multiply a vector by a scalar: represent scalar multiplication graphically by scaling vectors and possibly reversing their direction, and perform scalar multiplication component-wise; compute the magnitude of a scalar multiple using ||cv|| = |c|v, and determine that the direction of the scalar multiple is along the original vector when the scalar is positive and against it when the scalar is negative. across representations and contexts.

Component progression

  1. 1I can draw the result of multiplying a vector by a positive or negative scalar, showing the change in length and, when negative, the reversed direction.
  2. 2I can compute the components of a scalar multiple of a vector by multiplying each component by the scalar.
  3. 3I can compute the magnitude of a scalar multiple of a vector using ||cv|| = |c| times the original magnitude, and determine whether the result points along the original vector or against it based on the scalar's sign.
Vectors Matrices · CCSS.Math.Content.HS.N-VM.C.6

Use matrices to represent data

Independently use matrices to represent and manipulate data, such as to represent payoffs or incidence relationships in a network. across representations and contexts.

Component progression

  1. 1I can organize real-world data, such as payoffs or a network's connections, into a matrix with correctly labeled rows and columns.
  2. 2I can identify the dimensions of a matrix and locate a specific entry by its row and column.
  3. 3I can interpret what a specific entry in a data matrix represents in the original real-world context.
Vectors Matrices · CCSS.Math.Content.HS.N-VM.C.7

Multiply matrices by scalars

Independently multiply matrices by scalars to produce new matrices, such as when computing double a matrix's entries. across representations and contexts.

Component progression

  1. 1I can multiply every entry of a matrix by a given scalar to produce a new matrix.
  2. 2I can interpret the result of scalar matrix multiplication in a real-world context, such as scaling all payoffs by a fixed factor.
  3. 3I can given a matrix and a scalar multiple of it, solve for an unknown scalar or unknown entry.
Vectors Matrices · CCSS.Math.Content.HS.N-VM.C.8

Add, subtract, and multiply matrices

Independently add, subtract, and multiply matrices of appropriate dimensions. across representations and contexts.

Component progression

  1. 1I can add or subtract two matrices of the same dimensions by combining corresponding entries.
  2. 2I can determine whether two matrices can be multiplied based on their dimensions, and find the dimensions of the product.
  3. 3I can multiply two matrices of compatible dimensions using the row-by-column process.
Vectors Matrices · CCSS.Math.Content.HS.N-VM.C.9

Understand properties of matrix multiplication

Independently understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties. across representations and contexts.

Component progression

  1. 1I can compute ab and ba for a specific pair of square matrices and show they are generally not equal.
  2. 2I can verify with an example that (ab)c = a(bc) for compatible matrices.
  3. 3I can verify with an example that a(b + c) = ab + ac for compatible matrices.
Vectors Matrices · CCSS.Math.Content.HS.N-VM.C.10

Understand the role of zero and identity matrices

Independently understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. the determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse. across representations and contexts.

Component progression

  1. 1I can write the zero matrix and identity matrix for a given dimension.
  2. 2I can verify that adding the zero matrix to any matrix of the same dimension leaves it unchanged, and that multiplying a square matrix by the identity matrix of matching dimension leaves it unchanged.
  3. 3I can determine whether a square matrix has a multiplicative inverse by checking whether its determinant is nonzero.
Vectors Matrices · CCSS.Math.Content.HS.N-VM.C.11

Multiply a vector by a matrix

Independently multiply a vector, regarded as a matrix with one column, by a matrix of suitable dimensions to produce another vector; work with matrices as transformations of vectors. across representations and contexts.

Component progression

  1. 1I can write a vector as a matrix with one column so it can be used in matrix multiplication.
  2. 2I can multiply a matrix by a compatible column vector to produce a new vector.
  3. 3I can interpret matrix-vector multiplication as the matrix transforming the input vector into an output vector.
Vectors Matrices · CCSS.Math.Content.HS.N-VM.C.12

Interpret 2×2 matrices as transformations

Independently work with 2×2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area. across representations and contexts.

Component progression

  1. 1I can apply a 2×2 matrix to the vertices of a figure to find the image of the figure under the transformation.
  2. 2I can compute the determinant of a 2×2 matrix.
  3. 3I can interpret the absolute value of a 2×2 matrix's determinant as the factor by which the transformation scales area.
Precalculus Functions · CCSS.Math.Content.HS.F-BF.B.4

Find inverses of more complex functions

Independently find inverse functions for more complex functions, including exponential, logarithmic, and simple trigonometric functions on a restricted domain; verify inverses by composition; and read values of an inverse function from a graph or table given that the function has an inverse. across representations and contexts.

Component progression

  1. 1I can find the inverse of an exponential function (a logarithmic function), a logarithmic function (an exponential function), or a trigonometric function restricted to its standard domain, by solving for the swapped variable.
  2. 2I can given that a function has an inverse, read values of the inverse function directly from the original function's graph or table.
  3. 3I can verify that a found inverse is correct by composing it with the original function and confirming the result is the identity.
Precalculus Functions · CCSS.Math.Content.HS.F-BF.B.5

Understand the inverse relationship between exponents and logarithms

Independently understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. across representations and contexts.

Component progression

  1. 1I can rewrite an equation between its exponential form and its equivalent logarithmic form.
  2. 2I can apply the product, quotient, and power properties of logarithms to simplify or expand a logarithmic expression.
  3. 3I can solve an equation involving exponents or logarithms by converting between the two forms.
Precalculus Functions · CCSS.Math.Content.HS.F-LE.A.4

Apply logarithmic solutions within complex modeling problems

Independently for exponential models with base 2, 10, or e, express as a logarithm the solution to ab^(ct) = d and evaluate the logarithm using technology, within multi-step precalculus modeling problems that combine exponential growth or decay with other functions or constraints. across representations and contexts.

Component progression

  1. 1I can set up an equation of the form ab^(ct) = d, with base b equal to 2, 10, or e, from a multi-step precalculus modeling scenario.
  2. 2I can isolate the exponential term, apply a logarithm to solve for the unknown, and evaluate the logarithm using technology, within the larger modeling problem.
  3. 3I can interpret the logarithmic solution in context and check it against any stated constraints of the model.
Precalculus Functions · CCSS.Math.Content.HS.F-TF.A.3

Determine special-triangle trig values and unit-circle symmetry relationships

Independently use special triangles to determine geometrically the values of sine, cosine, and tangent for π/3, π/4, and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π-x, π+x, and 2π-x in terms of their values for x, where x is any real number. across representations and contexts.

Component progression

  1. 1I can use 30-60-90 and 45-45-90 special right triangles to determine the exact sine, cosine, and tangent values for π/6, π/4, and π/3.
  2. 2I can use the unit circle to express the sine, cosine, and tangent of π-x and π+x in terms of their values for x.
  3. 3I can use the unit circle to express the sine, cosine, and tangent of 2π-x in terms of their values for x, for any real number x.
Precalculus Functions · CCSS.Math.Content.HS.F-TF.A.4

Explain and apply symmetry and periodicity in extended trigonometric analysis

Independently use the unit circle to explain the symmetry (odd and even behavior) and periodicity of trigonometric functions, and apply these properties to analyze and simplify more complex trigonometric expressions and equations encountered in precalculus modeling. across representations and contexts.

Component progression

  1. 1I can use the unit circle to explain why sine, cosine, and tangent exhibit even/odd symmetry and periodicity.
  2. 2I can use periodicity to rewrite a trigonometric function evaluated at a large or negative angle as an equivalent evaluation within a standard interval.
  3. 3I can use even/odd symmetry and periodicity together to simplify a trigonometric expression or solve a trigonometric equation, finding all solutions within a given interval.
Precalculus Functions · CCSS.Math.Content.HS.F-TF.B.6

Formally justify inverse trigonometric function construction

Independently formally justify why restricting each trigonometric function to a domain on which it is always increasing or always decreasing produces a valid, well-defined inverse, comparing the restricted domains across sine, cosine, and tangent. across representations and contexts.

Component progression

  1. 1I can explain, using the horizontal line test or repeated outputs, why sine, cosine, and tangent are not invertible over their full domains.
  2. 2I can compare the standard restricted domains used to define the inverses of sine, cosine, and tangent, and explain why each was chosen.
  3. 3I can explain why restricting a trig function to a domain where it is strictly increasing or decreasing guarantees a well-defined inverse.
Precalculus Functions · CCSS.Math.Content.HS.F-TF.B.7

Solve advanced modeling equations with inverse trig functions

Independently use inverse trigonometric functions to solve multi-step trigonometric equations arising from precalculus modeling contexts, evaluating the solutions using technology and interpreting them in terms of the context. across representations and contexts.

Component progression

  1. 1I can set up a trigonometric equation from a multi-step precalculus modeling scenario, such as a combined periodic and linear process.
  2. 2I can use an inverse trigonometric function together with periodicity and symmetry, evaluated with technology, to find all solutions within the modeling context's domain.
  3. 3I can interpret each solution in terms of the original context and discard any that don't make sense given the situation's constraints.
Precalculus Functions · CCSS.Math.Content.HS.F-TF.C.9

Prove and apply sum and difference formulas in precalculus proofs and problems

Independently prove the addition and subtraction formulas for sine, cosine, and tangent, and use them to solve problems within multi-step precalculus proofs, including deriving double-angle and half-angle formulas. across representations and contexts.

Component progression

  1. 1I can prove the addition formulas for sine and cosine, such as by using the unit circle or a geometric construction, and derive the corresponding subtraction formulas.
  2. 2I can derive the double-angle formulas for sine and cosine by applying the angle sum formulas to a + a, and derive a half-angle formula from a double-angle formula, determining the correct sign from the resulting angle's quadrant.
  3. 3I can use angle sum, difference, double-angle, or half-angle formulas together to prove a trigonometric identity or solve a multi-step problem.
Probability · CCSS.Math.Content.HS.S-CP.A.1

Describe events using set operations

Independently describe events as subsets of a sample space using characteristics of the outcomes, or as unions, intersections, or complements of other events. across representations and contexts.

Component progression

  1. 1I can define the sample space for a given scenario and describe an event as a subset of that sample space.
  2. 2I can describe the union and intersection of two events in terms of the outcomes each includes.
  3. 3I can describe the complement of an event as the set of all outcomes not in that event.
Probability · CCSS.Math.Content.HS.S-CP.A.2

Understand independence via the multiplication rule

Independently understand that two events a and b are independent if the probability of a and b occurring together is the product of their probabilities, and use this characterization to determine if they are independent. across representations and contexts.

Component progression

  1. 1I can state that events a and b are independent exactly when p(a and b) = p(a) times p(b).
  2. 2I can use given or computed probabilities to test whether two events satisfy the independence criterion.
  3. 3I can explain the difference between independent events and mutually exclusive events, and give an example of each.
Probability · CCSS.Math.Content.HS.S-CP.A.3

Understand conditional probability and independence

Independently understand the conditional probability of a given b as p(a and b)/p(b), and interpret independence of a and b as saying that the conditional probability of a given b is the same as the probability of a, and the conditional probability of b given a is the same as the probability of b. across representations and contexts.

Component progression

  1. 1I can compute the conditional probability of a given b using the formula p(a and b)/p(b).
  2. 2I can interpret a computed conditional probability in terms of the restricted sample space it represents.
  3. 3I can explain why two events are independent exactly when the conditional probability of one given the other equals its unconditional probability.
Probability · CCSS.Math.Content.HS.S-CP.A.4

Analyze independence using two-way frequency tables

Independently construct and interpret two-way frequency tables of data for two categorical variables; use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. across representations and contexts.

Component progression

  1. 1I can construct a two-way frequency table summarizing data for two categorical variables.
  2. 2I can use row or column totals in a two-way table to approximate a conditional probability.
  3. 3I can use approximated conditional and unconditional probabilities from a two-way table to decide whether two events are independent.
Probability · CCSS.Math.Content.HS.S-CP.A.5

Explain conditional probability and independence in context

Independently recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations, such as comparing the chance of having a medical condition given a risk factor to the chance of having the risk factor given the condition. across representations and contexts.

Component progression

  1. 1I can explain what a specific conditional probability means in the context of a real-world scenario, without relying on formal notation.
  2. 2I can using a real-world example such as a medical condition and a risk factor, explain why the chance of the condition given the risk factor is generally not the same as the chance of the risk factor given the condition.
  3. 3I can explain why a conditional probability differing from the unconditional probability shows an association but not necessarily a causal relationship.
Probability · CCSS.Math.Content.HS.S-CP.B.6

Compute conditional probability from outcome counts

Independently find the conditional probability of a given b as the fraction of b's outcomes that also belong to a, and interpret the answer in terms of the model. across representations and contexts.

Component progression

  1. 1I can identify all outcomes belonging to event b within a given sample space.
  2. 2I can compute p(a given b) as the fraction of b's outcomes that also satisfy a.
  3. 3I can interpret the computed conditional probability in terms of the original probability model.
Probability · CCSS.Math.Content.HS.S-CP.B.7

Apply the Addition Rule

Independently apply the addition rule, p(a or b) = p(a) + p(b) - p(a and b), and interpret the answer in terms of the model. across representations and contexts.

Component progression

  1. 1I can determine p(a and b), the probability of the overlap between two events, from a given scenario.
  2. 2I can use the addition rule to compute p(a or b) given p(a), p(b), and p(a and b).
  3. 3I can apply the addition rule to mutually exclusive events, recognizing the overlap term is zero.
Probability · CCSS.Math.Content.HS.S-CP.B.8

Apply the general Multiplication Rule

Independently apply the general multiplication rule in a uniform probability model, p(a and b) = p(a)p(b|a) = p(b)p(a|b), and interpret the answer in terms of the model. across representations and contexts.

Component progression

  1. 1I can state the general multiplication rule for p(a and b) in terms of a conditional probability, and explain why it reduces to the independence formula only when events are independent.
  2. 2I can use the general multiplication rule to compute p(a and b) given a marginal and a conditional probability.
  3. 3I can use the general multiplication rule to solve for an unknown conditional probability given the joint and marginal probabilities.
Probability · CCSS.Math.Content.HS.S-CP.B.9

Use permutations and combinations in probability

Independently use permutations and combinations to compute probabilities of compound events and solve problems. across representations and contexts.

Component progression

  1. 1I can determine whether a counting problem calls for a permutation (order matters) or a combination (order doesn't matter).
  2. 2I can compute the number of permutations or combinations for a given counting problem.
  3. 3I can use permutation or combination counts to compute the probability of a compound event.
Decision Statistics · CCSS.Math.Content.HS.S-MD.A.1

Define and graph random variables

Independently define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space, and graph the corresponding probability distribution using the same graphical displays as for data distributions. across representations and contexts.

Component progression

  1. 1I can define a random variable by assigning a numerical value to each outcome in a sample space.
  2. 2I can list the possible values of a random variable together with their probabilities, verifying the probabilities sum to 1.
  3. 3I can graph the probability distribution of a random variable using an appropriate display.
Decision Statistics · CCSS.Math.Content.HS.S-MD.A.2

Calculate and interpret expected value

Independently calculate the expected value of a random variable, and interpret it as the mean of the probability distribution. across representations and contexts.

Component progression

  1. 1I can compute the expected value of a random variable by summing each value times its probability.
  2. 2I can interpret an expected value as the long-run average outcome over many repetitions, not a guaranteed single result.
  3. 3I can compare the expected values of two different random variables or distributions in context.
Decision Statistics · CCSS.Math.Content.HS.S-MD.A.3

Develop a theoretical probability distribution

Independently develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated, and find its expected value. across representations and contexts.

Component progression

  1. 1I can determine the theoretical probability of each outcome in a sample space, accounting for whether outcomes are equally likely.
  2. 2I can organize the theoretical probabilities into a complete probability distribution for a defined random variable.
  3. 3I can use the developed theoretical distribution to compute the random variable's expected value.
Decision Statistics · CCSS.Math.Content.HS.S-MD.A.4

Develop an empirical probability distribution

Independently develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically, using collected or simulated data, and find the expected value. across representations and contexts.

Component progression

  1. 1I can use collected or simulated data to assign an empirical probability to each outcome of a random variable.
  2. 2I can organize empirically assigned probabilities into a complete probability distribution.
  3. 3I can use the developed empirical probability distribution to compute the random variable's expected value.
Decision Statistics · CCSS.Math.Content.HS.S-MD.B.5

Weigh decisions using expected value

Independently weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values, such as in evaluating a game of chance or an insurance decision. across representations and contexts.

Component progression

  1. 1I can identify the possible outcomes of a decision and assign a payoff value and probability to each.
  2. 2I can compute the expected value of a decision from its payoff-probability pairs.
  3. 3I can compare two or more decision options by their expected values and explain which is more favorable and why.
Decision Statistics · CCSS.Math.Content.HS.S-MD.B.6

Use probability to make fair decisions

Independently use probabilities to make fair decisions, such as drawing by lots or using a random number generator. across representations and contexts.

Component progression

  1. 1I can design a random selection method, such as drawing lots, that gives each outcome its intended probability.
  2. 2I can verify whether a proposed selection method actually gives each option an equal (or intended) probability.
  3. 3I can apply a fair random process to make a decision in a real-world scenario, justifying why the process is fair.
Decision Statistics · CCSS.Math.Content.HS.S-MD.B.7

Analyze decisions and strategies using probability

Independently analyze decisions and strategies using probability concepts, such as product testing, medical testing, or evaluating strategies in a game of chance. across representations and contexts.

Component progression

  1. 1I can identify the probabilities and conditional probabilities relevant to analyzing a real-world decision or strategy.
  2. 2I can use expected value or conditional probability to evaluate the effectiveness of a given strategy or decision rule.
  3. 3I can use the probability analysis to justify a recommended decision or strategy, acknowledging its limitations.
Calculus Readiness · EEP-MATH-12-CALC.1

Analyze limits and continuity

Independently analyze limits and continuity of functions numerically, graphically, and algebraically. across representations and contexts.

Component progression

  1. 1I can estimate the limit of a function at a point using a table of values approaching the point and the function's graph.
  2. 2I can evaluate a limit algebraically using direct substitution, factoring, or rationalizing when direct substitution fails.
  3. 3I can determine whether a function is continuous at a point by comparing the function's value, its limit, and whether they agree.
Calculus Readiness · EEP-MATH-12-CALC.2

Interpret derivatives as rates of change

Independently interpret derivatives as instantaneous rates of change and as slopes of tangent lines to a function's graph. across representations and contexts.

Component progression

  1. 1I can interpret the derivative of a function at a point as the instantaneous rate of change of the function at that point.
  2. 2I can interpret the derivative of a function at a point as the slope of the line tangent to the function's graph at that point.
  3. 3I can estimate the value of a derivative at a point using the slope of a secant line from a graph or table of values.
Calculus Readiness · EEP-MATH-12-CALC.3

Apply differentiation rules across function families

Independently apply differentiation rules to polynomial, exponential, logarithmic, and trigonometric functions. across representations and contexts.

Component progression

  1. 1I can apply the power rule and sum/difference rule to differentiate polynomial functions.
  2. 2I can apply the differentiation rules for exponential and logarithmic functions.
  3. 3I can apply the differentiation rules for sine, cosine, and tangent functions.
Calculus Readiness · EEP-MATH-12-CALC.4

Apply derivatives to optimization and motion

Independently use derivatives for optimization, motion analysis, and mathematical modeling. across representations and contexts.

Component progression

  1. 1I can use a derivative to find the maximum or minimum value of a function in an optimization problem.
  2. 2I can use a derivative to find the velocity or acceleration of an object from its position function.
  3. 3I can use a derivative to analyze the rate of change in a mathematical model of a real-world situation.
Calculus Readiness · EEP-MATH-12-CALC.5

Interpret definite integrals as accumulation and area

Independently interpret definite integrals as accumulation of a rate of change and as signed area between a function's graph and the x-axis. across representations and contexts.

Component progression

  1. 1I can interpret a definite integral as the total accumulated change of a quantity over an interval.
  2. 2I can interpret a definite integral as the signed area between a function's graph and the x-axis over an interval.
  3. 3I can approximate a definite integral using a riemann sum with a table of values or a graph.
Calculus Readiness · EEP-MATH-12-CALC.6

Apply the Fundamental Theorem of Calculus

Independently apply the fundamental theorem of calculus to evaluate definite integrals and solve contextual accumulation problems. across representations and contexts.

Component progression

  1. 1I can state the fundamental theorem of calculus connecting differentiation and integration.
  2. 2I can use the fundamental theorem of calculus to evaluate a definite integral by finding an antiderivative.
  3. 3I can apply the fundamental theorem of calculus to solve a real-world problem involving accumulated change.
Advanced Calculus Ab · AP.MATH.AP.CALCULUS.AB.1

Advanced Calculus I: 1

Independently analyze limits and continuity of functions using numerical, graphical, and algebraic methods, including one-sided limits and limits involving indeterminate forms. across representations and contexts.

Component progression

  1. 1I can estimate the limit of a function at a point using a table of values and the function's graph, including one-sided limits.
  2. 2I can evaluate limits algebraically, including indeterminate forms, using factoring, rationalizing, or other algebraic techniques.
  3. 3I can determine whether a function is continuous at a point and classify a discontinuity as removable, jump, or infinite.
Advanced Calculus Ab · AP.MATH.AP.CALCULUS.AB.2

Advanced Calculus I: 2

Independently define the derivative as the limit of a difference quotient, and interpret it as an instantaneous rate of change and as the slope of a tangent line. across representations and contexts.

Component progression

  1. 1I can define the derivative of a function at a point as the limit of its difference quotient as the interval shrinks to zero.
  2. 2I can interpret a derivative's value as both the instantaneous rate of change of the function and the slope of its tangent line at that point.
  3. 3I can explain why differentiability at a point implies continuity there, and give an example of a continuous function that is not differentiable at a point.
Advanced Calculus Ab · AP.MATH.AP.CALCULUS.AB.3

Advanced Calculus I: 3

Independently apply the power, product, quotient, and chain rules to differentiate polynomial, rational, exponential, logarithmic, and trigonometric functions, including implicit differentiation. across representations and contexts.

Component progression

  1. 1I can differentiate polynomial and rational functions using the power, product, and quotient rules.
  2. 2I can differentiate composed functions using the chain rule, and apply implicit differentiation to equations not solved for y.
  3. 3I can apply the differentiation rules for exponential, logarithmic, and trigonometric functions, combining them with the chain rule when needed.
Advanced Calculus Ab · AP.MATH.AP.CALCULUS.AB.4

Advanced Calculus I: 4

Independently use derivatives to analyze motion, find extrema and concavity, and solve related-rates and optimization problems. across representations and contexts.

Component progression

  1. 1I can use the first and second derivatives of a position function to find velocity, speed, and acceleration and interpret the motion they describe.
  2. 2I can use the first derivative test and second derivative test to find a function's extrema and determine intervals of concavity.
  3. 3I can set up and solve a related-rates problem or an optimization problem using derivatives.
Advanced Calculus Ab · AP.MATH.AP.CALCULUS.AB.5

Advanced Calculus I: 5

Independently interpret a definite integral as a limit of riemann sums, approximate it using riemann sum methods, and interpret it as net accumulation and as signed area. across representations and contexts.

Component progression

  1. 1I can approximate a definite integral using left, right, midpoint, and trapezoidal riemann sums from a table of values or a graph.
  2. 2I can interpret a definite integral of a rate function as the net accumulated change of the original quantity over an interval.
  3. 3I can interpret a definite integral as the signed area between a function's graph and the x-axis, accounting for regions below the axis.
Advanced Calculus Ab · AP.MATH.AP.CALCULUS.AB.6

Advanced Calculus I: 6

Independently apply the fundamental theorem of calculus to differentiate accumulation functions, evaluate definite integrals, and solve contextual accumulation problems. across representations and contexts.

Component progression

  1. 1I can use the fundamental theorem of calculus, part 1, to differentiate an accumulation function, including one with a variable upper bound requiring the chain rule.
  2. 2I can use the fundamental theorem of calculus, part 2, to evaluate a definite integral by finding and evaluating an antiderivative.
  3. 3I can apply the fundamental theorem of calculus to solve a real-world problem involving total accumulated change.
Advanced Calculus Ab · AP.MATH.AP.CALCULUS.AB.7

Advanced Calculus I: 7

Independently sketch and interpret slope fields, solve separable differential equations, and model exponential growth and decay. across representations and contexts.

Component progression

  1. 1I can sketch a slope field for a given differential equation and use it to sketch an approximate solution curve through a given point.
  2. 2I can solve a separable differential equation by separating variables and integrating both sides.
  3. 3I can use a separable differential equation to model and solve an exponential growth or decay problem.
Advanced Calculus Ab · AP.MATH.AP.CALCULUS.AB.8

Advanced Calculus I: 8

Independently use integration to find the area between curves, the volume of a solid of revolution, and the average value of a function. across representations and contexts.

Component progression

  1. 1I can set up and evaluate a definite integral to find the area between two curves.
  2. 2I can set up and evaluate a definite integral to find the volume of a solid of revolution using the disk, washer, or shell method.
  3. 3I can compute the average value of a function over an interval using a definite integral, and interpret it in context.
Advanced Calculus Bc · AP.MATH.AP.CALCULUS.BC.1

Advanced Calculus II: 1

Independently apply integration by parts, partial fraction decomposition, and evaluate improper integrals. across representations and contexts.

Component progression

  1. 1I can evaluate an integral using integration by parts, choosing u and dv appropriately.
  2. 2I can decompose a rational function into partial fractions and integrate the resulting terms.
  3. 3I can evaluate an improper integral with an infinite bound or a discontinuous integrand by taking an appropriate limit.
Advanced Calculus Bc · AP.MATH.AP.CALCULUS.BC.2

Advanced Calculus II: 2

Independently differentiate parametric equations, find arc length of parametric and polar curves, and find area enclosed by a polar curve. across representations and contexts.

Component progression

  1. 1I can find dy/dx for a curve defined parametrically using the parametric differentiation formula.
  2. 2I can set up and evaluate a definite integral to find the arc length of a curve defined parametrically or in polar form.
  3. 3I can set up and evaluate a definite integral to find the area enclosed by a polar curve.
Advanced Calculus Bc · AP.MATH.AP.CALCULUS.BC.3

Advanced Calculus II: 3

Independently differentiate and integrate vector-valued functions, and find velocity, acceleration, and speed for planar motion. across representations and contexts.

Component progression

  1. 1I can differentiate and integrate a vector-valued function component-wise.
  2. 2I can find the velocity and acceleration vectors for an object whose position is given by a vector-valued function.
  3. 3I can find an object's speed as the magnitude of its velocity vector, and find its displacement over an interval.
Advanced Calculus Bc · AP.MATH.AP.CALCULUS.BC.4

Advanced Calculus II: 4

Independently apply convergence tests to determine whether an infinite series converges, and determine the radius and interval of convergence of a power series. across representations and contexts.

Component progression

  1. 1I can apply the nth-term, geometric, p-series, ratio, and integral tests to determine whether a given series converges or diverges.
  2. 2I can use the ratio test to determine the radius of convergence of a power series, then test the endpoints to determine the full interval of convergence.
  3. 3I can determine whether an alternating series converges and estimate the error of a partial sum using the alternating series error bound.
Advanced Calculus Bc · AP.MATH.AP.CALCULUS.BC.5

Advanced Calculus II: 5

Independently construct taylor and maclaurin series for functions, use them to approximate function values, and estimate error using the lagrange error bound. across representations and contexts.

Component progression

  1. 1I can construct the taylor series (or maclaurin series, centered at 0) for a given function using its derivatives at the center point.
  2. 2I can use a taylor polynomial of a specified degree to approximate a function's value near the center point.
  3. 3I can use the lagrange error bound to estimate the maximum error of a taylor polynomial approximation.
Advanced Statistics · AP.MATH.AP.STATISTICS.1

Advanced Statistics: 1

Independently describe the shape, center, and spread of a one-variable distribution, construct appropriate graphical displays, and compute summary statistics. across representations and contexts.

Component progression

  1. 1I can describe the shape, center, and spread of a one-variable distribution using appropriate vocabulary and statistics.
  2. 2I can construct and interpret a dotplot, histogram, or boxplot for one-variable data.
  3. 3I can compute and interpret summary statistics for one-variable data, including standard deviation and z-scores.
Advanced Statistics · AP.MATH.AP.STATISTICS.2

Advanced Statistics: 2

Independently construct and interpret scatterplots and correlation, fit and interpret a least-squares regression line, and use residual plots to assess model fit. across representations and contexts.

Component progression

  1. 1I can construct a scatterplot for two-variable data and interpret the correlation coefficient's strength and direction.
  2. 2I can fit a least-squares regression line to two-variable data and interpret its slope and intercept in context.
  3. 3I can construct and analyze a residual plot to assess whether a linear model is an appropriate fit for the data.
Advanced Statistics · AP.MATH.AP.STATISTICS.3

Advanced Statistics: 3

Independently compare sampling methods and identify bias, distinguish observational studies from experiments, and apply principles of experimental design. across representations and contexts.

Component progression

  1. 1I can compare simple random, stratified, and cluster sampling methods, and identify sources of bias in a described sampling method.
  2. 2I can distinguish an observational study from a controlled experiment and explain what conclusions each design supports.
  3. 3I can apply the principles of randomization, control, and replication to design or critique an experiment.
Advanced Statistics · AP.MATH.AP.STATISTICS.4

Advanced Statistics: 4

Independently apply probability rules including conditional probability, define discrete random variables, compute their mean and variance, and apply the binomial and geometric distributions. across representations and contexts.

Component progression

  1. 1I can apply basic probability rules, including the addition rule and conditional probability, to compute probabilities of events.
  2. 2I can compute the mean and variance of a discrete random variable from its probability distribution.
  3. 3I can determine when the binomial or geometric distribution applies to a scenario and use it to compute a probability.
Advanced Statistics · AP.MATH.AP.STATISTICS.5

Advanced Statistics: 5

Independently describe the sampling distribution of a sample proportion and a sample mean, and apply the central limit theorem to approximate a sampling distribution. across representations and contexts.

Component progression

  1. 1I can describe the shape, center, and spread of the sampling distribution of a sample proportion.
  2. 2I can describe the shape, center, and spread of the sampling distribution of a sample mean.
  3. 3I can use the central limit theorem to approximate a sampling distribution as approximately normal when sample-size conditions are met.
Advanced Statistics · AP.MATH.AP.STATISTICS.6

Advanced Statistics: 6

Independently construct and interpret confidence intervals for a population proportion and a population mean, and interpret confidence level and margin of error. across representations and contexts.

Component progression

  1. 1I can construct and interpret a confidence interval for a population proportion, checking the required conditions.
  2. 2I can construct and interpret a confidence interval for a population mean, checking the required conditions.
  3. 3I can correctly interpret what a stated confidence level and margin of error mean about the estimation procedure.
Advanced Statistics · AP.MATH.AP.STATISTICS.7

Advanced Statistics: 7

Independently state null and alternative hypotheses, compute test statistics and p-values, make conclusions, and explain type i and type ii errors. across representations and contexts.

Component progression

  1. 1I can state appropriate null and alternative hypotheses for a scenario and compute the corresponding test statistic.
  2. 2I can compute a p-value from a test statistic and use it, with a significance level, to make a conclusion about the null hypothesis.
  3. 3I can explain what a type i error and a type ii error would mean in a given context, and their consequences.
Advanced Statistics · AP.MATH.AP.STATISTICS.8

Advanced Statistics: 8

Independently communicate statistical conclusions in context, identify limitations of a study or inference procedure, and evaluate generalizability and causal claims. across representations and contexts.

Component progression

  1. 1I can state a statistical conclusion in plain language connected to the original real-world context.
  2. 2I can identify limitations of a described study's design, sample, or inference procedure.
  3. 3I can evaluate whether a study's conclusion can be generalized to a broader population and whether it supports a causal claim.
Advanced Linear Algebra · EEP-ADV.MATH.LINEAR.ALGEBRA.1

Introductory Linear Algebra: 1

Independently represent a system of linear equations as an augmented matrix, apply gaussian elimination to row-reduce it, and interpret the resulting solution set. across representations and contexts.

Component progression

  1. 1I can write a system of linear equations as an augmented matrix.
  2. 2I can apply gaussian elimination using elementary row operations to row-reduce a matrix to echelon form.
  3. 3I can interpret a row-reduced matrix to determine whether the system has a unique solution, infinitely many solutions, or no solution.
Advanced Linear Algebra · EEP-ADV.MATH.LINEAR.ALGEBRA.2

Introductory Linear Algebra: 2

Independently verify whether a set satisfies the vector space axioms, determine whether a subset is a subspace, and determine the span of a set of vectors. across representations and contexts.

Component progression

  1. 1I can verify whether a given set with defined operations satisfies the vector space axioms.
  2. 2I can determine whether a subset of a vector space is a subspace by checking closure under addition and scalar multiplication and the presence of the zero vector.
  3. 3I can determine the span of a given set of vectors and describe it geometrically or algebraically.
Advanced Linear Algebra · EEP-ADV.MATH.LINEAR.ALGEBRA.3

Introductory Linear Algebra: 3

Independently test a set of vectors for linear independence, find a basis for a vector space or subspace, and determine its dimension. across representations and contexts.

Component progression

  1. 1I can test whether a set of vectors is linearly independent by row-reducing a matrix formed from them.
  2. 2I can find a basis for a given vector space or subspace.
  3. 3I can determine the dimension of a vector space or subspace from the number of vectors in a basis.
Advanced Linear Algebra · EEP-ADV.MATH.LINEAR.ALGEBRA.4

Introductory Linear Algebra: 4

Independently verify whether a transformation is linear, find its standard matrix, and determine its kernel and range. across representations and contexts.

Component progression

  1. 1I can verify whether a given transformation satisfies both additivity and homogeneity, the defining properties of a linear transformation.
  2. 2I can find the standard matrix representing a linear transformation by applying it to the standard basis vectors.
  3. 3I can determine the kernel and range of a linear transformation from its standard matrix.
Advanced Linear Algebra · EEP-ADV.MATH.LINEAR.ALGEBRA.5

Introductory Linear Algebra: 5

Independently compute the determinant of a matrix using cofactor expansion, use it to determine invertibility, and interpret it as a scale factor for area or volume. across representations and contexts.

Component progression

  1. 1I can compute the determinant of a matrix using cofactor expansion along a row or column.
  2. 2I can use a matrix's determinant to determine whether it is invertible.
  3. 3I can interpret the absolute value of a matrix's determinant as the factor by which the corresponding transformation scales area or volume.
Advanced Linear Algebra · EEP-ADV.MATH.LINEAR.ALGEBRA.6

Introductory Linear Algebra: 6

Independently find the eigenvalues of a matrix using its characteristic polynomial, find corresponding eigenvectors, and use them to interpret transformations and diagonalize a matrix. across representations and contexts.

Component progression

  1. 1I can find the eigenvalues of a matrix by solving the characteristic equation formed from its determinant.
  2. 2I can find the eigenvectors corresponding to a given eigenvalue by solving the resulting homogeneous system.
  3. 3I can interpret eigenvalues and eigenvectors geometrically as directions a transformation only scales, and use them to diagonalize a matrix when possible.
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