Curriculum (United States)

Mathematics — Grade 12

12th grade math according to the Common Core State Standards, usually Precalculus: complex numbers in polar form, vectors, matrices and systems of equations, trigonometric values, expected value and binomial probability. Every topic comes with printable worksheets.

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What do students learn in 12th grade math?

In 12th grade students write complex numbers in polar form and find their modulus and conjugate, add and scale vectors and compute dot products, and multiply and invert 2×2 matrices to solve systems of equations. They find exact trigonometric values and compute expected values and binomial probabilities. Most of these are (+) standards for students heading to calculus.

About the standards: the Common Core high school standards are organized by conceptual category (Number and Quantity, Algebra, Functions, Geometry, Statistics and Probability), not by grade. Assigning a topic to a grade is our decision, following the common course sequence Algebra I, Geometry, Algebra II and Precalculus. Standards marked (+) go beyond the college and career readiness core.

Page status: the topics of this grade are ready as plain exercises (instruction and formula); problems with diagrams and word problems are being added next.

Curriculum scope

Curriculum point HSN-CN · Teaching content · checked against the act

The Complex Number System

CCSS Mathematics (2010), High School, domain The Complex Number System (HSN-CN)
Curriculum point HSN-CN.A · Teaching content · checked against the act

Perform arithmetic operations with complex numbers.

CCSS Mathematics (2010), High School, The Complex Number System, cluster A
Curriculum point HSN-CN.B · Teaching content · checked against the act

Represent complex numbers and their operations on the complex plane.

CCSS Mathematics (2010), High School, The Complex Number System, cluster B
Curriculum point HSN-CN.A.3 · Teaching content · checked against the act

(+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of c omplex numbers.

CCSS Mathematics (2010), High School, The Complex Number System, standard HSN-CN.A.3

Complex Conjugates, Modulus, and Division (we teach in grade 12)
Curriculum point HSN-CN.B.4 · Teaching content · checked against the act

(+) Represent complex numbers on the complex plane in rectangular and polar f orm (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.

CCSS Mathematics (2010), High School, The Complex Number System, standard HSN-CN.B.4

Polar and Rectangular Forms of Complex Numbers (we teach in grade 12)
Curriculum point HSN-CN.B.5 · Teaching content · checked against the act

(+) Represent addition, subtraction, multiplication, and conjugation of c omplex numbers geometrically on the complex plane; use properties of this representation for computation. For example, (–1 + √3 i)3 = 8 because (–1 + √3 i) has modulus 2 and argument 120°.

CCSS Mathematics (2010), High School, The Complex Number System, standard HSN-CN.B.5
Curriculum point HSN-CN.B.6 · Teaching content · checked against the act

(+) Calculate the distance between numbers in the complex plane as the modulus of the diff erence, and the midpoint of a segment as the average of the numbers at its endpoints.

CCSS Mathematics (2010), High School, The Complex Number System, standard HSN-CN.B.6

Complex Conjugates, Modulus, and Division (we teach in grade 12)
Curriculum point HSN-VM · Teaching content · checked against the act

Vector and Matrix Quantities

CCSS Mathematics (2010), High School, domain Vector and Matrix Quantities (HSN-VM)
Curriculum point HSN-VM.A · Teaching content · checked against the act

Represent and model with vector quantities.

CCSS Mathematics (2010), High School, Vector and Matrix Quantities, cluster A
Curriculum point HSN-VM.B · Teaching content · checked against the act

Perform operations on vectors.

CCSS Mathematics (2010), High School, Vector and Matrix Quantities, cluster B
Curriculum point HSN-VM.C · Teaching content · checked against the act

Perform operations on matrices and use matrices in applications.

CCSS Mathematics (2010), High School, Vector and Matrix Quantities, cluster C
Curriculum point HSN-VM.A.1 · Teaching content · checked against the act

(+) Recognize vector quantities as having both magnitude and direction. R epresent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v).

CCSS Mathematics (2010), High School, Vector and Matrix Quantities, standard HSN-VM.A.1
Curriculum point HSN-VM.A.2 · Teaching content · checked against the act

(+) Find the components of a vector by subtracting the coordinates of an initial point fr om the coordinates of a terminal point.

CCSS Mathematics (2010), High School, Vector and Matrix Quantities, standard HSN-VM.A.2

Vector Lengths and Linear Combinations (we teach in grade 12)
Curriculum point HSN-VM.A.3 · Teaching content · checked against the act

(+) Solve problems involving velocity and other quantities that can be r epresented by vectors.

CCSS Mathematics (2010), High School, Vector and Matrix Quantities, standard HSN-VM.A.3
Curriculum point HSN-VM.B.4 · Teaching content · checked against the act

(+) Add and subtract vectors. a. Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes. b. Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum. c. Understand vector subtraction v – w as v + (–w), where –w is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise.

CCSS Mathematics (2010), High School, Vector and Matrix Quantities, standard HSN-VM.B.4

Vector Lengths and Linear Combinations (we teach in grade 12)
Curriculum point HSN-VM.B.5 · Teaching content · checked against the act

(+) Multiply a vector by a scalar. a. Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(vx, vy) = (cvx, cvy). b. Compute the magnitude of a scalar multiple cv using ||cv|| = |c|v. Compute the direction of cv knowing that when |c|v ≠ 0, the direction of cv is either along v (for c > 0) or against v (for c < 0).

CCSS Mathematics (2010), High School, Vector and Matrix Quantities, standard HSN-VM.B.5

Vector Lengths and Linear Combinations (we teach in grade 12)
Curriculum point HSN-VM.C.6 · Teaching content · checked against the act

(+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidenc e relationships in a network.

CCSS Mathematics (2010), High School, Vector and Matrix Quantities, standard HSN-VM.C.6
Curriculum point HSN-VM.C.7 · Teaching content · checked against the act

(+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the pa yoffs in a game are doubled.

CCSS Mathematics (2010), High School, Vector and Matrix Quantities, standard HSN-VM.C.7

Matrix Operations and Multiplying Vectors (we teach in grade 12)
Curriculum point HSN-VM.C.8 · Teaching content · checked against the act

(+) Add, subtract, and multiply matrices of appropriate dimensions.

CCSS Mathematics (2010), High School, Vector and Matrix Quantities, standard HSN-VM.C.8

Matrix Operations and Multiplying Vectors (we teach in grade 12)
Curriculum point HSN-VM.C.9 · Teaching content · checked against the act

(+) Understand that, unlike multiplication of numbers, matrix multiplication for squar e matrices is not a commutative operation, but still satisfies the associative and distributive properties.

CCSS Mathematics (2010), High School, Vector and Matrix Quantities, standard HSN-VM.C.9

Matrix Operations and Multiplying Vectors (we teach in grade 12)
Curriculum point HSN-VM.C.10 · Teaching content · checked against the act

(+) Understand that the zero and identity matrices play a role in matrix addition and multiplica tion 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.

CCSS Mathematics (2010), High School, Vector and Matrix Quantities, standard HSN-VM.C.10

Finding the Determinant and Inverse of a 2×2 Matrix (we teach in grade 12)
Curriculum point HSN-VM.C.11 · Teaching content · checked against the act

(+) Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions t o produce another vector. Work with matrices as transformations of vectors.

CCSS Mathematics (2010), High School, Vector and Matrix Quantities, standard HSN-VM.C.11

Matrix Operations and Multiplying Vectors (we teach in grade 12)
Curriculum point HSN-VM.C.12 · Teaching content · checked against the act

(+) Work with 2 × 2 matrices as transformations of the plane, and interpret the absolut e value of the determinant in terms of area.

CCSS Mathematics (2010), High School, Vector and Matrix Quantities, standard HSN-VM.C.12

Finding the Determinant and Inverse of a 2×2 Matrix (we teach in grade 12)
Curriculum point HSA-REI · Teaching content · checked against the act

Reasoning with Equations and Inequalities

CCSS Mathematics (2010), High School, domain Reasoning with Equations and Inequalities (HSA-REI)
Curriculum point HSA-REI.C · Teaching content · checked against the act

Solve systems of equations

CCSS Mathematics (2010), High School, Reasoning with Equations and Inequalities, cluster C
Curriculum point HSA-REI.C.8 · Teaching content · checked against the act

(+) Represent a system of linear equations as a single matrix equation in a vector variable.

CCSS Mathematics (2010), High School, Reasoning with Equations and Inequalities, standard HSA-REI.C.8

Solving Systems of Equations with Matrices (we teach in grade 12)
Curriculum point HSA-REI.C.9 · Teaching content · checked against the act

(+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).

CCSS Mathematics (2010), High School, Reasoning with Equations and Inequalities, standard HSA-REI.C.9

Solving Systems of Equations with Matrices (we teach in grade 12) · Finding the Determinant and Inverse of a 2×2 Matrix (we teach in grade 12)
Curriculum point HSF-BF · Teaching content · checked against the act

Building Functions

CCSS Mathematics (2010), High School, domain Building Functions (HSF-BF)
Curriculum point HSF-BF.B · Teaching content · checked against the act

Build new functions from existing functions

CCSS Mathematics (2010), High School, Building Functions, cluster B
Curriculum point HSF-BF.B.5 · Teaching content · checked against the act

(+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

CCSS Mathematics (2010), High School, Building Functions, standard HSF-BF.B.5
Curriculum point HSF-TF · Teaching content · checked against the act

Trigonometric Functions

CCSS Mathematics (2010), High School, domain Trigonometric Functions (HSF-TF)
Curriculum point HSF-TF.A · Teaching content · checked against the act

Extend the domain of trigonometric functions using the unit circle

CCSS Mathematics (2010), High School, Trigonometric Functions, cluster A
Curriculum point HSF-TF.B · Teaching content · checked against the act

Model periodic phenomena with trigonometric functions

CCSS Mathematics (2010), High School, Trigonometric Functions, cluster B
Curriculum point HSF-TF.A.3 · Teaching content · checked against the act

(+) Use special triangles to determine geometrically the values of sine, cosine, tangent f or 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.

CCSS Mathematics (2010), High School, Trigonometric Functions, standard HSF-TF.A.3

Exact Values of Sine, Cosine, and Tangent (we teach in grade 12)
Curriculum point HSF-TF.A.4 · Teaching content · checked against the act

(+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

CCSS Mathematics (2010), High School, Trigonometric Functions, standard HSF-TF.A.4
Curriculum point HSF-TF.B.6 · Teaching content · checked against the act

(+) Understand that restricting a trigonometric function to a domain on which it is alw ays increasing or always decreasing allows its inverse to be constructed.

CCSS Mathematics (2010), High School, Trigonometric Functions, standard HSF-TF.B.6
Curriculum point HSF-TF.B.7 · Teaching content · checked against the act

(+) Use inverse functions to solve trigonometric equations that arise in modeling c ontexts; evaluate the solutions using technology, and interpret them in terms of the context.★

CCSS Mathematics (2010), High School, Trigonometric Functions, standard HSF-TF.B.7
Curriculum point HSG-GPE · Teaching content · checked against the act

Expressing Geometric Properties with Equations

CCSS Mathematics (2010), High School, domain Expressing Geometric Properties with Equations (HSG-GPE)
Curriculum point HSG-GPE.A · Teaching content · checked against the act

Translate between the geometric description and the equation for a conic section

CCSS Mathematics (2010), High School, Expressing Geometric Properties with Equations, cluster A
Curriculum point HSG-GPE.A.3 · Teaching content · checked against the act

(+) Derive the equations of ellipses and hyperbolas given the foci, using the fact tha t the sum or difference of distances from the foci is constant.

CCSS Mathematics (2010), High School, Expressing Geometric Properties with Equations, standard HSG-GPE.A.3
Curriculum point HSG-GMD · Teaching content · checked against the act

Geometric Measurement and Dimension

CCSS Mathematics (2010), High School, domain Geometric Measurement and Dimension (HSG-GMD)
Curriculum point HSG-GMD.A · Teaching content · checked against the act

Explain volume formulas and use them to solve problems

CCSS Mathematics (2010), High School, Geometric Measurement and Dimension, cluster A
Curriculum point HSG-GMD.A.2 · Teaching content · checked against the act

(+) Give an informal argument using Cavalieri’s principle for the formulas for the v olume of a sphere and other solid figures.

CCSS Mathematics (2010), High School, Geometric Measurement and Dimension, standard HSG-GMD.A.2
Curriculum point HSS-MD · Teaching content · checked against the act

Using Probability to Make Decisions

CCSS Mathematics (2010), High School, domain Using Probability to Make Decisions (HSS-MD)
Curriculum point HSS-MD.A · Teaching content · checked against the act

Calculate expected values and use them to solve problems

CCSS Mathematics (2010), High School, Using Probability to Make Decisions, cluster A
Curriculum point HSS-MD.B · Teaching content · checked against the act

Use probability to evaluate outcomes of decisions

CCSS Mathematics (2010), High School, Using Probability to Make Decisions, cluster B
Curriculum point HSS-MD.A.1 · Teaching content · checked against the act

(+) Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions.

CCSS Mathematics (2010), High School, Using Probability to Make Decisions, standard HSS-MD.A.1
Curriculum point HSS-MD.A.2 · Teaching content · checked against the act

(+) Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.

CCSS Mathematics (2010), High School, Using Probability to Make Decisions, standard HSS-MD.A.2

Finding Expected Value from a Probability Table (we teach in grade 10-12)
Curriculum point HSS-MD.A.3 · Teaching content · checked against the act

(+) Develop a probability distribution for a random variable defined for a sample spac e in which theoretical probabilities can be calculated; find the expected value. For example, find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes.

CCSS Mathematics (2010), High School, Using Probability to Make Decisions, standard HSS-MD.A.3

Finding Expected Value from a Probability Table (we teach in grade 10-12) · Binomial Distributions and Expected Value (we teach in grade 12)
Curriculum point HSS-MD.A.4 · Teaching content · checked against the act

(+) Develop a probability distribution for a random variable defined for a sample spac e in which probabilities are assigned empirically; find the expected value. For example, find a current data distribution on the number of TV sets per household in the United States, and calculate the expected number of sets per household. How many TV sets would you expect to find in 100 randomly selected households?

CCSS Mathematics (2010), High School, Using Probability to Make Decisions, standard HSS-MD.A.4

Finding Expected Value from a Probability Table (we teach in grade 10-12)
Curriculum point HSS-MD.B.5 · Teaching content · checked against the act

(+) Weigh the possible outcomes of a decision by assigning probabilities to pa yoff values and finding expected values. a. Find the expected payoff for a game of chance. For example, find the expected winnings from a state lottery ticket or a game at a fast-food restaurant. b. Evaluate and compare strategies on the basis of expected values. For example, compare a high-deductible versus a low-deductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident.

CCSS Mathematics (2010), High School, Using Probability to Make Decisions, standard HSS-MD.B.5

Finding Expected Value from a Probability Table (we teach in grade 10-12)

Skills step by step

Complex Conjugates, Modulus, and Division

Students learn to find complex conjugates, calculate modulus and geometric distance, and use conjugates to divide complex numbers.

Curriculum point HSN-CN.A.3 (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of c omplex numbers.
Curriculum point HSN-CN.B.6 (+) Calculate the distance between numbers in the complex plane as the modulus of the diff erence, and the midpoint of a segment as the average of the numbers at its endpoints.

At this level, students work with complex numbers written in standard form with real and imaginary parts. They learn to determine the complex conjugate by reversing the sign of the imaginary term, and they use this value to calculate a number's modulus—its absolute value or distance from zero. Students also use conjugates to divide complex numbers, clearing the imaginary unit from the denominator, and find the geometric distance between two points in the complex plane by taking the modulus of their difference.

A frequent stumbling block occurs when finding the conjugate: students often mistakenly change the sign of the real part rather than the imaginary part, converting 3 + 4i into -3 + 4i instead of 3 - 4i. When dividing, students also frequently make sign mistakes when multiplying terms involving the imaginary unit, forgetting that i multiplied by i equals -1.

Practice tasks prompt students to solve direct exercises across four core formats:

Complex conjugate: writing the conjugate for a given complex value.
Modulus of a complex number: computing the absolute value of an expression.
Dividing complex numbers: multiplying numerator and denominator by the conjugate to simplify a fraction into standard form.
Distance in the complex plane: finding the distance between two complex coordinates by calculating the modulus of their difference.

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Task types: Modulus of a complex number · Complex conjugate · Distance in the complex plane · Dividing complex numbers

Matrix Operations and Multiplying Vectors

Students learn to scale, add, subtract, and multiply matrices, and multiply matrices by column vectors to model transformations.

Curriculum point HSN-VM.C.11 (+) Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions t o produce another vector. Work with matrices as transformations of vectors.
Curriculum point HSN-VM.C.7 (+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the pa yoffs in a game are doubled.
Curriculum point HSN-VM.C.8 (+) Add, subtract, and multiply matrices of appropriate dimensions.
Curriculum point HSN-VM.C.9 (+) Understand that, unlike multiplication of numbers, matrix multiplication for squar e matrices is not a commutative operation, but still satisfies the associative and distributive properties.

Students compute with matrices and vectors of appropriate dimensions. They perform scalar multiplication by scaling every entry, combine matrices through addition and subtraction, and multiply matrices together. They also multiply a matrix by a single-column vector to produce a new vector, while recognizing that matrix multiplication is associative and distributive, but not commutative.

A frequent error occurs during matrix multiplication when students try to multiply corresponding entries directly across, rather than multiplying each row's entries by a column's entries and adding the results. Students also commonly treat matrix multiplication like everyday number multiplication, mistakenly assuming that changing the order of the matrices will yield the exact same answer or failing to check whether the inner dimensions match.

In practice, students solve problems across three specific formats:

Linear combination of matrices, where they multiply matrices by numerical factors and add or subtract the resulting arrays;
Product of matrices, where they check that the column count of the first matrix matches the row count of the second before computing the final entries;
Matrix times vector, where they apply a transformation matrix to a single column vector to produce the resulting output vector.

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Task types: Linear combination of matrices · Matrix times vector · Product of matrices

Binomial Distributions and Expected Value

Students calculate theoretical probabilities for repeated success-or-failure events and determine the expected value or mean outcome across those trials.

Curriculum point HSS-MD.A.3 (+) Develop a probability distribution for a random variable defined for a sample spac e in which theoretical probabilities can be calculated; find the expected value. For example, find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes.

Students learn to model scenarios involving repeated independent events that have two outcomes, known as Bernoulli trials. Using situations like guessing on a five-question multiple-choice test with four options per question, students build probability distributions for the total number of successes and find the expected value under different scoring rules.

A typical error occurs when students calculate the probability of a specific outcome, such as getting two questions right out of five. They often multiply the probabilities of success and failure together but forget to account for the number of different orderings in which those successes can occur, leaving out the combination factor.

In practice, tasks draw from two main templates. In Bernoulli trials, students identify single-trial probabilities and compute the exact likelihood of getting a specific number of successes across repeated trials. In Mean of a binomial distribution, students use the number of trials and success probabilities to calculate the average expected outcome or expected grade over time.

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Task types: Mean of a binomial distribution · Bernoulli trials

Vector Lengths and Linear Combinations

Students learn to find the magnitude of a vector and combine vectors algebraically using scalar multiplication, addition, and subtraction.

Curriculum point HSN-VM.A.2 (+) Find the components of a vector by subtracting the coordinates of an initial point fr om the coordinates of a terminal point.
Curriculum point HSN-VM.B.4 (+) Add and subtract vectors. a. Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes. b. Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum. c. Understand vector subtraction v – w as v + (–w), where –w is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise.
Curriculum point HSN-VM.B.5 (+) Multiply a vector by a scalar. a. Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(vx, vy) = (cvx, cvy). b. Compute the magnitude of a scalar multiple cv using ||cv|| = |c|v. Compute the direction of cv knowing that when |c|v ≠ 0, the direction of cv is either along v (for c > 0) or against v (for c < 0).

At this stage, students work with two-dimensional vectors expressed in component form. They compute the length (magnitude) of a vector and create linear combinations by scaling vectors—multiplying components by a scalar factor c to produce (cv_x, cv_y)—and combining them through component-wise addition and subtraction.

A common misconception occurs when finding the magnitude of a combined vector: students often assume the length of a sum is equal to the sum of the individual lengths, which is typically not true. Another frequent mistake involves negative scalars. When multiplying by a negative number, students sometimes drop negative signs during component subtraction or forget that magnitude must remain positive, scaled by the absolute value of the scalar.

Worksheet tasks reinforce these skills through two specific formats:

Length of a vector: Students determine the numerical magnitude from a vector's given coordinate components.
Linear combination of vectors: Students are given two or more vectors and must scale and combine them to find the resulting vector components.

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Task types: Length of a vector · Linear combination of vectors

Polar and Rectangular Forms of Complex Numbers

Students learn to find the argument of a complex number and convert between polar and rectangular forms using special angles on the complex plane.

Curriculum point HSN-CN.B.4 (+) Represent complex numbers on the complex plane in rectangular and polar f orm (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.

At this stage, students learn to represent complex numbers—including real and purely imaginary numbers—on the complex plane in both rectangular and polar forms. They connect coordinates in standard a + bi form with the distance from the origin and the angle of rotation, using special trigonometric angles to find exact values without relying on rounded approximations.

A common mistake occurs when determining the angle, or argument, of a complex number outside the first quadrant. Students often find the correct reference angle using trigonometric ratios but fail to account for negative signs in the real or imaginary parts, placing the resulting angle in the wrong quadrant of the complex plane.

Practice tasks target this skill through two main exercises:

Argument of a complex number: Students identify the exact direction angle for a given complex value plotted on the complex plane.
Polar form to rectangular form: Students use exact values for special angles to convert polar expressions back into standard rectangular coordinates.

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Task types: Argument of a complex number · Polar form to rectangular form

Finding the Determinant and Inverse of a 2×2 Matrix

Students learn to calculate the determinant of a 2×2 matrix and use it to find the matrix's multiplicative inverse.

Curriculum point HSA-REI.C.9 (+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).
Curriculum point HSN-VM.C.10 (+) Understand that the zero and identity matrices play a role in matrix addition and multiplica tion 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.
Curriculum point HSN-VM.C.12 (+) Work with 2 × 2 matrices as transformations of the plane, and interpret the absolut e value of the determinant in terms of area.

Students work by hand with 2×2 square matrices to compute their determinants and determine whether an inverse exists. They learn that a matrix has a multiplicative inverse only when its determinant is not zero. When the determinant is nonzero, students use it alongside the rearranged matrix elements to calculate the exact inverse matrix.

A common error occurs with negative signs when calculating the determinant, where students add the diagonal products instead of subtracting the off-diagonal product from the main diagonal product. Students also frequently misplace signs or swap the wrong pairs of entries when writing out the inverse matrix, or they attempt to calculate an inverse for a matrix whose determinant equals zero.

Practice tasks come in two standard worksheet formats: Determinant of a matrix, where students evaluate the numerical value of a 2×2 grid of numbers, and Inverse matrix, where students first check if an inverse is possible and then write out the resulting inverse matrix.

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Task types: Determinant of a matrix · Inverse matrix

Finding Expected Value from a Probability Table

Students learn to calculate the expected value of a discrete random variable using a probability table and interpret it as the average outcome over time.

Curriculum point HSS-MD.A.2 (+) Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.
Curriculum point HSS-MD.A.3 (+) Develop a probability distribution for a random variable defined for a sample spac e in which theoretical probabilities can be calculated; find the expected value. For example, find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes.
Curriculum point HSS-MD.A.4 (+) Develop a probability distribution for a random variable defined for a sample spac e in which probabilities are assigned empirically; find the expected value. For example, find a current data distribution on the number of TV sets per household in the United States, and calculate the expected number of sets per household. How many TV sets would you expect to find in 100 randomly selected households?
Curriculum point HSS-MD.B.5 (+) Weigh the possible outcomes of a decision by assigning probabilities to pa yoff values and finding expected values. a. Find the expected payoff for a game of chance. For example, find the expected winnings from a state lottery ticket or a game at a fast-food restaurant. b. Evaluate and compare strategies on the basis of expected values. For example, compare a high-deductible versus a low-deductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident.
Curriculum point HSS-MD.B.6 (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number gener ator).
Curriculum point HSS-MD.B.7 (+) Analyze decisions and strategies using probability concepts (e.g., product t esting, medical testing, pulling a hockey goalie at the end of a game).

In Grade 12, students work with discrete random variables displayed in probability distribution tables. They calculate the expected value by multiplying each possible numerical outcome by its matching probability and adding all the products together. Through this process, they learn to interpret the resulting number as the mean or weighted center of the probability distribution.

A common error occurs when students treat all outcomes equally. Instead of calculating a weighted sum using the given probabilities, they may simply add up the possible outcome values and divide by the number of outcomes, confusing a basic arithmetic mean with expected value. Another frequent slip is accidentally multiplying probabilities by each other rather than pairing each outcome with its corresponding probability.

Worksheet tasks present a completed distribution table showing a row or column of numerical outcomes alongside their respective probabilities as fractions or decimals. Students compute the product for each pair, sum those values, and state the expected value of the random variable.

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Task types: Expected value from a distribution table · Is the game fair?

Exact Values of Sine, Cosine, and Tangent

Students find the exact values of sine, cosine, and tangent for key angles like π/6, π/4, and π/3, and use the unit circle to evaluate related angles across different quadrants.

Curriculum point HSF-TF.A.3 (+) Use special triangles to determine geometrically the values of sine, cosine, tangent f or 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.

In Grade 12, students determine the exact values of the sine, cosine, and tangent functions without relying on a calculator. They use special right triangles to find geometric values for benchmark angles—specifically π/6, π/4, and π/3 radians. Students then use the geometry of the unit circle to relate these values to other quadrants, rewriting expressions of the form π – x, π + x, and 2π – x in terms of an angle x.

A frequent stumbling block is getting the positive or negative sign wrong. Students often determine the correct numerical ratio from the reference angle but forget to check which quadrant the full angle lies in, mistakenly writing a positive value for cosine in the second quadrant or for sine in the third quadrant.

Worksheet tasks focusing on the Exact value of a trigonometric function present students with specific expressions such as finding the exact value of cos(5π/6) or tan(4π/3). Students identify the underlying special triangle, apply the reduction formula to locate the angle on the unit circle, and write the final solution as a simplified fraction or radical rather than a rounded decimal.

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Solving Systems of Equations with Matrices

Students learn to write linear systems as matrix equations and solve them using matrix inverses, using technology for systems of size 3 × 3 or larger.

Curriculum point HSA-REI.C.8 (+) Represent a system of linear equations as a single matrix equation in a vector variable.
Curriculum point HSA-REI.C.9 (+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).

Students learn to rewrite a system of linear equations as a single matrix equation in the form Ax = b, where A is the coefficient matrix and x is the variable vector. They determine whether the coefficient matrix has an inverse and multiply by that inverse to solve for the unknown values. While students can compute inverses for simple systems by hand, they use technology to find inverses and solve systems with matrices of dimension 3 × 3 or greater.

A common error involves the order of matrix multiplication. Because matrix multiplication is not commutative, multiplying the constants by the inverse in the wrong order—calculating bA^-1 instead of A^-1b—leads to dimension mismatches or incorrect answers. Students also frequently overlook missing variables in an equation, forgetting to insert a 0 into the coefficient matrix for that position.

In practice, tasks like Solving a system with a matrix present students with two or more equations. Students first organize the equations into matrix form, find the inverse of the coefficient matrix either algebraically or using a graphing calculator, and then multiply to find the exact coordinates or variable values that satisfy the entire system.

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Exact value of a trigonometric function

Solving Systems of Equations with Matrices

Solving a system with a matrix

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