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Grade 11 Math

37 curriculum outcomes with explicit teaching progressions

Select a record to inspect its progression

Complex Numbers · CCSS.Math.Content.HS.N-CN.A.1

Understand the complex number system

Independently know there is a complex number i such that i² = -1, and every complex number has the form a + bi with a and b real. across representations and contexts.

Component progression

  1. 1I can state that i is defined by the property i² = -1 and use it to simplify expressions like i³ and i⁴.
  2. 2I can given a complex number a + bi, identify its real part a and imaginary part b.
  3. 3I can rewrite an expression involving a square root of a negative number in the standard a + bi form.
Complex Numbers · CCSS.Math.Content.HS.N-CN.A.2

Add, subtract, and multiply complex numbers

Independently use the relation i² = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. across representations and contexts.

Component progression

  1. 1I can add and subtract complex numbers by combining real parts and imaginary parts separately.
  2. 2I can multiply two complex numbers using the distributive property and simplify using i² = -1.
  3. 3I can simplify an expression that combines addition, subtraction, and multiplication of complex numbers into standard a + bi form.
Complex Numbers · CCSS.Math.Content.HS.N-CN.A.3

Use conjugates to find moduli and quotients

Independently find the conjugate of a complex number and use conjugates to find moduli and quotients of complex numbers. across representations and contexts.

Component progression

  1. 1I can given a complex number a + bi, write its conjugate a - bi.
  2. 2I can compute the modulus of a complex number as √(a² + b²).
  3. 3I can divide two complex numbers by multiplying the numerator and denominator by the denominator's conjugate, and simplify to standard form.
Complex Numbers · CCSS.Math.Content.HS.N-CN.C.7

Solve quadratics with complex solutions

Independently solve quadratic equations with real coefficients that have complex solutions. across representations and contexts.

Component progression

  1. 1I can use the discriminant to determine when a quadratic equation's solutions are complex rather than real.
  2. 2I can apply the quadratic formula to a quadratic with a negative discriminant, expressing the result as a complex conjugate pair.
  3. 3I can verify that a complex solution satisfies the original quadratic equation by substitution.
Complex Numbers · CCSS.Math.Content.HS.N-CN.C.8

Extend polynomial identities to complex numbers

Independently extend polynomial identities to the complex numbers, such as factoring x² + 4 as (x + 2i)(x - 2i). across representations and contexts.

Component progression

  1. 1I can factor an expression such as x² + 4 into complex conjugate linear factors.
  2. 2I can multiply complex linear factors back together to verify they produce the original real polynomial.
  3. 3I can use a complex factorization to find all solutions, real and complex, of a polynomial equation.
Complex Numbers · CCSS.Math.Content.HS.N-CN.C.9

Apply the Fundamental Theorem of Algebra

Independently know the fundamental theorem of algebra, and show that it is true for quadratic polynomials. across representations and contexts.

Component progression

  1. 1I can state that a degree-n polynomial has exactly n roots when counted with multiplicity, allowing complex roots.
  2. 2I can show, using the quadratic formula, that every quadratic polynomial has exactly two roots (real or complex, counting multiplicity).
  3. 3I can determine the total number of roots, counted with multiplicity, of a given polynomial from its degree.
Advanced Algebra · CCSS.Math.Content.HS.A-SSE.B.4

Derive and apply the finite geometric series formula

Independently derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems such as calculating mortgage payments. across representations and contexts.

Component progression

  1. 1I can derive the formula for the sum of a finite geometric series by multiplying the series by the common ratio and subtracting.
  2. 2I can use the derived formula to compute the sum of a specified finite geometric series.
  3. 3I can use the finite geometric series formula to solve a real-world problem such as computing total payments in a savings or loan scenario.
Advanced Algebra · CCSS.Math.Content.HS.A-APR.B.2

Apply the Remainder Theorem

Independently know and apply the remainder theorem: for a polynomial p(x) and a number a, the remainder on division by (x - a) is p(a), so p(a) = 0 if and only if (x - a) is a factor of p(x). across representations and contexts.

Component progression

  1. 1I can use the remainder theorem to find the remainder when a polynomial is divided by (x - a) by evaluating p(a).
  2. 2I can use the remainder theorem to determine whether (x - a) is a factor of a given polynomial.
  3. 3I can use the remainder theorem to test candidate values and identify roots of a polynomial.
Advanced Algebra · CCSS.Math.Content.HS.A-APR.B.3

Identify zeros and sketch polynomial graphs

Independently identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. across representations and contexts.

Component progression

  1. 1I can identify all zeros of a polynomial given in factored form, including their multiplicities.
  2. 2I can determine whether the graph crosses or touches the x-axis at each zero based on whether its multiplicity is odd or even.
  3. 3I can use the zeros, their multiplicities, and the polynomial's end behavior to sketch a rough graph of the function.
Advanced Algebra · CCSS.Math.Content.HS.A-APR.C.4

Prove and apply polynomial identities

Independently prove polynomial identities and use them to describe numerical relationships, such as the identity (x² + y²)² = (x² - y²)² + (2xy)², which can be used to generate pythagorean triples. across representations and contexts.

Component progression

  1. 1I can prove a given polynomial identity by expanding both sides and showing they are equal for all values of the variables.
  2. 2I can use a proven polynomial identity, such as the sum-of-squares identity, to generate a numerical pattern such as pythagorean triples.
  3. 3I can apply a known polynomial identity to simplify or evaluate an expression more efficiently than direct expansion.
Advanced Algebra · CCSS.Math.Content.HS.A-APR.C.5

Apply the Binomial Theorem

Independently know and apply the binomial theorem for the expansion of (x + y)ⁿ in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined by pascal's triangle. across representations and contexts.

Component progression

  1. 1I can generate the binomial coefficients for a given power n using pascal's triangle or the combination formula.
  2. 2I can use the binomial theorem to expand (x + y)ⁿ for a specified positive integer n.
  3. 3I can use the binomial theorem to find a specific term of a binomial expansion without writing out the full expansion.
Advanced Algebra · CCSS.Math.Content.HS.A-APR.D.6

Rewrite rational expressions using polynomial division

Independently rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for more complicated examples, a computer algebra system. across representations and contexts.

Component progression

  1. 1I can rewrite a simple rational expression a(x)/b(x) in q(x) + r(x)/b(x) form by inspection when the numerator's structure makes the division obvious.
  2. 2I can divide a(x) by b(x) using polynomial long division to find the quotient q(x) and remainder r(x), for cases too complex to rewrite by inspection.
  3. 3I can verify a rewritten rational expression by multiplying q(x) by b(x), adding r(x), and confirming the result equals a(x).
Advanced Algebra · CCSS.Math.Content.HS.A-APR.D.7

Understand rational expressions as a closed system

Independently understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions. across representations and contexts.

Component progression

  1. 1I can add and subtract rational expressions by rewriting them with a common denominator.
  2. 2I can multiply and divide rational expressions by factoring and canceling common factors.
  3. 3I can identify values excluded from the domain of a rational expression because they make a denominator zero.
Advanced Algebra · CCSS.Math.Content.HS.A-REI.A.2

Solve rational and radical equations, checking for extraneous solutions

Independently solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. across representations and contexts.

Component progression

  1. 1I can solve a rational equation in one variable by clearing denominators and solving the resulting polynomial equation.
  2. 2I can solve a radical equation in one variable by isolating the radical and raising both sides to an appropriate power.
  3. 3I can check candidate solutions in the original equation, identify any extraneous solutions, and explain why they arose.
Advanced Algebra · CCSS.Math.Content.HS.A-REI.B.4

Solve quadratics by completing the square, connecting to complex roots

Independently solve quadratic equations in one variable using inspection, taking square roots, factoring, completing the square, and the quadratic formula, as appropriate to the equation's initial form; derive the quadratic formula by completing the square on the general quadratic equation, and recognize when it gives complex solutions, writing them in the form a ± bi for real numbers a and b. across representations and contexts.

Component progression

  1. 1I can solve a quadratic equation by inspection, taking square roots, or factoring when the equation's initial form makes one of these methods more efficient than the quadratic formula.
  2. 2I can solve a quadratic equation in one variable by completing the square, including cases with a leading coefficient other than 1, and use that process to derive the quadratic formula.
  3. 3I can apply the quadratic formula to solve a quadratic equation, recognizing a negative discriminant and writing the resulting complex solutions in the form a ± bi.
Advanced Functions · CCSS.Math.Content.HS.F-IF.C.7

Graph polynomial, rational, exponential, and logarithmic functions

Independently graph functions expressed symbolically, including polynomial, rational, exponential, and logarithmic functions, and show key features of the graph such as intercepts, asymptotes, and end behavior. across representations and contexts.

Component progression

  1. 1I can graph a polynomial or rational function, identifying intercepts, asymptotes (for rational functions), and end behavior.
  2. 2I can graph an exponential or logarithmic function, identifying intercepts, asymptotes, and end behavior.
  3. 3I can given the graph of a function from one of these families, identify its intercepts, asymptotes, and end behavior.
Advanced Functions · CCSS.Math.Content.HS.F-IF.C.8

Reveal properties by rewriting a function in equivalent forms

Independently write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. across representations and contexts.

Component progression

  1. 1I can rewrite a quadratic function in factored form to reveal its zeros, or in vertex form to reveal its maximum or minimum.
  2. 2I can rewrite an exponential function to reveal a percent rate of change per unit interval.
  3. 3I can explain what property of the function is made explicit by a given equivalent form.
Advanced Functions · CCSS.Math.Content.HS.F-IF.C.9

Compare functions across representations

Independently compare properties of two functions, from an expanded set of function families, each represented in a different way — algebraically, graphically, numerically in tables, or by verbal descriptions. across representations and contexts.

Component progression

  1. 1I can determine a specific property of a function — such as growth rate, an asymptote, or an extreme value — from an algebraic, graphical, tabular, or verbal representation.
  2. 2I can compare a specific property between two functions given in different representations.
  3. 3I can draw and justify a conclusion, such as which function grows faster, based on the comparison.
Advanced Functions · CCSS.Math.Content.HS.F-BF.A.1

Build functions modeling combined and composed relationships

Independently write a function that describes a relationship between two quantities, including combining standard function types using arithmetic operations and composing functions. across representations and contexts.

Component progression

  1. 1I can build a function modeling a relationship by adding, subtracting, multiplying, or dividing two given functions.
  2. 2I can build a composed function f(g(x)) from two given functions and evaluate it at specified values.
  3. 3I can construct a function describing a real-world relationship between two quantities using combination or composition.
Advanced Functions · CCSS.Math.Content.HS.F-BF.B.4

Find inverse functions

Independently find inverse functions, including solving an equation for the inverse of a simple function and verifying that two functions are inverses by composing them. across representations and contexts.

Component progression

  1. 1I can find the inverse of a simple function by swapping variables and solving for the new dependent variable.
  2. 2I can verify that two functions are inverses of each other by showing that composing them in either order produces the identity.
  3. 3I can interpret the inverse of a function that models a real-world relationship in terms of the original context.
Advanced Functions · CCSS.Math.Content.HS.F-LE.A.4

Express exponential equation solutions as logarithms

Independently for exponential models, express as a logarithm the solution to ab^(ct) = d, where a, c, and d are numbers and the base b is 2, 10, or e, and evaluate the logarithm using technology. across representations and contexts.

Component progression

  1. 1I can rewrite an equation of the form ab^(ct) = d to isolate the exponential expression b^(ct).
  2. 2I can apply the appropriate logarithm to both sides of the isolated equation and solve for t.
  3. 3I can evaluate the resulting logarithmic expression using technology to find a decimal approximation for t.
Trigonometry · CCSS.Math.Content.HS.F-TF.A.1

Understand radian measure

Independently understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle. across representations and contexts.

Component progression

  1. 1I can explain that an angle's radian measure equals the length of the arc it subtends on the unit circle.
  2. 2I can convert an angle measure between degrees and radians.
  3. 3I can identify the radian measures of common angles such as π/6, π/4, π/3, π/2, and π.
Trigonometry · CCSS.Math.Content.HS.F-TF.A.2

Extend trig functions using the unit circle

Independently explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle. across representations and contexts.

Component progression

  1. 1I can locate the terminal point of a given angle, including angles beyond 2π or negative angles, on the unit circle.
  2. 2I can define the sine and cosine of an angle as the y- and x-coordinates of its terminal point on the unit circle.
  3. 3I can evaluate sine, cosine, and tangent for any real-number angle measure using the unit circle.
Trigonometry · CCSS.Math.Content.HS.F-TF.A.3

Use special triangles on the unit circle

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 triangles to geometrically determine sine, cosine, and tangent for π/6, π/4, and π/3.
  2. 2I can use a reference angle and quadrant sign to find sine, cosine, and tangent for angles like π - x, π + x, and 2π - x from the value at x.
  3. 3I can use known special-angle values and reference-angle relationships to evaluate trigonometric expressions without a calculator.
Trigonometry · CCSS.Math.Content.HS.F-TF.A.4

Use the unit circle to explain symmetry and periodicity

Independently use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. across representations and contexts.

Component progression

  1. 1I can use the unit circle to explain why sine and cosine repeat every 2π and tangent repeats every π.
  2. 2I can use unit-circle symmetry to determine whether sine, cosine, and tangent are even, odd, or neither.
  3. 3I can use periodicity and even/odd symmetry to simplify or evaluate a trigonometric expression.
Trigonometry · CCSS.Math.Content.HS.F-TF.B.5

Model periodic phenomena with trig functions

Independently choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. across representations and contexts.

Component progression

  1. 1I can identify the amplitude, period, and midline of a periodic phenomenon from a verbal description or data.
  2. 2I can construct a sine or cosine function with the correct amplitude, period, and midline to model the phenomenon.
  3. 3I can use the constructed trigonometric model to predict a value or answer a question about the real-world phenomenon.
Trigonometry · CCSS.Math.Content.HS.F-TF.B.6

Understand inverse trigonometric functions

Independently understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed. across representations and contexts.

Component progression

  1. 1I can explain why sine, cosine, and tangent are not one-to-one over their full domains and why this prevents a direct inverse.
  2. 2I can identify the standard restricted domain used for each trig function to construct its inverse.
  3. 3I can construct the inverse function on the restricted domain and describe its range.
Trigonometry · CCSS.Math.Content.HS.F-TF.B.7

Use inverse trig functions to solve modeling equations

Independently use inverse functions to solve trigonometric equations that arise in modeling contexts, evaluate the solutions using technology, and interpret the solutions in terms of the context. across representations and contexts.

Component progression

  1. 1I can use an inverse trigonometric function to solve a trigonometric equation for one solution, evaluating the result with technology when it isn't a special angle.
  2. 2I can determine all solutions of a trigonometric equation within a specified domain, using periodicity and symmetry.
  3. 3I can interpret the solutions of a trigonometric equation in terms of the real-world periodic phenomenon being modeled.
Trigonometry · CCSS.Math.Content.HS.F-TF.C.8

Prove and apply the Pythagorean identity

Independently prove the pythagorean identity sin²θ + cos²θ = 1, and use it to find sin θ, cos θ, or tan θ given one value and the quadrant of the angle. across representations and contexts.

Component progression

  1. 1I can prove sin²θ + cos²θ = 1 using the unit-circle definitions of sine and cosine.
  2. 2I can use the pythagorean identity to find sin θ or cos θ given the other value.
  3. 3I can use the given quadrant of the angle to determine the correct sign of the missing trig value.
Trigonometry · CCSS.Math.Content.HS.F-TF.C.9

Prove and apply angle sum and difference formulas

Independently prove the addition and subtraction formulas for sine, cosine, and tangent, and use them to solve problems. across representations and contexts.

Component progression

  1. 1I can use the angle sum and difference formulas for sine and cosine to evaluate a trigonometric expression at a non-special angle by decomposing it into special angles.
  2. 2I can derive the tangent addition formula from the sine and cosine addition formulas.
  3. 3I can use an angle sum or difference formula to solve a problem or simplify an expression involving a sum or difference of angles.
Statistical Inference · CCSS.Math.Content.HS.S-ID.C.6

Model relationships between two quantitative variables

Independently represent data on two quantitative variables on a scatter plot, and describe how the variables are related, including fitting a function to the data and assessing the fit. across representations and contexts.

Component progression

  1. 1I can construct a scatter plot for bivariate data and describe the type of relationship it suggests (linear, quadratic, exponential, or none).
  2. 2I can fit an appropriate function to a scatter plot's pattern and use it to make predictions.
  3. 3I can compute and examine residuals to assess how well a fitted function models the data.
Statistical Inference · CCSS.Math.Content.HS.S-IC.A.1

Understand statistical inference from random samples

Independently understand statistics as a process for making inferences about population parameters based on a random sample from that population. across representations and contexts.

Component progression

  1. 1I can distinguish between a statistic computed from a sample and the corresponding parameter of the full population.
  2. 2I can explain why random sampling supports valid inference about a population parameter.
  3. 3I can describe, for a given scenario, how a sample statistic would be used to make an inference about the population parameter.
Statistical Inference · CCSS.Math.Content.HS.S-IC.A.2

Evaluate whether a model is consistent with data

Independently decide if a specified model is consistent with results from a given data-generating process, using simulation. across representations and contexts.

Component progression

  1. 1I can design a simulation that reflects the assumptions of a specified probability model, such as a fair coin.
  2. 2I can run a simulation and compare its results to observed data from an actual process.
  3. 3I can use the simulation results to decide whether the observed data is consistent with the specified model or suggests the model is wrong.
Statistical Inference · CCSS.Math.Content.HS.S-IC.B.3

Distinguish sample surveys, experiments, and observational studies

Independently recognize the purposes of and differences among sample surveys, experiments, and observational studies, and explain how randomization relates to each. across representations and contexts.

Component progression

  1. 1I can identify whether a given study is a sample survey, an experiment, or an observational study based on its purpose and design.
  2. 2I can explain how random sampling is used in surveys and random assignment is used in experiments, and why observational studies use neither.
  3. 3I can determine whether a given study design supports a claim of association or a claim of causation.
Statistical Inference · CCSS.Math.Content.HS.S-IC.B.4

Estimate a population mean or proportion with margin of error

Independently use data from a sample survey to estimate a population mean or proportion, and develop a margin of error through the use of simulation models for random sampling. across representations and contexts.

Component progression

  1. 1I can compute a sample mean or sample proportion from survey data as a point estimate of the population parameter.
  2. 2I can use a simulation of repeated random sampling to model how the sample estimate varies from sample to sample.
  3. 3I can use the simulated sampling variability to develop a margin of error and interpret what it says about the estimate's precision.
Statistical Inference · CCSS.Math.Content.HS.S-IC.B.5

Compare treatments using randomized experiments and simulation

Independently use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant. across representations and contexts.

Component progression

  1. 1I can compute the observed difference between two treatment groups' outcomes from a randomized experiment.
  2. 2I can use simulation to model the distribution of differences expected if the treatments actually had no effect.
  3. 3I can compare the observed difference to the simulated distribution to decide whether it is large enough to be considered significant.
Statistical Inference · CCSS.Math.Content.HS.S-IC.B.6

Evaluate reports based on data

Independently evaluate reports based on data, assessing the study design, the appropriateness of the statistical methods used, and the validity of the conclusions drawn. across representations and contexts.

Component progression

  1. 1I can identify whether a report's data came from a survey, experiment, or observational study, and evaluate whether that design supports its claims.
  2. 2I can assess whether the statistical methods described in a report, such as an average or a margin of error, were used and interpreted appropriately.
  3. 3I can judge whether a report's stated conclusions are actually supported by its data and methods, identifying any overreach.
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