allenai/achieve-the-core huggingface dataset:
8.EE.C.8b (Sub-standard, major cluster): Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.
Aspects: Procedural Skill and Fluency
8.EE.C.8c (Sub-standard, major cluster): Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.
Aspects: Application
8.F.A (Cluster, major cluster): Define, evaluate, and compare functions.
Aspects: conceptual understanding
8.F.A.1 (Standard, major cluster): Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. (Function notation is not required in Grade 8.)
Aspects: conceptual understanding
8.F.A.2 (Standard, major cluster): Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.
Aspects: conceptual understanding
8.F.A.3 (Standard, major cluster): Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.
Aspects: conceptual understanding
8.F.B (Cluster, major cluster): Use functions to model relationships between quantities.
Aspects: Application
8.F.B.4 (Standard, major cluster): Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
Aspects: Application
8.F.B.5 (Standard, major cluster): Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.
Aspects: Application
8.G.A (Cluster, major cluster): Understand congruence and similarity using physical models, transparencies, or geometry software.
8.G.A.1 (Standard, major cluster): Verify experimentally the properties of rotations, reflections, and translations:
- Lines are taken to lines, and line segments to line segments of the same length.
- Angles are taken to angles of the same measure.
- Parallel lines are taken to parallel lines.
8.G.A.1a (Sub-standard, major cluster): Lines are taken to lines, and line segments to line segments of the same length.
8.G.A.1b (Sub-standard, major cluster): Angles are taken to angles of the same measure.
8.G.A.1c (Sub-standard, major cluster): Parallel lines are taken to parallel lines.
8.G.A.2 (Standard, major cluster): Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.
8.G.A.3 (Standard, major cluster): Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
8.G.A.4 (Standard, major cluster): Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.
8.G.A.5 (Standard, major cluster): Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.
8.G.B (Cluster, major cluster): Understand and apply the Pythagorean Theorem.
8.G.B.6 (Standard, major cluster): Explain a proof of the Pythagorean Theorem and its converse.
8.G.B.7 (Standard, major cluster): Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
8.G.B.8 (Standard, major cluster): Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
8.G.C (Cluster, additional cluster): Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.
8.G.C.9 (Standard, additional cluster): Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.
8.SP.A (Cluster, supporting cluster): Investigate patterns of association in bivariate data.
8.SP.A.1 (Standard, supporting cluster): Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.
8.SP.A.2 (Standard, supporting cluster): Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.
8.SP.A.3 (Standard, supporting cluster): Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.
8.SP.A.4 (Standard, supporting cluster): Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?
N-RN.A (Cluster, widely applicable prerequisite): Extend the properties of exponents to rational exponents.
N-RN.A.1 (Standard, widely applicable prerequisite): Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 5^(1/3) to be the cube root of 5 because we want (5^(1/3))^(3) = 5(^(1/3))^(3) to hold, so (5^(1/3))^(3) must equal 5.
Aspects: conceptual understanding
N-RN.A.2 (Standard, widely applicable prerequisite): Rewrite expressions involving radicals and rational exponents using the properties of exponents.
N-RN.B (Cluster, widely applicable prerequisite): Use properties of rational and irrational numbers.
N-RN.B.3 (Standard, widely applicable prerequisite): Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.
N-Q.A (Cluster, widely applicable prerequisite): Reason quantitatively and use units to solve problems.
Aspects: Application
N-Q.A.1 (Standard, widely applicable prerequisite): Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
Aspects: Application
N-Q.A.2 (Standard, widely applicable prerequisite): Define appropriate quantities for the purpose of descriptive modeling.
Aspects: Application
N-Q.A.3 (Standard, widely applicable prerequisite): Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
Aspects: Application
N-CN.A.1 (Standard, none): Know there is a complex number i such that i^(2) = –1, and every complex number has the form a + bi with a and b real.
N-CN.A.2 (Standard, none): Use the relation i^(2) = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.
N-CN.A.3 (Standard, none): Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.
N-CN.B (Cluster, none): Represent complex numbers and their operations on the complex plane.
N-CN.B.4 (Standard, none): Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.
N-CN.B.5 (Standard, none): Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. For example, (-1 + Ö3i)^(3) = 8 because (-1 + Ö3i) has modulus 2 and argument 120°.
N-CN.B.6 (Standard, none): Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.
N-CN.C (Cluster, none): Use complex numbers in polynomial identities and equations.
N-CN.C.7 (Standard, none): Solve quadratic equations with real coefficients that have complex solutions.
N-CN.C.8 (Standard, none): Extend polynomial identities to the complex numbers. For example, rewrite x^(2) + 4 as (x + 2i)(x – 2i).
N-CN.C.9 (Standard, none): Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
N-VM.A.1 (Standard, none): 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 (e.g., v, |v|, ||v||, v).
N-VM.A.2 (Standard, none): Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.
N-VM.A.3 (Standard, none): Solve problems involving velocity and other quantities that can be represented by vectors.
N-VM.B.4a (Sub-standard, none): 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.
N-VM.B.4b (Sub-standard, none): Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
N-VM.B.4c (Sub-standard, none): 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.
N-VM.B.5 (Standard, none): Multiply a vector by a scalar.
N-VM.B.5a (Sub-standard, none): Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(v_(x), v_(y)) = (cv_(x), cv_(y)).
N-VM.B.5b (Sub-standard, none): 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).
N-VM.C (Cluster, none): Perform operations on matrices and use matrices in applications.
N-VM.C.6 (Standard, none): Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.
N-VM.C.7 (Standard, none): Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.
N-VM.C.8 (Standard, none): Add, subtract, and multiply matrices of appropriate dimensions.
N-VM.C.9 (Standard, none): Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.
N-VM.C.10 (Standard, none): 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.
N-VM.C.11 (Standard, none): 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.
N-VM.C.12 (Standard, none): Work with 2 × 2 matrices as a transformations of the plane, and interpret the absolute value of the determinant in terms of area.
A-SSE.A (Cluster, widely applicable prerequisite): Interpret the structure of expressions.
A-SSE.A.1 (Standard, widely applicable prerequisite): Interpret expressions that represent a quantity in terms of its context.
A-SSE.A.1a (Sub-standard, widely applicable prerequisite): Interpret parts of an expression, such as terms, factors, and coefficients.
A-SSE.A.1b (Sub-standard, widely applicable prerequisite): Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)^(n )as the product of P and a factor not depending on P.
Aspects: Procedural Skill and Fluency
A-SSE.A.2 (Standard, widely applicable prerequisite): Use the structure of an expression to identify ways to rewrite it. For example, see x^(4) – y^(4 )as (x^(2))^(2 )– (y^(2))^(2), thus recognizing it as a difference of squares that can be factored as (x^(2 )– y^(2))(x^(2 )+ y^(2)).
A-SSE.B (Cluster, widely applicable prerequisite): Write expressions in equivalent forms to solve problems.
A-SSE.B.3 (Standard, widely applicable prerequisite): Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
Aspects: Application
A-SSE.B.3a (Sub-standard, widely applicable prerequisite): Factor a quadratic expression to reveal the zeros of the function it defines.
Aspects: Application
A-SSE.B.3b (Sub-standard, widely applicable prerequisite): Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
Aspects: Application
A-SSE.B.3c (Sub-standard, widely applicable prerequisite): Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15^(t) can be rewritten as (1.15^(1/12))^(12)^(t )≈1.012^(12t) to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.
Aspects: Application
A-SSE.B.4 (Standard, widely applicable prerequisite): 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. For example, calculate mortgage payments.
A-APR.A (Cluster, widely applicable prerequisite): Perform arithmetic operations on polynomials.
A-APR.A.1 (Standard, widely applicable prerequisite): Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
A-APR.B (Cluster, widely applicable prerequisite): Understand the relationship between zeros and factors of polynomials.
Aspects: conceptual understanding
A-APR.B.2 (Standard, widely applicable prerequisite): 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).
Aspects: conceptual understanding
A-APR.B.3 (Standard, widely applicable prerequisite): 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.
Aspects: conceptual understanding
A-APR.C (Cluster, widely applicable prerequisite): Use polynomial identities to solve problems.
A-APR.C.4 (Standard, widely applicable prerequisite): Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x^(2) + y^(2))^(2) = (x^(2 )– y^(2))^(2) + (2xy)^(2 )can be used to generate Pythagorean triples.
A-APR.C.5 (Standard, widely applicable prerequisite): 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 for example by Pascal’s Triangle. (The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.)
A-APR.D (Cluster, widely applicable prerequisite): Rewrite rational expressions.
A-APR.D.6 (Standard, widely applicable prerequisite): 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 the more complicated examples, a computer algebra system.
A-APR.D.7 (Standard, widely applicable prerequisite): 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.
A-CED.A (Cluster, widely applicable prerequisite): Create equations that describe numbers or relationships
A-CED.A.1 (Standard, widely applicable prerequisite): Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
A-CED.A.2 (Standard, widely applicable prerequisite): Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
A-CED.A.3 (Standard, widely applicable prerequisite): Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.
A-CED.A.4 (Standard, widely applicable prerequisite): Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.
A-REI.A (Cluster, widely applicable prerequisite): Understand solving equations as a process of reasoning and explain the reasoning.
A-REI.A.1 (Standard, widely applicable prerequisite): Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
A-REI.A.2 (Standard, widely applicable prerequisite): Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
A-REI.B (Cluster, widely applicable prerequisite): Solve equations and inequalities in one variable.
A-REI.B.3 (Standard, widely applicable prerequisite): Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
A-REI.B.4 (Standard, widely applicable prerequisite): Solve quadratic equations in one variable.
A-REI.B.4a (Sub-standard, widely applicable prerequisite): Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)^(2) = q that has the same solutions. Derive the quadratic formula from this form.
A-REI.B.4b (Sub-standard, widely applicable prerequisite): Solve quadratic equations by inspection (e.g., for x^(2) = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.
A-REI.C (Cluster, widely applicable prerequisite): Solve systems of equations.
A-REI.C.5 (Standard, widely applicable prerequisite): Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
A-REI.C.6 (Standard, widely applicable prerequisite): Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
A-REI.C.7 (Standard, widely applicable prerequisite): Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = –3x and the circle x^(2 )+ y^(2) = 3.
A-REI.C.8 (Standard, widely applicable prerequisite): Represent a system of linear equations as a single matrix equation in a vector variable.
A-REI.C.9 (Standard, widely applicable prerequisite): 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).
A-REI.D (Cluster, widely applicable prerequisite): Represent and solve equations and inequalities graphically.
A-REI.D.10 (Standard, widely applicable prerequisite): Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).
A-REI.D.11 (Standard, widely applicable prerequisite): Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
A-REI.D.12 (Standard, widely applicable prerequisite): Graph the solutions to a linear inequality in two variables as a half plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
F-IF.A (Cluster, widely applicable prerequisite): Understand the concept of a function and use function notation.
F-IF.A.1 (Standard, widely applicable prerequisite): Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
F-IF.A.2 (Standard, widely applicable prerequisite): Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
F-IF.A.3 (Standard, widely applicable prerequisite): Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ³ 1.