allenai/achieve-the-core huggingface dataset:
6.EE.B.8 (Standard, major cluster): Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.
6.EE.C (Cluster, major cluster): Represent and analyze quantitative relationships between dependent and independent variables.
6.EE.C.9 (Standard, major cluster): Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.
Aspects: Application
6.G.A (Cluster, supporting cluster): Solve real-world and mathematical problems involving area, surface area, and volume.
6.G.A.1 (Standard, supporting cluster): Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
6.G.A.2 (Standard, supporting cluster): Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.
6.G.A.3 (Standard, supporting cluster): Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.
6.G.A.4 (Standard, supporting cluster): Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.
6.SP.A.1 (Standard, additional cluster): Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. For example, “How old am I?” is not a statistical question, but “How old are the students in my school?” is a statistical question because one anticipates variability in students’ ages.
6.SP.A.2 (Standard, additional cluster): Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.
6.SP.A.3 (Standard, additional cluster): Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.
6.SP.B.4 (Standard, additional cluster): Display numerical data in plots on a number line, including dot plots, histograms, and box plots.
6.SP.B.5 (Standard, additional cluster): Summarize numerical data sets in relation to their context, such as by:
- Reporting the number of observations.
- Describing the nature of the attribute under investigation, including how it was measured and its units of measurement.
- Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.
- Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.
6.SP.B.5a (Sub-standard, additional cluster): Reporting the number of observations.
6.SP.B.5b (Sub-standard, additional cluster): Describing the nature of the attribute under investigation, including how it was measured and its units of measurement.
6.SP.B.5c (Sub-standard, additional cluster): Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.
6.SP.B.5d (Sub-standard, additional cluster): Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.
7.RP.A (Cluster, major cluster): Analyze proportional relationships and use them to solve real-world and mathematical problems.
Aspects: Application
7.RP.A.1 (Standard, major cluster): Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction 1/2/1/4 miles per hour, equivalently 2 miles per hour.
Aspects: Application
7.RP.A.2 (Standard, major cluster): Recognize and represent proportional relationships between quantities.
Aspects: Application
7.RP.A.2a (Sub-standard, major cluster): Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.
Aspects: Application
7.RP.A.2b (Sub-standard, major cluster): Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.
Aspects: Application
7.RP.A.2c (Sub-standard, major cluster): Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn.
Aspects: Application
7.RP.A.2d (Sub-standard, major cluster): Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.
Aspects: Application
7.RP.A.3 (Standard, major cluster): Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.
Aspects: Application
7.NS.A (Cluster, major cluster): Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
Aspects: conceptual understanding, Procedural Skill and Fluency
7.NS.A.1 (Standard, major cluster): Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.
Aspects: conceptual understanding, Procedural Skill and Fluency
7.NS.A.1a (Sub-standard, major cluster): Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged.
Aspects: conceptual understanding, Procedural Skill and Fluency
7.NS.A.1b (Sub-standard, major cluster): Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts.
Aspects: conceptual understanding, Procedural Skill and Fluency
7.NS.A.1c (Sub-standard, major cluster): Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.
Aspects: conceptual understanding, Procedural Skill and Fluency
7.NS.A.1d (Sub-standard, major cluster): Apply properties of operations as strategies to add and subtract rational numbers.
Aspects: conceptual understanding, Procedural Skill and Fluency
7.NS.A.2 (Standard, major cluster): Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
Aspects: conceptual understanding, Procedural Skill and Fluency
7.NS.A.2a (Sub-standard, major cluster): Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (–1)(–1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts.
Aspects: conceptual understanding, Procedural Skill and Fluency
7.NS.A.2b (Sub-standard, major cluster): Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then –(p/q) = (–p)/q = p/(–q). Interpret quotients of rational numbers by describing real world contexts.
Aspects: conceptual understanding, Procedural Skill and Fluency
7.NS.A.2c (Sub-standard, major cluster): Apply properties of operations as strategies to multiply and divide rational numbers.
Aspects: conceptual understanding, Procedural Skill and Fluency
7.NS.A.2d (Sub-standard, major cluster): Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.
Aspects: conceptual understanding, Procedural Skill and Fluency
7.NS.A.3 (Standard, major cluster): Solve real-world and mathematical problems involving the four operations with rational numbers. (Computations with rational numbers extend the rules for manipulating fractions to complex fractions.)
Aspects: conceptual understanding, Application, Procedural Skill and Fluency
7.EE.A (Cluster, major cluster): Use properties of operations to generate equivalent expressions.
Aspects: conceptual understanding
7.EE.A.1 (Standard, major cluster): Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.
Aspects: conceptual understanding, Procedural Skill and Fluency
7.EE.A.2 (Standard, major cluster): Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. For example, a + 0.05a = 1.05a means that “increase by 5%” is the same as “multiply by 1.05.”
Aspects: conceptual understanding
7.EE.B (Cluster, major cluster): Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
7.EE.B.3 (Standard, major cluster): Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.
Aspects: Application
7.EE.B.4 (Standard, major cluster): Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
7.EE.B.4a (Sub-standard, major cluster): Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?
Aspects: Procedural Skill and Fluency
7.EE.B.4b (Sub-standard, major cluster): Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. For example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions.
7.G.A (Cluster, additional cluster): Draw, construct, and describe geometrical figures and describe the relationships between them.
7.G.A.1 (Standard, additional cluster): Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.
7.G.A.2 (Standard, additional cluster): Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.
7.G.A.3 (Standard, additional cluster): Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.
7.G.B (Cluster, additional cluster): Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.
7.G.B.4 (Standard, additional cluster): Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.
7.G.B.5 (Standard, additional cluster): Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.
7.G.B.6 (Standard, additional cluster): Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.
7.SP.A (Cluster, supporting cluster): Use random sampling to draw inferences about a population.
7.SP.A.1 (Standard, supporting cluster): Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.
7.SP.A.2 (Standard, supporting cluster): Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be.
7.SP.B (Cluster, additional cluster): Draw informal comparative inferences about two populations.
7.SP.B.3 (Standard, additional cluster): Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable.
7.SP.B.4 (Standard, additional cluster): Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. For example, decide whether the words in a chapter of a seventh-grade science book are generally longer than the words in a chapter of a fourth-grade science book.
7.SP.C (Cluster, supporting cluster): Investigate chance processes and develop, use, and evaluate probability models.
7.SP.C.5 (Standard, supporting cluster): Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.
7.SP.C.6 (Standard, supporting cluster): Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.
7.SP.C.7 (Standard, supporting cluster): Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.
7.SP.C.7a (Sub-standard, supporting cluster): Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected.
7.SP.C.7b (Sub-standard, supporting cluster): Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-end down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies?
7.SP.C.8 (Standard, supporting cluster): Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.
7.SP.C.8a (Sub-standard, supporting cluster): Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.
7.SP.C.8b (Sub-standard, supporting cluster): Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., “rolling double sixes”), identify the outcomes in the sample space which compose the event.
7.SP.C.8c (Sub-standard, supporting cluster): Design and use a simulation to generate frequencies for compound events. . For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood?
8.NS.A (Cluster, supporting cluster): Know that there are numbers that are not rational, and approximate them by rational numbers.
8.NS.A.1 (Standard, supporting cluster): Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.
8.NS.A.2 (Standard, supporting cluster): Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., p^(2)). For example, by truncating the decimal expansion of Ö2, show that Ö2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.
8.EE.A (Cluster, major cluster): Work with radicals and integer exponents.
8.EE.A.1 (Standard, major cluster): Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 3^(2) × 3^(–5) = 3^(–3) = 1/3^(3) = 1/27.
8.EE.A.2 (Standard, major cluster): Use square root and cube root symbols to represent solutions to equations of the form x^(2) = p and x^(3) = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that Ö2 is irrational.
8.EE.A.3 (Standard, major cluster): Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 × 108 and the population of the world as 7 × 109, and determine that the world population is more than 20 times larger.
8.EE.A.4 (Standard, major cluster): Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.
8.EE.B (Cluster, major cluster): Understand the connections between proportional relationships, lines, and linear equations.
Aspects: conceptual understanding
8.EE.B.5 (Standard, major cluster): Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.
Aspects: conceptual understanding
8.EE.B.6 (Standard, major cluster): Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.
Aspects: conceptual understanding
8.EE.C (Cluster, major cluster): Analyze and solve linear equations and pairs of simultaneous linear equations.
8.EE.C.7 (Standard, major cluster): Solve linear equations in one variable.
Aspects: Procedural Skill and Fluency
8.EE.C.7a (Sub-standard, major cluster): Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).
Aspects: Procedural Skill and Fluency
8.EE.C.7b (Sub-standard, major cluster): Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.
Aspects: Procedural Skill and Fluency
8.EE.C.8 (Standard, major cluster): Analyze and solve pairs of simultaneous linear equations.
8.EE.C.8a (Sub-standard, major cluster): Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.