allenai/achieve-the-core huggingface dataset:
5.OA.A.2 (Standard, additional cluster): Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation “add 8 and 7, then multiply by 2” as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product.
5.OA.B.3 (Standard, additional cluster): Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule “Add 3” and the starting number 0, and given the rule “Add 6” and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.
5.NBT.A (Cluster, major cluster): Understand the place value system.
Aspects: conceptual understanding
5.NBT.A.1 (Standard, major cluster): Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.
Aspects: conceptual understanding
5.NBT.A.2 (Standard, major cluster): Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.
Aspects: conceptual understanding
5.NBT.A.3 (Standard, major cluster): Read, write, and compare decimals to thousandths.
Aspects: conceptual understanding
5.NBT.A.3a (Sub-standard, major cluster): Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.g., 347.392 = 3 × 100 + 4 × 10 + 7 × 1 + 3 × (1/10) + 9 × (1/100) + 2 × (1/1000).
Aspects: conceptual understanding
5.NBT.A.3b (Sub-standard, major cluster): Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.
Aspects: conceptual understanding
5.NBT.A.4 (Standard, major cluster): Use place value understanding to round decimals to any place.
Aspects: conceptual understanding
5.NBT.B (Cluster, major cluster): Perform operations with multi-digit whole numbers and with decimals to hundredths.
Aspects: conceptual understanding
5.NBT.B.5 (Standard, major cluster): Fluently multiply multi-digit whole numbers using the standard algorithm.
Aspects: conceptual understanding, Procedural Skill and Fluency
5.NBT.B.6 (Standard, major cluster): Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
Aspects: conceptual understanding
5.NBT.B.7 (Standard, major cluster): Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.
Aspects: conceptual understanding
5.NF.A (Cluster, major cluster): Use equivalent fractions as a strategy to add and subtract fractions.
5.NF.A.1 (Standard, major cluster): Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.)
5.NF.A.2 (Standard, major cluster): Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2.
5.NF.B (Cluster, major cluster): Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
Aspects: conceptual understanding
5.NF.B.3 (Standard, major cluster): Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?
Aspects: conceptual understanding
5.NF.B.4 (Standard, major cluster): Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
Aspects: conceptual understanding
5.NF.B.4a (Sub-standard, major cluster): Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)
Aspects: conceptual understanding
5.NF.B.4b (Sub-standard, major cluster): Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.
Aspects: conceptual understanding
5.NF.B.5 (Standard, major cluster): Interpret multiplication as scaling (resizing), by:
- Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n×a)/(n×b) to the effect of multiplying a/b by 1.
- Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.
Aspects: conceptual understanding
5.NF.B.5a (Sub-standard, major cluster): Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.
Aspects: conceptual understanding
5.NF.B.5b (Sub-standard, major cluster): Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n×a)/(n×b) to the effect of multiplying a/b by 1.
Aspects: conceptual understanding
5.NF.B.6 (Standard, major cluster): Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.
Aspects: conceptual understanding, Application
5.NF.B.7 (Standard, major cluster): Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. (Students able to multiply fractions in general can develop strategies to divide fractions in general, by reasoning about the relationship between multiplication and division. But division of a fraction by a fraction is not a requirement at this grade.)
Aspects: conceptual understanding
5.NF.B.7a (Sub-standard, major cluster): Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3.
Aspects: conceptual understanding
5.NF.B.7b (Sub-standard, major cluster): Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4.
Aspects: conceptual understanding
5.NF.B.7c (Sub-standard, major cluster): Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?
Aspects: conceptual understanding, Application
5.MD.A (Cluster, supporting cluster): Convert like measurement units within a given measurement system.
5.MD.A.1 (Standard, supporting cluster): Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems.
5.MD.B.2 (Standard, supporting cluster): Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots. For example, given different measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally.
5.MD.C (Cluster, major cluster): Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.
5.MD.C.3 (Standard, major cluster): Recognize volume as an attribute of solid figures and understand concepts of volume measurement.
5.MD.C.3a (Sub-standard, major cluster): A cube with side length 1 unit, called a “unit cube,” is said to have “one cubic unit” of volume, and can be used to measure volume.
5.MD.C.3b (Sub-standard, major cluster): A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units.
5.MD.C.4 (Standard, major cluster): Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.
5.MD.C.5 (Standard, major cluster): Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.
5.MD.C.5a (Sub-standard, major cluster): Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole-number products as volumes, e.g., to represent the associative property of multiplication.
5.MD.C.5b (Sub-standard, major cluster): Apply the formulas V = l × w × h and V = b × h for rectangular prisms to find volumes of right rectangular prisms with whole number edge lengths in the context of solving real world and mathematical problems.
5.MD.C.5c (Sub-standard, major cluster): Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems.
5.G.A (Cluster, additional cluster): Graph points on the coordinate plane to solve real-world and mathematical problems.
5.G.A.1 (Standard, additional cluster): Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., x-axis and x-coordinate, y-axis and y-coordinate).
5.G.A.2 (Standard, additional cluster): Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.
5.G.B (Cluster, additional cluster): Classify two-dimensional figures into categories based on their properties.
5.G.B.3 (Standard, additional cluster): Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles.
5.G.B.4 (Standard, additional cluster): Classify two-dimensional figures in a hierarchy based on properties.
6.RP.A (Cluster, major cluster): Understand ratio concepts and use ratio reasoning to solve problems.
Aspects: conceptual understanding
6.RP.A.1 (Standard, major cluster): Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.”
Aspects: conceptual understanding
6.RP.A.2 (Standard, major cluster): Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠0, and use rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.” (Expectations for unit rates in this grade are limited to non-complex fractions.)
Aspects: conceptual understanding
6.RP.A.3 (Standard, major cluster): Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
Aspects: conceptual understanding, Application
6.RP.A.3a (Sub-standard, major cluster): Make tables of equivalent ratios relating quantities with whole number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.
Aspects: conceptual understanding, Application
6.RP.A.3b (Sub-standard, major cluster): Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?
Aspects: conceptual understanding, Application
6.RP.A.3c (Sub-standard, major cluster): Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.
Aspects: conceptual understanding, Application
6.RP.A.3d (Sub-standard, major cluster): Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.
Aspects: conceptual understanding, Application
6.NS.A (Cluster, major cluster): Apply and extend previous understandings of multiplication and division to divide fractions by fractions.
6.NS.A.1 (Standard, major cluster): Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?
Aspects: Application
6.NS.B (Cluster, additional cluster): Compute fluently with multi-digit numbers and find common factors and multiples.
6.NS.B.2 (Standard, additional cluster): Fluently divide multi-digit numbers using the standard algorithm.
Aspects: Procedural Skill and Fluency
6.NS.B.3 (Standard, additional cluster): Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.
Aspects: Procedural Skill and Fluency
6.NS.B.4 (Standard, additional cluster): Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4 (9 + 2).
6.NS.C (Cluster, major cluster): Apply and extend previous understandings of numbers to the system of rational numbers.
6.NS.C.5 (Standard, major cluster): Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.
6.NS.C.6 (Standard, major cluster): Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.
6.NS.C.6a (Sub-standard, major cluster): Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., –(–3) = 3, and that 0 is its own opposite.
6.NS.C.6b (Sub-standard, major cluster): Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.
6.NS.C.6c (Sub-standard, major cluster): Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.
6.NS.C.7 (Standard, major cluster): Understand ordering and absolute value of rational numbers.
6.NS.C.7a (Sub-standard, major cluster): Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. For example, interpret –3 > –7 as a statement that –3 is located to the right of –7 on a number line oriented from left to right.
6.NS.C.7b (Sub-standard, major cluster): Write, interpret, and explain statements of order for rational numbers in real-world contexts. For example, write –3 ^(o)C > –7 ^(o)C to express the fact that –3 ^(o)C is warmer than –7 ^(o)C.
6.NS.C.7c (Sub-standard, major cluster): Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. For example, for an account balance of –30 dollars, write |–30| = 30 to describe the size of the debt in dollars
6.NS.C.7d (Sub-standard, major cluster): Distinguish comparisons of absolute value from statements about order. For example, recognize that an account balance less than –30 dollars represents a debt greater than 30 dollars.
6.NS.C.8 (Standard, major cluster): Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.
6.EE.A (Cluster, major cluster): Apply and extend previous understandings of arithmetic to algebraic expressions.
Aspects: Procedural Skill and Fluency
6.EE.A.1 (Standard, major cluster): Write and evaluate numerical expressions involving whole-number exponents.
Aspects: Procedural Skill and Fluency
6.EE.A.2 (Standard, major cluster): Write, read, and evaluate expressions in which letters stand for numbers.
Aspects: Procedural Skill and Fluency
6.EE.A.2a (Sub-standard, major cluster): Write expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation “Subtract y from 5” as 5 – y.
Aspects: Procedural Skill and Fluency
6.EE.A.2b (Sub-standard, major cluster): Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. For example, describe the expression 2 (8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms.
Aspects: Procedural Skill and Fluency
6.EE.A.2c (Sub-standard, major cluster): Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas V = s^(3) and A = 6 s^(2) to find the volume and surface area of a cube with sides of length s = 1/2.
Aspects: Procedural Skill and Fluency
6.EE.A.3 (Standard, major cluster): Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.
Aspects: conceptual understanding, Procedural Skill and Fluency
6.EE.A.4 (Standard, major cluster): Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for.
Aspects: Procedural Skill and Fluency
6.EE.B (Cluster, major cluster): Reason about and solve one-variable equations and inequalities.
6.EE.B.5 (Standard, major cluster): Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.
6.EE.B.6 (Standard, major cluster): Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.
6.EE.B.7 (Standard, major cluster): Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.
Aspects: Application