Arizona State Standards

Count to 100 by ones and by tens.

Count forward from a given number other than one, within the known sequence (e.g., 'Starting at the number 5, count up to 11.').

Write numbers from 0 to 20. Represent a number of objects with a written numeral 0 to 20 (with 0 representing a count of no objects).

Understand the relationship between numbers and quantities; connect counting to cardinality.

Count to answer 'How many?' questions when 20 or fewer objects are arranged in a line, rectangular array, circle, or scattered configuration; given a number 1–20, count out that many objects.

Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group (up to 10 objects).

Compare two numbers between 0 and 10 presented as written numerals.

Represent addition and subtraction concretely.

Solve addition and subtraction word problems and add and subtract within 10.

Decompose numbers less than or equal to 10 into pairs in more than one way (e.g., using fingers, objects, symbols, tally marks, drawings, expressions).

For any number from 1 to 9, find the number that makes 10 when added to the given number.

Fluently add and subtract within 5.

Compose and decompose numbers from 11 to 19 into ten ones and additional ones using objects, drawings and/or equations. Understand these numbers are composed of ten ones and one, two, ..., nine ones (e.g., 18 = 10 + 8).

Demonstrate understanding of addition and subtraction within 10 using place value.

Describe measurable attributes of a single object (e.g., length and weight).

Directly compare two objects with a measurable attribute in common to see which has 'more of' or 'less of' the attribute, and describe the difference.

Classify objects into given categories; count the number in each category and sort the categories by count (limit category counts ≤ 10).

Describe objects in the environment using names of shapes, and describe their relative positions using terms such as above, below, beside, in front of, behind, and next to.

Correctly name shapes regardless of their orientation or overall size (e.g., circle, triangle, square, rectangle, rhombus, trapezoid, hexagon, cube, cone, cylinder, sphere).

Identify shapes as two-dimensional (lying in a plane, flat) or three-dimensional (solid).

Analyze and compare two-dimensional and three-dimensional shapes in different sizes and orientations, using informal language to describe similarities, differences, parts (e.g., number of sides and vertices/corners), and other attributes.

Model shapes in the world by building shapes from components (e.g., use sticks and clay balls) and drawing shapes.

Use simple shapes to form composite shapes. For example, 'Can you join these two triangles with full sides touching to make a rectangle?'

Partition circles and rectangles into two and four equal shares; describe the shares using the words halves, fourths, and quarters. Describe the whole as two of, or four of the shares.

Use addition and subtraction within 20 to solve word problems with unknowns in all positions, using objects, drawings, and/or equations with a symbol for the unknown.

Solve word problems that call for addition of three whole numbers whose sum is ≤ 20.

Apply properties of operations (commutative and associative properties of addition) as strategies to add and subtract through 20.

Understand subtraction as an unknown-addend problem within 20 (e.g., subtract 10 - 8 by finding the number that makes 10 when added to 8).

Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).

Fluently add and subtract within 10.

Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false.

Determine the unknown whole number in an addition or subtraction equation relating three whole numbers.

Count to 120 by 1's, 2's, and 10's starting at any number less than 100. Read and write numerals and represent a number of objects with a written numeral.

Understand that the two digits of a two-digit number represent amounts of tens and ones.

Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results with >, =, and <.

Demonstrate understanding of addition within 100, connecting objects or drawings to strategies based on place value (including multiples of 10), properties of operations, and/or the relationship between addition and subtraction.

Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count.

Subtract multiples of 10 in the range 10 to 90 (positive or zero differences), using objects or drawings and strategies based on place value.

Order three objects by length; compare the lengths of two objects indirectly by using a third object.

Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object end to end.

Tell and write time in hours and half-hours using analog and digital clocks.

Identify coins by name and value (pennies, nickels, dimes and quarters).

Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than another.

Distinguish between defining attributes versus non-defining attributes for two-dimensional shapes; build and draw shapes that possess defining attributes.

Compose two-dimensional shapes or three-dimensional shapes to create a composite shape.

Read and write numbers up to 1000 using base-ten numerals, number names, and expanded form.

Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols.

Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

Add up to three two-digit numbers using strategies based on place value and properties of operations.

Demonstrate understanding of addition and subtraction within 1000, connecting objects or drawings to strategies based on place value.

Mentally add 10 or 100 to a given number 100–900, and mentally subtract 10 or 100 from a given number 100–900.

Explain why addition and subtraction strategies work, using place value and the properties of operations.

Measure the length of an object by selecting and using appropriate tools (e.g., ruler, meter stick, yardstick, measuring tape).

Measure the length of an object twice, using different standard length units; describe how the two measurements relate to the size of the unit chosen.

Estimate lengths using units of inches, feet, centimeters, and meters.

Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit.

Use addition and subtraction within 100 to solve word problems involving lengths given in the same unit.

Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points; represent whole-number sums and differences within 100 on a number line diagram.

Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.

Solve word problems involving collections of money, including dollar bills, quarters, dimes, nickels, and pennies. Record the total using $ and ¢ appropriately.

Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot.

Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and compare problems using information presented in the graph.

Identify and describe specified attributes of two-dimensional and three-dimensional shapes according to number and shape of faces, number of angles, and number of sides and/or vertices. Draw two-dimensional shapes based on specified attributes.

Partition a rectangle into rows and columns of same-size squares and count to find the total number of squares.

Partition circles and rectangles into two, three, or four equal shares; describe the shares using words halves, thirds, fourths.

Count within 1000; skip count by 5's, 10's and 100's.

Use addition and subtraction within 100 to solve one- and two-step word problems. Represent a word problem as an equation with a symbol for the unknown.

Fluently add and subtract within 20. By the end of Grade 2, know from memory all sums of two one-digit numbers.

Determine whether a group of objects (up to 20) has an odd or even number of members (e.g., by pairing objects or counting them by 2's).

Use addition to find the total number of objects arranged in rectangular arrays (with up to 5 rows and 5 columns). Write an equation to express the total as a sum of equal addends.

Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones.

Interpret products of whole numbers as the total number of objects in equal groups (e.g., 5 × 7 as 5 groups of 7 objects each).

Interpret whole-number quotients of whole numbers (e.g., 56 ÷ 8 as the number of objects in each group when 56 objects are partitioned into 8 equal groups).

Use multiplication and division within 100 to solve word problems involving equal groups, arrays, and measurement quantities.

Determine the unknown whole number in a multiplication or division equation relating three whole numbers.

Apply properties of operations as strategies to multiply and divide. Properties include commutative, associative, and distributive properties.

Understand division as an unknown-factor problem (e.g., find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8).

Fluently multiply and divide within 100. By the end of Grade 3, know from memory all multiplication products through 10 × 10.

Solve two-step word problems using the four operations. Represent these problems using equations with a letter for the unknown. Apply Order of Operations when there are no parentheses.

Identify patterns in the addition table and the multiplication table and explain them using properties of operations.

When solving problems, assess the reasonableness of answers using mental computation and estimation strategies including rounding.

Use place value understanding to round whole numbers to the nearest 10 or 100.

Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.

Multiply one-digit whole numbers by multiples of 10 in the range 10 to 90 using strategies based on place value and properties of operations.

Understand a fraction 1/b as the quantity formed by one part when a whole is partitioned into b equal parts; understand a/b as the quantity formed by a parts of size 1/b.

Understand a fraction as a number on the number line; represent fractions on a number line diagram.

Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.

Tell and write time to the nearest minute and measure time intervals in minutes. Solve word problems involving addition and subtraction of time intervals in minutes.

Solve word problems involving money through $20.00, using symbols $, '.', ¢.

Measure and estimate liquid volumes and masses of objects using metric units. Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes given in the same units.

Create a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step problems using information presented in scaled bar graphs.

Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch to the nearest quarter-inch. Show the data by making a line plot.

Understand area as an attribute of plane figures and understand concepts of area measurement.

Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units).

Relate area to the operations of multiplication and addition.

Solve real-world and mathematical problems involving perimeters of plane figures and areas of rectangles, including finding an unknown side length.

Understand that shapes in different categories may share attributes, and that shared attributes can define a larger category. Recognize rhombuses, rectangles, and squares as examples of quadrilaterals.

Partition shapes into b parts with equal areas. Express the area of each part as a unit fraction 1/b of the whole.

Solve multistep word problems using the four operations, including problems in which remainders must be interpreted. Understand how the remainder is a fraction of the divisor.

Find all factor pairs for a whole number in the range 1 to 100 and understand that a whole number is a multiple of each of its factors.

Generate a number pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself.

When solving problems, assess the reasonableness of answers using mental computation and estimation strategies including rounding.

Apply concepts of place value, multiplication, and division to understand that a digit in one place represents ten times what it represents in the place to its right.

Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers.

Use place value understanding to round multi-digit whole numbers to any place.

Fluently add and subtract multi-digit whole numbers using a standard algorithm.

Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and properties of operations.

Demonstrate understanding of division by finding whole-number quotients and remainders with up to four-digit dividends and one-digit divisors.

Explain why a fraction a/b is equivalent to (n×a)/(n×b) using visual fraction models. Use this principle to understand and generate equivalent fractions.

Compare two fractions with different numerators and different denominators (e.g., by creating common denominators or comparing to a benchmark fraction).

Understand a fraction a/b with a > 1 as a sum of unit fractions. Add and subtract mixed numbers with like denominators.

Build fractions from unit fractions. Understand a/b as a multiple of 1/b. Multiply a whole number by a fraction.

Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100.

Use decimal notation for fractions with denominators 10 or 100, and locate these decimals on a number line.

Compare two decimals to hundredths by reasoning about their size. Record results with >, =, or <.

Know relative sizes of measurement units within one system (km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec). Express measurements in a larger unit in terms of a smaller unit and vice versa.

Use the four operations to solve word problems involving distances, time intervals, liquid volumes, masses of objects, and money, including fractions and decimals.

Apply area and perimeter formulas for rectangles in mathematical and real-world problems including problems with unknown side lengths.

Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Solve problems involving addition and subtraction of fractions using information presented in line plots.

Recognize angles as geometric shapes formed wherever two rays share a common endpoint, and understand concepts of angle measurement.

Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.

Understand angle measures as additive. Solve addition and subtraction problems to find unknown angles on a diagram.

Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures.

Classify two-dimensional figures based on presence or absence of parallel or perpendicular lines, or presence or absence of angles of a specified size.

Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry.

Represent verbal statements of multiplicative comparisons as multiplication equations. Interpret a multiplication equation as a comparison.

Multiply or divide within 1000 to solve word problems involving multiplicative comparison.

Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area.

Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for variability in the answers.

Understand that a set of data collected to answer a statistical question has a distribution described by its center, spread, and overall shape.

Recognize that a measure of center summarizes all values with a single number, while a measure of variation describes the spread of the data set.

Display and interpret numerical data by creating plots on a number line including histograms, dot plots, and box plots.

Summarize numerical data sets in relation to their context by reporting number of observations, nature of the attribute, quantitative measures of center and variability, and overall patterns.

Understand solving an equation or inequality as a process of reasoning to find the value(s) of the variables that make the equation or inequality true.

Use variables to represent numbers and write expressions when solving problems; understand that a variable can represent an unknown number or any number in a specified set.

Solve mathematical and real-world problems by writing and solving equations of the form x+p=q, x-p=q, px=q, and x/p=q.

Write an inequality of the form x>c, x<c, x≥c, or x≤c to represent a constraint or condition. Recognize that inequalities have infinitely many solutions; represent solutions on number lines.

Use variables to represent two quantities that change in relationship to each other. Write an equation to express one quantity in terms of the other. Analyze the relationship using graphs and tables.

Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing or decomposing into rectangles or triangles.

Find the volume of a right rectangular prism with fractional edge lengths. Understand and use the formula V = B·h.

Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find side lengths.

Understand the concept of a ratio as comparing two quantities multiplicatively or composing them in a way that preserves a multiplicative relationship. Use ratio language to describe a ratio relationship.

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.

Use ratio and rate reasoning to solve mathematical and real-world problems using tables, tape diagrams, double number lines, or equations.

Interpret and compute quotients of fractions to solve problems involving division of fractions by fractions using visual fraction models and equations.

Fluently divide multi-digit numbers using a standard algorithm.

Fluently add, subtract, multiply, and divide multi-digit decimals using a standard algorithm for each operation.

Find the greatest common factor (GCF) and least common multiple (LCM). Use the distributive property to express a sum of two whole numbers with a common factor.

Understand that positive and negative numbers describe quantities having opposite directions or values. Use positive and negative numbers to represent quantities in real-world context, explaining the meaning of 0.

Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes to represent points with negative number coordinates.

Understand ordering and absolute value of rational numbers.

Solve problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points.

Write and evaluate numerical expressions involving whole-number exponents.

Write, read, and evaluate algebraic expressions.

Apply the properties of operations to generate equivalent expressions.

Identify when two expressions are equivalent.

Compute unit rates associated with ratios involving both simple and complex fractions, including ratios of quantities measured in like or different units.

Recognize and represent proportional relationships between quantities in tables, graphs, equations, diagrams, and verbal descriptions.

Use proportional relationships to solve multi-step ratio and percent problems (e.g., simple interest, tax, markups and markdowns, gratuities, percent increase and decrease, percent error).

Add and subtract integers and other rational numbers; represent addition and subtraction on a number line diagram.

Multiply and divide integers and other rational numbers.

Solve real-world and mathematical problems involving the four operations with rational numbers.

Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.

Rewrite an expression in different forms and understand the relationship between the different forms and their meanings in a problem context.

Solve multi-step problems involving positive and negative rational numbers in any form. Convert between forms and assess reasonableness.

Use variables to represent quantities and construct simple equations and inequalities to solve problems. Solve equations of the form px+q=r and p(x+q)=r; solve inequalities of the form px+q>r or px+q<r.

Solve problems involving scale drawings of geometric figures, such as computing actual lengths and areas from a scale drawing.

Draw geometric shapes with given conditions, focusing on constructing triangles from three measures of angles or sides.

Describe the two-dimensional figures that result from slicing three-dimensional figures.

Understand and use the formulas for the area and circumference of a circle to solve problems; give an informal derivation of the relationship between circumference and area.

Use facts about supplementary, complementary, vertical, and adjacent angles in multi-step problems to write and solve simple equations for an unknown angle.

Solve mathematical and real-world problems involving area of two-dimensional objects composed of triangles and quadrilaterals, and volume and surface area of three-dimensional objects composed of cubes and right prisms.

Understand that statistics can be used to gain information about a population by examining a sample. Understand that random sampling tends to produce representative samples.

Use data from a random sample to draw inferences about a population with an unknown characteristic. Generate multiple samples to gauge variation in estimates.

Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between centers by expressing it as a multiple of a measure of variability.

Use measures of center and variability for numerical data from random samples to draw informal comparative inferences about two populations.

Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring.

Approximate the probability of a chance event by collecting data on the chance process and observing its long-run relative frequency.

Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies.

Understand the Pythagorean Theorem and its converse.

Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

Understand and use formulas for volumes of cones, cylinders and spheres and use them to solve real-world and mathematical problems.

Construct and interpret scatter plots for bivariate measurement data to investigate patterns such as clustering, outliers, positive/negative association, linear association, and nonlinear association.

Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots suggesting linear association, informally fit a straight line and assess the model fit.

Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.

Understand that patterns of association can be seen in bivariate categorical data by displaying frequencies in a two-way table. Construct and interpret a two-way table.

Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.

Understand and apply the properties of integer exponents to generate equivalent numerical expressions.

Use square root and cube root symbols to represent solutions to equations x²=p and x³=p. Know that √2 is irrational. Evaluate square roots of perfect squares ≤ 225 and cube roots of perfect cubes ≤ 1000.

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 express how many times larger or smaller one is than the other.

Perform operations with numbers expressed in scientific notation including problems where both decimal and scientific notation are used.

Graph proportional relationships, interpreting the unit rate as the slope. Compare two proportional relationships represented in different ways.

Use similar triangles to explain why the slope m is the same between any two points on a non-vertical line. Derive the equations y=mx and y=mx+b.

Fluently solve linear equations and inequalities in one variable, including cases with one solution, infinitely many solutions, or no solution.

Analyze and solve pairs of simultaneous linear equations algebraically and graphically, including cases of no solution and infinite solutions.

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 of an input and the corresponding output.

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or verbally).

Interpret the equation y = mx + b as defining a linear function; give examples of functions that are not linear.

Given a description of a situation, generate a function to model a linear relationship. Determine rate of change and initial value from a table, graph, or description.

Describe qualitatively the functional relationship between two quantities by analyzing a graph (where increasing, decreasing, linear, nonlinear). Sketch a graph that exhibits described qualitative features.

Verify experimentally the properties of rotations, reflections, and translations: lines taken to lines, segments to segments of same length, angles to angles of same measure, parallel lines to parallel lines.

Understand that a two-dimensional figure is congruent to another if one can be obtained from the other by a sequence of rotations, reflections, and translations.

Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

Understand that a two-dimensional figure is similar to another if one can be obtained from the other by a sequence of rotations, reflections, translations, and dilations.

Use informal arguments to establish facts about angle sum and exterior angle of triangles, angles when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.

Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion. Numbers whose decimal expansions do not terminate or repeat are called irrational.

Use rational approximations of irrational numbers to compare the size of irrational numbers. Locate them approximately on a number line and estimate their values.

Understand that given any two distinct rational numbers, a < b, there exist a rational number c and an irrational number d such that a < c < b and a < d < b.

Create equations and inequalities in one variable and use them to solve problems.

Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

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.

Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents.

Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data.

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.

For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities.

Prove theorems about lines and angles.

Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

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.

Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. Fit a function to the data; use functions fitted to data to solve problems in the context of the data.

Use coordinates to prove simple geometric theorems algebraically.

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.

Represent data with plots on the real number line (dot plots, histograms, and box plots).

Write a function that describes a relationship between two quantities.

Use coordinates to prove simple geometric theorems algebraically.

Create equations and inequalities in one variable and use them to solve problems.

Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents.