New York State Next Generation Mathematics Learning Standards

Count to 100 by ones and by tens.

Count to 100 by ones beginning from any given number (instead of beginning at 1).

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

Understand the relationship between numbers and quantities up to 20; connect counting to cardinality.

When counting objects, say the number names in the standard order, pairing each object with one and only one number name...

Understand that the last number name said tells the number of objects counted, (cardinality). The number of objects is t...

Understand the concept that each successive number name refers to a quantity that is one larger.

Understand the concept of ordinal numbers (first through tenth) to describe the relative position and magnitude of whole...

Answer counting questions using as many as 20 objects arranged in a line, a rectangular array, and a circle. Answer coun...

Given a number from 1–20, count out that many objects.

Identify whether the number of objects in one group is greater than (more than), less than (fewer than), or equal to (th...

Compare two numbers between 1 and 10 presented as written numerals.

Describe objects in the environment using names of shapes, and describe the relative positions of these objects using te...

Name shapes regardless of their orientation or overall size.

Understand the difference between two-dimensional (lying in a plane, "flat") and three-dimensional ("solid") shapes.

Analyze, compare, and sort two-and three-dimensional shapes, in different sizes and orientations, using informal languag...

Model objects in their environment by building and/or drawing shapes.

Compose larger shapes from simple shapes.

Describe measurable attributes of an object(s), such as length or weight, using appropriate vocabulary.

Directly compare two objects with a common measurable attribute and describe the difference.

Classify objects into given categories; count the objects in each category and sort the categories by count.

Explore coins (pennies, nickels, dimes, and quarters) and begin identifying pennies and dimes.

Compose and decompose the numbers from 11 to 19 into ten ones and one, two, three, four, five, six, seven, eight, or nin...

Represent addition and subtraction using objects, fingers, pennies, drawings, sounds, acting out situations, verbal expl...

Add and subtract within 10.

Solve addition and subtraction word problems within 10.

Decompose numbers less than or equal to 10 into pairs in more than one way.

Find the number that makes 10 when given a number from 1 to 9.

Fluently add and subtract within 5.

Duplicate, extend, and create simple patterns using concrete objects.

Distinguish between defining attributes versus non-defining attributes for a wide variety of shapes. Build and/or draw s...

Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-...

Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, an...

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

Measure the length of an object using same-size "length units" placed end to end with no gaps or overlaps. Express the l...

Tell and write time in hours and half-hours using analog and digital clocks. Develop an understanding of common terms, s...

Recognize and identify coins (penny, nickel, dime, and quarter) and their value and use the cent symbol (¢) appropriatel...

Count a mixed collection of dimes and pennies and determine the cent value (total not to exceed 100 cents).

Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of ...

Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of obj...

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

Understand 10 can be thought of as a bundle of ten ones, called a "ten".

Understand the numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine o...

Understand the numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or ni...

Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with t...

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

Subtract multiples of 10 from multiples of 10 in the range 10-90 using concrete models or drawings, and strategies based...

Use addition and subtraction within 20 to solve one step word problems involving situations of adding to, taking from, p...

Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20.

Apply properties of operations as strategies to add and subtract.

Understand subtraction as an unknown-addend problem within 20.

Relate counting to addition and subtraction.

Add and subtract within 20. Use strategies such as: counting on, making ten, decomposing a number leading to a ten, usin...

Fluently add and subtract within 10.

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

Determine the unknown whole number in an addition or subtraction equation with the unknown in all positions.

Classify two-dimensional figures as polygons or non-polygons.

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

Partition circles and rectangles into two, three, or four equal shares. Describe the shares using the words halves, thir...

Measure the length of an object to the nearest whole by selecting and using appropriate tools such as rulers, yardsticks...

Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve ...

Measure the length of an object twice, using different "length units" for the two measurements; describe how the two mea...

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...

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

Represent whole numbers as lengths from 0 on a number line with equally spaced points corresponding to the numbers 0, 1,...

Tell and write time from analog and digital clocks in five minute increments, using a.m. and p.m. Develop an understandi...

Count a mixed collection of coins whose sum is less than or equal to one dollar.

Solve real world and mathematical problems within one dollar involving quarters, dimes, nickels, and pennies, using the ...

Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measu...

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

Understand 100 can be thought of as a bundle of ten tens, called a "hundred."

Understand the numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eig...

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

Read and write numbers 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 to r...

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

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

Add and subtract within 1000, using concrete models or drawings, and strategies based on place value, properties of oper...

Understand that in adding or subtracting up to three-digit numbers, one adds or subtracts hundreds and hundreds, tens an...

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.

Use addition and subtraction within 100 to solve one-step word problems involving situations of adding to, taking from, ...

Use addition and subtraction within 100 to develop an understanding of solving two-step word problems involving situatio...

Fluently add and subtract within 20 using mental strategies. Strategies could include: counting on, making ten, decompos...

Know from memory all sums within 20 of two one-digit numbers.

Determine whether a group of objects (up to 20) has an odd or even number of members.

Write an equation to express an even number as a sum of two equal addends.

Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns. W...

Recognize and classify polygons based on the number of sides and vertices (triangles, quadrilaterals, pentagons, and hex...

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

Tell and write time to the nearest minute and measure time intervals in minutes. Solve one-step word problems involving ...

Measure and estimate liquid volumes and masses of objects using grams (g), kilograms (kg), and liters (l).

Add, subtract, multiply, or divide to solve one-step word problems involving masses or liquid volumes that are given in ...

Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one-and two-st...

Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by ...

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

Recognize a square with side length 1 unit, called "a unit square," is said to have "one square unit" of area, and can b...

Recognize a plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n sq...

Measure areas by counting unit squares.

Relate area to the operations of multiplication and addition.

Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be...

Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real world an...

Use tiling to show in a concrete case that the area of a rectangle with whole-number side length a and side length b + c...

Recognize area as additive. Find areas of figures composed of non-overlapping rectangles, and apply this technique to so...

Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the s...

Identify rectangles with the same perimeter and different areas or with the same area and different perimeters.

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

Fluently add and subtract within 1,000 using strategies and algorithms based on place value, properties of operations, a...

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

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

Read and write four digit numbers using base-ten numerals, number names, and expanded form.

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

Explain equivalence of fractions and compare fractions by reasoning about their size.

Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.

Recognize and generate equivalent fractions. Explain why the fractions are equivalent.

Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.

Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comp...

Interpret products of whole numbers.

Interpret whole-number quotients of whole numbers.

Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and meas...

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.

Understand division as an unknown-factor problem.

Fluently solve single-digit multiplication and related divisions, using strategies such as the relationship between mult...

Know from memory all products of two one-digit numbers.

Solve two-step word problems posed with whole numbers and having whole-number answers using the four operations.

Represent these problems using equations or expressions with a letter standing for the unknown quantity.

Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

Identify and extend arithmetic patterns (including patterns in the addition table or multiplication table).

Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify t...

Identify and name triangles based on angle size (right, obtuse, acute).

Identify and name all quadrilaterals with 2 pairs of parallel sides as parallelograms.

Identify and name all quadrilaterals with four right angles as rectangles.

Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded...

Know relative sizes of measurement units: ft., in.; km, m, cm • Know the conversion factor and use it to convert measure...

Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects...

Solve problems involving fractions or decimals, and problems that require expressing measurements given in a larger unit...

Represent measurement quantities using diagrams that feature a measurement scale, such as number lines.

Apply the area and perimeter formulas for rectangles in real world and mathematical problems.

Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts ...

Recognize an angle that turns through n one-degree angles is said to have an angle measure of n degrees.

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

Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the wh...

Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place ...

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

Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the r...

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 stra...

Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies base...

Compare two fractions with different numerators and different denominators. Recognize that comparisons are valid only wh...

Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.

Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decompositio...

Add and subtract mixed numbers with like denominators.

Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denomina...

Apply and extend previous understandings of multiplication to multiply a whole number by a fraction.

Solve word problems involving multiplication of 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...

Use decimal notation for fractions with denominators 10 or 100.

Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when two dec...

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

Multiply or divide to solve word problems involving multiplicative comparison, distinguishing multiplicative comparison ...

Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, includ...

Represent these problems using equations or expressions with a letter standing for the unknown quantity.

Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

Find all factor pairs for a whole number in the range 1–100. Recognize that a whole number is a multiple of each of its ...

Generate a number or shape pattern that follows a given rule. Identify and informally explain apparent features of the p...

Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines...

Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and int...

Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that c...

Classify two-dimensional figures in a hierarchy based on properties.

Convert among different-sized standard measurement units within a given measurement system when the conversion factor is...

Recognize volume as an attribute of solid figures and understand concepts of volume measurement.

Recognize that a cube with side length 1 unit, called a "unit cube," is said to have "one cubic unit" of volume, and can...

Recognize that a solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume o...

Measure volumes by counting unit cubes, using cubic cm, cubic in., cubic ft., and improvised units.

Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving ...

Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that...

Apply the formulas V = l × w × h and V = B × h for rectangular prisms to find volumes of right rectangular prisms with w...

Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by ...

Use whole-number exponents to denote powers of 10. Explain patterns in the number of zeros of the product when multiplyi...

Read, write, and compare decimals to thousandths.

Read and write decimals to thousandths using base-ten numerals, number names, and expanded form.

Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record t...

Use place value understanding to round decimals to any place.

Fluently multiply multi-digit whole numbers using a standard algorithm.

Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies ba...

Using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship...

Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivale...

Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unli...

Find the area of a rectangle with fractional side lengths by tiling it with rectangles of the appropriate unit fraction ...

Interpret multiplication as scaling (resizing).

Compare the size of a product to the size of one factor on the basis of the size of the other factor, without performing...

Solve real world problems involving multiplication of fractions and mixed numbers.

Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit...

Interpret division of a unit fraction by a non-zero whole number, and compute such quotients.

Interpret division of a whole number by a unit fraction, and compute such quotients.

Solve real-world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers b...

Apply the order of operations to evaluate numerical expressions.

Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating t...

Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form...

Write and evaluate numerical expressions involving whole-number exponents.

Write, read, and evaluate expressions in which letters stand for numbers.

Write expressions that record operations with numbers and with letters standing for numbers.

Identify parts of an expression using mathematical terms (term, coefficient, sum, difference, product, factor, and quoti...

Evaluate expressions given specific values for their variables. Include expressions that arise from formulas in real-wor...

Apply the properties of operations to generate equivalent expressions.

Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if...

Use variables to represent numbers and write expressions when solving a real-world or mathematical problem. Understand t...

Write an inequality of the form x > c, x ≥ c, x ≤ c, or x < c to represent a constraint or condition in a real-world or ...

Use variables to represent two quantities in a real-world problem that change in relationship to one another. Given a ve...

Find area of triangles, trapezoids, and other polygons by composing into rectangles or decomposing into triangles and qu...

Find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathemati...

Draw polygons in the coordinate plane given coordinates for the vertices. Use coordinates to find the length of a side j...

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

Use area and volume models to explain perfect squares and perfect cubes.

Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions.

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 of two whole numbers less than or equal to 100. Use the distributive property to express...

Understand that positive and negative numbers are used together to describe quantities having opposite directions or val...

Understand a rational number as a point on the number line. Use number lines and coordinate axes to represent points on ...

Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line. Recognize that th...

Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane. Recognize tha...

Find and position integers and other rational numbers on a horizontal or vertical number line. Find and position pairs o...

Understand ordering and absolute value of rational numbers.

Interpret statements of inequality as statements about the relative position of two numbers on a number line.

Write, interpret, and explain statements of order for rational numbers in real-world contexts.

Understand the absolute value of a rational number as its distance from 0 on the number line. Interpret absolute value a...

Distinguish comparisons of absolute value from statements about order.

Solve real-world and mathematical problems by graphing points on a coordinate plane. Include use of coordinates and abso...

Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.

Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0 (b not equal to zero), and use rate lan...

Use ratio and rate reasoning to solve real-world and mathematical problems.

Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, ...

Solve unit rate problems.

Find a percent of a quantity as a rate per 100. Solve problems that involve finding the whole given a part and the perce...

Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or divid...

Recognize that a statistical question is one that anticipates variability in the data related to the question and accoun...

Understand that statistics can be used to gain information about a population by examining a sample of the population; g...

Understand that the method and sample size used to collect data for a particular question is intended to reduce the diff...

Understand that a set of quantitative data collected to answer a statistical question has a distribution which can be de...

Recognize that a measure of center for a quantitative data set summarizes all of its values with a single number while a...

Display quantitative data in plots on a number line, including dot plots, and histograms.

Summarize quantitative data sets in relation to their context.

Report the number of observations.

Describe the nature of the attribute under investigation, including how it was measured and its units of measurement.

Calculate range and measures of center, as well as describe any overall pattern and any striking deviations from the ove...

Relate the range and the choice of measures of center to the shape of the data distribution and the context in which the...

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

Approximate the probability of a simple event by collecting data on the chance process that produces it and observing it...

Develop a probability model and use it to find probabilities of simple events. Compare probabilities from a model to obs...

Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine proba...

Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process

Add, subtract, factor, and expand linear expressions with rational coefficients by applying the properties of operations...

Understand that rewriting an expression in different forms in real-world and mathematical problems can reveal and explai...

Solve multi-step real-world and mathematical problems posed with positive and negative rational numbers in any form (who...

Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequa...

Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are rational numbers...

Solve word problems leading to inequalities of the form px + q > r, px + q ≥ r, px + q ≤ r, or px + q < r, where p, q, a...

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

Draw triangles when given measures of angles and/or sides, noticing when the conditions determine a unique triangle, mor...

Describe the two-dimensional shapes that result from slicing three-dimensional solids parallel or perpendicular to the b...

Apply the formulas for the area and circumference of a circle to solve problems.

Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve s...

Solve real-world and mathematical problems involving area of two-dimensional objects composed of triangles and trapezoid...

Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers. Represent add...

Describe situations in which opposite quantities combine to make 0.

Understand addition of rational numbers; p + q is the number located a distance |q| from p, in the positive or negative ...

Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that the distance betw...

Apply properties of operations as strategies to add and subtract rational numbers.

Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational...

Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to s...

Apply properties of operations as strategies to multiply and divide rational numbers.

Convert a fraction to a decimal using long division; know that the decimal form of a rational number terminates in 0s or...

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

Compute unit rates associated with ratios of fractions.

Recognize and represent proportional relationships between quantities.

Decide whether two quantities are in a proportional relationship.

Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of ...

Represent a proportional relationship using an equation.

Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special at...

Use proportional relationships to solve multistep ratio and percent problems.

Construct and interpret box-plots, find the interquartile range, and determine if a data point is an outlier.

Informally assess the degree of visual overlap of two quantitative data distributions.

Use measures of center and measures of variability for quantitative data from random samples or populations to draw info...

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

Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the samp...

Represent sample spaces for compound events using methods such as organized lists, sample space tables, and tree diagram...

Design and use a simulation to generate frequencies for compound events.

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

Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a po...

Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small qu...

Perform multiplication and division with numbers expressed in scientific notation, including problems where both standar...

Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportion...

Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in t...

Solve linear equations in one variable.

Recognize when linear equations in one variable have one solution, infinitely many solutions, or no solutions. Give exam...

Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressi...

Analyze and solve pairs of simultaneous linear equations.

Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of t...

Solve systems of two linear equations in two variables with integer coefficients: graphically, numerically using a table...

Solve real-world and mathematical problems involving systems of two linear equations in two variables with integer coeff...

Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set o...

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tabl...

Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line. Recognize examples of f...

Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial val...

Describe qualitatively the functional relationship between two quantities by analyzing a graph. Sketch a graph that exhi...

Verify experimentally the properties of rotations, reflections, and translations.

Verify experimentally lines are mapped to lines, and line segments to line segments of the same length.

Verify experimentally angles are mapped to angles of the same measure.

Verify experimentally parallel lines are mapped to parallel lines.

Know that a two-dimensional figure is congruent to another if the corresponding angles are congruent and the correspondi...

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

Know that a two-dimensional figure is similar to another if the corresponding angles are congruent and the corresponding...

Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created ...

Understand a proof of the Pythagorean Theorem and its converse.

Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical proble...

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

Given the formulas for the volume of cones, cylinders, and spheres, solve mathematical and real-world problems.

Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion ev...

Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately o...

Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two ...

Understand that straight lines are widely used to model relationships between two quantitative variables. For scatter pl...

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

Add, subtract, and multiply polynomials and recognize that the result of the operation is also a polynomial. This forms ...

Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solut...

Rewrite formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plan...

Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a str...

Explain each step when solving a linear or quadratic equation as following from the equality of numbers asserted at the ...

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

Solve quadratic equations in one variable.

Solve systems of linear equations in two variables both algebraically and graphically.

Solve a system, with rational solutions, consisting of a linear equation and a quadratic equation (parabolas only) in tw...

Write the standard form of a given polynomial and identify the terms, coefficients, degree, leading coefficient, and con...

Interpret expressions by viewing one or more of their parts as a single entity.

Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by t...

Use the properties of exponents to rewrite exponential expressions.

Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notati...

Determine the domain of a function from its graph and, where applicable, identify the appropriate domain for a function ...

Graph functions and show key features of the graph by hand and by using technology where appropriate.

Graph square root, and piecewise-defined functions, including step functions and absolute value functions and show key f...

Write a function in different but equivalent forms to reveal and explain different properties of the function.

For a quadratic function, use an algebraic process to find zeros, maxima, minima, and symmetry of the graph, and interpr...

Distinguish between situations that can be modeled with linear functions and with exponential functions.

Justify that a function is linear because it grows by equal differences over equal intervals, and that a function is exp...

Recognize situations in which one quantity changes at a constant rate per unit interval relative to another, and therefo...

Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linear...

Identify zeros of polynomial functions when suitable factorizations are available.

Create equations and inequalities in one variable to represent a real-world context.

Explain each step when solving rational or radical equations as following from the equality of numbers asserted at the p...

Solve a system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.

Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by t...

Use the properties of exponents to rewrite exponential expressions.

Write a function that describes a relationship between two quantities.

Determine a function from context. Determine an explicit expression, a recursive process, or steps for calculation from ...

Combine standard function types using arithmetic operations.

Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, an...

Recognize that a sequence is a function whose domain is a subset of the integers.

Graph functions and show key features of the graph by hand and using technology when appropriate.

Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

Graph cube root, exponential and logarithmic functions, showing intercepts and end behavior; and trigonometric functions...

Write a function in different but equivalent forms to reveal and explain different properties of the function.

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tabl...

Interpret the parameters in a linear or exponential function in terms of a context.

Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

Apply concepts of the unit circle in the coordinate plane to calculate the values of the six trigonometric functions giv...

Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, horizontal shift, and mi...

Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes...

Interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the...

Determine if a value for a sample proportion or sample mean is likely to occur based on a given simulation.

Recognize the purposes of and differences among surveys, experiments, and observational studies. Explain how randomizati...

Given a simulation model based on a sample proportion or mean, construct the 95% interval centered on the statistic (+/-...

Use the tools of statistics to draw conclusions from numerical summaries.

Recognize whether or not a normal curve is appropriate for a given data set.

If appropriate, determine population percentages using a graphing calculator for an appropriate normal curve.

Represent bivariate data on a scatter plot, and describe how the variables' values are related.

Fit a function to real-world data; use functions fitted to data to solve problems in the context of the data.

Use properties and operations to understand the different forms of rational and irrational numbers.

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

Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of e...

Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

Calculate (using technology) and interpret the correlation coefficient of a linear fit.

Prove that all circles are similar.

Identify, describe and apply relationships between the angles and their intercepted arcs of a circle.

Identify, describe and apply relationships among radii, chords, tangents, and secants of a circle.

Know precise definitions of angle, circle, perpendicular lines, parallel lines, and line segment, based on the undefined...

Make and justify the constructions for inscribing an equilateral triangle, a square and a regular hexagon in a circle.

Represent transformations as geometric functions that take points in the plane as inputs and give points as outputs. Com...

Develop definitions of rotations, reflections, and translations in terms of points, angles, circles, perpendicular lines...

Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a ...

Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corre...

Explain how the criteria for triangle congruence (ASA, SAS, SSS, AAS and HL (Hypotenuse Leg)) follow from the definition...

Explore the proof for the relationship between slopes of parallel and perpendicular lines;

Determine if lines are parallel, perpendicular, or neither, based on their slopes; and

Apply properties of parallel and perpendicular lines to solve geometric problems.

Use geometric shapes, their measures, and their properties to describe objects.

Apply concepts of density based on area and volume of geometric figures in modeling situations.

Verify experimentally the properties of dilations given by a center and a scale factor.

Verify experimentally that dilation takes a line not passing through the center of the dilation to a parallel line, and ...

Verify experimentally that the dilation of a line segment is longer or shorter in the ratio given by the scale factor.

Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar...

Solve problems algebraically and geometrically

Prove relationships in geometric figures.

Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to d...

Explain and use the relationship between the sine and cosine of complementary angles.