Practice Every NYS Math Standard

Count to 100 by ones and by tens.

Counting and Cardinality

Count forward beginning from a given number within the known sequence (instead of having to begin at 1).

Counting and Cardinality

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

Counting and Cardinality

Understand the relationship between numbers and quantities; connect counting to cardinality. When counting objects say the number names in standard order; understand the last number said tells the number of objects counted; understand each successive number name refers to a quantity that is one larger.

Counting and Cardinality

Count to answer how many? questions about as many as 20 things arranged in a line or array or as many as 10 things in a scattered configuration.

Counting and Cardinality

Identify whether the number of objects in one group is greater than less than or equal to the number of objects in another group.

Counting and Cardinality

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

Counting and Cardinality

Represent addition and subtraction with objects fingers mental images drawings sounds acting out situations verbal explanations expressions or equations.

Operations and Algebraic Thinking

Add and subtract within 10 to solve problems by using objects or drawings to represent the problem.

Operations and Algebraic Thinking

Decompose numbers less than or equal to 10 into pairs in more than one way and record each decomposition by a drawing or equation.

Operations and Algebraic Thinking

For any number from 1 to 9 find the number that makes 10 when added to the given number and record the answer with a drawing or equation.

Operations and Algebraic Thinking

Fluently add and subtract within 5.

Operations and Algebraic Thinking

Compose and decompose numbers from 11 to 19 into ten ones and some further ones and record each composition or decomposition by a drawing or equation.

Number and Operations in Base Ten

Describe measurable attributes of objects such as length or weight. Describe several measurable attributes of a single object.

Measurement and Data

Directly compare two objects with a measurable attribute in common to see which object has more or less of the attribute.

Measurement and Data

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

Measurement and Data

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

Geometry

Correctly name shapes regardless of their orientations or overall size.

Geometry

Identify shapes as two-dimensional (flat) or three-dimensional (solid).

Geometry

Analyze and compare two- and three-dimensional shapes in different sizes and orientations using informal language to describe their similarities differences parts and other attributes.

Geometry

Model shapes in the world by building shapes from components and drawing shapes.

Geometry

Compose simple shapes to form larger shapes.

Geometry

Use addition and subtraction within 20 to solve word problems involving situations of adding to taking from putting together taking apart and comparing with unknowns in all positions.

Operations and Algebraic Thinking

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

Operations and Algebraic Thinking

Apply properties of operations as strategies to add and subtract. Understand commutative and associative properties of addition.

Operations and Algebraic Thinking

Understand subtraction as an unknown-addend problem. Solve subtraction problems by thinking about what number adds to the subtrahend to get the minuend.

Operations and Algebraic Thinking

Relate counting to addition and subtraction. Count on to add and count back to subtract.

Operations and Algebraic Thinking

Add and subtract within 20 demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on making ten decomposing a number leading to a ten and using the relationship between addition and subtraction.

Operations and Algebraic Thinking

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

Operations and Algebraic Thinking

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

Operations and Algebraic Thinking

Count to 120 starting at any number less than 120. In this range read and write numerals and represent a number of objects with a written numeral.

Number and Operations in Base Ten

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.

Number and Operations in Base Ten

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

Number and Operations in Base Ten

Add within 100 including adding a two-digit number and a one-digit number and adding a two-digit number and a multiple of 10.

Number and Operations in Base Ten

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

Number and Operations in Base Ten

Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90.

Number and Operations in Base Ten

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

Measurement and Data

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

Measurement and Data

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

Measurement and Data

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

Measurement and Data

Distinguish between defining attributes versus non-defining attributes; build and draw shapes to possess defining attributes.

Geometry

Compose two-dimensional shapes or three-dimensional shapes to create a composite shape and compose new shapes from the composite shape.

Geometry

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

Geometry

Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to taking from putting together taking apart and comparing with unknowns in all positions.

Operations and Algebraic Thinking

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

Operations and Algebraic Thinking

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

Operations and Algebraic Thinking

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

Operations and Algebraic Thinking

Understand that the three 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.

Number and Operations in Base Ten

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

Number and Operations in Base Ten

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

Number and Operations in Base Ten

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

Number and Operations in Base Ten

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

Number and Operations in Base Ten

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

Number and Operations in Base Ten

Add and subtract within 1000 using concrete models or drawings and strategies based on place value properties of operations and the relationship between addition and subtraction.

Number and Operations in Base Ten

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

Number and Operations in Base Ten

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

Number and Operations in Base Ten

Measure the length of an object by selecting and using appropriate tools such as rulers yardsticks meter sticks and measuring tapes.

Measurement and Data

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

Measurement and Data

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

Measurement and Data

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

Measurement and Data

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

Measurement and Data

Solve word problems involving dollar bills quarters dimes nickels and pennies using $ and ¢ symbols appropriately.

Measurement and Data

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 in the graphs.

Measurement and Data

Recognize and draw shapes having specified attributes such as a given number of angles or a given number of equal faces. Identify triangles quadrilaterals pentagons hexagons and cubes.

Geometry

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

Geometry

Partition circles and rectangles into two three or four equal shares and describe the shares using the words halves thirds half of a third of etc. Recognize that equal shares of identical wholes need not have the same shape.

Geometry

Interpret products of whole numbers as the total number of objects in a group. Understand multiplication as equal groups.

Operations and Algebraic Thinking

Interpret whole-number quotients of whole numbers. Understand division as equal sharing or equal grouping.

Operations and Algebraic Thinking

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

Operations and Algebraic Thinking

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

Operations and Algebraic Thinking

Apply properties of operations as strategies to multiply and divide: commutative associative and distributive properties.

Operations and Algebraic Thinking

Understand division as an unknown-factor problem. Divide by thinking about what number multiplied by the divisor equals the dividend.

Operations and Algebraic Thinking

Fluently multiply and divide within 100 using strategies and know from memory all products of two one-digit numbers.

Operations and Algebraic Thinking

Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity.

Operations and Algebraic Thinking

Identify arithmetic patterns (including patterns in the addition table or multiplication table) and explain them using properties of operations.

Operations and Algebraic Thinking

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

Number and Operations in Base Ten

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

Number and Operations in Base Ten

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

Number and Operations in Base Ten

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

Number and Operations - Fractions

Understand a fraction as a number on the number line; represent fractions on a number line diagram. Represent a fraction 1/b on a number line by partitioning the interval from 0 to 1.

Number and Operations - Fractions

Explain equivalence of fractions in special cases and compare fractions by reasoning about their size. Understand two fractions as equivalent if they are the same size or the same point on a number line.

Number and Operations - Fractions

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.

Measurement and Data

Measure and estimate liquid volumes and masses of objects using standard units of grams kilograms and liters. Solve one-step word problems involving masses or volumes that are given in the same units.

Measurement and Data

Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step how many more and how many less problems.

Measurement and Data

Recognize area as an attribute of plane figures and understand concepts of area measurement. A unit square with side length 1 unit has one square unit of area.

Measurement and Data

Measure areas by counting unit squares.

Measurement and Data

Relate area to the operations of multiplication and addition. Find area of rectangles by multiplying side lengths. Find areas of rectilinear figures by decomposing them.

Measurement and Data

Solve real world and mathematical problems involving perimeters of polygons including finding the perimeter given the side lengths and finding an unknown side length.

Measurement and Data

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

Geometry

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

Geometry

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

Operations and Algebraic Thinking

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

Operations and Algebraic Thinking

Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations including problems in which remainders must be interpreted.

Operations and Algebraic Thinking

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 factors. Determine whether a given whole number is prime or composite.

Operations and Algebraic Thinking

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

Operations and Algebraic Thinking

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

Number and Operations in Base Ten

Read and write multi-digit whole numbers using base-ten numerals number names and expanded form. Compare two multi-digit numbers using > = and < symbols.

Number and Operations in Base Ten

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

Number and Operations in Base Ten

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

Number and Operations in Base Ten

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 the properties of operations.

Number and Operations in Base Ten

Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors using strategies based on place value.

Number and Operations in Base Ten

Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models. Generate equivalent fractions.

Number and Operations - Fractions

Compare two fractions with different numerators and different denominators by creating common denominators or numerators or by comparing to a benchmark fraction such as 1/2.

Number and Operations - Fractions

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

Number and Operations - Fractions

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

Number and Operations - Fractions

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.

Number and Operations - Fractions

Use decimal notation for fractions with denominators 10 or 100.

Number and Operations - Fractions

Compare two decimals to hundredths by reasoning about their size. Record the results of comparisons with the symbols > = or <.

Number and Operations - Fractions

Know relative sizes of measurement units within one system of units. Express larger units in terms of smaller units. Record measurement equivalents in a two-column table.

Measurement and Data

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

Measurement and Data

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

Measurement and Data

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 by using information in a line plot.

Measurement and Data

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

Measurement and Data

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

Measurement and Data

Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts the angle measure of the whole is the sum of the angle measures of the parts.

Measurement and Data

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

Geometry

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

Geometry

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.

Geometry

Use parentheses brackets or braces in numerical expressions and evaluate expressions with these symbols.

Operations and Algebraic Thinking

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

Operations and Algebraic Thinking

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.

Operations and Algebraic Thinking

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.

Number and Operations in Base Ten

Explain patterns in the number of zeros of the product when multiplying a number by powers of 10. Explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10.

Number and Operations in Base Ten

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

Number and Operations in Base Ten

Use place value understanding to round decimals to any place.

Number and Operations in Base Ten

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

Number and Operations in Base Ten

Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors using strategies based on place value and the properties of operations.

Number and Operations in Base Ten

Add subtract multiply and divide decimals to hundredths using concrete models or drawings and strategies based on place value properties of operations and the relationship between operations.

Number and Operations in Base Ten

Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions that have a like denominator.

Number and Operations - Fractions

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

Number and Operations - Fractions

Interpret a fraction as division of the numerator by the denominator. Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers.

Number and Operations - Fractions

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

Number and Operations - Fractions

Interpret multiplication as scaling (resizing) by 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.

Number and Operations - Fractions

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

Number and Operations - Fractions

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

Number and Operations - Fractions

Convert among different-sized standard measurement units within a given measurement system and use these conversions in solving multi-step real world problems.

Measurement and Data

Make a line plot to display a data set of measurements in fractions of a unit. Use operations on fractions for this grade to solve problems involving information presented in line plots.

Measurement and Data

Recognize volume as an attribute of solid figures and understand concepts of volume measurement. A unit cube with side length 1 unit occupies exactly 1 cubic unit of volume.

Measurement and Data

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

Measurement and Data

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

Measurement and Data

Use a pair of perpendicular number lines called axes to define a coordinate system. Understand that the first number in an ordered pair indicates how far to travel from the origin in the x direction.

Geometry

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.

Geometry

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

Geometry

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

Geometry

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

Ratios and Proportional Relationships

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.

Ratios and Proportional Relationships

Use ratio and rate reasoning to solve real-world and mathematical problems including making tables of equivalent ratios finding missing values in tables plotting pairs of values on a coordinate plane and using unit rates.

Ratios and Proportional Relationships

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

The Number System

Fluently divide multi-digit numbers using the standard algorithm.

The Number System

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

The Number System

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 with a common factor as a multiple of a sum of two whole numbers with no common factor.

The Number System

Understand that positive and negative numbers are used together to describe quantities having opposite directions or values. Use positive and negative numbers to represent quantities in real-world contexts.

The Number System

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

The Number System

Understand ordering and absolute value of rational numbers. Interpret absolute value as magnitude for a positive or negative quantity in a real-world situation.

The Number System

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.

The Number System

Write and evaluate numerical expressions involving whole-number exponents.

Expressions and Equations

Write read and evaluate expressions in which letters stand for numbers. Include writing expressions that record operations with numbers and variables and identifying parts of an expression.

Expressions and Equations

Apply the properties of operations to generate equivalent expressions including combining like terms.

Expressions and Equations

Identify when two expressions are equivalent (i.e. when the two expressions name the same number regardless of which value is substituted into them).

Expressions and Equations

Understand solving an equation or inequality as a process of answering 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.

Expressions and Equations

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.

Expressions and Equations

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.

Expressions and Equations

Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions.

Expressions and Equations

Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity in terms of the other quantity.

Expressions and Equations

Find the area of right triangles other triangles special quadrilaterals and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.

Geometry

Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths. Apply the formulas V = l w h and V = b h.

Geometry

Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate.

Geometry

Represent three-dimensional figures using nets made up of rectangles and triangles and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.

Geometry

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

Statistics and Probability

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

Statistics and Probability

Recognize that a measure of center for a numerical data set summarizes all of its values with a single number while a measure of variation describes how its values vary with a single number.

Statistics and Probability

Display numerical data in plots on a number line including dot plots histograms and box plots.

Statistics and Probability

Summarize numerical data sets in relation to their context such as by reporting the number of observations describing the nature of the attribute under investigation and giving quantitative measures of center and variability.

Statistics and Probability

Compute unit rates associated with ratios of fractions including ratios of lengths areas and other quantities measured in like or different units.

Ratios and Proportional Relationships

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.

Ratios and Proportional Relationships

Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest tax markups and markdowns gratuities and commissions fees percent increase and decrease percent error.

Ratios and Proportional Relationships

Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.

The Number System

Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. Understand that multiplication is extended from fractions to rational numbers by requiring operations to continue to satisfy the properties of operations.

The Number System

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

The Number System

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

Expressions and Equations

Understand that rewriting an expression in different forms can shed light on the problem and how the quantities in it are related.

Expressions and Equations

Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers fractions and decimals) using tools strategically.

Expressions and Equations

Use variables to represent quantities in a problem and construct simple equations and inequalities to solve problems by reasoning about the quantities. Solve word problems leading to equations of the form px + q = r and p(x + q) = r.

Expressions and Equations

Solve problems involving scale drawings of geometric figures including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.

Geometry

Draw geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides noticing when the conditions determine a unique triangle more than one triangle or no triangle.

Geometry

Describe the two-dimensional figures that result from slicing three-dimensional figures such as right rectangular prisms and right rectangular pyramids.

Geometry

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

Geometry

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

Geometry

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

Geometry

Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population.

Statistics and Probability

Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples of the same size to gauge the variation in estimates or predictions.

Statistics and Probability

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

Statistics and Probability

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

Statistics and Probability

Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. A probability near 0 indicates an unlikely event and near 1 indicates a likely event.

Statistics and Probability

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

Statistics and Probability

Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good explain possible sources of the discrepancy.

Statistics and Probability

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

Statistics and Probability

Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats into a rational number.

The Number System

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

The Number System

Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example 3² × 3⁻⁵ = 3⁻³ = 1/27.

Expressions and Equations

Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes.

Expressions and Equations

Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities and to express how many times as much one is than the other.

Expressions and Equations

Perform operations with numbers expressed in scientific notation including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities.

Expressions and Equations

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

Expressions and Equations

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

Expressions and Equations

Solve linear equations in one variable including equations with rational number coefficients and equations that require using the distributive property and combining like terms.

Expressions and Equations

Analyze and solve pairs of simultaneous linear equations by graphing substitution and elimination. Understand that solutions are the intersection points of the graphs.

Expressions and Equations

Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.

Functions

Compare properties of two functions each represented in a different way (algebraically graphically numerically in tables or by verbal descriptions).

Functions

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

Functions

Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x y) values including reading these from a table or from a graph.

Functions

Describe qualitatively the functional relationship between two quantities by analyzing a graph. Sketch a graph that exhibits the qualitative features of a function that has been described verbally.

Functions

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

Geometry

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

Geometry

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

Geometry

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

Geometry

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

Geometry

Explain a proof of the Pythagorean Theorem and its converse.

Geometry

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

Geometry

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

Geometry

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

Geometry

Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering outliers positive or negative association linear association and nonlinear association.

Statistics and Probability

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

Statistics and Probability

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

Statistics and Probability

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

Statistics and Probability

Interpret parts of an expression such as terms factors and coefficients in context.

Seeing Structure in Expressions

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

Seeing Structure in Expressions

Use the structure of an expression to identify ways to rewrite it. For example see x⁴ - y⁴ as (x²)² - (y²)² and factor accordingly.

Seeing Structure in Expressions

Factor a quadratic expression to reveal the zeros of the function it defines.

Seeing Structure in Expressions

Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

Seeing Structure in Expressions

Understand that polynomials form a system analogous to the integers; add subtract and multiply polynomials.

Arithmetic with Polynomial Expressions

Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions and simple rational and exponential functions.

Creating Equations

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

Creating Equations

Represent constraints by equations or inequalities and by systems of equations and/or inequalities and interpret solutions as viable or non-viable options in a modeling context.

Creating Equations

Rearrange formulas to highlight a quantity of interest using the same reasoning as in solving equations. For example rearrange Ohm's law V = IR to highlight resistance R.

Creating Equations

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.

Reasoning with Equations and Inequalities

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

Reasoning with Equations and Inequalities

Solve quadratic equations in one variable using inspection (e.g. x² = 49) taking square roots completing the square the quadratic formula and factoring as appropriate to the initial form of the equation.

Reasoning with Equations and Inequalities

Prove that given a system of two equations in two variables replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

Reasoning with Equations and Inequalities

Solve systems of linear equations exactly and approximately focusing on pairs of linear equations in two variables.

Reasoning with Equations and Inequalities

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

Reasoning with Equations and Inequalities

Explain why the x-coordinates of the points where the graphs of equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x).

Reasoning with Equations and Inequalities

Graph the solutions to a linear inequality in two variables as a half-plane and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

Reasoning with Equations and Inequalities

Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain then f(x) denotes the output of f corresponding to the input x.

Interpreting Functions

Use function notation evaluate functions for inputs in their domains and interpret statements that use function notation in terms of a context.

Interpreting Functions

Recognize that sequences are functions sometimes defined recursively whose domain is a subset of the integers.

Interpreting Functions

For a function that models a relationship between two quantities interpret key features of graphs and tables in terms of the quantities and sketch graphs showing key features given a verbal description of the relationship.

Interpreting Functions

Relate the domain of a function to its graph and where applicable to the quantitative relationship it describes.

Interpreting Functions

Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

Interpreting Functions

Graph linear and quadratic functions and show intercepts maxima and minima.

Interpreting Functions

Graph exponential and logarithmic functions showing intercepts and end behavior and trigonometric functions showing period midline and amplitude.

Interpreting Functions

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

Interpreting Functions

Write a function that describes a relationship between two quantities. Determine an explicit expression a recursive process or steps for calculation from a context.

Building Functions

Write arithmetic and geometric sequences both recursively and with an explicit formula use them to model situations and translate between the two forms.

Building Functions

Identify the effect on the graph of replacing f(x) by f(x) + k k·f(x) f(kx) and f(x + k) for specific values of k; find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology.

Building Functions

Distinguish between situations that can be modeled with linear functions and with exponential functions. Recognize that linear functions grow by equal differences over equal intervals and exponential functions grow by equal factors over equal intervals.

Linear and Exponential Models

Construct linear and exponential functions including arithmetic and geometric sequences given a graph a description of a relationship or two input-output pairs.

Linear and Exponential Models

Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly quadratically or (more generally) as a polynomial function.

Linear and Exponential Models

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

Linear and Exponential Models

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

Interpreting Categorical and Quantitative Data

Use statistics appropriate to the shape of the data distribution to compare center (median mean) and spread (interquartile range standard deviation) of two or more different data sets.

Interpreting Categorical and Quantitative Data

Interpret differences in shape center and spread in the context of the data sets accounting for possible effects of extreme data points (outliers).

Interpreting Categorical and Quantitative Data

Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data including joint marginal and conditional relative frequencies.

Interpreting Categorical and Quantitative Data

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.

Interpreting Categorical and Quantitative Data

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

Interpreting Categorical and Quantitative Data

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

Interpreting Categorical and Quantitative Data

Know precise definitions of angle circle perpendicular line parallel line and line segment based on the undefined notions of point line distance along a line and distance around a circular arc.

Congruence

Represent transformations in the plane; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not.

Congruence

Given a rectangle parallelogram trapezoid or regular polygon describe the rotations and reflections that carry it onto itself.

Congruence

Develop definitions of rotations reflections and translations in terms of angles circles perpendicular lines parallel lines and line segments.

Congruence

Given a geometric figure and a rotation reflection or translation draw the transformed figure using appropriate tools. Specify a sequence of transformations that will carry a given figure onto another.

Congruence

Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures use the definition of congruence in terms of rigid motions to decide if they are congruent.

Congruence

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

Congruence

Explain how the criteria for triangle congruence (ASA SAS SSS AAS and HL) follow from the definition of congruence in terms of rigid motions.

Congruence

Prove theorems about lines and angles: vertical angles are congruent; when a transversal crosses parallel lines alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector are equidistant from the segment's endpoints.

Congruence

Prove theorems about triangles: the triangle angle-sum theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; medians of a triangle meet at a point.

Congruence

Prove theorems about parallelograms: opposite sides are congruent; opposite angles are congruent; diagonals of a parallelogram bisect each other and conversely rectangles are parallelograms with congruent diagonals.

Congruence

Make formal geometric constructions with a variety of tools and methods (compass and straightedge string reflective devices paper folding dynamic geometric software etc.).

Congruence

Construct an equilateral triangle a square and a regular hexagon inscribed in a circle.

Congruence

Verify experimentally the properties of dilations given by a center and a scale factor. A dilation takes a line not passing through the center of the dilation to a parallel line and leaves a line passing through the center unchanged. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

Similarity Right Triangles and Trigonometry

Given two figures use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using the definition of similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

Similarity Right Triangles and Trigonometry

Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

Similarity Right Triangles and Trigonometry

Prove theorems about triangles. The ratio of the areas of two similar triangles is the square of the ratio of their corresponding sides. The Pythagorean Theorem can be proven using triangle similarity.

Similarity Right Triangles and Trigonometry

Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

Similarity Right Triangles and Trigonometry

Understand that by similarity side ratios in right triangles are properties of the angles in the triangle leading to definitions of trigonometric ratios for acute angles.

Similarity Right Triangles and Trigonometry

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

Similarity Right Triangles and Trigonometry

Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

Similarity Right Triangles and Trigonometry

Identify and describe relationships among inscribed angles radii and chords. The measure of an inscribed angle is half the measure of the intercepted arc; an angle inscribed in a semicircle is a right angle.

Circles

Construct the inscribed and circumscribed circles of a triangle and prove properties of angles for a quadrilateral inscribed in a circle.

Circles

Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius and define the radian measure of the angle as the constant of proportionality.

Circles

Derive the equation of a circle given center (h k) and radius r using the Pythagorean Theorem. Complete the square to find the center and radius of a circle given by an equation.

Geometric Properties with Equations

Derive the equation of a parabola given a focus and directrix.

Geometric Properties with Equations

Use coordinates to prove simple geometric theorems algebraically. For example prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle.

Geometric Properties with Equations

Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g. find the equation of a line parallel or perpendicular to a given line that passes through a given point).

Geometric Properties with Equations

Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

Geometric Properties with Equations

Use coordinates to compute perimeters of polygons and areas of triangles and rectangles. For example find the area of a triangle given the coordinates of its vertices.

Geometric Properties with Equations

Give an informal argument for the formulas for the circumference of a circle area of a circle volume of a cylinder pyramid and cone. Use dissection arguments Cavalieri's principle and informal limit arguments.

Geometric Measurement and Dimension

Use volume formulas for cylinders pyramids cones and spheres to solve problems.

Geometric Measurement and Dimension

Identify the shapes of two-dimensional cross-sections of three-dimensional objects and identify three-dimensional objects generated by rotations of two-dimensional objects.

Geometric Measurement and Dimension

Use geometric shapes their measures and their properties to describe objects (e.g. modeling a tree trunk or a human torso as a cylinder).

Modeling with Geometry

Apply concepts of density based on area and volume in modeling situations (e.g. persons per square mile BTUs per cubic foot).

Modeling with Geometry

Apply geometric methods to solve design problems (e.g. designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).

Modeling with Geometry

Use the structure of an expression to identify ways to rewrite it. Apply to polynomial and rational expressions in the context of Algebra II.

Seeing Structure in Expressions

Derive the formula for the sum of a finite geometric series and use the formula to solve problems.

Seeing Structure in Expressions

Know and apply the Remainder Theorem: for a polynomial p(x) and a number a the remainder on division by x - a is p(a). If p(a) = 0 then (x - a) is a factor of p(x).

Arithmetic with Polynomial Expressions

Identify zeros of polynomials when suitable factorizations are available and use the zeros to construct a rough graph of the function defined by the polynomial.

Arithmetic with Polynomial Expressions

Prove polynomial identities and use them to describe numerical relationships.

Arithmetic with Polynomial Expressions

Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x) where a(x) b(x) q(x) and r(x) are polynomials with the degree of r(x) less than the degree of b(x).

Arithmetic with Polynomial Expressions

Understand that rational expressions form a system analogous to the rational numbers; add subtract multiply and divide rational expressions.

Arithmetic with Polynomial Expressions

Solve simple rational and radical equations in one variable and give examples showing how extraneous solutions may arise.

Reasoning with Equations and Inequalities

Explain why the x-coordinates of the points where the graphs of y = f(x) and y = g(x) intersect are the solutions of f(x) = g(x); find the solutions approximately using technology. Include cases where f(x) and/or g(x) are polynomial rational exponential logarithmic and trigonometric functions.

Reasoning with Equations and Inequalities

For a function that models a relationship between two quantities interpret key features of graphs and tables and sketch graphs showing key features. Apply to polynomial rational exponential logarithmic and trigonometric functions.

Interpreting Functions

Graph functions expressed symbolically and show key features of the graph. Graph polynomial functions identifying zeros and end behavior. Graph rational functions identifying zeros and asymptotes. Graph exponential and logarithmic functions showing intercepts and end behavior.

Interpreting Functions

Find inverse functions. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse.

Building Functions

Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

Building Functions

For exponential models express as a logarithm the solution to ab^(ct) = d where a c and d are numbers and the base b is 2 10 or e; evaluate the logarithm using technology.

Linear Quadratic and Exponential Models

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

Trigonometric Functions

Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers interpreted as radian measures of angles traversed counterclockwise around the unit circle.

Trigonometric Functions

Choose trigonometric functions to model periodic phenomena with specified amplitude frequency and midline.

Trigonometric Functions

Prove the Pythagorean identity sin²(θ) + cos²(θ) = 1 and use it to find sin(θ) cos(θ) or tan(θ) given sin(θ) cos(θ) or tan(θ) and the quadrant of the angle.

Trigonometric Functions

Understand statistics as a process for making inferences about population parameters based on a random sample from that population.

Making Inferences and Justifying Conclusions

Decide if a specified model is consistent with results from a given data-generating process (e.g. using simulation).

Making Inferences and Justifying Conclusions

Recognize the purposes of and differences among sample surveys experiments and observational studies; explain how randomization relates to each.

Making Inferences and Justifying Conclusions

Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.

Making Inferences and Justifying Conclusions

Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.

Making Inferences and Justifying Conclusions

Evaluate reports based on data by considering the source of the data the design of the study and the way the data are analyzed and displayed.

Making Inferences and Justifying Conclusions

Describe events as subsets of a sample space using characteristics of the outcomes or as unions intersections or complements of other events.

Conditional Probability and the Rules of Probability

Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities and use this characterization to determine if they are independent.

Conditional Probability and the Rules of Probability

Understand the conditional probability of A given B as P(A and B)/P(B) and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A.

Conditional Probability and the Rules of Probability

Apply the Addition Rule P(A or B) = P(A) + P(B) - P(A and B) and interpret the answer in terms of the model.

Conditional Probability and the Rules of Probability