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K12 Mathematical Practice Standards 
1 Make sense of problems and persevere in solving them. (174) 
2 Reason abstractly and quantitatively. (87) 
3 Construct viable arguments and critique the reasoning of others. (31) 
4 Model with mathematics. (42) 
5 Use appropriate tools strategically. (29) 
6 Attend to precision. (29) 
7 Look for and make use of structure. (28) 
8 Look for and express regularity in repeated reasoning. (26) 
K  Counting and Cardinality 
Know number names and the count sequence. 
Count to tell the number of objects. 
4. Understand the relationship between numbers and quantities; connect counting to cardinality. 
a. When counting objects, say the number names in the standard order, pairing each object with one and only one number name and each number name with one and only one object. (116) 
b. Understand that the last number name said tells the number of objects counted. The number of objects is the same regardless of their arrangement or the order in which they were counted. (110) 
c. Understand that each successive number name refers to a quantity that is one larger. (109) 
Compare numbers. 
K5 Operations and Algebraic Thinking 
Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from. 
1. Represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations. (116) 
2. Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem. (112) 
4. For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation. (107) 
5. Fluently add and subtract within 5. (38) 
Represent and solve problems involving addition and subtraction. 
Understand and apply properties of operations and the relationship between addition and subtraction. 
4. Understand subtraction as an unknownaddend problem. For example, subtract 10 – 8 by finding the number that makes 10 when added to 8. (96) 
Add and subtract within 20. 
2. Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two onedigit numbers. (80) 
5. Relate counting to addition and subtraction (e.g., by counting on 2 to add 2). (93) 
Work with equal groups of objects to gain foundations for multiplication. 
4. 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. (12) 
Represent and solve problems involving multiplication and division. 
Understand properties of multiplication and the relationship between multiplication and division. 
6. Understand division as an unknownfactor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8. (90) 
Multiply and divide within 100. 
Solve problems involving the four operations, and identify and explain patterns in arithmetic. 
Work with addition and subtraction equations. 
Analyze patterns and relationships. 
Use the four operations with whole numbers to solve problems. 
Gain familiarity with factors and multiples. 
Write and interpret numerical expressions. 
1. Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. (47) 
Generate and analyze patterns. 
K5 Number and Operations in Base Ten 
Work with numbers 11–19 to gain foundations for place value. 
Extend the counting sequence. 
Understand place value. 
2. Count within 1000; skipcount by 5s, 10s, and 100s. (33) 
3. Compare two twodigit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <. (88) 
3. Read and write numbers to 1000 using baseten numerals, number names, and expanded form. (39) 
4. Compare two threedigit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons. (30) 
Use place value understanding and properties of operations to add and subtract. 
5. Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction. (53) 
5. Given a twodigit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used. (88) 
6. Add up to four twodigit numbers using strategies based on place value and properties of operations. (50) 
8. Mentally add 10 or 100 to a given number 100–900, and mentally subtract 10 or 100 from a given number 100–900. (43) 
9. Explain why addition and subtraction strategies work, using place value and the properties of operations. (49) 
Generalize place value understanding for multidigit whole numbers. 
3. Use place value understanding to round multidigit whole numbers to any place. (56) 
Understand the place value system. 
1. Recognize that in a multidigit 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. (5) 
3. Read, write, and compare decimals to thousandths. 
a. Read and write decimals to thousandths using baseten numerals, number names, and expanded form, e.g., 347.392 = 3 × 100 + 4 × 10 + 7 × 1 + 3 × (1/10) + 9 × (1/100) + 2 × (1/1000). (3) 
b. Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons. (4) 
4. Use place value understanding to round decimals to any place. (6) 
Perform operations with multidigit whole numbers and with decimals to hundredths. 
5. Fluently multiply multidigit whole numbers using the standard algorithm. (50) 
Use place value understanding and properties of operations to perform multidigit arithmetic. 
1. Use place value understanding to round whole numbers to the nearest 10 or 100. (31) 
2. Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction. (48) 
3. Multiply onedigit whole numbers by multiples of 10 in the range 10–90 (e.g., 9 × 80, 5 × 60) using strategies based on place value and properties of operations. (87) 
4. Fluently add and subtract multidigit whole numbers using the standard algorithm. (47) 
K5 Measurement and Data 
Describe and compare measurable attributes. 
1. Describe measurable attributes of objects, such as length or weight. Describe several measurable attributes of a single object. (40) 
Classify objects and count the number of objects in each category. 
Measure lengths indirectly and by iterating length units. 
1. Order three objects by length; compare the lengths of two objects indirectly by using a third object. (55) 
Tell and write time. 
Measure and estimate lengths in standard units. 
1. Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes. (99) 
2. 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. (25) 
3. Estimate lengths using units of inches, feet, centimeters, and meters. (49) 
4. Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit. (54) 
Work with time and money. 
7. Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m. (16) 
8. Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately. Example: If you have 2 dimes and 3 pennies, how many cents do you have? (56) 
Relate addition and subtraction to length. 
Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects. 
Represent and interpret data. 
Convert like measurement units within a given measurement system. 
Geometric measurement: understand concepts of area and relate area to multiplication and to addition. 
5. Recognize area as an attribute of plane figures and understand concepts of area measurement. 
6. Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units). (46) 
7. Relate area to the operations of multiplication and addition. 
a. Find the area of a rectangle with wholenumber side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths. (49) 
Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures. 
Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit. 
Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition. 
3. Recognize volume as an attribute of solid figures and understand concepts of volume measurement. 
4. Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units. (22) 
5. Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume. 
Geometric measurement: understand concepts of angle and measure angles. 
6. Measure angles in wholenumber degrees using a protractor. Sketch angles of specified measure. (18) 
K8 Geometry 
Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres). 
1. Describe objects in the environment using names of shapes, and describe the relative positions of these objects using terms such as above, below, beside, in front of, behind, and next to. (85) 
2. Correctly name shapes regardless of their orientations or overall size. (53) 
3. Identify shapes as twodimensional (lying in a plane, “flat”) or threedimensional (“solid”). (61) 
Reason with shapes and their attributes. 
1. 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. (91) 
2. Partition a rectangle into rows and columns of samesize squares and count to find the total number of them. (17) 
Analyze, compare, create, and compose shapes. 
5. Model shapes in the world by building shapes from components (e.g., sticks and clay balls) and drawing shapes. (69) 
6. Compose simple shapes to form larger shapes. For example, “Can you join these two triangles with full sides touching to make a rectangle?” (39) 
Draw and identify lines and angles, and classify shapes by properties of their lines and angles. 
1. Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in twodimensional figures. (117) 
Graph points on the coordinate plane to solve realworld and mathematical problems. 
2. 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. (12) 
Classify twodimensional figures into categories based on their properties. 
4. Classify twodimensional figures in a hierarchy based on properties. (1) 
Draw, construct, and describe geometrical figures and describe the relationships between them. 
1. 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. (58) 
3. Describe the twodimensional figures that result from slicing threedimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids. (2) 
Solve reallife and mathematical problems involving angle measure, area, surface area, and volume. 
4. 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. (53) 
5. Use facts about supplementary, complementary, vertical, and adjacent angles in a multistep problem to write and solve simple equations for an unknown angle in a figure. (87) 
Solve realworld and mathematical problems involving area, surface area, and volume. 
Understand and apply the Pythagorean Theorem. 
6. Explain a proof of the Pythagorean Theorem and its converse. (34) 
7. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in realworld and mathematical problems in two and three dimensions. (39) 
8. Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. (20) 
Understand congruence and similarity using physical models, transparencies, or geometry software. 
3. Describe the effect of dilations, translations, rotations, and reflections on twodimensional figures using coordinates. (24) 
Solve realworld and mathematical problems involving volume of cylinders, cones, and spheres. 
35 Number and Operations—Fractions 
Develop understanding of fractions as numbers. 
1. 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. (40) 
2. Understand a fraction as a number on the number line; represent fractions on a number line diagram. 
3. Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size. 
a. Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line. (34) 
b. Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3). Explain why the fractions are equivalent, e.g., by using a visual fraction model. (41) 
Use equivalent fractions as a strategy to add and subtract fractions. 
Apply and extend previous understandings of multiplication and division to multiply and divide fractions. 
4. Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. 
6. Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem. (66) 
7. Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. 
Extend understanding of fraction equivalence and ordering. 
Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. 
3. Understand a fraction a/b with a > 1 as a sum of fractions 1/b. 
a. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. (21) 
4. Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. 
a. Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4). (11) 
Understand decimal notation for fractions, and compare decimal fractions. 
6. Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram. (38) 
67 Ratios and Proportional Relationships 
Analyze proportional relationships and use them to solve realworld and mathematical problems. 
2. Recognize and represent proportional relationships between quantities. 
b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. (54) 
d. Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate. (50) 
Understand ratio concepts and use ratio reasoning to solve problems. 
3. Use ratio and rate reasoning to solve realworld and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. 
c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent. (52) 
d. Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities. (39) 
68 The Number System 
Apply and extend previous understandings of multiplication and division to divide fractions by fractions. 
Compute fluently with multidigit numbers and find common factors and multiples. 
2. Fluently divide multidigit numbers using the standard algorithm. (11) 
3. Fluently add, subtract, multiply, and divide multidigit decimals using the standard algorithm for each operation. (29) 
Apply and extend previous understandings of numbers to the system of rational numbers. 
6. Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates. 
c. Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane. (25) 
7. Understand ordering and absolute value of rational numbers. 
b. Write, interpret, and explain statements of order for rational numbers in realworld contexts. For example, write –3 °C > –7 °C to express the fact that –3 °C is warmer than –7 °C. (26) 
d. Distinguish comparisons of absolute value from statements about order. For example, recognize that an account balance less than –30 dollars represents a debt greater than 30 dollars. (24) 
Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers. 
1. 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. 
a. Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged. (28) 
d. Apply properties of operations as strategies to add and subtract rational numbers. (39) 
2. Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. 
c. Apply properties of operations as strategies to multiply and divide rational numbers. (32) 
d. Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats. (17) 
3. Solve realworld and mathematical problems involving the four operations with rational numbers. (39) 
Know that there are numbers that are not rational, and approximate them by rational numbers. 
68 Expressions and Equations 
Apply and extend previous understandings of arithmetic to algebraic expressions. 
1. Write and evaluate numerical expressions involving wholenumber exponents. (19) 
2. Write, read, and evaluate expressions in which letters stand for numbers. 
a. Write expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation “Subtract y from 5” as 5 – y. (58) 
Reason about and solve onevariable equations and inequalities. 
7. Solve realworld 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. (115) 
Represent and analyze quantitative relationships between dependent and independent variables. 
Use properties of operations to generate equivalent expressions. 
1. Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. (20) 
Solve reallife and mathematical problems using numerical and algebraic expressions and equations. 
4. Use variables to represent quantities in a realworld or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. 
Work with radicals and integer exponents. 
1. Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 3^{2} × 3^{–5} = 3^{–3} = 1/3^{3} = 1/27. (34) 
Understand the connections between proportional relationships, lines, and linear equations. 
Analyze and solve linear equations and pairs of simultaneous linear equations. 
7. Solve linear equations in one variable. 
b. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. (18) 
8. Analyze and solve pairs of simultaneous linear equations. 
8 Functions 
Define, evaluate, and compare functions. 
1. 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. (31) 
Use functions to model relationships between quantities. 
68 Statistics and Probability 
Use random sampling to draw inferences about a population. 
Draw informal comparative inferences about two populations. 
Investigate chance processes and develop, use, and evaluate probability models. 
7. 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. 
8. Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation. 
a. Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs. (44) 
Develop understanding of statistical variability. 
2. 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. (122) 
Investigate patterns of association in bivariate data. 
Summarize and describe distributions. 
4. Display numerical data in plots on a number line, including dot plots, histograms, and box plots. (115) 
High School — Number and Quantity 
The Real Number System 
Extend the properties of exponents to rational exponents. 
2. Rewrite expressions involving radicals and rational exponents using the properties of exponents. (19) 
Use properties of rational and irrational numbers. 
Quantities^{★} 
Reason quantitatively and use units to solve problems. 
2. Define appropriate quantities for the purpose of descriptive modeling. (8) 
3. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. (7) 
The Complex Number System 
Perform arithmetic operations with complex numbers. 
1. Know there is a complex number i such that i^{2} = –1, and every complex number has the form a + bi with a and b real. (15) 
2. Use the relation i^{2} = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. (18) 
3. (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers. (7) 
Represent complex numbers and their operations on the complex plane. 
6. (+) Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints. (3) 
Use complex numbers in polynomial identities and equations. 
7. Solve quadratic equations with real coefficients that have complex solutions. (24) 
8. (+) Extend polynomial identities to the complex numbers. For example, rewrite x^{2} + 4 as (x + 2i)(x – 2i). (29) 
9. (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. (4) 
Vector and Matrix Quantities 
Represent and model with vector quantities. 
2. (+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point. (15) 
3. (+) Solve problems involving velocity and other quantities that can be represented by vectors. (18) 
Perform operations on vectors. 
4. (+) Add and subtract vectors. 
a. Add vectors endtoend, componentwise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes. (14) 
b. Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum. (13) 
5. (+) Multiply a vector by a scalar. 
Perform operations on matrices and use matrices in applications. 
11. (+) Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors. (6) 
12. (+) Work with 2 × 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area. (2) 
6. (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network. (2) 
7. (+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled. (2) 
8. (+) Add, subtract, and multiply matrices of appropriate dimensions. (7) 
High School — Algebra 
Seeing Structure in Expressions 
Interpret the structure of expressions 
1. Interpret expressions that represent a quantity in terms of its context.^{★} 
Write expressions in equivalent forms to solve problems 
3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.^{★} 
a. Factor a quadratic expression to reveal the zeros of the function it defines. (57) 
b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. (13) 
4. Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments.^{★} (6) 
Arithmetic with Polynomials and Rational Expressions 
Perform arithmetic operations on polynomials 
Understand the relationship between zeros and factors of polynomials 
2. 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), so p(a) = 0 if and only if (x – a) is a factor of p(x). (17) 
3. 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. (35) 
Use polynomial identities to solve problems 
Rewrite rational expressions 
Creating Equations^{★} 
Create equations that describe numbers or relationships 
1. 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. (52) 
2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. (37) 
4. 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. (25) 
Reasoning with Equations and Inequalities 
Understand solving equations as a process of reasoning and explain the reasoning 
2. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. (42) 
Solve equations and inequalities in one variable 
3. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. (55) 
4. Solve quadratic equations in one variable. 
Solve systems of equations 
5. 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. (24) 
6. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. (41) 
8. (+) Represent a system of linear equations as a single matrix equation in a vector variable. (3) 
9. (+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater). (9) 
Represent and solve equations and inequalities graphically 
10. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). (49) 
High School — Functions 
Interpreting Functions 
Understand the concept of a function and use function notation 
2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. (24) 
Interpret functions that arise in applications in terms of the context 
6. 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.^{★} (25) 
Analyze functions using different representations 
7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.^{★} 
a. Graph linear and quadratic functions and show intercepts, maxima, and minima. (50) 
b. Graph square root, cube root, and piecewisedefined functions, including step functions and absolute value functions. (23) 
c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. (33) 
d. (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. (9) 
e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. (43) 
8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. 
a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. (50) 
Building Functions 
Build a function that models a relationship between two quantities 
1. Write a function that describes a relationship between two quantities.^{★} 
a. Determine an explicit expression, a recursive process, or steps for calculation from a context. (59) 
2. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.^{★} (14) 
Build new functions from existing functions 
4. Find inverse functions. 
a. 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. For example, f(x) =2 x3 or f(x) = (x+1)/(x–1) for x ≠ 1. (26) 
b. (+) Verify by composition that one function is the inverse of another. (22) 
c. (+) Read values of an inverse function from a graph or a table, given that the function has an inverse. (28) 
d. (+) Produce an invertible function from a noninvertible function by restricting the domain. (20) 
5. (+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. (35) 
Linear, Quadratic, and Exponential Models^{★} 
Construct and compare linear, quadratic, and exponential models and solve problems 
1. Distinguish between situations that can be modeled with linear functions and with exponential functions. 
a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. (36) 
b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. (22) 
c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. (24) 
3. Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. (36) 
4. 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. (25) 
Interpret expressions for functions in terms of the situation they model 
Trigonometric Functions 
Extend the domain of trigonometric functions using the unit circle 
1. Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle. (6) 
4. (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. (6) 
Model periodic phenomena with trigonometric functions 
5. Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.^{★} (14) 
6. (+) Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed. (3) 
7. (+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.^{★} (23) 
Prove and apply trigonometric identities 
High School — Geometry 
Congruence 
Experiment with transformations in the plane 
3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. (24) 
4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. (23) 
Understand congruence in terms of rigid motions 
7. 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. (4) 
8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. (4) 
Prove geometric theorems 
Make geometric constructions 
13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. (19) 
Similarity, Right Triangles, and Trigonometry 
Understand similarity in terms of similarity transformations 
3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. (7) 
Prove theorems involving similarity 
5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. (12) 
Define trigonometric ratios and solve problems involving right triangles 
6. 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. (19) 
7. Explain and use the relationship between the sine and cosine of complementary angles. (17) 
8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.^{★} (49) 
Apply trigonometry to general triangles 
10. (+) Prove the Laws of Sines and Cosines and use them to solve problems. (26) 
11. (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and nonright triangles (e.g., surveying problems, resultant forces). (25) 
9. (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. (17) 
Circles 
Understand and apply theorems about circles 
1. Prove that all circles are similar. (20) 
3. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. (19) 
4. (+) Construct a tangent line from a point outside a given circle to the circle. (17) 
Find arc lengths and areas of sectors of circles 
Expressing Geometric Properties with Equations 
Translate between the geometric description and the equation for a conic section 
1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. (30) 
2. Derive the equation of a parabola given a focus and directrix. (8) 
3. (+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant. (37) 
Use coordinates to prove simple geometric theorems algebraically 
6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. (14) 
7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.^{★} (25) 
Geometric Measurement and Dimension 
Explain volume formulas and use them to solve problems 
2. (+) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures. (2) 
3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.^{★} (10) 
Visualize relationships between twodimensional and threedimensional objects 
Modeling with Geometry 
Apply geometric concepts in modeling situations 
1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).^{★} (9) 
2. Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).^{★} (23) 
High School — Statistics and Probability^{★} 
Interpreting Categorical and Quantitative Data 
Summarize, represent, and interpret data on a single count or measurement variable 
1. Represent data with plots on the real number line (dot plots, histograms, and box plots). (84) 
2. 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. (90) 
3. Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). (58) 
Summarize, represent, and interpret data on two categorical and quantitative variables 
6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. 
b. Informally assess the fit of a function by plotting and analyzing residuals. (23) 
c. Fit a linear function for a scatter plot that suggests a linear association. (26) 
Interpret linear models 
Making Inferences and Justifying Conclusions 
Understand and evaluate random processes underlying statistical experiments 
1. Understand statistics as a process for making inferences about population parameters based on a random sample from that population. (24) 
Make inferences and justify conclusions from sample surveys, experiments, and observational studies 
3. Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. (40) 
4. 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. (30) 
5. Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant. (21) 
6. Evaluate reports based on data. (66) 
Conditional Probability and the Rules of Probability 
Understand independence and conditional probability and use them to interpret data 
Use the rules of probability to compute probabilities of compound events in a uniform probability model 
6. Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model. (10) 
7. 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. (7) 
8. (+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(BA) = P(B)P(AB), and interpret the answer in terms of the model. (10) 
9. (+) Use permutations and combinations to compute probabilities of compound events and solve problems. (17) 
Using Probability to Make Decisions 
Calculate expected values and use them to solve problems 
2. (+) Calculate the expected value of a random variable; interpret it as the mean of the probability distribution. (6) 
Use probability to evaluate outcomes of decisions 
5. (+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values. 
a. Find the expected payoff for a game of chance. For example, find the expected winnings from a state lottery ticket or a game at a fastfood restaurant. (2) 
6. (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). (10) 
7. (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game). (21) 
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Michigan Online Resources for Educators is supported by a Title II, Part D, Enhancing Education Through Technology Grant (EETT). This project is made possible by grant funds from the U.S. Institute of Museum and Library Services (IMLS) administered by the State of Michigan through the Library of Michigan. Additional funding provided by the State of Michigan. 
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