Common Core State Standards for Mathematics

Common Core State StandardS

Grade 10 Mathematics Testing in Washington

This document highlights, in yellow, the mathematical practices and high school content standards, clusters, and domains included for 10th grade testing starting in spring 2018. Students should receive instruction toward the entirety of the high school standards, not just the tested standards, throughout their high school experience.

Further information about 10th grade testing is available in the Grade 10 Mathematics and ELA Testing and Content document and the Grade 10 Mathematics Testing: Further Focus document. Common Core State StandardS for matHematICS

mathematics | Standards for mathematical Practice

The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy).

1 Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

2 Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing

symbols as if they have a life of their own, without necessarily attending to | StandardS for matHematICal praCtICe their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

3 Construct viable arguments and critique the reasoning of others. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, 6 Common Core State StandardS for matHematICS

communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

4 Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

5 Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external | StandardS for matHematICal praCtICe mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.

6 Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. 7 Common Core State StandardS for matHematICS

7 Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive 2 property. In the expression x + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.

8 Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.

Connecting the Standards for Mathematical Practice to the Standards for Mathematical Content The Standards for Mathematical Practice describe ways in which developing student practitioners of the discipline of mathematics increasingly ought to engage with the subject matter as they grow in mathematical maturity and expertise throughout the elementary, middle and high school years. Designers of curricula, assessments, and professional development should all attend to the need to connect the mathematical practices to mathematical content in mathematics instruction.

The Standards for Mathematical Content are a balanced combination of procedure and understanding. Expectations that begin with the word “understand” are often especially good opportunities to connect the practices to the content. Students

who lack understanding of a topic may rely on procedures too heavily. Without | StandardS for matHematICal praCtICe a flexible base from which to work, they may be less likely to consider analogous problems, represent problems coherently, justify conclusions, apply the mathematics to practical situations, use technology mindfully to work with the mathematics, explain the mathematics accurately to other students, step back for an overview, or deviate from a known procedure to find a shortcut. In short, a lack of understanding effectively prevents a student from engaging in the mathematical practices.

In this respect, those content standards which set an expectation of understanding are potential “points of intersection” between the Standards for Mathematical Content and the Standards for Mathematical Practice. These points of intersection are intended to be weighted toward central and generative concepts in the school mathematics curriculum that most merit the time, resources, innovative energies, and focus necessary to qualitatively improve the curriculum, instruction, assessment, professional development, and student achievement in mathematics. 8 HIGH SCHool | 57

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Make Make sense of problems solving and them. persevere in Reason abstractly and quantitatively. Construct viable arguments and the critique reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express reasoning. regularity in repeated Common Core State StandardS for matHematICS matHematICS for StandardS State Core Common mathematical Practices Practices mathematical 1. 2. 3. 4. 5. 6. 7. 8. reason quantitatively and use units to solve solve and use units to quantitatively reason problems with complex arithmetic operations Perform numbers numbers and their complex represent plane the complex on operations numbers in polynomial identities Use complex and equations quantities. and model with vector represent on vectors. operations Perform and use on matrices operations Perform in applications. matrices extend the properties of exponents to rational rational to of exponents the properties extend exponents and irrational of rational Use properties numbers. Quantities • The Complex Number System • • • Vector and Matrix Quantities • • • � overview and Quantity number System Number The Real • • HIGH SCHool — nUmber and qUantIty | 60 n n-Q n-Q

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g(x)

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2 find the points of intersection between the line y the line between of intersection find the points or greater). Graph the solutions to a plane linear (excluding inequality the in boundary two in graph variables the the as case solution a of set half- a to strict a inequality), system and of linear inequalities in two variables Understand Understand that the graph of all an its equation solutions in plotted two in (which variables the could is coordinate be the plane, a set often line). of forming a Explain curve why the the equations (+) Represent a system of in linear a equations vector as variable. a single matrix (+) equation Find the inverse of of a linear matrix equations if (using it technology exists for and matrices use of it dimension to 3 solve systems equation technology to graph the functions, successive make approximations. tables Include of cases values, where or find Solve systems of linear equationsgraphs), exactly focusing and on approximately pairs (e.g., of with linear equations Solve in a two simple variables. system consisting equation of in a two linear variables equation algebraically and and a graphically. quadratic y are are linear, polynomial, rational, absolute logarithmic value, functions. exponential, and as the intersection of the corresponding half-planes. 8. 12. 10. 10. 11. 9. 9. 6. 7. Represent and solve equations and inequalities graphically graphically and inequalities equations and solve Represent HIGH SCHool — fUnCtIonS | 67

v; the rule

Common Core State StandardS for matHematICS matHematICS for StandardS State Core Common

;bx or by a recursive rule. The graph

a +

f(x) =

finding inputsthat yieldabouta when given two involvesoutput functions have equations, solving the whose an same solutions Questions equation. value can for be Because the visualized functions same from describe input the relationships lead intersection between used to of quantities, in their they modeling. are graphs. Sometimes frequently functions can are be defined displayed effectively by using a recursive a process, spreadsheet which or other technology. Connections to Expressions, Equations, Modeling, and Coordinates. Modeling, and Coordinates. Equations, Expressions, to Connections of a function is often models, a and useful manipulating way a of mathematicalon visualizing expression the for the function’s a relationship properties. function of can the throw function Functions light presented as expressions can important model families many of important functions phenomena. characterizedwhich Two by grow laws at of a growth constant percent are rate, rate. linear and Linear functions, exponential functions functions, with relationships. which a grow constant at term a of constant zero describe A proportional graphing utility or a properties computer of algebra these system functions can and of be their functions, used graphs including to and recursively experiment defined to with build functions. computational models T(v) = v expresses 100/ this relationship name algebraically is and defines a function whose The set of inputs toall a inputs function for is which called thefunction its expression makes defining domain. sense We in often a function a infer given the has context. A domainvalue, a function to can be be or for describeda which the in seismograph); by various a ways, such verbal city;” as rule, by as by an a in, algebraic graph “I’ll expression (e.g., like give the you trace a of state, you give me the capital mathematics | High School—functions | High School—functions mathematics Functions describe situations where one the quantity return determines on another. $10,000 For invested function example, at of an the annualized length percentage of theories rate time about of the dependencies 4.25% money between is is quantities are a invested. in important Because nature tools we and in continually society, the make functions construction of In mathematical school models. mathematics, functions usually often defined have numerical by inputs an and algebraica expression. outputs car and to example, For are drive 100 the miles is time a in function hours of takes for it the car’s speed in miles per hour, Determining an output value for a particular input involves evaluating an expression; HIGH SCHool — fUnCtIonS | 68

mathematical Practices Practices mathematical 1. 2. 3. 4. 5. 6. 7. 8. Build a function that models a relationship models a relationship Build a function that quantities two between functions existing functions from Build new and quadratic, linear, and compare Construct problems models and solve exponential of functions in terms for expressions Interpret model they the situation functions the domain of trigonometric extend using the unit circle model periodic phenomena with trigonometric functions and apply trigonometric identities Prove Understand the concept of a function and use a function and of concept the Understand function notation in arise in applications that functions Interpret of the context terms functions using different analyze representations

Building Functions Building Functions • • and Exponential Models Quadratic, Linear, • • Functions Trigonometric • • • � overview functions Functions Interpreting • • • HIGH SCHool — fUnCtIonS | 69

f-If f-If

, and t/10

Common Core State StandardS for matHematICS matHematICS for StandardS State Core Common ).

x is an element

( ★ f x , y = (1.2) For example, the example, For 12t =

corresponding corresponding to the

, y = (1.01) t ★

For example, if the function if the example, For

is a function and

f , y = (0.97)

For example, identify percent rate of change of rate identify percent example, For

is the graph of the equation

) denotes the output of f x ( f Key features include: intercepts; intervals where the where intervals include: intercepts; features Key ★

★

. The graph of

in functions such as y = (1.02) classify them as representing exponential growth or decay. or decay. growth exponential them as representing classify Use the properties of exponents exponential to functions. interpret expressions for Use the process of factoring quadratic and function completing to the show square zeros, of in extreme the a values, graph, and and symmetry interpret these in terms of a context. Graph Graph polynomial functions, identifying zeros factorizations when are suitable available, and showing end behavior. (+) Graph rational functions, identifying when zeros suitable and factorizations asymptotes are available, behavior. and showing end Graph exponential and logarithmic functions, and showing end intercepts behavior, and trigonometric midline, functions, and showing amplitude. period, Graph Graph linear and quadratic functions maxima, and and show minima. intercepts, Graph square root, cube root, including and step piecewise-defined functions functions, and absolute value functions.

b. a. c. d. e. a. b. h(n) gives the number of person-hours it takes to assemble n engines in a assemble to person-hours it takes the number of h(n) gives the domain for be an appropriate would integers then the positive factory, function. function is increasing, decreasing, positive, or negative; relative maximums relative or negative; positive, decreasing, function is increasing, and periodicity. end behavior; and minimums; symmetries; Write a function defined by forms to expressionan reveal and different in explain different equivalent but properties of the function. Calculate Calculate and interpret the average (presented rate symbolically of or change as of a a Estimate table) the function over rate a of specified change interval. from a graph. Graph functions expressed symbolically and the show graph, key by features hand of in complicated simple cases. cases and using technology for more Relate Relate the domain of a the function quantitative to relationship its it graph describes. and, where applicable, to For For a function that models interpret a key features relationship of between two graphsand quantities, and sketch tables graphs in showing terms keyof of features the the given relationship. quantities, a verbal description Use function notation, evaluate functionsand for interpret inputs statements in that their use context. domains, function notation in terms of a Recognize that sequences are functions, recursively, sometimes whose defined domain is a subset of the integers. Understand Understand that a function from another one set set (called (called the the range) exactly domain) assigns one to to element each of element the of range. the If domain Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + by f(0) = f(1) = 1, f(n+1) recursively is defined sequence Fibonacci ≥ 1. n for f(n-1) input of its domain, then 8. 7. 7. 5. 2. 3. 1. 6. 4. � functions Interpreting Analyze functions using different representations representations functions using different Analyze Interpret functions that arise in applications in terms of the context the context of in terms arise in applications functions that Interpret Understand the concept of a function and use function notation use function notation of a function and the concept Understand HIGH SCHool — fUnCtIonS | 70

★

f-Le f-Le f-Bf f-Bf For For

), For For x ( . k f

, Common Core State StandardS for matHematICS matHematICS for StandardS State Core Common k

) +

x ( f

) by x (

(both positive and negative); ★

For example, if T(y) is the temperature in if T(y) is the temperature example, For ★

or f(x) = (x+1)/(x–1) for x ≠ 1. for or f(x) = (x+1)/(x–1) 3

given given the graphs. Experiment with cases and

k ) for specific values of

For example, given a graph of one quadratic function and function one quadratic of a graph given example, For

(

example, f(x) =2 x example, the atmosphere as a function of height, and h(t) is the height of a h(t) is the height of height, and of as a function the atmosphere temperature then T(h(t)) is the time, of balloon as a function weather time. as a function of balloon the weather of the location at example, build a function that models the temperature of a cooling a cooling of the temperature models function that build a example, and exponential, decaying a function to a constant adding body by the model. these functions to relate (+) Verify by composition that another. one function is the inverse of (+) Read values of an given inverse that function the from function a has graph an or inverse. a (+) table, Produce an invertible function by from restricting a the non-invertible domain. function (+) Compose functions. Determine Determine an explicit expression, a calculation recursive from process, a or context. steps for Combine standard function types using arithmetic operations. Recognize Recognize situations in which one rate quantity per changes unit at interval a relative to constant another. Recognize situations in which a constant quantity percent grows rate or per decays unit by interval a relative to another Prove Prove that linear functions grow intervals, by and equal that differences exponential over functions equal over grow equal by intervals. equal factors Solve Solve an equation of the that form has f(x) an = inverse c and for write a an simple expression for function the f inverse.

), and

kx c. a. b. b. c. a. b. c. d. a. ( Find inverse functions. Distinguish Distinguish between situations that can functions be and modeled with with exponential linear functions. Include recognizing even and odd functions from their graphs and their graphs and odd functions from even Include recognizing them. for expressions algebraic Identify the effect on the f graph of replacing Write Write arithmetic and geometric sequences with both an recursively explicit and formula, use between them the to two model forms. situations, and translate Write Write a function that describes a relationship between two quantities. Compare Compare properties of two functions way each (algebraically, graphically, represented numerically in in a tables, descriptions). different or by verbal

(+) Understand the inverse relationship logarithms between and exponents use and this relationship logarithms to and solve exponents. problems involving an algebraic expression for another, say which has the larger maximum. maximum. has the larger which say another, for expression an algebraic find the value of illustrate illustrate an explanation of the effects on the graph using technology. 4. 1. 3. 1. 9. 9. 2. 5. Linear, Quadratic, and exponential models exponential and Quadratic, Linear, � functions Building Construct and compare linear, quadratic, and exponential models and exponential quadratic, linear, and compare Construct problems and solve Build new functions from existing functions existing functions from Build new Build a function that models a relationship between two quantities quantities two between models a relationship that Build a function HIGH SCHool — fUnCtIonS | 71

f-tf f-tf

; e

Common Core State StandardS for matHematICS matHematICS for StandardS State Core Common

is 2, 10, or

π+x, and 2π–x in terms (θ) = 1 and use it to find 2

★ π–x,

★

(θ) + cos 2

/6, π/6, and use the unit circle to express

x is any real number.

are are numbers and the base

/4 π/4 and d

π/3, x, where , and

c ,

where where

of their values for Prove Prove the Pythagorean identity sin (+) Prove the addition and tangent subtraction and formulas use for them sine, to cosine, solve and problems. (+) Understand that restricting a on trigonometric which function it to is a always to domain increasing be or constructed. always decreasing allows its (+) inverse Use inverse functions to in solve modeling trigonometric contexts; equations evaluate that the interpret arise solutions them using in technology, terms and of the context. (+) Use the unit circleperiodicity to of explain trigonometric symmetry functions. (odd and even) and Choose trigonometric functions to model specified periodic amplitude, phenomena frequency, and with midline. Explain how the unit circle extension in of the trigonometric coordinate functions plane to radian enables all measures the real of numbers, angles interpreted traversed circle. as counterclockwise around the unit (+) Use special triangles to cosine, determine tangent geometrically for the values the of values sine, of sine, cosine, and tangent for Interpret Interpret the parameters in a a linear context. or exponential function in terms of Understand radian measure of an unit angle circle as subtended the by length the of angle. the arc on the For For exponential models, express as ab a logarithm the solution to Construct Construct linear and exponential functions, geometric including sequences, arithmetic given and a graph, two a input-output description pairs of (include a reading relationship, these or from Observe a using table). graphs and tables exponentially that eventually a exceeds a quantity quantity increasing quadratically, increasing or linearly, (more generally) as a polynomial function. sin(θ), θ), cos( or tan()θ given of sin( θ), the θ), cos( angle. or tan(θ) and the quadrant evaluate evaluate the logarithm using technology. 8. 6. 7. 4. 5. 2. 3. 5. 1. 4. 2. 3. 9. 9. � functions trigonometric Prove and apply trigonometric identities and apply Prove Model periodic phenomena with trigonometric functions Model periodic phenomena with Extend the domain of trigonometric functions using the unit circle trigonometric functions using the unit circle the domain of Extend Interpret expressions for functions in terms of the situation they they of the situation terms functions in for expressions Interpret model HIGH SCHool — modelInG | 72

Common Core State StandardS for matHematICS matHematICS for StandardS State Core Common

Estimating how much water andrelief food in is a needed devastated city for distributed. of emergency 3 million people, and howPlanning it a might table be tennis tournamenttables, for where 7 each players player at plays against a each club Designing other with the player. 4 layout of themuch stalls money in as a possible. school fair so Analyzing as stopping to distance for raise as a car. Modeling savings account balance, bacterial investment colony growth. growth, or Engaging in critical path analysis, aircraft at e.g., an applied airport. to turnaround of an Analyzing risk in situations such and as terrorism. extreme sports, pandemics, Relating population statistics to individual predictions.

• • • • • • • •

mathematics | High School—modeling | High School—modeling mathematics Modeling links classroom mathematics and decision-making. statistics Modeling to is everyday the life, process mathematics work, of and and choosing statistics and to analyze using better, and empirical appropriate to situations, improve to decisions. understand economic, Quantities them public and policy, their social, relationships and mathematical in everyday and physical, situations statistical can methods. be When technology modeled making is using mathematical valuable models, for varying comparing assumptions, predictions exploring with consequences, data. and A model can be very and simple, number such bought, as or writing using a total a coin. cost Even geometric as such shape a simple to to product models describe model of involve a a making unit physical coin choices. price object as disk It like a works is three-dimensional well up cylinder, enough to or for route, us whether our a whether a purposes. production two-dimensional schedule, Other or situations—modeling elaborate a a models comparison delivery that of use loan other situations amortizations—need tools are more from not the organized mathematical and representing sciences. labeled such Real-world for models, analysis; and formulating analyzing Like tractable them every models, such is process, appropriately this a depends creative process. on Some acquired examples expertise of as such well situations as might creativity. include: involves involves (1) identifying variables in the situation and selecting In situations like these, the precise models an devised answer depend do on we need want a to or number understand, need? of control, What factors: or have? aspects How The optimize? What of range of resources the of models situation the time that do limitations and we we of tools can most our do create mathematical, to and we statistical, recognize analyze significant and is variables technical alsoand skills, various constrained relationships and kinds, by among our spreadsheets them. ability and Diagrams other for of technology, understanding and and algebra solving are problems situations. powerful drawn tools from different types of real-world One of the insights providedmathematical by or mathematical statistical modeling structure is can situations. that sometimes Models essentially model can the seemingly also same different shed the light mathematical structures on themselves, for example, as when a growth model makes more of vivid bacterial the growth explosive of the exponential function. The basic modeling cycle is summarized in the diagram. those It that represent essential features, a (2) model formulating by creating and tabular, algebraic, selecting or geometric, statistical graphical, representations relationships that between describe the variables, (3) on analyzing these and relationships performing to operations draw mathematics conclusions, in (4) terms interpreting of the the comparing results original them of situation, with the (5) the validating situation, the and conclusions then by either improving the model or, if it HIGH SCHool — modelInG | 73

Common Core State StandardS for matHematICS matHematICS for StandardS State Core Common ).

★ over over time. 2

Modeling is best interpreted not as a collection of isolated isolated of a collection as not interpreted Modeling is best

Analytic modeling seeks to explain albeit data with on parameters the that basis are of empirically of bacterial based; deeper colonies for theoretical (until example, ideas, cut-off intervene) exponential follows mechanisms growth from such a as constant pollution tool reproduction or for rate. starvation analyzing Functions such are problems. an important Graphing utilities, spreadsheets, computer algebra software systems, are and powerful dynamic tools geometry that phenomena can (e.g., be the used behavior to of model polynomials) purely as mathematical well as physical phenomena. topics but rather in relation to other standards. Making mathematical models is Making mathematical standards. other to in relation but rather topics appear standards modeling and specific Practice, Mathematical for a Standard ( symbol a star by indicated the high school standards throughout modeling Standards modeling Standards is acceptable, (6) reporting on Choices, the assumptions, conclusions and and approximations are the present reasoning throughout behind In this them. descriptive cycle. modeling, a model them simply in describes a the compact phenomena form.for or Graphs example, graphs of summarizes of observations global are a temperature and familiar atmospheric descriptive CO model— HIGH SCHool — Geometry | 74

Common Core State StandardS for matHematICS matHematICS for StandardS State Core Common

The The correspondence between numerical coordinates

Connections to Equations. Equations. to Connections mathematics | High School—Geometry | High School—Geometry mathematics An understanding of the attributes applied and in relationships diverse of contexts—interpreting a geometric of schematic objects wood drawing, can needed estimating be to the frame sewing amount a pattern sloping for roof, the rendering most computer efficient graphics, useAlthough or there designing material. of are a many types to of plane geometry, Euclidean school geometry, mathematics studied analytically is both (with devoted synthetically coordinates). primarily (without Euclidean coordinates) geometry by and is the characterized Parallel most Postulate, importantly that parallel through line. a (Spherical point geometry, not in on contrast, has aDuring no given high parallel line school, lines.) there students is beginelementary exactly to and one formalize middle their school, geometry using proofs. experiences Later more from in precise definitions college some and from developing students a careful develop small Euclidean set and of other axioms. geometries The carefully concepts of congruence, similarity, the and perspective symmetry of can geometric be transformation. translations, understood Fundamental rotations, from are reflections, the and combinations rigid assumed motions: to of these, preserve distance all and and of angles rotations which (and each are here therefore explain shapes a object generally). particular offer Reflections type insight of into symmetry, its triangle and attributes—as assures the when that symmetries the its of reflective base an symmetry angles of are In congruent. the an isosceles approach taken here, is two a geometric sequence figures of are rigid of defined motions superposition. that For to carries triangles, congruent be one congruence pairs means onto of if there the the sides equality other. and This of all experiences is all drawing corresponding the triangles corresponding pairs principle from of given angles. enough conditions, During measures students the in notice middle a ways grades, triangle congruent. to through to Once specify ensure these that triangle all congruence using triangles criteria rigid (ASA, drawn motions, with SAS, they and those can and SSS) measures be other are are used geometric established to figures. prove theorems about Similarity triangles, transformations quadrilaterals, (rigid motions followedin by the dilations) same define way similarity that similarity rigid ideas motions of define "same congruence, shape" These thereby transformations and formalizing lead "scale the to factor" the developed corresponding in criterion angles the for are triangle middle congruent. similarity grades. that two The pairs definitions of of sine, cosine,triangles and and tangent similarity, and, with for real-world the acute and Pythagorean angles theoretical are Theorem, situations. are founded Theright fundamental on right Pythagorean triangles in Theorem by many is the generalized Law the to of triangle non- Cosines. congruence Together, criteria the for to Laws the completely of cases solve Sines a where and three triangle. Cosines the pieces Furthermore, embody ambiguous these of case, laws information illustrating yield suffice that two Side-Side-Angle possible is Analytic solutions not geometry in a connects congruence algebra criterion. and of geometry, analysis resulting and in problem powerful solving. locations methods Just in as one the dimension, number a with line pair locations associates of in numbers perpendicular two with dimensions. axes and associates This geometric pairs correspondence points between of allows numerical numbers methods versa. coordinates from The algebra solution to set be of a applied an tool to equation for geometry becomes doing and a and equations, vice geometric understanding making curve, algebra. algebraic making Geometric manipulation visualization shapes intomodeling, can a and be tool proof. Geometric for described geometric transformations by to of understanding, algebraic the changes graphs in of their equations equations. correspond Dynamic geometry environments provide students tools with that experimental allow and them modeling to computer investigate algebra geometric systems phenomena allow in them much to experiment the with same algebraic way phenomena. as and geometric points allows methods versa. from The algebra solution to set be of a applied an tool to equation for geometry becomes doing and a and equations, vice geometric understanding making curve, algebra. algebraic making Geometric manipulation visualization shapes intomodeling, can a and be tool proof. for described geometric by understanding, HIGH SCHool — Geometry | 75

mathematical Practices Practices mathematical 1. 2. 3. 4. 5. 6. 7. 8. Understand similarity in terms of similarity in terms similarity Understand transformations similarity involving theorems Prove problems and solve ratios define trigonometric right triangles involving triangles general apply trigonometry to about circles and apply theorems Understand circles of of sectors lengths and areas find arc the geometric description between translate section a conic for and the equation simple geometric prove to Use coordinates algebraically theorems solve and use them to formulas explain volume problems two- between relationships Visualize objects dimensional and three-dimensional in modeling apply geometric concepts situations experiment with transformations in the plane in the plane with transformations experiment of rigid in terms congruence Understand motions theorems geometric Prove constructions geometric make

Similarity, Right Triangles, and Trigonometry and Trigonometry Right Triangles, Similarity, • • • • Circles • • with Equations Geometric Properties Expressing • • and Dimension Geometric Measurement • • Modeling with Geometry • � overview Geometry Congruence • • • • HIGH SCHool — Geometry | 76 G-Co G-Co

Common Core State StandardS for matHematICS matHematICS for StandardS State Core Common

Copying a segment; Copying

Theorems include: vertical include: vertical Theorems Theorems include: opposite include: opposite Theorems

Theorems include: measures of interior interior of include: measures Theorems

copying an angle; bisecting a segment; bisecting an angle; constructing an angle; bisecting a segment; bisecting an angle; constructing copying a line segment; of perpendicular lines, including the perpendicular bisector on the point not a line through a given to a line parallel and constructing line. Construct Construct an equilateral triangle, a inscribed square, in and a a circle. regular hexagon Make formal geometric constructions withmethods a (compass variety and of straightedge, tools string,paper and reflective folding, devices, dynamic geometric software, etc.). Prove Prove theorems about parallelograms. Prove Prove theorems about triangles. Explain how the criteria for follow triangle from congruence the (ASA, definition SAS, of and congruence SSS) terms in of rigid motions. Prove theorems about lines and angles. Use geometric descriptions of rigid to motions predict to the transform effect figures of two and a figures, given use rigid the motion definition to on decide a of congruence if given they figure; in are given congruent. terms Use of rigid motions the definition of congruence that two terms in triangles are of congruent sides rigid motions if and and corresponding to show only pairs if of corresponding angles pairs are of congruent. Develop Develop definitions rotations,of reflections,of angles, and translations circles, perpendicular in terms lines, parallel lines, Given and a line geometric segments. figure and draw the rotation, a transformed reflection, figure using,geometry or translation, e.g., software. graph Specify paper, a tracingpaper, sequence carry or of a transformations given that figure will onto another. Represent Represent transformations in the plane and using, geometry e.g., software; transparencies describe transformations take as points functions in that the plane Compare as transformations inputs that and preserve distance give that and other do angle points not to as (e.g., those outputs. translation versus horizontalGiven a stretch). rectangle, parallelogram, trapezoid, describe or the regular rotations polygon, and reflections that carry onto it itself. Know Know precise definitions of angle, line, circle, and perpendicular line line, segment, parallel based distance on along the a undefined line, notions and distance of point, around line, a circular arc. angles of a triangle sum to 180°; base angles of isosceles triangles are triangles are isosceles 180°; base angles of sum to a triangle angles of a triangle is sides of two midpoints of the segment joining congruent; a triangle side and half the length; the medians of the third to parallel a point. meet at sides are congruent, opposite angles are congruent, the diagonals congruent, are angles opposite congruent, sides are are rectangles and conversely, bisect each other, a parallelogram of diagonals. � with congruent parallelograms angles are congruent; when a transversal crosses parallel lines, alternate lines, alternate parallel crosses when a transversal congruent; angles are congruent; angles are and corresponding congruent angles are interior those exactly a line segment are of points on a perpendicular bisector endpoints. � the segment’s from equidistant 13. 11. 12. 8. 9. 6. 7. 4. 5. 2. 3. 1. 10. 10. � Congruence Make geometric constructions geometric constructions Make Prove geometric theorems geometric theorems Prove Understand congruence in terms of rigid motions of rigid in terms congruence Understand Experiment with transformations in the plane in the with transformations Experiment HIGH SCHool — Geometry | 77 t G-C G-C G-Sr

Common Core State StandardS for matHematICS matHematICS for StandardS State Core Common

dilations given by a center and

) for the area of a triangle by C

sin( Theorems include: a line parallel to one to include: a line parallel Theorems

ab ★

= 1/2 A

Include the relationship between central, inscribed, and inscribed, central, between Include the relationship

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

a. b. circumscribed angles; inscribed angles on a diameter are right angles; are angles on a diameter angles; inscribed circumscribed the radius the tangent where is perpendicular to a circle of the radius the circle. intersects Construct Construct the inscribed and circumscribed prove circles properties of of a angles triangle, for and a quadrilateral (+) inscribed Construct in a a tangent circle. line circle. from a point outside a given circle to the Prove Prove that all circles are similar. Identify and describe relationships among and inscribed chords. angles, radii, (+) Prove the Laws of problems. Sines and Cosines and use them(+) to Understand solve and apply the to Law find of unknown Sines measurements and surveying the in right problems, Law resultant of and non-right forces). Cosines triangles (e.g., Use trigonometric ratios and thetriangles Pythagorean in Theorem applied to problems. solve right (+) Derive the formula Understand Understand that by similarity, sideproperties ratios of in the right angles triangles in trigonometric are the ratios triangle, for leading acute to angles. definitions of Explain and use the relationship complementary between angles. the sine and cosine of Prove Prove theorems about triangles. Use congruence and similarity criteriaand for to triangles prove to relationships solve in problems geometric figures. Use the properties of similaritycriterion transformations for to two establish triangles the to AA be similar. a scale factor: Given two figures, use the transformations definition to decide of similarity if they transformations in are the similar; meaning similarity terms of explain of using similarityof similarity for all triangles corresponding as pairs the of corresponding equality angles pairs and of the sides. proportionality of all Verify experimentally the properties of side of a triangle divides the other two proportionally, and conversely; the and conversely; proportionally, two divides the other a triangle side of similarity. using triangle proved Pythagorean Theorem drawing drawing an auxiliary line from side. a vertex perpendicular to the opposite 3. 5. 4. 1. 2. 10. 10. 11. 8. 9. 6. 7. 4. 3. 2. 1. � Circles � trigonometry and triangles, right Similarity, Understand and apply theorems about circles about circles and apply theorems Understand Apply trigonometry to general triangles general Apply trigonometry to Define trigonometric ratios and solve problems involving right involving problems and solve ratios Define trigonometric triangles Prove theorems involving similarity involving theorems Prove Understand similarity in terms of similarity transformations transformations of similarity similarity in terms Understand HIGH SCHool — Geometry | 78 d

For For ★

G-mG G-mG G-GPe G-GPe

G-Gm Use ★

Common Core State StandardS for matHematICS matHematICS for StandardS State Core Common ★ ★

★

Derive the equation of a parabola given a focus and directrix. example, prove or disprove that a figure defined by four given points in the in points given four by defined figure a that disprove or prove example, point (1, √3) lies the that or disprove prove plane is a rectangle; coordinate 2). the point (0, the origin and containing at centered on the circle Apply geometric methods to solve an design object problems or (e.g., structure designing to working satisfy with physical typographic constraints grid or systems minimize based cost; on ratios). Use geometric shapes, their measures, objects and (e.g., their modeling properties a to tree describe trunk orApply a concepts human of torso density as based situations a on (e.g., cylinder). area persons and per volume square in mile, modeling BTUs per cubic foot). Use volume formulas for cylinders, solve pyramids, problems. cones, and spheres to Identify the shapes of two-dimensional dimensional cross-sections objects, of and three- identify three-dimensional by objects rotations generated of two-dimensional objects. Give Give an informal argument for a the circle, formulas area for of the a circumference circle, of volume of a cylinder, pyramid, and (+) cone. Give an informal argument formulas using for Cavalieri’s the principle volume for of the a sphere and other solid figures. Find the point on a that directed partitions line the segment segment between in two a given given Use points ratio. coordinates to compute perimeters triangles of and polygons rectangles, and e.g., areas using of the distance formula. Prove Prove the slope criteria for them parallel to and solve perpendicular geometric lines problems parallel and (e.g., or use find perpendicular the to equation a point). given line a line of that passes through a given (+) Derive the equations of using ellipses the and fact hyperbolas that given the constant. the sum foci, or difference of distances from the foci is Use coordinates to prove simple geometric theorems algebraically. Derive Derive the equation of a Pythagorean circle Theorem; of complete given the center squareradius and to of radius find a using the circle the center given and by an . equation Derive Derive using similarity the fact by that an the angle length is of proportional measure the to of arc the the intercepted radius, angle and as formula define the for the constant the radian of area proportionality; of derive a the sector.

dissection arguments, Cavalieri’s principle, and informal limit arguments. arguments. limit and informal principle, Cavalieri’s arguments, dissection 5. 3. 1. 2. 3. 4. 1. 6. 7. 2. 3. 4. 1. 5. 2. � Geometry with modeling � dimension and measurement Geometric � equations with Properties Geometric expressing Apply geometric concepts in modeling situations in modeling situations Apply geometric concepts Visualize relationships between two-dimensional and three- two-dimensional between relationships Visualize dimensional objects Explain volume formulas and use them to solve problems problems solve and use them to formulas Explain volume Use coordinates to prove simple geometric theorems algebraically algebraically simple geometric theorems prove to Use coordinates Translate between the geometric description and the equation for a for and the equation the geometric description between Translate section conic Find arc lengths and areas of sectors of circles of circles of sectors and areas lengths Find arc HIGH SCHool — StatIStICS | 79

Common Core State StandardS for matHematICS matHematICS for StandardS State Core Common

. Functions may be used to describe

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mathematical Practices Practices mathematical 1. 2. 3. 4. 5. 6. 7. 8. Understand and evaluate random processes processes random and evaluate Understand experiments underlying statistical from conclusions and justify inferences make and observational experiments sample surveys, studies and conditional independence Understand data interpret and use them to probability compute to Use the rules of probability in a uniform events of compound probabilities model probability to and use them values expected Calculate problems solve of outcomes evaluate to Use probability decisions Summarize, represent, and interpret data on a data and interpret represent, Summarize, variable measurement or single count on data interpret and represent, Summarize, variables quantitative and categorical two models linear Interpret

Making Inferences and Justifying Conclusions Conclusions and Justifying Making Inferences • • - and the Rules of Prob Probability Conditional ability • • Decisions Make to Using Probability • • Interpreting Categorical and Quantitative Data Data Quantitative and Categorical Interpreting • • • � overview Probability and Statistics HIGH SCHool — StatIStICS | 81

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Common Core State StandardS for matHematICS matHematICS for StandardS State Core Common

For example, a model example, For

Use given functions or choose Use given

a function suggested by the context. Emphasize linear, quadratic, and quadratic, linear, Emphasize the context. by a function suggested models. exponential Fit a function to the problems data; in use the functions context fitted of to the data data. solve to Informally assess the fit residuals. of a function by plotting analyzing and Fit a linear function for association. a scatter plot that suggests a linear

a. b. c. Recognize Recognize the purposes of and experiments, differences and among observational sample studies; surveys, explain relates how to randomization each. Understand statistics as a process population for parameters making based inferences on about a population. random sample from that Decide if a specified model data-generating process, e.g., consistentis using results with simulation. a given from Distinguish Distinguish between correlation and causation. Interpret Interpret the slope (rate of of change) a and linear the model intercept in (constant the term) context Compute of (using the technology) data. and interpret of the a correlation linear coefficient fit. Summarize Summarize categorical data for two tables. categories Interpret in relative two-way frequencies frequency in (including the joint, context marginal, of and the conditional data Recognize relative possible frequencies). associations and trends in the Represent data. data on two quantitative describe variables how on the a variables scatter are plot, related. and Use the mean and standard distribution deviation and of to a estimate data population there set percentages. are to Recognize data fit that sets it for Use which calculators, to such spreadsheets, a normal a and procedure tables normal is to curve. not estimate appropriate. areas under the Use statistics appropriate to the compare shape center of (median, the mean) data and standard distribution spread deviation) to (interquartile of range, two or more different Interpret data differences sets. in shape, center, the and data spread sets, in accounting the for context (outliers). of possible effects of extreme data points Represent data with plots on histograms, the and real box number plots). line (dot plots, says a spinning coin falls heads up with probability 0.5. Would a result of 5 of a result Would 0.5. heads up with probability falls a spinning coin says the model? question to cause you tails in a row 3. 1. 2. 9. 9. 7. 7. 8. 5. 6. 4. 2. 3. 1. � Conclusions Justifying and Inferences making � data Quantitative and Categorical Interpreting Make inferences and justify conclusions from sample surveys, sample surveys, from conclusions and justify inferences Make studies and observational experiments, Understand and evaluate random processes underlying statistical underlying statistical processes random and evaluate Understand experiments Interpret linear models linear Interpret Summarize, represent, and interpret data on two categorical and categorical on two data and interpret represent, Summarize, variables quantitative Summarize, represent, and interpret data on a single count or a single count on data and interpret represent, Summarize, variable measurement HIGH SCHool — StatIStICS | 82 S-CP S-CP S-md S-md

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Understand Understand the conditional probability of � a smoker are if you lung cancer having of the chance compare example, � lung cancer. have if you being a smoker of with the chance in terms of the model. the model. (+) Calculate the expected value the of mean a of random the variable; probability interpret distribution. it as (+) Define random a variablea for numerical value a quantity to each corresponding interest of event probability in distribution assigning by a using sample the displays space; same as graph graphical for the data distributions. Apply the Addition Rule, (+) Apply the general Multiplication model, Rule in a uniform probability (+) Use permutations and combinations compound to events compute and probabilities solve of problems. Find the conditional probability of Recognize Recognize and explain the concepts independence of in conditional everyday probability language and and everyday situations. Construct Construct and interpret two-way frequency categories tables are of associated data with when each two-way two object table being as classified. a Use sample and the space to to approximate decide conditional if probabilities. events are independent Understand Understand that two events B Evaluate Evaluate reports based on data. Describe events as subsets of using a characteristics sample (or space categories) (the of intersections, set the or of outcomes, complements or outcomes) of as other unions, events (“or,” “and,” “not”). Use data from a sample proportion; survey develop to a estimate margin a of models population error for mean through random or the sampling. use of simulation Use data from a randomized use experiment simulations to to compare decide two if treatments; significant. differences between parameters are use this characterization to determine if they are independent. of interpret interpret the answer in terms of the model. conditional conditional probability of of data from a random sample of students in your school on their favorite school on their favorite in your students sample of a random from data a that the probability and English. Estimate science, subject among math, that given science school will favor your from student selected randomly subjects and compare other Do the same for grade. is in tenth the student the results. outcomes outcomes that also belong to probability probability of 5. 2. 1. 7. 7. 8. 9. 6. 4. 2. 3. 6. 1. 4. 5. � decisions make to Probability Using Conditional Probability and the rules of Probability Probability of rules the and Probability Conditional Calculate expected values and use them to solve problems problems solve and use them to values expected Calculate Use the rules of probability to compute probabilities of compound of compound probabilities compute to Use the rules of probability model probability in a uniform events Understand independence and conditional probability and use them probability and conditional independence Understand data interpret to HIGH SCHool — StatIStICS | 83

Common Core State StandardS for matHematICS matHematICS for StandardS State Core Common

For example, find example, For

For example, find the theoretical probability probability find the theoretical example, For

For example, find a current data distribution on the distribution data find a current example, For

the expected winnings from a state lottery ticket or a game at a fast- or a game at ticket lottery a state winnings from the expected restaurant. food For example, compare a high-deductible versus a low-deductible a low-deductible a high-deductible versus compare example, For of chances but reasonable, various, using policy insurance automobile a minor or a major accident. having Find the expected payoff for a game of chance. Evaluate and compare strategies on the basis of expected values.

a. b. (+) Use probabilities to makea fair random decisions number (e.g., generator). drawing by lots, using (+) Analyze decisions and strategies product using testing, probability medical concepts testing, (e.g., pulling a a game). hockey goalie at the end of (+) Weigh the possible outcomes probabilities of to a payoff decision values by and assigning finding expected values. (+) Develop a probability distribution for for a a sample random space variable in the defined which expected probabilities value. are assigned empirically; find (+) Develop a probability distribution for for a a sample random space variable in find defined which the theoretical expected probabilities value. can be calculated; distribution for the number of correct answers obtained by guessing on guessing by obtained answers correct of the number for distribution four has question each where test of a multiple-choice questions all five schemes. grading various under grade expected and find the choices, number of TV sets per household in the United States, and calculate the the and calculate States, the United TV sets per household in number of you TV sets would many per household. How sets of number expected households? randomly selected in 100 find to expect 4. 7. 7. 5. 3. 6. Use probability to evaluate outcomes of decisions outcomes evaluate to Use probability