Finding diagonal distances on the coordinate plane. Linear Inequalities. Permutation & Combinations. Standard Deviation

Transcription

1 Introduction of Surface Area Measurement Unit Conversion Volume of Right Rectangular Prisms Statistical Variability (MAD and IQR) Multiplication of Fractions Scientific Notation Introduction of Transformations Area and Circumference of Circles Introduction to Probability Constant of Proportionality Slope- Intercept Form of a Line Finding diagonal distances on the coordinate plane Congruence & Similarity Volume of Cones, Cylinders, and Spheres Linear Inequalities Angle- Angle Criterion Rational Exponents Completing the Square Distance Formula Midpoint Parallel and Perpendicular Lines Literal Equations Radian Measure Unit Circle Matrices Permutation & Combinations Laws of Sines & Cosines Direct Variation Standard Deviation

2 Alignment Activity Answer Key Concept CCSS CCSS Grade/Courses 2003 Grade/Course(s) Standard Introduction of Surface Area 6.G.4 6 th Grade 7 th Grade Measurement Unit Conversions 6.G.3d 6 th Grade 5 th Grade Volume of Right Rectangular Prisms 6.G.2 6 th Grade (fractional edge lengths whole number edges in 5 th grade) 7 th Grade Statistical Variability (MAD and IQR) 6.SP.5c 6 th Grade MAD not in 2003 NCSCOS IQR (7 th grade) Multiplication of Fractions 5 th Grade 5 th Grade 6 th Grade Scientific Notation 8.EE.3 8 th Grade 6 th Grade Introduction of Transformations 8.G.3 8 th Grade 6 th Grade Area and Circumference of 7.G.4, 7 th Grade, Geometry, 7 th Grade Circles G.GMD.1 IM II Introduction to Probability 7.SP.5 7 th Grade 6 th Grade 7.SP.8 Constant of Proportionality 7.RP.2b 7 th Grade Algebra I Slope-Intercept Form of a Line 8.EE.6 8 th Grade 7 th Grade Volumes of Cones, Cylinders, and Spheres 8.G.9, G.GMD.3 8 th Grade, Geometry, IM II Finding diagonal distances on the 8.G.8 8 th Grade Algebra I coordinate plane Linear Inequalities A.REI.12 Algebra I, IM I 8 th Grade Congruence & Similarity 8.G.2, 8.G.4, G.CO.6, G.CO.7, G.CO.8, G.SRT.2 8 th Grade, Geometry, IM I G.SRT.2 in is Geometry and IM II 7 th Grade Cylinders (7 th grade) Cones (Geometry) Spheres (Pre-Calculus) Angle-Angle Criterion 8.G.5, 8 th Grade, Geometry, Geometry G.SRT.3 IM II Rational Exponents N.RN.1 Algebra I, IM II Algebra II Completing the Square A.SSE.3 Algebra I, IM II Algebra II Distance Formula G.GPE.7 Geometry, IM I Algebra I Midpoint G.GPE.6 Geometry, IM II Algebra I Parallel and Perpendicular Lines G.GPE.5 Geometry, IM I Algebra I Literal Equations A.CED.4 Algebra I, IM I Not in 2003 NCSCOS Radian Measure F.TF.1 Algebra II, IM III Pre-Calculus Unit Circle F.TF.2 Algebra II, IM III Pre-Calculus Matrices N.VM.6 N.CM.12 4 th year courses Algebra I, Geometry, Algebra II Permutations & combinations S.CP.9 Geometry, IM II 3 rd and 6 th Grade Laws of Sines & Cosines G.SRT.10 Geometry, IM III Pre-Calculus Direct Variation 7.RP.2 7 th Grade Algebra I Standard Deviation S.ID.2 Algebra I, IM I 4 th year courses

3 Summary Time to Reflect

4 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 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, 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

5 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 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. 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 , in preparation for learning about the distributive property. In the expression x 2 + 9x + 14, older students can see the 14 as 2 7 and the 9 as 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)(x 2 + x + 1), and (x 1)(x 3 + 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.

6 What do teachers do in their classrooms to cause these effects on students? Teacher Actions (Cause) Student Practice (Effect) 1. Make sense of problems and persevere in solving them. DO STUDENTS: Use multiple representations (verbal descriptions, symbolic, tables, graphs, etc.)? Check their answers using different methods? Continually ask, Does this make sense? Understand the approaches of others and identify correspondences between different approaches? Teacher Actions (Cause) Student Practice (Effect) 3. Construct viable arguments and critique the reasoning of others. DO STUDENTS: Make conjectures and build a logical progression of statements to explore the truth of their conjectures? Analyze situations and recognize and use counter examples? Justify their conclusions, communicate them to others, and respond to arguments of others? Hear or read arguments of others and decide whether they make sense, and ask useful questions to clarify or improve the argument?

7 What do teachers do in their classrooms to cause these effects on students? Teacher Actions (Cause) Student Practice (Effect) 2. Reason abstractly and quantitatively. DO STUDENTS: Make sense of quantities and their relationships in problem situations? Decontextualize a problem? Contextualize a problem? Create a coherent representation of the problem, consider the units involved, and attend to the meaning of quantities? Teacher Actions (Cause) Student Practice (Effect) 6. Attend to precision. DO STUDENTS: Communicate precisely to others? Use clear definitions? Use the equal sign consistently and appropriately? Calculate accurately and efficiently?

8 What do teachers do in their classrooms to cause these effects on students? Teacher Actions (Cause) Student Practice (Effect) 4. Model with mathematics. DO STUDENTS: Apply the mathematics they know to solve problems in everyday life? Apply what they know and make assumptions and approximations to simplify a complicated situation as an initial approach? Identify important quantities in a practical situation? Analyze relationships mathematically to draw conclusions? Interpret their mathematical results in the context of the situation and reflect on whether the results make sense? Teacher Actions (Cause) Student Practice (Effect) 5. Use appropriate tools strategically. DO STUDENTS: Consider the available tools when solving mathematical problems? Know the tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful? Identify relevant external mathematical resources and use them to pose or solve problems? Use technological tools to explore and deepen their understanding of concepts?

9 What do teachers do in their classrooms to cause these effects on students? Teacher Actions (Cause) Student Practice (Effect) 7. Look for and make use of structure. DO STUDENTS: Look closely to determine a pattern or structure? Utilize properties? Decompose and recombine numbers and expressions? Teacher Actions (Cause) Student Practice (Effect) 8. Look for and express regularity in repeated reasoning. DO STUDENTS: Notice if calculations are repeated, and look both for general methods and for shortcuts? Maintain oversight of the process, while attending to the details? Continually evaluate the reasonableness of their intermediate result?

10 Cloze Reading Activity Make sense of problems and persevere in solving them. Mathematically students start by to themselves the meaning of a and looking for points to its solution. They givens, constraints, relationships and goals. They make about the form and meaning of the solution and plan a solution rather than simply jumping into a solution attempt. Construct viable arguments and critique the reasoning of others. Mathematically students understand and stated assumptions, definitions, and previously established results in constructing. They make conjectures and build a logical progression of to explore the of their conjectures. Model with Mathematics Mathematically proficient students can the mathematics they know to problems arising in everyday, society, and the workplace. In early grades, this might be as writing an addition equation to a situation. In middle grades, a student might apply reasoning to plan a school event or analyze a problem in the.

11 Teaching Aid Master 2 Area of a Circle Centimeter Graph Paper Prentice-Hall, Inc. All rights reserved. Multiple-Use Classroom Resources 2

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13 6 th Grade At A Glance New to 6 th Grade: Unit rate (6.RP.3b) Measurement unit conversions (6.RP 3d) Number line opposites and absolute value (6.NS.6a, 6.NS.7c) Vertical and horizontal distances on the coordinate plane (6.NS.8) Distributive property and factoring (6.EE.3) Introduction of independent and dependent variables (6.NS.9) Volume of right rectangular prisms with fractional edges (6.G.2) Surface area with nets (only triangle and rectangle faces) (6.G.4) Dot plots, histograms, box plots (6.SP.4) Statistical variability (M.A.D. and Interquartile Range) (6.G.5c) Moved from 6 th Grade: Multiplication of fractions (moved to 5 th grade) Scientific notation (moved to 8 th grade) Transformations (moved to 8 th grade) Area and circumference of circles (moved to 7 th grade) Probability (moved to 7 th grade) Two-step equations (moved to 7 th grade) Solving one- and two-step inequalities (moved to 7 th grade) Notes: Topics may appear to be similar between the CCSS and the 2003 NCSCOS; however, the CCSS may be presented at a higher cognitive demand. Equivalent fractions, decimals and percents are in 6 th grade but as conceptual representations (see 6.RP.2c). Use of the number line (building on elementary foundations) is also encouraged. For more detailed information, see the crosswalks. 6.NS. 2 is the final check for student understanding of place value.

14 7 th Grade At A Glance New to 7 th Grade: Constant of proportionality (7.RP.2b) Percent of error (7.RP.3) Factoring to create equivalent expressions (7.EE.1) Triangle side lengths (7.G.2) Area and circumference of circles (7.G.4) Angles (supplementary, complementary, vertical) (7. G.5) Surface area and volume of pyramids (7.G.6) Probability (7.SP.5 7.SP.8) Moved from 7 th Grade: Similar and congruent polygons (moved to 8 th grade) Surface area and volume of cylinders (moved to 8 th grade volume only) Box plots and histograms (as in the 2003 standards used in CCSS to compare two data sets) (creation of box plots and histograms moved to 6 th grade) Linear relations and functions (y-intercept moved to 8 th grade) Views from 3-Dimensional figures (removed from CCSS) Statistical measures (moved to 6 th grade) Notes: Topics may appear to be similar between the CCSS and the 2003 NCSCOS; however, the CCSS may be presented at a higher cognitive demand. Proportionality in 7 th grade now includes identifying proportional relationships from tables and graphs; writing equations to represent proportional relationships. Using a number line for rational number operations is emphasized in CCSS. For more detailed information, see the crosswalks.

15 8 th Grade At A Glance New to 8 th Grade: Integer exponents with numerical bases (8.EE.1) Scientific notation, including multiplication and division (8.EE.3 and 8.EE.4) Unit rate as slope (8.EE.5) Qualitative graphing (8.F.5) Transformations (8.G.1 and 8.G.3) Congruent and similar figures (characterized through transformations) (8.G.2 and 8.G.4) Angles (exterior angles, parallel cut by transversal, angle-angle criterion) (8.G.5) Finding diagonal distances on a coordinate plane using the Pythagorean Theorem (8.G.8) Volume of cones, cylinders and spheres (8.G.9) Two-way tables (8.SP.4) Moved from 8 th Grade: Indirect measurement (embedded throughout) Linear inequalities (moved to high school) Effect of dimension changes (moved to high school) Misuses of data (embedded throughout) Function notation (moved to high school) Point-slope form (moved to high school) and standard form of a linear equation (not in CCSS) Notes: Topics may appear to be similar between the CCSS and the 2003 NCSCOS; however, the CCSS may be presented at a higher cognitive demand. For more detailed information, see the crosswalks.

16 Algebra I At A Glance New to Algebra I: Rational Exponents (N.RN.1) Radicals as Rational Exponents (N.RN.2) Closure Properties with Rational and Irrational Numbers (N.RN.3) Interpreting Parts of an Expression (A.SSE.1) Reveal and Explain Properties of a Quantity Represented by an Expression (A.SSE.3) Completing the Square (A.SSE.3.b) Transform Expressions for Exponential Functions (A.SSE.3.c) Use Units as a Way to Understand Problems and Guide the Solution (N.Q.1) Define Quantities for Descriptive Modeling (N.Q.2) Choosing Level of Accuracy Appropriate to Limitations on Measurement (N.Q.3) Represent Data with Plots on the Number Line (S.ID.1) Interquartile Range and Standard Deviation (S.ID.2) Interpret Shape, Center, and Spread in the Context of Data Sets (S.ID.3) Summarize Categorical Data for Two Categories in Two-way Frequency Tables (S.ID.5) Compute and Interpret the Correlation Coefficient of a Linear fit (S.ID.8) Distinguish Between Correlation and Causation (S.ID.9) Understand Solving Equations as a Process of Reasoning and Explain the Reasoning (A.REI.1) Solve Equations and Inequalities in One Variable with Coefficients Represented by Letters (A.REI.3) Understand the Concept of a Function and use Function Notation (F.IF.1) Calculate and Interpret the Average Rate of Change of a Function (F.IF.6) Graph and Interpret Square Root, Cube Root, and Piecewise Functions (F.IF.7.b) Use the Properties of Exponents to Interpret Expressions for Exponential Functions (F.IF.8.b) Compare Properties of Two Functions, each represented in a different way (F.IF.9) Combine Standard Function Types using Arithmetic Operations (F.BF.1) Build New Functions From Existing Functions (F.BF.3) Find the Inverse of a Function (F.BF.4) Construct and Compare Linear, Quadratic, and Exponential Models and Solve Problems (F.LE.3) Moved From Algebra I Laws of Exponents (8 th Grade, Integer Exponents and Numerical Bases) Direct Variation (7 th Grade) Distance Formula and Midpoint (Geometry CCSS) Parallel and Perpendicular Lines (Geometry CCSS) Matrices (4 th Courses) Notes: Topics may appear to be similar between the CCSS and the 2003 NCSCoS, however the CCSS may be presented at a higher cognitive demand The CCSS are a Continuation of Foundational Objectives from the Middle School CCSS Identifying and Interpreting Linear, Quadratic and Exponential Functions are the intention at this level

17 Algebra II At A Glance New to Algebra II Extend Polynomial Identities to the Complex Numbers (N.CN.8) Fundamental Theorem of Algebra (N.CN.9) Interpret Parts of an Expression (A.SSE.1) Use Polynomial Identities to Solve Problems (A.APR.4) Literal Equations (A.CED.4) Interpret Key Features of graphs and Tables and Sketch Graphs Showing Key Features Given a Verbal Description of the Relationship (F.IF.4) Relate the Domain of a Function to its Graph and to the Quantitative Relationship it Describes ((F.IF.5) Calculate and Interpret the Average Rate of Change of a Function (F.IF.6) Analyze Functions Using Different Representations (F.IF.9) Build a Functions That Models a Relationship Between Two Quantities (F.BF.1) Derive the Formula for the Sum of a Finite Geometric Series (A.SSE.4) Explain why the x-coordinates of the Points Where the graphs of the equations intersect are the solutions of the equations A.REI.11) Build New Functions From Existing Functions (F.BF.3) Understand Radian Measure of an angle (F.TF.1) Explain How the Unit Circle Enables the Extension of Trigonometric Functions to all Real Numbers (F.TF.2) Model Periodic Phenomena with Trigonometric Functions (F.TF.5) Prove and Apply Trigonometric Identities (F.TF.8) Summarize, Represent, and Interpret Data on a Single Count or Measurement Variable (S.ID.4) Understand Statistics as a Process for Making Inferences about Population Parameters (S.IC.1) Decide if a Specified Model is Consistent with Results form a Given Data-Generated Process (S.IC.2) Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each (S.IC.3) 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 (S.IC.4) Use data from a randomized experiment to compare two treatments (S.IC.5) Evaluate Reports Based on Data (S.IC.6) Use Probabilities to Make Fair Decisions (S.MD.6) Analyze Decisions and Strategies Using Probability Concepts (S.MD.7) Moved From Algebra II Operate with Matrices to Model and Solve Problems (4 th Math) Model and Solve Problems using Direct, Inverse, Combined and Joint Variation (Direct Variation in MS, Others Out of CCSS) Create and use Best-Fit Mathematical Models (Algebra I CCSS) Equations of Parabolas and Circles to Model and Solve Problems (Geometry CCSS) Notes: Topics may appear to be similar between the CCSS and the 2003 NCSCoS, however the CCSS may be presented at a higher cognitive demand Polynomial, Rational, Radical, Absolute Value, Exponential, and Logarithmic Functions are the Expectation at This Level Understanding Closure Properties of the Integers and the Rationals, not divided by zero, is expected at this level Selection of Appropriate Models or Functions is the Expectation at This Level Comparing Functions is New to CCSS Using Composition of Functions to Verify Functions is not in the CCSS Probability Concepts Build on the Foundations from MS and Geometry with more complex situation

18 Geometry At A Glance New to Geometry: Apply Geometric Methods to Solve Design Problems (G.MG.3) Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. (G.C.5) 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. (G.GPE.1) Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. (G.CO.6) 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. (G.CO.7) Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. (G.CO.8) Make formal geometric constructions with a variety of tools and methods. (G.CO.12) Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. (G.CO.13) 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. (G.SRT.9) Prove the Laws of Sines and Cosines and use them to solve problems. (G.SRT.10) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles. (G.SRT.11) Prove that all circles are similar. (G.C.1) Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle (G.C.3) Construct a tangent line from a point outside a given circle to the circle. (G.C.4) 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. (G.GPE.1) Derive the equation of a parabola given a focus and directrix (G.GPE.2) Use coordinates to prove simple geometric theorems algebraically (G.GPE.6) Visualize relationships between two-dimensional and thee-dimensional objects (G.GMD.4) Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ( or, and, not ) (S.CP.1) Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent (S.CP.2) Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B (S.CP.3) Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities (S.CP.4) Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations (S.CP.5) 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 (S.CP.6) 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 (S.CP.7) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B A) = P(B)P(A B), and interpret the answer in terms of the model (S.CP.8)

19 Use permutations and combinations to compute probabilities of compound events and solve problems (S.CP.9) Use probabilities to make fair decisions (S.MD.6) Analyze decisions and strategies using probability concepts (S.MD.7) Moved from Geometry: Use length, area, and volume to model and solve problems involving probability Use matrix operations to describe the transformation of polygons in the coordinate plane. Notes: Topics may appear to be similar between the CCSS and the 2003 NCSCoS, however the CCSS may be presented at a higher cognitive demand Use logic and deductive reasoning to draw conclusions is throughout the CCSS via the Mathematical Practices In the past, area and volume were typically applied in context individually, not in the context of density. Density is new in the CCSS. Similarity is defined in terms of size transformation Congruence is defined in terms of ridged motion Congruence is a special case of similarity The expectation of Using Probability to Make Decisions is the extension of the application of the counting rules from Middle School

20 Integrated Mathematics I At a Glance New to Integrated Mathematics I: Interpret parts of an expression such as terms, factors, and coefficients (A.SSE.1a) Interpret complicated expressions by viewing one or more of their parts as a single entity (A.SSE.1b) Use units as a way to understand problems and to guide the solution of multi-step problems (N.Q.1) Define appropriate quantities for the purpose of descriptive modeling (N.Q.2) Choose a level of accuracy appropriate to limitations on measurement when reporting quantities (N.Q.3) Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment (G.CO.1) Represent transformations in the plane (G.CO.2) Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself (G.CO.3) Develop definitions of rotations, reflections, and translations (G.CO.4) Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure (G.CO.5) Given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent (G.CO.6) Use the definition of congruence in terms of rigid motions to show that two triangles are congruent (G.CO.7) Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions (G.CO.8) Make formal geometric constructions with a variety of tools and methods (G.CO.12) Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle (G.CO.13) Use coordinates to prove simple geometric theorems algebraically (G.GPE.4) Prove the slope criteria for parallel and perpendicular lines; use them to solve geometric problems (G.GPE.5) Represent data with plots on the real number line (S.ID.1) 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 (S.ID.2) Interpret differences in shape, center, and spread in the context of the data sets (S.ID.3) Summarize categorical data for two categories in two-way frequency tables (S.ID.5) Compute (using technology) and interpret the correlation coefficient of a linear fit (S.ID.8) Distinguish between correlation and causation (S.ID.9) Create equations and inequalities in one variable and use them to solve problems (A.CED.1) Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales (A.CED.2) Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context (A.CED.3) Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations (A.CED.4) For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship (F.IF.4) Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes (F.IF.5) 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 (F.IF.6) Analyze Functions Using Different Representations (F.IF.9) Build a Function From Existing Functions (F.BF.3) Recognize situations in which one quantity changes at a constant rate per unit interval relative to another (F.LE.1b) Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another (F.LE.1c)

21 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function (F.LE.3) Moved From Integrated Mathematics I: Apply Laws of exponents (integer) moved to 8 th grade in the CCSS Operate with Polynomials moved to Integrated Mathematics II in the CCSS Factor Polynomials has moved to Integrated Mathematics II and III in the CCSS Develop and apply properties of solids to solve problems moved to Integrated Mathematics II in the CCSS Using Graph Theory is not in the CCSS Use Theoretical and Experimental Probability to Model and Solve Problems has moved to Integrated Mathematics II and III in the CCSS. Notes: Topics may appear to be similar between the CCSS and the 2003 NCSCoS, however the CCSS may be presented at a higher cognitive demand Linear and Exponential (integer inputs) functions are the expectation at this level Equations of parallel lines have moved from middle school and equations of perpendicular lines have moved from Integrated Mathematics II. Interquartile Range has moved from middle school and Standard Deviation is new to the CCSS Calculating and interpreting the average rate of change of a function moved from Integrated IV

22 Integrated Mathematics II At A Glance New to Integrated Mathematics II: Use properties of rational and irrational numbers (N.RN.3) Construct and compare linear, quadratic, and exponential models and solve problems (F.LE.3) Understand similarity in terms of similarity transformations (G.SRT.3) Prove theorems about triangles (G.SRT.4) Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures (G.SRT.4) Prove theorems about lines and angles (G.CO.9) Prove theorems about triangles (G.CO.10) 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 (G.GPE.1) Derive the equation of a parabola given a focus and directrix (G.GPE.2) Prove that all circles are similar (G.C.1) Identify and describe relationships among inscribed angles, radii, and chords (G.C.2) Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle (G.C.3) Construct a tangent line from a point outside a given circle to the circle (G.C.4) Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector (G.C.5) Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone (G.GMD.1) Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems (G.GMD.3) Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ( or, and, not ) (S.CP.1) Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent (S.CP.2) Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B (S.CP.3) Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified (S.CP.4) Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations (S.CP.5) 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 (S.CP.6) 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 (S.CP.7) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B A) = P(B)P(A B), and interpret the answer in terms of the model (S.CP.8) Use permutations and combinations to compute probabilities of compound events and solve problems (S.CP.9) Use probabilities to make fair decisions (S.MD.6) Analyze decisions and strategies using probability concepts (S.MD.7) Know there is a complex number i such that i2 = 1, and every complex number has the form a + bi with a and b real (N.CN.1) Use the relation i2 = 1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers (N.CN.2)

23 Solve quadratic equations with real coefficients that have complex solutions (N.CN.7) Extend polynomial identities to the complex numbers (N.CN.8) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials (N.CN.9) Prove and apply trigonometric identities (F.TF.8) For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship (F.IF.4) Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes (F.IF.5) 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 (F.IF.6) Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function (F.IF.8) Compare properties of two functions each represented in a different way (F.IF.9) Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs (F.BF.3) Find inverse functions (F.BF.4) Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales (A.CED.2) Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations (A.CED.4) Solve systems of equations (A.REI.7) Moved from Integrated Mathematics II: Model and solve problems using direct variation moved to middle school CCSS Operate with matrices to model and solve problems moved to fourth courses in CCSS Describe data to solve problems moved to CCSS Math I Create and use, for sets of data, calculator-generated models of linear, exponential, and quadratic functions to solve problems moved to CCSS Math I Use systems of linear equations or inequalities in two variables to model and solve problems. Solve using tables, graphs, and algebraic properties; justify steps used moved to Math I Notes: Topics may appear to be similar between the CCSS and the 2003 NCSCoS, however the CCSS may be presented at a higher cognitive demand In analyzing functions, linear, quadratic, exponential, absolute value, step, and piecewise-defined functions are the expectations at this level. Simple circle theorems are the expectations at this level. The distance formula and measures of three-dimensional figures are built on foundations laid in the middle school CCSS. Applying introductory counting rules is the expectation at this level

24 Integrated Mathematics III At A Glance New to Integrated Mathematics III: Apply trigonometry to general triangles (G.SRT.9) Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle (F.TF.1) Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers (F.TF.2) Visualize relationships between two-dimensional and three- dimensional objects (G.GMD.4) Use geometric shapes, their measures, and their properties to describe objects (G.MG.1) Apply concepts of density based on area and volume in modeling situations (G.MG.2) Apply geometric methods to solve design problems (G.MG.3) Create equations that describe numbers or relationships (A.CED.1) Represent and solve equations and inequalities graphically (A.REI.11) Summarize, represent, and interpret data on a single count or measurement variable (S.ID.4) Understand statistics as a process for making inferences about population parameters based on a random sample from that population (S.IC.1) Decide if a specified model is consistent with results from a given data-generating process (S.IC.2) Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each (S.IC.3) 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 (S.IC.4) Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant (S.IC.5) Evaluate reports based on data (S.IC.6) Use probabilities to make fair decisions (S.MD.6) Analyze decisions and strategies using probability concepts (S.MD.7) Extend polynomial identities to the complex numbers (N.CN.8) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials (N.CN.9) Perform arithmetic operations on polynomials (A.APR.1) Know and apply the Remainder Theorem (A.APR.2) 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 (A.APR.3) Prove polynomial identities and use them to describe numerical relationships (A.APR.4) Know and apply the Binomial Theorem (A.APR.5) Rewrite rational expressions (A.APR.7) Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales (A.CED.2) Rearrange formulas to highlight a quantity of interest (A.CED.4) For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship (F.IF.4) Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes (F.IF.5) Calculate and interpret the average rate of change (F.IF.6) 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 (F.IF.8a) Use the properties of exponents to interpret expressions for exponential functions (F.IF.8b) Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions) (F.IF.9) Model periodic phenomena with trigonometric functions (F.TF.5) Build a function that models a relationship between two quantities (F.BF.1)

25 Build new functions from existing functions (F.BF.4) Moved from Integrated III: Direct variation is in middle school, however inverse variation, joint variation, and combined variation are not in the CCSS Logic and deductive reasoning are addressed in the Mathematical Practices and embedded throughout the CCSS Vertical and adjacent angles, and parallel lines are in the middle school CCSS, however applying properties, definitions, and theorems of angles and lines to solve problems and write proofs are foundational to CCSS Math II. Similarity and Congruence are in Math II CCSS Using quadratic functions and inequalities to model and solve problems has moved to IM I and IM II in the CCSS Using equations and inequalities with absolute value has moved to IM II of the CCSS Notes: Topics may appear to be similar between the CCSS and the 2003 NCSCoS, however the CCSS may be presented at a higher cognitive demand Equations of all types are the expectation at this level; polynomial, root, rational, radical, absolute value, exponential, and logarithmic Many statistics standards are taken from the 2003 fourth math courses The selection of the appropriate model to represent a given function is the intention at this level.