Common Core Mathematics

Transcription

1 Common Core Mathematics Dr. Jere Confrey Joseph D. Moore Distinguished University Professor, Mathematics Education North Carolina State University Dr. Ruth Steidinger Senior Director of Middle School Programs Wake County Public Schools

2 Introduction of Staff John Williams Sr. Director of High School Programs Wendy Carlyle Director of AIG Michelle Tucker Sr. Administrator, Elementary Mathematics 4 12 Christina Zukowski Sr. Administrator, Middle School Mathematics Susan Shell Coordinating Teacher, Middle School Mathematics Sonia Dupree Sr. Administrator, Middle School Mathematics

3 Today s Presentation Changes in Standards Common Core Naming of middle and high school courses Math sequence options Criteria for placement Direction for board policy 4 12

4 Since last we met. Partnered with math colleagues in the Triangle High Five districts to determine common naming and sequencing to present to DPI. SAS supported our efforts in hosting our bi-monthly meetings Dr. Jere Confrey consulted with the group as we developed the names and sequences. Presented proposal to five superintendents to garner their support in approaching DPI. Obtained full support from DPI provided five districts with course codes.

5 Our Goals Taking into consideration the rigor of the common core curriculum, our goals are: To create consistent sequences/course offerings for mathematical study, and 4 12 To create criteria for placement into those sequences/courses that is fair, equitable, and applied consistently for all students.

6 Then NCTM Standards (1989) and AAAS and NRC (Science) Standards state-by-state Beginning of NCLB Disaggregation of data Nascent technology applications, graphing calculators Early internet, minimal access in schools Increasingly mobile student and teacher populations Common Core State Standards, 43 states New designs for ESEA needed Wireless networking and Cloud Computing (shared services model) Social Networking everywhere Data-intensive empirical research changing the way science is done, and different demands on quantitative modeling and literacy. Analytics (SAS) Now Knowledge Intensive Industries and STEM Increasingly mobile student and teacher populations Increasing gaps associated with race and poverty

7 1. More demanding standards and new topics 2. New assessments measuring more complex reasoning 3. Implementation factors 4. Urgency to address performance gaps 5. Expanded data and new technologies transform instruction

8 A. Eight Mathematical Practices B. Changes by Grade Bands (K-5, 6-8, 9-12) C. Structure and Learning Progressions D. Interdisciplinary Content: Reading and Writing in Science and Technical Subjects 8

9 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. CCSS-Math,

10 Practices are always interrelated with content. They must be addressed in tandem. Practices may provide a way to observe classrooms to see how the content is made understandable, challenging and engaging to students. 10

11 Numeration and operation intensified, and introduced earlier Early place value foundations in grade K Regrouping as composing / decomposing in grade 2 Decimals to hundredths in grade 4 All three types of measurement simultaneously Non-standard, English and Metric Emphasis on fractions as numbers Emphasis on number line as visualization / structure 11

12 Ratio and Proportion focused on in grade 6 Ratio, unit rates, converting measurement, tables of values, graphing, missing value problems Percents introduced in grade 6 Statistics introduced in grade 6 Statistical variability (measures of central tendency, distributions, interquartile range, mean and absolute deviation, data shape) Rational numbers in grade 7 Grade 8: One-third of high-school algebra for all students 12

13 Much higher expectations at middle grades, where our capacity, and student performance, are weakest 13

14 Supports both/either (a) continuing an integrated approach or (b) a traditional siloed approach (Algebra I, Geometry, Algebra II)--or new models that synthesize these two. All students must master some topics traditionally from algebra 2, or beyond Simple periodic functions Polynomials, Radicals More probability and statistics (correlation, random processes) Introduction to mathematical modeling 14

15 One promise of common state standards is that over time, they will allow research on learning progressions to inform and improve the design of Standards to a much greater extent than is possible today. CCSS 2010, p.5 15

16 Confrey (2006) Design Studies Chapter Cambridge Handbook of the Learning Sciences 16

17 To obtain posters, Order at www. wirelessgeneration.com/posters17

18 Use trig ratios and the Pythagorean theorem to solve right triangles in context Count (starting at any number < 120 ), read, and write numerals <= NBT.1 For 2-digit numbers, add or subtract 10 without counting 1.NBT.5 Compare two 2- digit numbers using place value 1.NBT.3 Subtract multiples of 10 (<=90) and explain strategies 1.NBT.6 Solve word problems with + of 3 whole numbers whose sum is <=20 1.OA.2 Relate counting to addition and subtraction 1.OA.5 Apply properties (commutative property of addition, associative property of addition) as strategies 1.OA.3 Add and subtract within 20 using strategies (near doubles, pairs for 10, etc) 1.OA.6 Understand subtraction as an unknown addend problem 1.OA.4 Add within 100 including with composing 10s using strategies 1.NBT.4 Use + and within 20 to solve word problems 1.OA.1 Understand the meaning of equal sign in addition subtraction equations 1.OA.7 Find unknown in addition and subtraction equations 1.OA.8 Equipartition circles and rectangles into 2 and 4 fair shares and name 1.G.3 Equipartition rectangles into rows and columns to produce different numbers of squares confirmed by counting 2.G.2 Count within 1000; skip-count by 5s, 10s, and 100s 2.NBT.2 Read and write to 1000 using expanded form 2.NBT.3 Fluently add and subtract within 100 using strategies, place value, inverse operations and properties 2.NBT.5 Add up to 4 2-digit numbers using place value and properties 2.NBT.6 Compare numbers (0 < n < 1000) using place value and >, =, and < 2.NBT.4 Mentally add and subtract 10 or 100 (100 < n < 900) 2.NBT.8 Equipartition circles and rectangles into 3 and 4 fair shares and name. Note fair shares need not be congruent 2.G.3 Fluently add and subtract within 20 using mental strategies 2.OA.2 Solve 1 and 2-step word problems within 100 using addition and subtraction 2.OA.1 Explain addition and subtraction strategies using place value and properties 2.NBT.9 Add and subtract to 1000 including composition and decomposition based on place value, inverse operations, and properties 2.NBT.7 Identify groups up to 20 as odd or even, showing all even numbers as doubles 2.OA.3 Find totals of arrays up to 5x5 2.OA.4 Construct number line of whole numbers and show + and - 2.MD.6 Equipartition a whole into n parts and name as a unit fraction (1/n) 3.G.2 Fluently add and subtract within 1000 using strategies and algorithms 3.NBT.2 Round whole numbers to the nearest 10 or 100 using place value 3.NBT.1 Interpret a x b as the total number of objects in a groups of b objects 3.OA.1 Understand a fraction a/b can be viewed as a parts of size 1/b (for b=2,3,4,6,8) 3.NF.1 2 fractions are equal if they are the same size, or the same point on a number line. 3.NF.3a Apply properties of operations (commutative property of x, associative property of x, and distributive property) 3.OA.5 Apply properties of operations (commutative property of x, associative property of x, and distributive property) 3.OA.5 Apply properties of operations (commutative property of x, associative property of x, and distributive property) 3.OA.5 Apply properties of operations (commutative property of x, associative property of x, and distributive property) 3.OA.5 Apply properties of operations (commutative property of x, associative property of x, and distributive property) 3.OA.5 Understand division as an unknown-factor problem 3.OA.6 Interpret a b=c (a,b,c whole numbers) as a equipartitioned among b people giving c per person or as a measured by length b, c times 3.OA.2 Apply properties of operations (commutative property of x, associative property of x, and distributive property) 3.OA.5 Apply properties of operations (commutative property of x, associative property of x, and distributive property) 3.OA.5 Solve word problems (x, ) within 100 involving equal groups, arrays, and measurement 3.OA.3 Multiply 1-digit numbers by multiples of 10 3.NBT.3 Apply properties of operations (commutative property of x, associative property of x, and distributive property) 3.OA.5 Apply properties of operations (commutative property of x, associative property of x, and distributive property) 3.OA.5 Solve 2-step whole number word problems applying order of operations for +, -, x, and 3.OA.8 Fluently multiply and divide within 100 using strategies and properties of operations 3.OA.7 Find unknowns in a multiplication or division problem involving 3 whole numbers 3.OA.4 Recognize a digit in one place represents ten times what it represents in the place to the right 4.NBT.1 Use place value to round multi-digit whole numbers to any place 4.NBT.3 Interpret a = b x c as a multiplicative comparison using times as many or times as much 4.OA.1 Solve multistep word problems including interpreting remainders 4.OA.3 Solve word problems involving multiplicative comparisons 4.OA.2 Identify all factors for numbers from distinguishing primes and composites 4.OA.4 Multiply whole numbers < by 1-digit numbers, and 2 2-digit numbers and explain 4.NBT.5 Divide 4-digit numbers by 1-digit numbers with remainders and explain 4.NBT.6 Find equivalent fractions symbolically 4.NF.1 Compare fractions using benchmark fractions, common denominators/ numerators, and reference units 4.NF.2 Read, write, and compare numbers using expanded form and place value 4.NBT.2 Fluently add and subtract multi-digit whole numbers using standard algorithm 4.NBT.4 Add 2 fractions with denominators 10 and 100 by using equivalent fractions 4.NF.5 Divide 4-digit numbers by 2-digit numbers and explain 5.NF.NBT.6 Fluently multiply multidigit whole numbers using the standard algorithm 5.NBT.5 Interpret a fraction as a division of the numerator by the denominator and relate to fair shares and estimate bounds of responses 5.NF.3 Estimate and solve word problems using addition and subtraction of fractions with unlike denominators 5.NF.2 Add, subtract, multiply, and divide decimals to hundredths and explain 5.NBT.7 Recognize a digit in one place represents ten times what it represents in the place to the right and 1/10 th of what it represents to the left 5.NBT.1 Write numeric expressions and relate them to verbal statements 5.OA.2 Use place value to round decimals to any place 5.NBT.4 Add and subtract fractions with unlike denominators including mixed numbers 5.NF.1 Describe several measurable attributes of an object K.MD.1 Directly compare 2 objects as more of, less of, or find the difference K.MD.2 Order 3 objects directly and indirectly using 3 rd object 1.MD.1 Quantify length by iterating units with no gaps or overlaps 1.MD.2 Measure object length using tools (with inches, ft, yds, cms, and meters) 2.MD.1 Recognize length of object depends the size of the unit 2.MD.2 Measure lengths and describe difference in units 2.MD.4 Estimate lengths using inches, ft, yds, cms, and meters 2.MD.3 Solve word problems involving whole number lengths (0-100) using + and 2.MD.5 Measure and estimate liquid volumes and object masses using g, kg, and l and apply arithmetic operations to answer questions 3.MD.2 Measure volume by counting unit cubes (cubic cm, cubic in, and cubic ft) 5.MD.4 Solve problems involving area and perimeter and their relationship 3.MD.8 Measure area by counting unit squares (square cm, square m, square in, and square ft) 3.MD.6 Find unknown values in area and perimeter problems involving rectangles 4.MD.3 Know relative sizes of measurement units and generate conversion tables 4.MD.1 Solve multi-step real world problems using conversion of measurement units 5.MD.1 Solve word problems about distance, time, liquid volume, mass, and money using multiple operations and simple fractions/decimals 4.MD.2 Understand a fraction a/b as a multiple of 1/b 4.NF.4.a Interpret and compute division of a unit fraction by a non-zero whole number 5.NF.7.a Solve real world problems involving multiplication of fractions and mixed numbers 5.NF.6 Understand the concept of ratio as a relationship between 2 quantities 6.RP.1 Understand the concept of a unit ratio such that a:b :: a/b:1 6.RP.2 Find unit ratios equivalent to complex fractions (a/b : c/d :: a/b / c/d : 1) 7.RP.1 Use proportional relationships to solve multistep ratio problems (including percent increase and decrease) 7.RP.3 Fluently divide multidigit numbers using the standard algorithm 6.NS.2 Find GCF (n1, n2 <=100) and LCF (n1, n2 <=12). Use distributive property to write equivalent numeric expressions. 6.NS.4 Fluently add, subtract, multiply, and divide multi-digit decimals using standard algorithms. 6.NS.3 Interpret, solve, and explain division of fractions. 6.NS.1 Understand integers as indicating magnitude and direction in real world contexts. 6.NS.5 Solve real world problems using the coordinate plane. Find distance between points sharing one coordinate 6.NS.8 Examine proportional relationships using multiple representations (graphs, tables, and equations) and interpret slope 8.EE.5 Approximate irrationals with rationals and locate on a number line using approximations and/or constructions 8.NS.2 Understand that rational numbers are produced by the division of p by q (p and q integers, q 0) including real world contexts 7.NS.2.b Solve real world problems using the four operations with rational numbers (including complex fractions) 7.NS.3 Link rational numbers to decimal expansions and vice versa and introduce irrationals as nonrepeating non-terminating decimals 8.NS.1 Find volumes by packing unit cubes and deriving formula as v = lwh or V =bh including fractional edges including real world problems 6.G.2 Know the formulas for area and circumference of a circle, informally relate them and use to solve real world problems 7.G.4 Solve real world problems involving area, volume and surface area of 2D and 3D objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms 7.G.6 Solve simple equations for unknown angles using definitions of complimentary, vertical, and adjacent angles 7.G.5 Write and evaluate numerical expressions involving whole-number exponents 6.EE.1 Recognize equivalent expressions generate equal outputs for any given value of x 6.EE.4 Solve real world problems by using variables as missing values or as numbers that vary 6.EE.6 Solve equations of the form x + p = q and px = q (x, p, q >=0) including real world problems 6.EE.7 Write equations in 2 variables and represent as graphs and tables identifying dependent and independent variables 6.EE.9 Generate equivalent expressions by applying properties including distributivity and adding like terms. 6.EE.3 Understand solving equations or inequalities as identifying values yielding true statements 6.EE.5 Write inequalities as x>c or x<c and represent on number lines 6.EE.8 Solve multi-step real world problems coordinating among integers, fractions, decimals and percents 7.EE.3 Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients 7.EE.1 Understand different algebraic forms of equations reveal relationships among quantities 7.EE.2 Simplify expressions with integer exponents by applying properties of exponents 8.EE.1 Use simple scientific notation ( a single digit times a power of 10) to estimate very large and very small numbers and their relative size 8.EE.3 Measure, operate, and compare large and small numbers using scientific notation with and without technology 8.EE.4 Sketch qualitative graphs from verbal descriptions of functions, including increasing, decreasing, linear, or nonlinear 8.F.5 Model linear relationships including initial values and rates of change using multiple representations 8.F.4 Legend Kindergarten Grade Levels First Grade Second Grade Fourth Grade Content Strands Number and Operations Fifth Grade Geometry Measurement and Data Ratio and Proportion Third Grade Sixth Grade Seventh Grade Eighth Grade Statistics and Probability Expressions and Equations Functions Analyze closure with respect to adding and multiplying rational and/ or irrational numbers N-RN.3 Extend exponents to include rational values N-RN.1 Rewrite expressions involving radicals and rational exponents using properties of exponents N-RN.2 Introduce complex numbers of the form a + bi and know i 2 = -1 N-CN.1 Add, subtract, and multiply complex numbers using properties N-CN.2 Use conjugates of a complex number to find moduli and quotients of complex numbers N-CN.3 Represent complex numbers on the plane in rectangular and polar form and translate between them N-CN.4 Represent addition, subtraction, multiplication, and conjugation of complex numbers on the complex plane N-CN.5 Calculate the distance between numbers in the complex plane N-CN.6 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 (prime factorization) N-CN.9 Define and represent vector quantities and related symbols N-VM.1 Solve problems involving velocity and other quantities using vectors N-VM.3 Subtract 2 vectors N-VM.2 Use matrices to represent and manipulate data in context N-VM.6 Work with 2x2 matrices as transformations on the plane and interpret absolute value of the determinant as area N-VM.12 Multiply matrices by scalars N-VM.7 Add, subtract, and multiply matrices N-VM.8 Understand that matrix multiplication for square matrices is not commutative but is associative and distributive N-VM.9 Understand 0 and identity matrices in addition and multiplication and that only square matrices with multiplicative inverses have nonzero determinants N-VM.10 Apply matrix multiplication to vectors and use to represent transformation of vectors N-VM.11 Write arithmetic and geometric sequences explicitly and recursively and translate including in context F-BF.2 Derive the formula for the sum of a finite geometric series and solve problems in context A-SSE.4 Prove sin 2 (Θ) + cos 2 (Θ) = 1 and use to solve for unknowns including sin, cos, tan, and Θ if given quadrant F-TF.8 Prove the addition and subtraction formulas for sin, cos, and tangent and solve in context F-TF.9 Understand and determine when two events are independent S-CP.2 Construct and interpret 2- way frequency tables for categorical data to decide if events are independent and approximate conditional probabilities S-CP.4 Apply the addition rule: P(A or B) = P(A) + P(B) - P(A and B) and interpret S-CP.7 Understand the rules of conditional probability to determine if events are independent S-CP.3 Recognize and explain conditional probability and independence in context S-CP.5 Graph the probability distribution of a random variable for a quantity of interest S-CP.6 Apply the Multiplication Rule in a probability model and interpret S-CP.8 Compare probabilities of compound events using permutations and combinations S-CP.9 Define a random variable for an event in a sample space and graph its probability distribution S-MD.1 Calculate the expected value of a random variable and interpret as mean of the probability distribution S-MD.2 Find the expected value of a random variable from the theoretical probability of an event S-MD.3 Find the expected value of a random variable from an experiment where the probabilities can be assigned empirically S-MD.4 Use probabilities to make fair decisions S-MD.6 Analyze decisions and strategies using probability concepts S-MD.7 Recognize associations and trends for categorical data in 2-way frequency tables (including joint, marginal, and conditional relative frequencies) S-ID.5 Represent data with plots (dot plots, histograms, and box plots) on the number line S-ID.1 Compare the center (median, mean) and spread (interquartile range, standard deviation) of 2 or more data sets S-ID.2 Interpret differences in shape, center, and spread of data sets accounting for effects of outliers S-ID.3 Use x and sd to fit normal distribution when appropriate and estimate population percentages S-ID.4 Recognize the purpose, differences, and how randomization relates to sample surveys, experiments, and observational studies S-IC.3 Evaluate reports based on data Compare 2 treatments in a randomized experiment and use simulations to determine significance of parameters S-IC.5 Use sample data to estimate population mean and simulations to find margin of error S-IC.4 Understand statistics as a process for making inferences about populations based on random samples S-IC.1 Compare experimental probabilities (through simulations) to theoretical probabilities S-IC.2 Interpret the slope and intercept of a linear model in context S-ID.7 Compute (using technology) and interpret the correlation coefficient of a linear fit. S-ID.8 Distinguish between correlation and causation S-ID.9 Define angle, circle, perpendicular and parallel lines, and line segment based on undefined terms G-CO.1 Determine if 2 figures are congruent based on application of rigid motions (rotation, reflection, and translation) G-CO.6 Determine if 2 triangles are congruent based on corresponding pairs of sides and corresponding pairs of angles G-CO.7 Prove theorems about lines and angles including vertical angles, alternate interior and corresponding angles, and perpendicular bisectors G-CO.9 Determine if 2 figures are similar based on a dilation and recognize equality of corresponding angles and proportionality of corresponding sides G-SRT.2 Show triangle similarity by applying transformations to establish the AA criterion G-SRT.3 Understand trig ratios as ratios of sides of right triangles associated to their acute angles G-SRT.6 Explain and use the relationship between the sine and cosine of complementary angles G-SRT.7 Prove theorems about parallelograms G-CO.11 Undertake geometric constructs including copying segments and lines, bisection, perpendiculars, and parallels using various tools and technologies G-CO.12 Construct an equilateral triangle, square, and regular hexagon inscribed in a circle G-CO.13 Relate transformations to geometric objects including angles, circles, perpendicular and parallel lines, and line segments G-CO.4 Distinguish among transformations of functions geometrically and associate them with algebraic representations G-CO.2 Design an object or structure using geometry G-MG.3 For rectangles, parallelograms, trapezoids, or regular polygons identify the rotations and reflections that carry it onto itself G-CO.3 Model objects using geometric shapes, measures, and properties G-MG.1 Find the point on a directed line segment between two given points that partitions the segment in a given ratio G-GPE.6 Derive the equation of a circle given its center and radius and vice versa G-GPE.1 Derive the equation of a parabola given a focus and directrix G-GPE.2 Derive equations of ellipses and hyperbolas given the foci G-GPE.3 Informally explain various formulae for circles, cylinder, pyramid, and cone using dissections, Cavalieri s principle, and limit arguments G-GMD.1 Apply concepts of density based on area and volume in modeling G-MG.2 Identify 2D cross sections of 3D objects and 3D objects generated by rotations of 2D objects G-GMD.4 Give an informal argument using Cavalieri s principle for the formulas for the volume of a sphere and other solid figures G-GMD.2 Solve problems using volume formulas for cylinders, pyramids, cones, and spheres G-GMD.3 Prove the Law of Sines and Cosines and use to solve problems G-SRT.10 Understand radian measure F-TF.1 Use special triangles to determine sin, cos, and tan for π/3, π/4 and π/6 and related values F-TF.3 Extend trigonometric functions to all real numbers as radians using the unit circle F-TF.2 Choose trig functions to model periodic situations based on amplitude, frequency, and vertical translation F-TF.5 G-SRT.8 Derive the formula area (triangle ABC) = ½ ab sin (C) G-SRT.9 Understand that trig functions have inverses when their domains are restricted F-TF.6 Use the unit circle to explain symmetry and periodicity of trig functions F-TF.4 Model using inverse trig functions and equations, and evaluate F-TF.7 High School Levels Level 1 Level 5 Level 2 Level 6 Level 3 Level 7 Level 4 Level 8 Relate definition of congruence to ASA, SAS, and SSS G-CO.8 Prove theorems about triangles including sum of interior angles, base angles of isosceles triangles, and relative lengths of sides and segments joining midpoints G-CO.10 Prove theorems about triangles including effects of drawing parallel lines to a side and the Pythagorean Theorem using similarity G-SRT.4 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures G-SRT.5 Define radian measure using similarity and define the formula for area of a sector G-C.5 Prove that all circles are similar G-C.1 Identify and describe relationships among inscribed angles, radii, and chords G-C.2 Construct circumscribed circles of triangles, inscribed triangles and quadrilaterals and prove related properties G-C.3 Given a preimage and either a sequence of transformations or an image, find the other G-CO.5 Prove slope criteria for parallel and perpendicular lines and use to solve geometric problems G-GPE.5 Use coordinates to prove simple geometric theorems algebraically G-GPE.4 Conduct unit analysis in analyzing formula and multi-step problems and interpret scale and origin in graphs N-Q.1 Define appropriate quantities for the purpose descriptive modeling N-Q.2 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities N-Q.3 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles G-GPE.7 Construct a tangent line from a point outside a given circle to the circle G-C.4 Understand and apply the Law of Sines and Law of Cosines in context G-SRT.11 Understand and perform arithmetic operations on rational expressions as a closed system (like rational numbers) under +, -, x, and A-APR.7 Compare fractions with identical reference units and equal numerators or denominators through inspection 3.NF.3d 10 is a bundle of 10 ones 1.NBT.2.a 11 to 19 are composed of one ten and some ones 1.NBT.2.b The numbers 10, 20, 30,, 90 refer to 1, 2, 3,, 9 tens 1.NBT.2.c 100 is a bundle of 10 tens 2.NBT.1.a The numbers 100, 200, 300, 900 refer to 1, 2, 3,, 9 hundreds 2.NBT.1.b Represent a fraction 1/b on number line using equipartitioning 3.NF.2.a Define a unit square to measure area in square units 3.MD.5.a Understand addition and subtraction of fractions as joining and separating parts referring to the same whole 4.NF.3.a Add and subtract mixed numbers (like denominators) 4.NF.3.c Solve word problems involving + and - of fractions with like denominators using models and equations 4.NF.3.d Solve word problems involving multiplying a fraction by a whole number using visual models and equations 4.NF.4.c Read and write decimals to thousandths using expanded notation and place value 5.NBT.3.a Compare decimals to thousandths using values of digits 5.NBT.3.b Interpret the product (a/b) q as a parts of a partition of q into b equal parts and apply in context 5.NF.4a Relate rectilinear area model to tiling and multiplication of fractions 5.NF.4b Comparing the relative size of the product to one factor based on the size of the other factor 5.NF.5.a Explain why multiplication by a fraction >1 produces a larger number, <1 produces a smaller number, and =1 produces equivalent fractions 5.NF.5.b Explain patterns in the number of zeroes in the products or quotients when multiplying or dividing by powers of 10 5.NBT.2 Solve real world problems involving division of unit fractions using visual models and equations 5.NF.7.c Define a unit cube to measure volume in cubic units 5.MD.3.a Find volume of a right rectangular prism by: packing with unit cubes, multiplying edge lengths, and as lxwxh; use to show associate property 5.MD.5.a Solve ratio problems using tables of values with coordinate plots to find missing values and compare ratios. 6.RP.3.a Solve real world problems including unit pricing and constant speed. 6.RP.3.b Solve real world problems involving ratios by using tables, tape diagrams, and number lines. Extend to include percents. 6.RP.3.c Convert measures and transform units when multiplying or dividing by applying ratio reasoning. 6.RP.3d Plot integers on the number line and recognize -(-a) = a and 0 as its own opposite. 6.NS.6.a Extend the coordinate plane to all 4 quadrants and recognize reflections exhibited by (a, b), (-a, b), (-a, -b), (a, -b). 6.NS.6.b Graph pairs of rational numbers on the number line and coordinate plane. 6.NS.6.c Write, interpret, and explain ordering of rational numbers in context. 6.NS.7.b Write expressions with operations, numbers, and letters 6.EE.2.a Identify parts of an expression as sums, terms, products, factors, quotients, and coefficients. 6.EE.2.b Evaluate algebraic expressions and formula at given values and apply order of operations. 6.EE.2.c 6.SP.5.a 6.SP.5.c Decide whether two quantities are in proportional relationship using tables of values or coordinate graphs 7.RP.2.a Identify constant of proportionality in tables, graphs, equations, diagrams or text 7.RP.2.b Describe situations in which opposite quantities combine to make 0 7.NS.1.a Understand subtraction of rational numbers as adding the additive inverse, p q = p + (-q) and distance as the absolute value of their difference 7.NS.1.c Apply properties of operations as strategies to add and subtract rational numbers 7.NS.1.d Extend multiplication and division to apply to rational numbers including real world contexts 7.NS.2.a Apply properties of operations as strategies to multiply and divide rational numbers 7.NS.2.c Convert rational numbers to terminating decimals or repeating decimals using division 7.NS.2.d Solve real world problems of the form px + q = r and p(x+q) = r (p,q,r are rational numbers representing quantities) 7.EE.4.a Solve real world problems of the form px + q > r or px+q<r (p, q, r are rational numbers representing quantities) 7.EE.4.b Understand relationship between a solution to simultaneous linear equations and intersection of graphs 8.EE.8.a Add vectors end-toend, component-wise, and by the parallelogram rule. N-VM.4.a Given two vectors, determine the magnitude and direction of their sum N-VM.4.b Understand vector subtraction v - w as v + (- w) with the same magnitude as w and pointing in the opposite direction including graphically N-VM.4.c Represent scalar multiplication graphically N-VM.5.a Compute the magnitude of a scalar multiple cv using cv = c v N-VM.5.b Verify experimentally that dilations map a line to itself or a parallel line G-SRT.1.a Fit a function (linear, quadratic, exponential) to data and interpret in context S-ID.6.a Find the expected payoff for a game of chance S-MD.5.a Compare 2 numbers between 1 and 10 presented as written numerals K.CC.7 Count to 100 by ones and tens K.CC.1 Count forward from any number K.CC.2 Count objects to answer how many, n <=20 K.CC.5 Write numbers from 0 to 20 K.CC.3 Compare size of 2 groups <= 10 as greater than, less than, or equal K.CC.6 Fluently add and subtract within 5 K.OA.5 Compose and decompose number from 1 to 19 as 10s and 1s K.NBT.1 Represent addition and subtraction through activity K.OA.1 Decompose numbers <= 10 into pairs K.OA.3 Find number pair equaling 10 K.OA.4 Solve word problems (+ and - <= 10) K.OA.2 Count objects and demonstrate 1-1 correspondence K.CC.4.a Associate last number name with number of objects K.CC.4.b Understand each successive number is one larger K.CC.4.c Relate tiling of rectangle to multiplication of side lengths as area 3.MD.7.a Find area of rectangles using multiplication of side lengths in context 3.MD.7.b Use area models and tiling to represent the distributive property 3.MD.7.c Recognize area as additive through decomposition and composition in context 3.MD.7.d Measure and sketch angles in whole number degrees 4.MD.6 Recognize angle measures as additive to answer questions 4.MD.7 Define a 1-degree angle as 1/360th of a turn of a circle 4.MD.5.a Define an n-degree angle as n 1- degree angles 4.MD.5.b Tell and write time in hours and halfhours using analog and digital clocks 1.MD.3 Solve word problems involving money (dollars, quarters, dimes, nickels, and pennies) including symbols 2.MD.8 Read and write time (digital and analog) to nearest 5 minutes 2.MD.7 Read and write time to nearest minute and calculate time intervals 3.MD.1 Use coordinate pairs as vertices to draw polygons and find side lengths if vertices share a common coordinate including real world problems Find areas of geometric figures (certain triangles and quadrilaterals, and polygons) by composing or decomposing into known figures including real world problems 6.G.1 Model shapes in the world by building shapes from components and drawing shapes K.G.5 Distinguish between defining and non-defining attributers 1.G.1 Compose 2 and 3- dimensional shapes from other shapes 1.G.2 Analyze and compare 2 and 3-dimensional shapes K.G.4 Find common geometric shapes in objects and describe relative position K.G.1 Correctly name shapes regardless of their orientations or overall size K.G.2 Compose simple shapes to form larger shapes K.G.6 Identify shapes as 2 or 3- dimensional K.G.3 Recognize triangles, quadrilaterals, pentagons, hexagons, and cubes in terms of faces and angles 2.G.1 Apply properties of operations (commutative property of x, associative property of x, and distributive property) 3.OA.5 Apply properties of operations (commutative property of x, associative property of x, and distributive property) 3.OA.5 Understand categories and subcategories of shapes among quadrilaterals, rhombuses, rectangles, and squares 3.G.1 Recognize and draw lines of symmetry for 2D figures 4.G.3 Classify 2D figures with respect to parallel or perpendicular lines and angle types 4.G.2 Draw and identify points, lines (parallel and perpindicular), line segments, rays, and angles 4.G.1 Recognize statistical questions as associated with variability in the data 6.SP.1 Understand distributions of data as described by center, spread, and overall shape 6.SP.2 Recognize measures of center and measures of variation as single values representing data sets 6.SP.3 Display numerical data in plots on a number line (dot plots, histograms, and box plots) 6.SP.4 Compare distributions of 2 data sets visually and explore different ways to quantify and compare the differences including using mean absolute deviation 7.SP.3 Use measures of center and variability for numerical data from random samples to draw informal comparative inferences about 2 populations 7.SP.4 Understand the relationship between samples and populations including random sampling, representative samples, and valid inference 7.SP.1 Generate or simulate multiple random samples of a given size to examine variation and informally discuss drawing inferences about the population 7.SP.2 Understand the probability of a chance event is a number between 0 and 1 and interpret the meaning of different values 7.SP.5 Use empirical data to estimate probability of a chance event examining the effects of conducting multiple trials 7.SP.6 Use scatter plots with bivariate data to investigate association (positive, negative, or none), clustering, outliers, and linear and nonlinear association 8.SP.1 Use the equation of a linear model to solve problems in the context of bivariate data, intepreting the slope and intercept 8.SP.3 Understand how to construct and interpret 2- way tables of bivariate categorical data for patterns of association based on relative frequencies 8.SP.4 Describe attribute, how it's measured, and units of measurement 6.SP.5.b Relate choice of measures of center and variability to the shape of the data distribution in context 6.SP.5.d Develop a uniform probability model and apply to events 7.SP.7.a Develop a probability model (may not be uniform) and observe frequencies from a chance process 7.SP.7.b Understand the probability of a compound event is the fraction of the outcomes in the sample space 7.SP.8.a Represent sample spaces for compound events using lists, tables, and tree diagrams 7.SP.8.b Design and use a simulation to generate frequencies for compound events. 7.SP.8.c Classify 2D figures in a hierarchy based on properties 5.G.4 Understand that attributes belonging to a category apply to all subcategories 5.G.3 Represent 3D figures as nets and apply to find surface area including real world problems 6.G.4 Graph coordinate pairs on the coordinate plane 5.G.1 Describe 2D figures from slicing 3D figures such as from right rectangular prisms and right rectangular pyramids 7.G.3 Use tools to construct geometric shapes identifying conditions for unique, multiple, or no solutions 7.G.2 Represent real world problems by graphing and interpreting points in the first quadrant 5.G.2 Know and apply formulas for cones, cylinders, and spheres 8.G.9 Describe the effect of dilations, translations, rotations, and reflections on 2D figures using coordinates 8.G.3 Understand 2 figures are congruent if the second is obtained from a sequence of rotations, reflections, and translations on the first 8.G.2 Understand 2 figures are similar if a sequence of rotations, reflections, translations, and dilations produces congruency 8.G.4 Explain a proof of the Pythagorean Theorem and its converse 8.G.6 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real world problems in 2 and 3D 8.G.7 Apply the Pythagorean Theorem to find the distance between 2 points in a coordinate system 8.G.8 Identify angle relationships including sum in a triangle, corresponding angles and alternate interior angles and establishing similarity 8.G.5 For transformations (rotations, reflections, and translations): Lines map to lines 8.G.1.a Report the number of observations 6.SP.5.a Describe measures of center (median, mean) and variability (interquartile range, mean absolute deviation) in relation to context 6.SP.5.c Describe events as subsets of a sample space using characteristics of the outcomes or as unions, intersections, or complements of other events S-CP.1 Represent a fraction a/b on number line as a lengths of 1/bth 3.NF.2.b Recognize and generate simple equivalent fractions and explain using models. 3.NF.3.b Express whole numbers as fractions, and fractions as whole numbers when appropriate 3.NF.3.c Decompose a fraction into a sum of fractions (same denominator) multiple ways and justify using models and equations 4.NF.3.b Multiply a fraction by a whole number 4.NF.4.b Use unit squares with no gaps or overlaps to refer to area 3.MD.5.b Apply volume formulas for right rectangular prisms including in context 5.MD.5.b Recognize volume as additive through decomposition and composition in context 5.MD.5.c Measure solid figures in unit cubes packed with no gaps or overlaps 5.MD.3.b Use decimals to represent fractions with denominators of 10 or 100 and locate on a number line 4.NF.6 Compare two decimals to hundredths using values of digits and justify with models 4.NF.7 Apply properties of operations (commutative property of x, associative property of x, and distributive property) 3.OA.5 Apply properties of operations (commutative property of x, associative property of x, and distributive property) 3.OA.5 Identify arithmetic patterns and explain using properties of operations such as in multiplication and addition tables 3.OA.9 Generate number patterns using a rule and explain number theoretic properties (oddness and eveness) 4.OA.5 Generate a 2-column table using a different numeric pattern in each column, graph ordered pairs, and explain relationships between corresponding values 5.OA.3 Use parentheses, brackets, or braces and evaluate using order of operations 5.OA.1 Interpret division of a whole number by a unit fraction, and compute such quotients 5.NF.7.b Understand absolute value as magnitude including in context 6.NS.7.c Distinguish comparisons of absolute value from statements about order 6.NS.7.d Represent proportional relationships by equations 7.RP.2.c Find and interpret unit rate and coordinate points for proportional relations 7.RP.2.d Understand addition of rational numbers as represented on number line including additive inverses, and in context 7.NS.1.b Solve scale drawing problems. 7.G.1 Translate between statements of inequality of two numbers and their relative position on a number line 6.NS.7.a For transformations (rotations, reflections, and translations): Angles map to angles of the same measure 8.G.1.b For transformations (rotations, reflections, and translations): Parallel lines map to parallel lines 8.G.1.c Generate and use line plots to show multiple length measures of one or more objects including measures of halves and quarters 3.MD.4 Represent categorical data using scaled picture and bar graphs and compare quantities. 3.MD.3 Generate and use line plots to show measurements of one or more objects including measures of ½, ¼, 1/8, and find sums and differences 4.MD.4 Categorize objects, count the number in each category and order by size K.MD.3 Organize, represent, and interpret data, <=3 categories 1.MD.4 Generate and use dot plots to show multiple length measures of one or more objects. 2.MD.9 Plot picture and bar graphs of data and answer questions by comparing, adding, and subtracting 2.MD.10 Generate and use line plots of measurements of objects including ½, ¼, and 1/8 and redistribute into equal shares 5.MD.2 Solve linear equations in 1 variable using distributivity and collecting like terms 8.EE.7.b Evaluate and compare strategies on the basis of expected values S-MD.5.b Verify experimentally that a dilation proportionally increases or decreases a line segment's length G-SRT.1.b Informally model linear relationships in scatter plots if appropriate and and assess fit 8.SP.2 Assess the fit of a function by plotting and analyzing residuals. S-ID.6.b Fit a linear function on a scatter plot and evaluate for a linear association S-ID.6.c Represent solutions to equations of the form x 2 = p and x 3 = p as square roots and cube roots and recognize common perfect squares and cubes 8.EE.2 Understand a function as a rule that assigns each input to exactly one output and graph as ordered pairs 8.F.1 Compare properties of 2 functions (algebraically, graphically, numerically in tables, or by verbal descriptions) 8.F.2 Rewrite expressions in equivalent forms including factoring A-SSE.2 Solve equations for any variable including formulae A-CED.4 Solve linear equations and inequalities in 1 variable A-REI.3 Identify and use zeroes of a polynomial and sketch graphs A-APR.3 Divide polynomial expressions using long division, synthetic division, or a computer algebra system A-APR.6 Know and apply the Remainder Theorem A-APR.2 Prove polynomial identities and use them to describe numerical relationships A-APR.4 Know and apply the binomial theorem and relate to Pascal s triangle A-APR.5 Solve systems of linear equations algebraically and graphically A-REI.6 Represent a system of linear equations using an augmented matrix format A-REI.8 Graph solutions to a linear inequality and to systems of linear inequalities A-REI.12 Model situations with constraints using systems of equations and inequalities and interpret and evaluate A-CED.3 Solve systems of linear equations using matrices and inverses with technology A-REI.9 Prove linear combination and substitution methods produce equivalent results A-REI.5 Understand equation solving as a proof process A-REI.1 Understand definition of a function, domain, range, and notation and link to graphs F-IF.1 Restrict the domain of functions based on context and/or mathematical restrictions F-IF.5 Interpret key features of functions including intercepts, increasing and decreasing intervals, positive or negative values of ranges, maxes and mins, symmetry, asymptotes, and periodicity F-IF.4 Identify the effect of algebraic transformations corresponding to translations and stretches (both vertical and horizontal) and use to identify odd and even functions F-BF.3 Understand the inverse relationship between exponents and logs and apply in context F-BF.5 Use function notation and evaluate for given values including in context F-IF.2 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers F-IF.3 Solve systems of a linear and a quadratic equation algebraically and graphically A-REI.7 Understand and perform arithmetic operations on polynomials as a closed system (like integers) under +, -, and x A-APR.1 Solve rational and radical equations in 1 variable including extraneous solutions A-REI.2 Find intersections (exact, successive approx) among functions (linear, poly, rat, abs, exp, and log) using tables, graphs, and equations A-REI.11 Calculate or estimate and interpret average rate of a function over an interval symbolically, in tables, and graphically F-IF.6 Factor a quadratic expression to reveal the zeroes of the function A-SSE.3a Link functions to situations and identify key parameters including in linear and exponentials F-LE.5 Compare properties of 2 functions in different representations F-IF.9 Observe the rate of change in exponential functions compared to in linear, quadratic, and polynomial functions F-LE.3 Solve exponential equations using logs (for base 2, 10, or e) and evaluate F-LE.4 Solve linear equations in 1 variable and classify as having unique, multiple, or no solutions 8.EE.7.a Interpret parts of an expression, such as terms, factors, and coefficients. A-SSE.1.a Factor expressions to find zeros of a function A-SSE.3.a Complete the square to transform a quadratic equation and derive quadratic formula A-REI.4.a Graph polynomial functions, identifying zeros, showing end behavior. F-IF.7.c Graph square root, cube root, and piecewise functions (step and absolute value) F-IF.7.b Manipulate different forms (general, standard, transformational) to reveal key properties (zeroes, extrema, symmetry) and explain in context F-IF.8.a Recognize if and when different algebraic forms of functions produce similar or different behaviors F-IF.8.b Describe a function using an explicit expression, a recursive process, or steps for calculation F-BF.1.a Find inverses for simple functions F-BF.4.a Graph exponential, logarithmic, and trigonometric functions, showing key features F-IF.7.e Create linear, exponential, quadratic, and rational equations and inequalities in one variable in context. A-CED.1 Solve simultaneous linear equations algebraically, graphically, and by inspection 8.EE.8.b Solve real world problems using simultaneous linear equations 8.EE.8.c Interpret complex expressions in terms of their parts A-SSE.1.b Define slope using ratios and similar triangles and derive the equation y = mx and y = mx+b 8.EE.6 Distinguish linear functions from simple nonlinear functions 8.F.3 Understand the graph of a function is the locus of points which satisfy the function A-REI.10 Verify by composition that one function is the inverse of another F-BF.4.b Read values of inverse functions from a graph or table F-BF.4.c Produce an invertible function from a noninvertible function by restricting the domain. F-BF.4.d Complete the square to find maximum or minimum of a function A-SSE.3.b Apply properties of exponents to exponential functions A-SSE.3.c Solve quadratic equations by inspection, square roots, completing the square, quadratic formula, and factoring and recognize complex solutions A-REI.4.b Graph rational functions, identifying zeros and asymptotes, showing end behavior. F-IF.7.d Combine standard function types using arithmetic operations F-BF.1.b Compose Functions F-BF.1.c Examine and compare rate of change of linear and exponential functions F-LE.1.a Recognize situations with constant additive rates of change F-LE.1.b Recognize situations with constant multiplicative rates of change F-LE.1.c Find linear or exponential functions given a graph, verbal description, or table of values when sufficient F-LE.2 Graph linear and quadratic functions and show intercepts, maxima, and minima F-IF.7.a Use completing a square of a quadratic function to produce transformational forms and identify maxes and mins A-SSE.3b Create and graph equations in 2 or more variables A-CED.2 by J. Confrey Hexagon map Wireless Generation 2011 Hexagon Map of Common Core K-12 Math Standards

19 For 2-digit Compare distributions of 2 data sets visually and explore different ways to quantify and compare the differences including using mean absolute deviation Use measures of center and variability for numerical data from random samples to draw informal comparative inferences about 2 populations Understand how to construct and interpret 2- way tables of bivariate categorical data for patterns of association based on relative frequencies Find volumes by packing unit cubes and deriving formula as v = lwh or V =bh including fractional edges including real world Find areas of geometric problems figures (certain triangles and quadrilaterals, and polygons) by composing or decomposing into known figures including real world problems Find volume of a right rectangular prism by: packing with unit cubes, multiplying edge lengths, and as lxwxh; use to show associate property Understand subtraction of rational numbers as adding the additive inverse, p q = p + (-q) and distance as the absolute value of their difference Explain why multiplication by a fraction >1 produces a larger number, <1 produces a smaller number, and =1 produces equivalent fractions Recognize a digit in one place represents ten times of what it represents to the left Learning Trajectories Special Fractions Place Value and Decimals Linear and Simultaneous Functions Counting Modeling Arithmetic Relations with Real Numbers Division and Multiplication Length, Area, and Volume Variation, Distribution, and Central Tendancy Addition and Subtraction Legend Decompose numbers <= 10 into pairs 7.SP.8.c 8.SP.3 Use the equation of a linear model to solve problems in the context of bivariate data, intepreting the slope and intercept Design and use a simulation to generate frequencies for 7.SP.8.b compound events. 8.SP.2 Represent sample Informally model linear spaces for compound relationships in scatter Chance events using lists, tables, plots if appropriate and 7.SP.7.b and tree diagrams 7.SP.8.a and assess fit Develop a probability Understand the model (may not be probability of a and uniform) and Probability observe compound event is the frequencies from a 7.SP.7.a fraction of the outcomes 8.SP.1 in the sample space chance process Use scatter plots with bivariate data to Develop a uniform investigate association probability model and (positive, negative, or 7.SP.5 apply to events 7.SP.6 none), clustering, outliers, and linear and Use empirical data to Understand the nonlinear association probability of a chance estimate probability of event is a number a chance event between 0 and 1 and examining the effects 7.SP.2 interpret the meaning of 8.SP.4 7.SP.3 of conducting multiple Generate or simulate different values multiple random samples trials of a given size to examine variation and informally discuss 6.SP.5.d 7.SP.1 7.SP.4 drawing inferences about Understand the Relate choice of the population relationship between measures of center and samples and populations variability to the shape of including random 6.SP.5.c the data distribution in sampling, representative context 6.SP.5.a 8.G.2 8.G.7 samples, and valid Describe measures of inference Understand 2 figures are Apply the Pythagorean Similarity, Congruence, center (median, mean) Report the congruent if the second is Theorem to determine and variability obtained from a unknown side lengths (interquartile range, number of sequence of rotations, in right triangles in real mean absolute deviation) 6.SP.3 observations 8.G.1.c reflections, and 8.G.3 world problems in 2 8.G.6 in relation to context Irrationals, and translations on the first Recognize measures of Describe the effect of and 3D For transformations center and measures of dilations, translations, (rotations, reflections, Explain a proof of the variation as single rotations, and and translations): Theorem 6.SP.2 values representing reflections on 2D 6.SP.5.b Parallel lines map to 8.G.1.b 8.G.5 and its converse data sets figures using Pythagorean 8.G.8 Theorem parallel lines Identify angle Understand Describe attribute, For transformations coordinates relationships including Apply the Pythagorean Modeling Arithmetic distributions of data as described by how it's measured, (rotations, reflections, sum in a triangle, Theorem to find the and translations): corresponding angles distance between 2 center, spread, and and units of 6.SP.4 Angles map to angles and alternate interior 7.NS.1.d overall shape measurement 8.G.4 8.G.9 points in a coordinate of the same measure angles and establishing system Display numerical data Understand 2 figures similarity Apply properties of Relations with Real in plots on a number are similar if a Know and apply operations as line (dot plots, sequence of rotations, formulas for cones, strategies to add and 6.SP.1 histograms, and box reflections, translations, cylinders, and 8.G.1.a 7.G.4 7.RP.2.d 6.NS.8 subtract rational 8.NS.2 plots) and dilations produces spheres Know the formulas for Solve real world numbers 8.EE.6 congruency Numbers Recognize statistical 6.SP.5.a Approximate irrationals For transformations area and circumference Find and interpret unit problems using the with rationals and Define slope using ratios questions as (rotations, reflections, of a circle, informally rate and coordinate coordinate plane. Find locate on a number line and similar triangles and associated with and translations): Lines relate them and use to points for proportional distance between 5.MD.2 7.G.6 derive the equation y = variability in the data using approximations map to lines 7.G.5 solve real world 6.NS.6.b relations 6.NS.6.c points sharing one 7.NS.1.c 8.EE.5 mx and y = mx+b 8.F.5 Generate and use line Solve real world problems Extend the coordinate and/or constructions coordinate Examine proportional Sketch qualitative plots of measurements problems involving area, Solve simple equations 6.SP.5.c plane to all 4 quadrants Graph pairs of rational relationships using graphs from verbal of objects including ½, volume and surface area for unknown angles using and recognize numbers on the multiple representations descriptions of ¼, and 1/8 and of 2D and 3D objects definitions of reflections exhibited by composed of triangles, complimentary, vertical, number line and (graphs, tables, and functions, including 4.MD.4 redistribute into equal 7.G.3 quadrilaterals, polygons, and adjacent angles (a, b), (-a, b), (-a, -b), 6.NS.7.b coordinate plane. 7.NS.1.b 7.RP.2.a equations) and interpret 8.F.4 increasing, decreasing, Generate and use line 7.G.2 shares cubes, and right prisms Use coordinate pairs as (a, -b). Understand addition slope Model linear linear, or nonlinear plots to show Describe 2D figures from Use tools to construct Write, interpret, and Decide whether two vertices to draw polygons of rational numbers measurements of one or relationships including slicing 3D figures such as geometric shapes quantities are in and find side lengths if more objects including explain ordering of from right rectangular as represented on initial values and rates identifying conditions proportional relationship vertices share a common of change using 3.MD.4 measures of ½, ¼, 1/8, 6.G.4 prisms and right rational numbers in number line including 7.G.1 for unique, multiple, or 5.G.2 6.G.2 using tables of values or coordinate including real 7.NS.1.a 7.RP.3 rectangular pyramids context. 7.RP.2.c 8.F.3 multiple 8.F.2 8.EE.2 and find sums and no solutions additive inverses, and coordinate graphs Generate and use line world problems differences Use proportional Represent 3D figures Represent real representations Represent solutions to in context plots to show multiple relationships to solve as nets and apply to world problems by Describe situations in Represent Distinguish linear Compare properties of 2 equations of the form x 2 = length measures of one functions (algebraically, Solve scale multistep ratio find surface area graphing and which opposite proportional functions from p and x 3 = p as square or more objects graphically, numerically roots and cube roots and problems (including 4.G.3 including real world 6.G.1 quantities combine to relationships by 3.MD.3 drawing problems. 5.G.3 5.G.1 interpreting points in simple nonlinear including measures of 6.NS.6.a 7.RP.2.b 6.RP.3d 8.NS.1 8.F.1 in tables, or by verbal percent increase and 8.EE.8.c recognize common problems make 0 equations halves and quarters the first quadrant functions descriptions) perfect squares and Represent categorical decrease) Link rational numbers to Understand that Identify constant of Convert measures and Understand a function cubes data using scaled Recognize and draw Graph coordinate Plot integers on the decimal expansions and Solve real world attributes belonging proportionality in transform units when as a rule that assigns picture and bar graphs number line and vice versa and introduce problems using lines of symmetry for pairs on the multiplying or dividing recognize -(-a) = a and tables, graphs, irrationals as nonrepeating non-terminating one output and graph 8.EE.8.b equations each input to exactly and compare to a category apply to simultaneous linear 2.MD.10 2D figures 5.G.4 6.NS.5 6.RP.3.c by applying ratio 7.RP.1 7.NS.2.b all subcategories 5.MD.5.c coordinate plane 5.MD.5.b 6.NS.7.d 0 as its own opposite. equations, diagrams quantities. or text Solve real world reasoning. decimals Understand that as ordered pairs Plot picture and bar Recognize volume as Distinguish rational numbers are Classify 2D figures Apply volume Understand integers as problems involving graphs of data and Find unit ratios additive through comparisons of produced by the Solve simultaneous linear answer questions by formulas for right indicating magnitude ratios by using tables, in a hierarchy based equivalent to complex equations algebraically, decomposition and division of p by q (p rectangular prisms absolute value from and direction in real 2.MD.9 comparing, adding, and tape diagrams, and fractions (a/b : c/d :: graphically, and by 3.OA.5 3.G.1 on properties 5.MD.3.b composition in 5.MD.4 5.MD.5.a 6.NS.7.c 6.RP.3.a and q integers, q 0) world contexts. number lines. Extend 5.NF.5.a 7.NS.2.d 8.EE.8.a subtracting including in context statements about a/b / c/d : 1) inspection context including real world Linear and Understand categories order Solve ratio problems to include percents. Convert rational Generate and use dot Apply properties of contexts Measure volume by Understand relationship and subcategories of Measure solid figures Understand Comparing the relative plots to show multiple operations (commutative using tables of values numbers to between a solution to shapes among in unit cubes packed counting unit cubes absolute value as size of the product to length measures of one property of x, associative with coordinate plots to terminating decimals simultaneous linear quadrilaterals, with no gaps or (cubic cm, cubic in, and one factor based on the 1.MD.4 2.G.1 property of x, and 5.MD.3.a cubic ft) 4.MD.2 6.RP.3.b find missing values and 5.NF.4b 7.NS.3 size of the other factor 7.NS.2.a or repeating decimals equations and rhombuses, rectangles, 4.G.2 magnitude including or more objects. distributive property) overlaps 6.NS.7.a 8.EE.7.a in context compare ratios. using division intersection of graphs Simultaneous and squares Solve word problems Solve real world Organize, Recognize triangles, Classify 2D figures Translate between Extend multiplication about distance, time, Solve real world Relate rectilinear area problems using the quadrilaterals, with respect to Define a unit cube to statements of inequality and division to apply Solve linear equations in represent, and liquid volume, mass, and problems including model to tiling and four operations with pentagons, hexagons, parallel or 1 variable and classify as measure volume in of two numbers and to rational numbers money using multiple unit pricing and multiplication of rational numbers having unique, multiple, K.MD.3 interpret data, <=3 and cubes in terms of cubic units 1.G.2 4.G.1 perpendicular lines 3.MD.2 operations and simple 5.MD.1 their relative position 6.RP.2 5.NF.2 5.NF.6 including real world constant speed. fractions 7.NS.2.c (including complex 8.EE.7.b or no solutionsequations 7.EE.4.a categories faces and angles and angle types 4.MD.7 Measure and estimate fractions/decimals on a number line Estimate and solve contexts Solve real world Apply properties of fractions) Solve real world Categorize objects, Compose 2 and 3- Draw and identify liquid volumes and Solve multi-step real Understand the word problems using problems of the form px points, lines (parallel Recognize angle problems involving operations as Solve linear equations in count the number dimensional object masses using g, world problems using addition and 1 variable using + q = r and p(x+q) = r concept of a unit ratio multiplication of in each category and perpindicular), measures as additive strategies to multiply shapes from other kg, and l and apply conversion of subtraction of fractions distributivity and (p,q,r are rational K.G.4 1.G.1 line segments, rays, to answer questions 4.MD.3 arithmetic operations to 4.MD.1 such that a:b :: a/b:1 measurement units 5.NF.1 fractions and mixed 6.NS.1 6.RP.1 with unlike 5.NF.4a and divide rational 4.MD.6 5.NF.7.b and order by size collecting like terms 7.EE.4.b shapes numbers representing and angles numbers Find unknown answer questions denominators numbers Know relative sizes Interpret the product quantities) Analyze and Distinguish Interpret division of values in area and Understand the Add and subtract Interpret, solve, Solve real world Measure and sketch (a/b) q as a parts of a whole number by compare 2 and between defining of measurement concept of ratio as a fractions with unlike and explain problems of the form px angles in whole perimeter a partition of q into b a unit fraction, and + q > r or px+q<r (p, q, units and generate relationship between denominators including K.G.3 3-dimensional and non-defining 3.MD.8 problems involving 4.NF.5 division of conversion tables 2 quantities mixed numbers equal parts and apply 5.NF.5.b 5.NF.7.c 7.EE.3 7.EE.1 K.G.6 4.MD.5.a number degrees 3.MD.7.b 4.NF.1 4.NF.4.b compute such r are rational numbers fractions. shapes attributers Apply properties of rectangles Add 2 fractions in context quotients representing quantities) Solve problems Solve real world Solve multi-step real operations as Identify shapes Compose simple Define a 1-degree Find area of Find equivalent with denominators Multiply a problems involving world problems strategies to add, involving area and rectangles using as 2 or 3- shapes to form angle as 1/360th of fractions 10 and 100 by fraction by a division of unit fractions coordinating among subtract, factor, and perimeter and their multiplication of side dimensional a turn of a circle 4.MD.5.b relationship using equivalent 4.NF.3.d 4.NF.4.c 5.NF.7.a using visual models 6.NS.2 integers, fractions, 7.EE.2 expand linear K.G.5 larger shapes 3.MD.7.a 4.NF.3.b lengths in context symbolically 4.NF.3.a whole number 6.EE.6 and equations decimals and percents expressions with Model shapes in the Decompose a fraction Understand addition fractions Solve word problems Solve word problems Interpret and Understand different rational coefficients Solve real world Relate tiling of world by building Define an n-degree into a sum of fractions and subtraction of involving + and - of involving multiplying a compute division of Fluently divide multidigit numbers using the equations reveal variables as missing algebraic forms of problems by using rectangle to (same denominator) fractions as joining and shapes from angle as n 1- fractions with like fraction by a whole a unit fraction by a multiplication of side multiple ways and separating parts K.G.2 components and 3.MD.6 3.NF.3a 4.NF.3.c denominators using 4.NF.4.a number using visual 5.NF.3 non-zero whole standard algorithm 6.NS.3 relationships among 3.MD.1 degree angles 3.NF.3.c 6.EE.9 values or as numbers lengths as area justify using models referring to the same 5.NBT.6 drawing shapes models and equations models and equations quantities Measure area by 2 fractions are equal Interpret a fraction as a number Write equations in 2 that vary Correctly name Read and write Express whole and equations whole Divide 4-digit Fluently add, subtract, counting unit squares if they are the same Add and subtract Understand a division of the numerator variables and represent shapes regardless time to nearest numbers as fractions, multiply, and divide by the denominator and numbers by 2-digit as graphs and tables minute and (square cm, square and fractions as size, or the same mixed numbers fraction a/b as a multi-digit decimals of their orientations relate to fair shares and identifying dependent K.G.1 2.MD.8 calculate time 3.MD.5.b m, square in, and 3.MD.7.c whole numbers when 3.NF.2.b point on a number 4.NF.2 (like denominators) 4.OA.2 numbers and multiple of 1/b 6.NS.4 estimate bounds of 5.NBT.5 5.NBT.7 using standard 6.EE.7 or overall size and independent 6.EE.4 square ft) line. Compare fractions Find GCF (n1, responses algorithms. Find common intervals appropriate explain n2 Add, subtract, variables Solve word problems geometric shapes Represent a using benchmark Solve word problems <=100) and LCF (n1, n2 Fluently multiply multidigit whole numbers form x + p = q and px = Solve equations of the Recognize equivalent involving money (dollars, Use unit squares with Use area models and multiply, and divide in objects and fraction a/b on fractions, common involving <=12). Use distributive quarters, dimes, nickels, no gaps or overlaps expressions generate tiling to represent the decimals to and pennies) including denominators/ multiplicative property to write using the standard q (x, p, q >=0) including equal outputs for any describe relative 2.MD.7 3.MD.5.a to refer to area 3.MD.7.d distributive property number line as a 3.NF.3.b 4.OA.1 4.OA.4 4.OA.3 numerators, and comparisons equivalent numeric algorithm 5.NBT.4 hundredths and 6.EE.8 3.NF.2.a real world problems 6.EE.3 symbols given value of x position lengths of 1/bth reference units Interpret a = b x c expressions. explain Recognize area as Represent a Recognize and as a multiplicative Identify all factors for Solve multistep word Generate equivalent Use place value to Write inequalities as Read and write time Define a unit square additive through generate simple expressions by fraction 1/b on numbers from problems including x>c or x<c and (digital and analog) to to measure area in decomposition and equivalent fractions comparison using round decimals to applying properties number line using distinguishing primes interpreting represent on number 1.MD.3 nearest 5 minutes square units composition in 2.MD.6 3.NF.3d and explain using 3.OA.5 3.OA.3 times as many or 3.OA.7 4.NBT.6 and composites remainders 5.NBT.3.a any place 6.EE.5 including distributivity 8.EE.4 equipartitioning lines 6.EE.2.c context Compare fractions models. and adding like terms. Solve Apply word properties problems times as much Measure, operate, Tell and write time of Fluently multiply and Divide 4-digit Understand solving with identical Read and write Evaluate algebraic Construct number line equations or and compare large operations (x, ) in hours and halfhours using analog equal numerators or identifying values within (commutative 100 divide within 100 numbers by 1-digit decimals to expressions and reference units and inequalities as of whole numbers 0- and small numbers property involving of x, associative equal using strategies and numbers with thousandths using formula at given values 2.MD and show + and - 3.NF.1 3.OA.5 3.OA.6 groups, property arrays, of x, and using scientific and 3.OA.5 properties of 4.NBT.5 remainders and 5.NBT.1 expanded notation and 5.NBT.2 denominators through 6.EE.2.b and apply order of 8.EE.3 and digital clocks distributive property) yielding true notation with and Understand a measurement operations explain place value operations. inspection Apply properties of Apply properties of Multiply whole Explain patterns in the statements Identify parts of an Use simple scientific without technology Estimate lengths fraction a/b can be number of zeroes in the operations Understand (commutative division operations (commutative numbers < by notation ( a single digit expression as sums, what it represents in the products or quotients times a power of 10) to using inches, ft, yds, viewed as a parts as property an unknown-factor of x, associative property of x, associative 1-digit numbers, and terms, products, place to the right and 1/10 th when multiplying or estimate very large and 2.MD.1 cms, and meters 2.MD.5 of size 1/b (for 3.OA.2 property problem of x, and 3.OA.4 property of x, and 3.OA.5 3.OA digit numbers and 4.NBT.3 5.NBT.3.b factors, quotients, and 8.EE.1 very small numbers and distributive property) distributive property) dividing by powers of 6.EE.2.a b=2,3,4,6,8) explain coefficients. Interpret a b=c (a,b,c their relative size Find unknowns in a Solve Apply 2-step properties whole 10 Measure object Solve word problems of whole numbers) as a Use place value to Simplify expressions multiplication or operations number (commutative word with integer length using tools involving whole Compare decimals to Write expressions with equipartitioned among b round multi-digit division problem property problems of x, applying associative thousandths using operations, numbers, exponents by (with inches, ft, yds, number lengths (0- people giving c per 2.MD.4 3.G.2 person or as a measured 3.OA.1 involving 3 whole 4.NBT.4 order property of operations of x, and whole numbers to cms, and meters) 100) using + and 4.NBT.2 4.NF.7 values of digits 5.OA.2 and letters 6.EE.1 applying properties of distributive property) by length b, c times numbers for +, -, x, and any place Fluently add and exponents Equipartition a Interpret a x b as Compare two subtract multi-digit Read, write, and Write numeric Measure lengths and Write and evaluate whole into n parts decimals to the total number of compare numbers expressions and numerical expressions describe difference in whole numbers hundredths using and name as a unit 2.OA.4 2.OA.3 objects in a groups using expanded form relate them to verbal involving whole-number 2.MD.2 units 3.NBT.2 using standard 4.NBT.1 fraction (1/n) of b objects and place value 4.NF.6 values of digits and statements exponents Fluently add and algorithm justify with models Recognize a digit in Use decimals to Identify groups up to Recognize length of subtract within Find totals of one place represents represent fractions 20 as odd or even, object depends the 1000 using ten times what it with denominators of arrays up to 5x5 showing all even 1.MD.2 size of the unit 2.G.3 2.OA.1 3.NBT.1 represents in the 10 or 100 and locate numbers as doubles 2.NBT.5 strategies and 5.OA.1 Fluently add and place to the right on a number line Equipartition circles algorithms Quantify length by Solve 1 and 2-step subtract within 100 Round whole Use parentheses, and rectangles into 3 iterating units with word problems within using strategies, numbers to the brackets, or braces and and 4 fair shares and no gaps or 100 using addition nearest 10 or MD.1 2.G.2 2.NBT.3 and subtraction 3.OA.5 3.OA.9 5.OA.3 evaluate using order of name. Note fair shares place value, inverse 2.NBT.9 2.NBT.7 overlaps using place value operations need not be congruent operations and Add and subtract to Identify arithmetic Generate a 2-column Order 3 objects Equipartition rectangles Explain addition properties 1000 including Apply properties of into rows and columns Read and write to patterns and explain table using a different and subtraction composition and operations (commutative numeric pattern in each directly and using properties of to produce different 1000 using indirectly using 3 rd strategies using decomposition based on property of x, associative column, graph ordered operations K.MD.2 numbers of squares expanded form 2.OA.2 2.NBT.6 3.NBT.3 property of such x, and as in 2.NBT.2 place value, inverse 4.OA.5 pairs, and explain 1.G.3 place value and object confirmed by counting operations, and distributive multiplication property) and relationships between Equipartition properties addition tables Generate number properties corresponding values Directly compare 2 Fluently add and Add up to 4 2-digit Multiply 1-digit patterns using a rule circles and Count within 1000; objects as more of, subtract within 20 numbers using and explain number skip-count by 5s, Hexagon map of K-8 rectangles into 2 using mental numbers by less of, or find the place value and theoretic properties K.MD.1 and 4 fair shares 1.OA.3 10s, and 100s 1.OA.2 strategies 2.NBT.4 2.NBT.8 multiples of 10 difference Apply properties properties (oddness and and name Solve word eveness) Describe several (commutative Compare numbers problems with + of Mentally add and measurable property of addition, (0 < n < 1000) 3 whole numbers subtract 10 or 100 attributes of an associative property 1.OA.1 using place value 1.NBT.1 of addition) as whose sum is 1.OA.8 2.NBT.1.b (100 < n < 900) object and >, =, and < strategies <=20 Count (starting at Find unknown in The numbers 100, Use + and within any number < 120 addition and 200, 300, to solve word ), read, and write subtraction refer to 1, 2, 3,, K.CC.3 1.OA.5 problems 1.NBT.4 equations 2.NBT.1.a 9 hundreds Ratio and numerals <= 120 Time and Relate counting Add within 100 Proportion and Money Write numbers including with 100 is a bundle of Percents to addition and from 0 to 20 composing 10s 1.OA.7 1.NBT.6 10 tens K.CC.5 K.OA.5 subtraction using strategies Understand the Count objects to meaning of equal Subtract multiples Fluently add and answer how many, sign in addition of 10 (<=90) and Similarity, subtract within 5 K.CC.2 n <=20 K.OA.2 1.OA.6 subtraction 1.NBT.5 explain strategies Number Lines, Congruence, Add and subtract equations Coordinate Irrationals, and Equipartitioning Solve word within 20 using Plane, and Count forward from numbers, add or problems (+ and strategies (near Transformations Pythagorean any number 1.OA.4 doubles, pairs for 1.NBT.3 subtract 10 K.CC.1 K.CC.4.c Theorem - <= 10) 10, etc) without counting Understand Compare two 2- Understand each Count to 100 by subtraction as an digit numbers successive number ones and tens unknown addend using place K.CC.4.b is one larger K.OA.4 problem 1.NBT.2.c value Chance and Expressions and The numbers 10, Associate last Probability Equations Find number pair 20, 30,, 90 number name with equaling 10 refer to 1, 2, 3, K.CC.4.a number of objects K.OA.1 1.NBT.2.b, 9 tens Represent 11 to 19 are Count objects and addition and composed of demonstrate 1-1 K.CC.7 subtraction one ten and correspondence 1.NBT.2.a some ones Compare 2 through activity Data and Modeling numbers between 10 is a bundle of 1 and ones K.CC.6 presented as K.NBT.1 written numerals Compare size of 2 Compose and groups <= 10 as decompose greater than, less K.OA.3 number from 1 to than, or equal 19 as 10s and 1s Data and Modeling Number Lines, Coordinate Plane, and Transformations Money and Time Counting Variation, Distribution, and Central Tendancy Length, Area, and Volume Equipartitioning Addition and Subtraction Place Value and Decimals Ratio and Proportion and Percents Fractions Multiplication and Division Expressions and Equations Common Core Math Standards With Learning Progressions Identified J. Confrey 2011

20 Go to: Describes a sense of the learning history (how students might have arrived at a particular standard) Describes underlying learning sciences and conceptual principles Articulates common misconceptions Identifies bridging standards needed Supports coherence within and between LTs 20

21 Goal: To build assessments worth teaching to; will include constructed response, performance assessments of connections, multi-step problems, relations between content and practices, and modeling Implementation in States: States must start preparing students now for new assessments

22 Partnership for Assessment of Readiness for College and Careers (PARCC) 24 states and the District of Columbia, more than 25 million students in K-12 SMARTER Balanced Assessment Consortium (SBAC) 29 states, over 23 million students in K

23 Course Naming Phasing in the Common Core Standards Reforming professional development Public awareness

24 Elementary: Common Core Math K-5 Middle School: Common Core 6, 7, 8 High School: Common Core Math I, II, III We need to name the high school courses to emphasize that the distinction between integrated and traditional math is no longer useful. There is one set of standards and with the addition of a significant chunk of statistics, all courses will have to cover a combination of algebra, geometry, statistics and probability 24

25 New assessments mean that students will be responsible for the content of the CCSS-M up to the year tested. One possible phasing plan: K-3, Grades 6 and 9: , Grades 4, 7, Grades 5, 8, Phasing is essential to ensure adequate professional development resources across grades. 25

26 Not just cross-walks or on-line training Obligatory and ongoing Delivered in professional development system Leverages higher education partners Includes teacher preparation Creates a career ladder for master/mentor teachers and coaches 26

27 Informed by research on teaching and learning and providing a strong foundation in subject content and methods of teaching. Integrated with district goals to improve education, guided by a coherent long-term plan, and driven by disaggregated data on student outcomes. Designed in response to teacher-identified needs and utilizing collaborative problem solving in which colleagues assist one another by discussing dilemmas and challenges. Primarily school-based, providing sufficient time and other resources, and enabling teachers to work with colleagues in their school building. Continuous and ongoing, incorporating principles of adult learning, and providing follow-up support for further learning. Evaluated ultimately on the basis of its impact on teacher effectiveness and student learning. 27

28 White Gr. 4, 2003 Black Hispanic Asian/Pacific AmerInd/Alaska Two or more White Gr. 4, 2011 Black Hispanic Asian/Pacific AmerInd/Alaska Two or more

29 White Gr. 4, 2003 Black Hispanic Asian/Pacific AmerInd/Alaska Two or more White Gr. 8, 2003 Black Hispanic Asian/Pacific AmerInd/Alaska Two or more

30 White Gr. 8, 2003 Black Hispanic Asian/Pacific AmerInd/Alaska Two or more White Gr. 8, 2011 Black Hispanic Asian/Pacific AmerInd/Alaska Two or more

31 White Gr. 4, 2011 Black Hispanic Asian/Pacific AmerInd/Alaska Two or more White Gr. 8, 2011 Black Hispanic Asian/Pacific AmerInd/Alaska Two or more

32 32

33 No more than one year compacting is advisable in middle grades due to the international quality of the standards and their academic rigor For some 5 th graders, supplementary materials can be provided for additional challenge, as richer and deeper extensions of grade-level standards Middle-grades MCC6 PLUS and MCC7 PLUS courses can cover MCC6 through MCC8 in 2 years 33

34 Compacting should be done by deliberately moving carefully-chosen components of learning trajectories into lower grades. 34

35 35

36 1. All students must be eligible. 2. Criteria for selection must be explicit and transparent, and monitored for fair implementation. 3. Active recruitment through information sessions is necessary to overcome a tendency towards bias (resulting from a lack of information, student hesitancy, a failure to understand the implications of decisions downstream, or low expectations). 36

37 4. Objective assessments for placement should be based solely on proficiency of students in prior material, taught in earlier grades. This focus on proficiency makes the criteria clear and equitably applied. 5. Students not deemed qualified should be given precise information about their weaknesses and other opportunities to challenge up should be made available. 6. All students should be provided an opportunity to appeal decisions and those data should be reviewed by an outside appeals body. This group should be empowered to monitor entry to and denial of entry to the programs. 37

38 Common Core Standards represent an opportunity and a challenge. They are a legacy to young teachers. New assessments will test more complex reasoning. Middle grades will be critical to student success. Implementation must consider naming, phasing, sequencing, obligatory and ongoing professional development, and public relations. Specificity of standards should increase equity, if students are provided adequate opportunities to learn 38

39 Our Goals Taking into consideration the rigor of the common core curriculum, our goals are: To create consistent sequences/course offerings for mathematical study, and 4 12 To create criteria for placement into those sequences/courses that is fair, equitable, and applied consistently for all students.

40 Review of Handouts 4 12

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