ࡱ> lnk}y $bjbj $t{{[" " ||8Ld%fHH^^^"""j%l%l%l%l%l%l%$Q'*J%"""""""%||^^J %$$$"v|^|8^j%$"j%$$$^ lb#$V%%0%$M*#"M*$M*$8""$"""""%%$"""%""""M*"""""""""" B: Practicing with the Shifts: The Common Core State Standards for Mathematics (Answer Document)  Shift One: Focus strongly where the Standards focus. In your groups, discuss ways to respond to the following question, Why focus? Theres so much math that students could be learning. Why limit them to just a few things? Engaging with the shift: What do you think belongs in the major work of each grade? GradeWhich 2 of the following represent areas of major focus for the indicated grade?KCompare numbersUse tally marks Tally marks are not a topic in the StandardsUnderstand meaning of addition and subtraction1Add and subtract within 20Measure lengths indirectly and by iterating length unitsCreate and extend patterns and sequences Patterns are a tool, not a topic. Pattern work is included in the standards with very specific aims, but simple creation and extension of patterns is not something that is required. Purposeful work with patterns begins in later grades.2Work with equal groups of objects to gain foundations for multiplicationUnderstand place valueIdentify line of symmetry in two dimensional figures Students will not be working with symmetry until 4th grade, where it is an additional cluster, not major work.3Multiply and divide within 100Identify the measures of central tendency and distribution Students will work with central tendency in middle school, when they have developed solid computational skills and can use authentic data.Develop understanding of fractions as numbers4Examine transformations on the coordinate plane Transformations are part of the major work of 8th grade. Generalize place value understanding for multi-digit whole numbersExtend understanding of fraction equivalence and ordering5Understand and calculate probability of single events Finding probability of an event occurs in 7th grade where it supports the major work of proportional reasoning.Understand the place value systemApply and extend previous understandings of multiplication and division to multiply and divide fractions6Understand ratio concepts and use ratio reasoning to solve problemsIdentify and utilize rules of divisibility Students are not required by the Standards to know the divisibility rules.Apply and extend previous understandings of arithmetic to algebraic expressions7Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbersUse properties of operations to generate equivalent expressionsGenerate the prime factorization of numbers to solve problems Students are not required by the Standards to generate the prime factorization of a number.8Standard form of a linear equation Though linear equations are a major focus of 8th grade, students are not required to use a standard form.Define, evaluate, and compare functionsUnderstand and apply the Pythagorean TheoremAlg.1Quadratic inequalities Quadratic inequalities will be addressed in Alg. 2Linear and quadratic functionsCreating equations to model situationsAlg.2Exponential and logarithmic functions Polar coordinates Polar coordinates are not required by the Standards for Alg. 2Using functions to model situations Shift Two: Coherence: Think across grades, link to major topics within grades In your groups, discuss what coherence in the math curriculum means to you. Be sure to address both elementscoherence within the grade and coherence across grades. Cite specific examples. Engaging with the shift: Investigate coherence in the standards with respect to fractions. In the space below, copy all of the standards related to multiplication and division of fractions and note how coherence is evident in these standards. Note also standards that are outside of the Number and OperationsFractions domain but are related to, or in support of, fractions. (Answers may vary) GradeStandardSummary of the Standard (If the standard has sub-parts, summarize each sub-part.) 3variousStandards that relate to the foundations of being able to multiply and divide fractions found in 3.OA and 3.NF 44.NF.1Recognize & generate equivalent fractions 44.NF.4 a, b & cApply and extend previous understandings of multiplication to multiply a fraction by a whole number4 4.MD.2Use 4 operations to solve word problems. involving simple fractions5 5.NF.3Interpreting a fraction as division of the numerator by the denominator.5 5.NF.4Apply & extend previous understandings of multiplication to multiply a fraction or whole number by a fraction5 5.NF.5Interpret multiplication as scaling (resizing) 55.NF.6 Solve real word problems involving multiplication of fractions and mixed numbers 55.NF.7Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions 55.MD.2Use operations on fractions to solve problems involving information presented in line plots 66.NS.1Apply & extend previous understandings of multiplication and division to divide fractions by fractions 66.G.2Find volume of a right rectangular prism with fractional edge lengths. 7variousStandards that relate to ratio and proportion (7.RP) and standards extending fractions to rational numbers 7.NS.1-3 Shift Three: Rigor: In major topics, pursue conceptual understanding, procedural skill and fluency and application with equal intensity In your groups, discuss ways to respond to one of the following comments: These standards are expecting that we just teach rote memorization. Seems like a step backwards to me. Or Im not going to spend time on fluencyit should just be a natural outcome of conceptual understanding. Engaging with the shift: Making a true statement: Rigor = _conceptual understanding__ + __procedural skill and fluency__+ ___application__ This shift requires a balance of three discrete components in math instruction. This is not a pedagogical option, but is required by the standards. Using grade 3 or 6 as a sample, find and copy in the space below standards which specifically set expectations for each component. Some standards such as 3.OA.4, 3.MD.6 , 3.G.2 and 6.EE.1, 6.EE.2 could be argued to require procedural skill and/or conceptual understanding. Not every Standard will necessarily fit neatly into one of these 3, however as seen by this exercise the majority of Standards specifically call for either fluency, conceptual understanding, or application. Grade 3 or 6 standards that require fluency: Grade 3: 3.OA.7; 3.NBT.2 Grade 6: 6.NS.2, 6.NS.3 Grade 3 or 6 standards that require deep conceptual understanding: Grade 3: 3.OA.1-2, 5-6, 9; 3.NBT.1, 3.NF.1-3, 3.MD.5, 7; 3.G.1 Grade 6: 6.RP.1-2; 6.NS.1, 4-7; 6.EE.3-5; 6.G.1,2; 6.SP.1-3 Grade 3 or 6 standards that require application: Grade 3: 3.OA.3,8; 3.MD.1-4,8 Grade 6: 6.RP.3; 6.NS.1, 8; 6.EE.6-9; 6.G.1-4; 6.SP.4-5      www.achievethecore.org  PAGE \* MERGEFORMAT 1 Module 2: Math Shifts K]^_`    B C    = > m ¸wwpf\OFhlQhu9^JhlQhu9B*^JphhlQhm7^JhlQhY7^J hlQhmhlQha^JhlQhm^JhlQh#X6^JhlQh2.Y5^JhlQh`%^JhlQh#X^JhlQhXx^JhlQh#X5^JhV Qh`%5:aJjhlQUmHnHuhT?5:CJ aJhV Qh#X5:CJ aJhV Q5:CJ aJ^`C Gkd$$Ifl0%Z# t0T&644 lapytlQ $$Ifa$gdlQ $Ifgd#Xgd#X dxgdV Q $dxa$gdV Q   > m n p ?kd$$Ifl\ <%(   t0T&644 lap(ytlQ $Ifgd#Xm n + A B v w  B F G 澵𵧙ugYLChlQh ^JhlQhu9CJ^JaJhlQhu97CJ^JaJhlQhY7CJ^JaJ$hlQhu9B*CJH*^JaJph!hlQhu9B*CJ^JaJphhlQhV Q7CJ^JaJhlQh77CJ^JaJhlQhm^J hlQhYhlQhu9B*^JphhlQhV Q7^JhlQhu97^JhlQhY7^JhlQhY^J hlQhm + B 6kdd$$Ifl\ <%(   t0T&644 lap(ytlQ $Ifgdc $Ifgd#XB  =44 $Ifgd#Xkd.$$Ifl\ <%(   t0T&644 lap(ytlQ $IfgdlQ p?4 $IfgdlQkd$$Ifl\ <%(   t0T&644 lap(ytlQ $Ifgd#X25dfo%'RT"#i01.1ʷʷʧwiʧ[hlQhB7CJ^JaJhlQhl7CJ^JaJhlQhu9CJ^JaJhlQhM7CJ^JaJ hlQh hlQh 7CJ^JaJhlQh ^J hlQhm$hlQhu9B*CJH*^JaJph!hlQhu9B*CJ^JaJphhlQhu97CJ^JaJhlQh77CJ^JaJhlQhm^J ?4 $IfgdlQkd$$Ifl\ <%(   t0T&644 lap(ytlQ $Ifgd#X"#%i?4 $IfgdlQkd$$Ifl\ <%(   t0T&644 lap(ytlQ $Ifgd#X013?kdV$$Ifl\ <%(   t0T&644 lap(ytlQ $Ifgd#XG=44 $Ifgd#Xkd $$Ifl\ <%(   t0T&644 lap(ytlQ $IfgdlQtu{   89:KLƿ|si`|WRM hXx5 h, 5hlQh#X^JhlQh g^JhlQhB7^JhlQh8^J hlQh3hlQhB^JhlQhz`6^JhlQh gB*^JphhlQh g7^JhlQhz`67^JhlQh3^J hlQh hlQh ^J$hlQh gB*CJH*^JaJph!hlQh gB*CJ^JaJphhlQh gCJ^JaJGtu{ ?kd$$Ifl\ <%(   t0T&644 lap(ytlQ $Ifgd#X  9:H????? 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