Common Core Mathematics
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Common Core Mathematics
Common Core MathematicsScope and Sequence
Christina Foran and Rebecca Candino, TSDIntroductionsChristina [email protected]: metateacher.netRebecca [email protected]: rcandino.edublogs.orgCore Math Wiki: metateacher.net/coremathAgendaOverview of the Content StandardsUsing the Common Core Navigator to develop standards-based IEP and differentiated learning laddersOverview of the Mathematical PracticesSelecting and modifying tasks for Constructed Response Assessment practice
Domains across Grade Level
IllustrativeMathematics.orgThe Common Core NavigatorStandards organized by topics called Learning PathwaysStandards organized developmentally across grade levelsHow to Use:Select your topic from the table of contents.Consider the grade-level standards.Drill down to your students level.Go even farther back with previous learning pathways.Standards-Based IEPsFreshman algebra student with a 1.5 grade level equivalencyGRADE LEVEL STANDARD: A-CED.1Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.GOAL: The student will use equations in one variable to solve problems.LEARNING PATHWAYS:Concepts of Equality and EquationsEquations and InequalitiesStandards-Based IEPsOBJECTIVES:Determine the unknown whole number in an addition or subtraction equation (1.OA.8).Determine the unknown whole number in a multiplication or division equation (3.OA.4)Represent two-step word problems in four operations using equations with a letter standing for the unknown quantity (3.OA.8).Apply properties of operations, including the distributive property, to generate equivalent expressions (6.EE.3)Solve problems by writing and solving equations in the form of x+p=q and px=q with nonnegative rational numbers (6.EE.7).Solve word problems leading to equations in the form of px+q=r and p(x+q)=r with rational numbers (7.EE.4).Standards-Based IEPsThird-grade student on a 1st grade levelGRADE LEVEL STANDARD: 3.OA.8Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity Assess the reasonableness of answers using mental computation and estimation strategies including rounding.GOAL: The student will represent two-step word problems in four operations using equations with a letter standing for the unknown quantity.LEARNING PATHWAYS:Concepts of Equality and EquationsStandards-Based IEPsOBJECTIVES:Apply commutative and associative properties of operations as strategies to add and subtract (1.OA.3).Understand the meaning of the equal sign (1.OA.7).Determine if equations involving addition and subtraction are true or false (1.OA.7).Determine the unknown whole number in an addition or subtraction equation (1.OA.8).Determine the unknown whole number in a multiplication or division equation (3.OA.4)Standards-Based IEPsThird-grade student on a 1st grade levelGRADE LEVEL STANDARD: 3.OA.3Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.GOAL: The student will multiply and divide within 100 to solve problems.LEARNING PATHWAYS:Basic Multiplication and DivisionCountingStandards-Based IEPsOBJECTIVES:Count to 120, starting at any number less than 120 (1.NBT.1).Skip count by 5s, 10s, and 100s (2.NBT.2).Use addition to find the total number of objects arranged in a rectangular array with up to 5 rows and 5 columns (2.OA.4).Interpret products of whole numbers as groups of objects (3.OA.1)Interpret whole-number quotients of whole numbers as being partitioned into equal shares (3.OA.2).Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities by using drawings and equations (3.OA.3).Fluently multiply and divide within 100, using strategies such as the relationship between multiplication or division or the properties of operations (3.OA.7).Differentiated Learning LaddersWhat are learning ladders?Lists of prerequisite skills to build up to a learning objectiveStated in positive, student-friendly language. Sample questions from each learning level could be used to make a pre- and post-assessment. Help teachers differentiate instruction. Help the students understand what they should be learning and lets them keep track of their progress toward the learning goal.Differentiated Learning LaddersLesson Topic: Dividing Fractions by FractionsClass: 6th grade with most students on 3rd-5th grade level and a couple on a first grade levelLearning Pathways:Dividing FractionsMultiplying FractionsRepresenting FractionsDifferentiated Learning LaddersI can divide fractions by fractions (6.NS.1).I can divide whole numbers by fractions and fractions by whole numbers (5.NF.7).I can multiply fractions by fractions (5.NS.4).I can draw rectangular arrays to model multiplication of fractions (5.NF.4).I can draw rectangular arrays to model multiplication problems (4.NBT.5).I can multiply fractions by whole numbers (4.NS.4).I can represent fractions as copies of the unit fraction (3.NF.1)I can draw unit fractions by dividing a shape into equal parts (3.G.2).I can draw thirds, fourths, and halves (2.G.3, 1.G.3).Task and Assessment ResourcesTNCore.orgparcconline.orgMasteryConnect.com IllustrativeMathematics.org by Bill McCallumInsideMathematics.orgMathematics Assessment Project - map.mathshell.orgteachingchannel.org
IllustrativeMathematics.orgCCSS for Mathematical PracticeBased on NCTM Process Standards and the National Research Council Adding it Up: Helping Children Learn Mathematics
Equal emphasis on conceptual understanding and procedural fluency
Seeing mathematics as sensible, useful, and worthwhile while believing in ones own skill and abilities
The 8 habits of highly successful problem solversCommon Core State Standards, NGA Center/CCSS), 2010
CCSS for Mathematical Practices1. 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.
Common Core State Standards for Mathematics, 2010, NGA Center/CCSSOSelecting and Modifying High Level Math TasksTasks form the basis for students opportunities to learn what mathematics is and how one does it [Tasks] are central to students learning, shaping not only their opportunity to learn, but their view of the subject matter.
Adding it Up, National Research Council, 2001, p. 335The Importance of High Level Instructional Tasks: The Hook
How Are Math Tasks Categorized?all tasks are not created equaldifferent tasksrequire different levels and kinds of student thinking.
The Mathematical Task Analysis Guide
Stein M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000). Implementing standards-based mathematicsinstruction: A casebook for professional development, p. 3. New York: Teachers College Press
Low level TasksHigh-Level TasksMemorization
Procedures without ConnectionsProcedures with Connections
Doing MathematicsExamples of Different Levels of TasksMemorization
Procedures With ConnectionsProcedures Without Connections
Instructional Task ModificationStudents create real-world stories for naked number problems.
Include a prompt that asks students to represent the information another way and to write about any noticings or wonderings about a math concept.
Eliminate too much scaffolding.
Include a prompt that asks students to notice a pattern or make a conjecture and test that conjecture.
Select numbers carefully to prevent use of rote memorization in responses.
Begin instruction of a new concept with a high level task before students have memorized rules and/or procedures.
Learning Research and Development Center, University of Pittsburgh, 2012Differentiating the Instructional TaskUse the Common Core Navigator to drill downGrade Level Standard: Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression (A-SSE.3). HSA-SSE.B.3 HSA-SSE.B.3 A-SSE.3
Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities by using drawings and equations (3.OA.3).
Questioning techniquesAssessing and Advancing Questions
Growth Occurs in the StruggleAllow students to grapple with high level tasks
The role of the teacher is to facilitate thinking and discussion
Be prepared to pull out of the task to teach prerequisite skills, then dive back in.
Practice, practice, practice
Intertwined Strands of ProficiencyConceptual UnderstandingProcedural FluencyCommon Core Content StandardsCommon Core Mathematical Practices
Adding It Up, National Research Council, 2001