Common Core Mathematics Scope and Sequence State-wide Conference Christina Foran and Rebecca Candino, TSD

Introductions Christina Foran Website: metateacher.netmetateacher.net Rebecca Candino Blog: rcandino.edublogs.orgrcandino.edublogs.org Core Math Wiki: metateacher.net/coremathmetateacher.net/coremath

Agenda Overview of the Content Standards Using the Common Core Navigator to develop standards-based IEP and differentiated learning ladders Overview of the Mathematical Practices Selecting and modifying tasks for Constructed Response Assessment practice

Domains across Grade Level IllustrativeMathematics.org

The Common Core Navigator Standards organized by topics called Learning Pathways Standards organized developmentally across grade levels How to Use: Select your topic from the table of contents. Consider the grade-level standards. Drill down to your student’s level. Go even farther back with previous learning pathways.

Standards-Based IEPs Freshman algebra student with a 1.5 grade level equivalency GRADE LEVEL STANDARD: A-CED.1 “Create 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 Equations Equations and Inequalities

Standards-Based IEPs OBJECTIVES: 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 IEPs Third-grade student on a 1st grade level GRADE LEVEL STANDARD: 3.OA.8 “Solve 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 Equations

Standards-Based IEPs OBJECTIVES: 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 IEPs Third-grade student on a 1st grade level GRADE LEVEL STANDARD: 3.OA.3 “Use 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 Division Counting

Standards-Based IEPs OBJECTIVES: 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 Ladders What are learning ladders? Lists of prerequisite skills to build up to a learning objective Stated 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 Ladders Lesson Topic: Dividing Fractions by Fractions Class: 6th grade with most students on 3rd-5th grade level and a couple on a first grade level Learning Pathways: Dividing Fractions Multiplying Fractions Representing Fractions

Differentiated Learning Ladders I 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 Resources TNCoreTNCore.org parcconline.org MasteryConnect.com IllustrativeMathematics.orgIllustrativeMathematics.org by Bill McCallum InsideMathematics.org Mathematics Assessment Project - map.mathshell.org map.mathshell.org teachingchannel.org

MasteryConnect.com

InsideMathematics.org

map.mathshell.org

IllustrativeMathematics.org

CCSS for Mathematical Practice Based 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 one’s own skill and abilities The “8 habits of highly successful problem solvers” Common Core State Standards, NGA Center/CCSS), 2010

CCSS for Mathematical Practices 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. Common Core State Standards for Mathematics, 2010, NGA Center/CCSSO

Selecting and Modifying High Level Math Tasks Tasks 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. 335

The Importance of High Level Instructional Tasks: “The Hook”

How Are Math Tasks Categorized? …all tasks are not created equal−different tasks require 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 mathematics instruction: A casebook for professional development, p. 3. New York: Teachers College Press Low –level TasksHigh-Level Tasks Memorization Procedures without Connections Procedures with Connections Doing Mathematics

Examples of Different Levels of Tasks MemorizationProcedures With Connections Procedures Without Connections Doing Mathematics

Instructional Task Modification Students 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, 2012

Differentiating the Instructional Task Use the Common Core Navigator to “drill down” Grade 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 techniques Assessing and Advancing Questions

Growth Occurs in the Struggle Allow 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 Proficiency Conceptual Understanding Procedural Fluency Common Core Content Standards Common Core Mathematical Practices Adding It Up, National Research Council, 2001

Thank You!!