Structure of the Common Core State Standards for Mathematics • Research-based learning progressions • Internationally benchmarked • Focused and coherent • Standards for Mathematical Practice • Identify important processes and proficiencies • Standards for Mathematical Content • Grade specific expectations
Structure of theCommon Core State Standards for Mathematics DOMAIN STANDARD CLUSTER Number and Operations in Base Ten 3.NBT Use place value understanding and properties of operations to perform multi-digit arithmetic. Use place value understanding to round whole numbers to the nearest 10 or 100. Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction. Multiply one-digit whole numbers by multiples of 10 in the range 10-90 (e.g., 9 x 80, 5 x 60) using strategies on place and properties of operations..
Cognitive Rigor and Depth of Knowledge • The level of complexity of the cognitive demand. • Level 1: Recall and Reproduction • Requires eliciting information such as a fact, definition, term, or a simple procedure, as well as performing a simple algorithm or applying a formula. • Level 2: Basic Skills and Concepts • Requires the engagement of some mental processing beyond a recall of information. • Level 3: Strategic Thinking and Reasoning • Requires reasoning, planning, using evidence, and explanations of thinking. • Level 4: Extended Thinking • Requires complex reasoning, planning, developing, and thinking most likely over an extended period of time.
Level 1 ExampleGrade 8 Select all of the expressions that have a value between 0 and 1. 87∙ 8–12 74 7–3 ∙ 1 3 2 1 3 9 (–5)6 (–5)10
Level 2 ExampleGrade 8 A cylindrical tank has a height of 10 feet and a radius of 4 feet. Jane fills this tank with water at a rate of 8 cubic feet per minute. How many minutes will it take Jane to completely fill the tank without overflowing at this rate? Round your answer to the nearest minute.
Level 3 ExampleGrade 8 • The total cost for an order of shirts from a company consists of the cost for each shirt plus a one-time design fee. The cost for each shirt is the same no matter how many shirts are ordered. • The company provides the following examples to customers to help them estimate the total cost for an order of shirts. • 50 shirts cost $349.50 • 500 shirts cost $2370 • Part A: Using the examples provided, what is the cost for each shirt, notincluding the one-time design fee? Explain how you found your answer. • Part B: What is the cost of the one-time design fee? Explain how you found your answer.
Level 4 ExampleGrade 8 During the task, the student assumes the role of an architect who is responsible for designing the best plan for a park with area and financial restraints. The student completes tasks in which he/she compares the costs of different bids, determines what facilities should be given priority in the park, and then develops a scale drawing of the best design for the park and an explanation of the choices made. This investigation is done in class using a calculator, an applet to construct the scale drawing, and a spreadsheet.
Cognitive Rigor Matrix This matrix from the Smarter Balanced Content Specifications for Mathematics draws from both Bloom’s (revised) Taxonomy of Educational Objectives and Webb’s Depth-of-Knowledge Levels below.
Mathematics Assessment Claims • Claim 1: Concepts and Procedures • Students can explain and apply mathematical concepts and interpret and carry out mathematical procedures with precision and fluency • Claim 2: Problem Solving • Students can solve a range of complex well-posed problems in pure and applied mathematics, making productive use of knowledge and problem solving strategies • Claim 3: Communicating Reasoning • Students can clearly and precisely construct viable arguments to support their own reasoning and to critique the reasoning of others • Claim 4: Modeling and Data Analysis • Students can analyze complex, real-world scenarios and can construct and use mathematical models to interpret and solve problems
Claim 1Concepts and Procedures Students can explain and apply mathematical concepts and interpret and carry out mathematical procedures with precision and fluency.
Claim 1 Concepts and Procedures Target A [m]: Use the four operations with whole numbers to solve problems. (DOK 1, 2) Tasks for this target will require students to use the four operations to solve straightforward, one-step contextual word problems in situations involving equal groups, arrays, and finding an unknown number, including problems where the remainder must be interpreted. Some of these tasks will draw on contexts in 4.MD Target I using measurement quantities such as time, liquid volume, and masses/weights of objects, and money (with decimal representations limited to those described in standards 4.NF.6 and 4.NF.7). Grade 4 Operations and Algebraic Thinking
Claims 2, 3, and 4 Assessment Targets for Claims 2, 3, and 4 are not divided into a grade-by-grade description. A general set of assessment targets applicable across grade levels.
Assessment TargetsClaim 2 – Problem Solving Claim 2: Students can solve a range of complex well-posed problems in pure and applied mathematics, making productive use of knowledge and problem solving strategies. • Apply mathematics to solve well-posed problems arising in everyday life, society, and the workplace • Select and use tools strategically • Interpret results in the context of the situation • Identify important quantities in a practical situation and map their relationships.
Assessment TargetsClaim 3 – Communicating Reason Claim 3: Students can clearly and precisely construct viable arguments to support their own reasoning and to critique the reasoning of others. • Test propositions or conjectures with specific examples. • Construct, autonomously, chains of reasoning that justify or refute propositions or conjectures. • State logical assumptions being used. • Use the technique of breaking an argument into cases. • Distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in the argument—explain what it is. • Base arguments on concrete referents such as objects, drawings, diagrams, and actions. • Determine conditions under which an argument does and does not apply.
Assessment TargetsClaim 4 – Modeling and Data Analysis Claim 4: Students can analyze complex, real-world scenarios and can construct and use mathematical models to interpret and solve problems. • Apply mathematics to solve problems arising in everyday life, society, and the workplace. • Construct, autonomously, chains of reasoning to justify mathematical models used, interpretations made, and solutions proposed for a complex problem. • State logical assumptions being used. • Interpret results in the context of a situation. • Analyze the adequacy of and make improvement to an existing model or develop a mathematical model of a real phenomenon. • Identify important quantities in a practical situation and map their relationships. • Identify, analyze, and synthesize relevant external resources to pose or solve problems.