Smarter Balanced
Assessment Consortium
Structure of the
Common Core State Standards for Mathematics
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Research-based learning progressions
Internationally benchmarked
Focused and coherent
Standards for Mathematical Practice
– Identify important processes and proficiencies
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Standards for Mathematical Content
– Grade specific expectations
Structure of the
Common 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.
1. Use place value understanding to round whole numbers to the nearest
10 or 100.
2. 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.
3. 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
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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 Example
Grade 8
Select all of the expressions that have a value between 0 and 1.
87  8–12
74
7–3
1
2
3
(–5)6
(–5)10

1
3
9
Selected Response
Traditional
Redesigned
Selected Response
Suggested Scoring Rubric
Selected Response
Level 2 Example
Grade 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 Example
Grade 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, not
including 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 Example
Grade 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
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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 1
Concepts and Procedures
Students can explain and apply mathematical concepts
and interpret and carry out mathematical procedures with
precision and fluency.
Grade Level
Number of
Assessment Targets
3
11
4
12
5
11
6
10
7
9
8
10
11
16
Claim 1
Concepts and Procedures
Grade 4
Operations and Algebraic Thinking
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).
Claims 2, 3, and 4
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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 Targets
Claim 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.
A. Apply mathematics to solve well-posed problems arising in
everyday life, society, and the workplace
B. Select and use tools strategically
C. Interpret results in the context of the situation
D. Identify important quantities in a practical situation and
map their relationships.
Assessment Targets
Claim 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.
A.
B.
C.
D.
E.
F.
G.
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 Targets
Claim 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.
A.
B.
C.
D.
E.
F.
G.
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.