Mathematics Content Specifications,
Item Specifications, and Depth of
Knowledge Training Module
Purpose of Training Module
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Introduce primary source materials
– Common Core State Standards for Mathematics
– Smarter Balanced Mathematics Content Specifications
– Smarter Balanced Mathematics Item Specifications
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Describe how these documents will be used
for item writing
Define Depth of Knowledge
Foundation for Item and Task Development
Items and
Performance Tasks
Smarter Balanced Item
and Task Specifications
Smarter Balanced
Content Specifications
Common Core
State Standards
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..
Structure of the Smarter Balanced Assessment
Consortium Mathematics Content Specifications
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Introduction and Background
General Considerations
Claims and Assessment Targets
Rationale and Evidence for Each Claim
Cognitive Rigor Matrix / Depth of Knowledge
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
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.
Smarter Balanced Assessment Consortium
Mathematics Content Specifications
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).
Claim 1 Assessment Targets
Grade 3
Operations and Algebraic Thinking
A. Represent and solve problems involving multiplication and
division.
B. Understand properties of multiplication and the relationship
between multiplication and division.
C. Multiply and divide within 100.
D. Solve problems involving the four operations, and identify and
explain patterns in arithmetic.
Number and Operations – Base 10
E.
Use place value understanding and properties of arithmetic to
perform multi-digit arithmetic.
Claim 1 Assessment Targets
Grade 3
Numbers and Operations – Fractions
F.
Develop understanding of fractions as numbers.
Measurement and Data
G. Solve problems involving measurement and estimation of intervals
of time, liquid volumes, and masses of objects.
H. Represent and interpret data.
I. Geometric measurement: understand concepts of area and relate
area to multiplication and to addition.
J. Geometric measurement: recognize perimeter as an attribute of
plane figures and distinguish between linear and area measures.
Geometry
K.
Reason with shapes and their attributes.
Claim 1 Assessment Targets
Grade 4
Operations and Algebraic Thinking
Use the four operations with whole numbers to solve problems.
Gain familiarity with factors and multiples.
Generate and analyze patterns.
Generalize place value understanding for multi-digit whole
numbers.
E. Use place value understanding and properties of operations to
perform multi-digit arithmetic.
A.
B.
C.
D.
Number and Operations – Fractions
F. Extend understanding of fraction equivalence and ordering.
G. Build fractions from unit fractions by applying and extending
previous understandings of operations on whole numbers.
H. Understand decimal notation for fractions, and compare
decimal fractions.
Claim 1 Assessment Targets
Grade 4
Measurement and Data
Solve problems involving measurement and conversion of
measurements from a larger unit to a smaller unit.
J. Represent and interpret data.
K. Geometric measurement: understand concepts of angle and
measure angles.
I.
Geometry
L.
Draw and identify lines and angles, and classify shapes by
properties of their lines and angles.
Claim 1 Assessment Targets
Grade 5
Operations and Algebraic Thinking
A. Write and interpret numerical expressions.
B. Analyze patterns and relationships.
Number and Operations – Base 10
C. Understand the place value system.
D. Perform operations with multi-digit whole numbers and with
decimals to hundredths.
Number and Operations – Fractions
Use equivalent fractions as a strategy to add and subtract
fractions.
F. Apply and extend previous understandings of multiplication
and division to multiply and divide fractions.
E.
Claim 1 Assessment Targets
Grade 5
Measurement and Data
G. Convert like measurement units within a given measurement
system.
H. Represent and interpret data.
I. Geometric measurement: understand concepts of volume and
relate volume to multiplication and to addition.
Geometry
Graph points on the coordinate plane to solve real-world and
mathematical problems.
K. Classify two-dimensional figures into categories based on their
properties.
J.
Claim 1 Assessment Targets
Grade 6
Ratios and Proportional Relationships
A. Understand ratio concepts and use ratio reasoning to solve
problems.
The Number System
Apply and extend previous understandings of multiplication and
division to divide fractions by fractions.
C. Compute fluently with multi-digit numbers and find common
factors and multiples.
D. Apply and extend previous understandings of numbers to the
system of rational numbers.
B.
Claim 1 Assessment Targets
Grade 6
Expressions and Equations
E. Apply and extend previous understandings of arithmetic to
algebraic expressions.
F. Reason about and solve one-variable equations and
inequalities.
G. Represent and analyze quantitative relationships between
dependent and independent variables.
Geometry
H. Solve real-world and mathematical problems involving area,
surface area, and volume.
Statistics and Probability
I.
J.
Develop understanding of statistical variability.
Summarize and describe distributions.
Claim 1 Assessment Targets
Grade 7
Ratios and Proportional Relationships
A. Analyze proportional relationships and use them to solve real-
world and mathematical problems.
The Number System
B.
Apply and extend previous understandings of operations with
fractions to add, subtract, multiply, and divide rational numbers.
Expressions and Equations
C. Use properties of operations to generate equivalent expressions.
D. Solve real-life and mathematical problems using numerical and
algebraic expressions and equations.
Claim 1 Assessment Targets
Grade 7
Geometry
Draw, construct and describe geometrical figures and describe the
relationships between them.
F. Solve real-life and mathematical problems involving angle
measure, area, surface area, and volume.
E.
Statistics and Probability
G. Use random sampling to draw inferences about a population.
H. Draw informal comparative inferences about two populations.
I. Investigate chance processes and develop, use, and evaluate
probability models.
Claim 1 Assessment Targets
Grade 8
The Number System
A. Know that there are numbers that are not rational, and
approximate them by rational numbers.
Expressions and Equations
Work with radicals and integer exponents.
Understand the connections between proportional relationships,
lines, and linear equations.
D. Analyze and solve linear equations and pairs of simultaneous linear
equations.
B.
C.
Functions
E.
F.
Define, evaluate, and compare functions.
Use functions to model relationships between quantities.
Claim 1 Assessment Targets
Grade 8
Geometry
G. Understand congruence and similarity using physical models,
transparencies, or geometry software.
H. Understand and apply the Pythagorean theorem.
I. Solve real-world and mathematical problems involving volume of
cylinders, cones and spheres.
Statistics and Probability
J.
Investigate patterns of association in bivariate data.
Claim 1 Assessment Targets
Grade 11
Number and Quantity
A. Extend the properties of exponents to rational exponents.
B. Use properties of rational and irrational numbers.
C. Reason quantitatively and use units to solve problems.
Algebra
D.
E.
F.
G.
H.
I.
J.
Interpret the structure of expressions.
Write expressions in equivalent forms to solve problems.
Perform arithmetic operations on polynomials.
Create equations that describe numbers or relationships.
Understand solving equations as a process of reasoning and
explain the reasoning.
Solve equations and inequalities in one variable.
Represent and solve equations and inequalities
graphically.
Claim 1 Assessment Targets
Grade 11
Functions
Understand the concept of a function and use function
notation.
L. Interpret functions that arise in applications in terms of a
context.
M. Analyze functions using different representations.
N. Build a function that models a relationship between two
quantities.
K.
Geometry
O.
Prove geometric theorems.
Statistics and Probability
P.
Summarize, represent and interpret data on a single count or
measurement variable.
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.
Structure of the Smarter Balanced
Assessment Consortium Mathematics
Item and Task Specifications
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Evidence Statements
Task Models
Sample Items
Mathematics Glossary
Structure of Item Specification Table
Grade 5 Evidence Statement
Assessment Target C:
Understand the place value system. (DOK 1, 2)
Evidence Statement:
The student explains patterns in the number of
zeros of the product/quotient when
multiplying/dividing a number by powers of 10.
Grade 5 Task Model
Evidence Statement:
The student explains patterns
in the number of zeros of
the product/quotient when
multiplying/dividing a number
by powers of 10.
Task Model
2. CR (DOK 1, 2)
Prompt Features 1: The student is prompted
to explain patterns in the number of zeros of
a product when multiplying a number by
powers of 10.
Stimuli: The student is presented with a
multiplication problem involving powers of
10.
Prompt Features 2: The student is prompted
to explain patterns in the number of zeros of
a quotient when dividing a number by powers
of 10.
Stimuli: The student is presented with a
division problem involving powers of 10.
Alignment to Claims and Assessment Targets
Module Summary
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Common Core State Standards: foundation for
claims and assessment targets
Smarter Balanced Content Specifications: define
claims and assessment targets
– Depth of Knowledge: Level of cognitive processes
applied by students
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Smarter Balanced Item and Task Specifications:
define evidence required and describe task models