Common Core State Standards in Mathematics Pre-Test
Grade Level
Operations and Algebraic Thinking
Domain
Reference
1.OA
Represent and solve problems involving addition and subtraction.
Cluster

Heading
CCSS.Math.Content.1.OA.A.1 Use addition and subtraction within 20 to solve wordStandard
problems involving
situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all
positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to
represent the problem.1
Standard
How to cite a standard:
1.OA.A.1
O
Grade Level
Domain
Cluster
Heading
Standard
Domains
Warm Up: How does the K-8 chart above convey the idea of the progression of knowledge across
1
grades? How is this distinctly different from the NJCCCS?
Common Core State Standards in Mathematics Pre-Test
QUESTION 1: CRITICAL AREAS (1e, 2d, 4d)
Not all of the content in a given grade is emphasized equally in the standards. Some clusters require
greater emphasis than the others based on the depth of the ideas, the time that they take to master,
and/or their importance to future mathematics or the demands of college and career readiness. In
addition, an intense focus on the most critical material at each grade allows depth in learning, which is
carried out through the Standards for Mathematical Practice.
What are the critical areas at each grade level (K-8)?
K–2:
3–5:
6:
7:
8:
2
Common Core State Standards in Mathematics Pre-Test
QUESTION 2: GRADE LEVEL CONTENT (1e)
In which grade level are the following topics either introduced or emphasized?
Grade: ____





Ratios and proportions
Probability of chance events
Statistics
Knowing formulas of area and
circumference of a circle
Measuring central tendency
Grade: ____





Multiplying and dividing whole
numbers within 100
Liquid volume
Understanding 1/b fractions
with denominators of
2,3,4,6,8
Introduction to area
Multiplying single digits by
multiples of 10
Grade: ____





+/- within 20
Count to 120 starting at any
number
Use place value
Tell and write time in hours
and half hours
Partition circles and triangles
Grade: ____




Grade: ____
Fluently +,-,x,÷ decimals
Unit rate
Identifying integers on the
number line
Evaluating Expressions
Grade: ____












+/- fractions with unlike
denominators
Graphing on the coordinate
plane
Rounding decimals
Grade: ____
Square root
Graphing linear relationships
Introduction to functions
Negative exponents
Congruence
Proving Pythagorean’s
Theorem
Scatter Plots
Grade: ____







Find factor pairs (1-100)
Fractional equivalence and
comparing fractions
(denominators of 2, 3, 4, 5, 6,
8, 10, 100)
Decimal notation for 10ths
and 100ths.
Angle measure
Grade: ____
+/- within 100
Determine odd and even up to
20
Skip count to 1000 by 5, 10,
100
Measure length using
appropriate measuring tools
Tell time using AM and PM
3





Counting to 100
<, >, =
Compare 2 numbers between
1 and 10
+/- within 10
Fluently +/- within 5
Common Core State Standards in Mathematics Pre-Test
QUESTION 3: SUPPORTING CLUSTERS (1e)
The Common Core State Standards for Mathematics denote content emphasis using 3 qualifying levels:
 Major Clusters (green)
 Supporting Clusters (blue)
 Additional Clusters (yellow)
Supporting Clusters work to support the Major Work of the grade. Describe how the Supporting
Cluster (5.MD.1) reinforces the Major Cluster (5.NBT.7).
5.NBT.A.7 Add, subtract, multiply, and divide decimals to hundredths, using concrete models or
drawings and strategies based on place value, properties of operations, and/or the relationship
between addition and subtraction; relate the strategy to a written method and explain the reasoning
used.
5.MD.B.1 Convert among different-sized standard measurement units within a given measurement
system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world
problems.
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Common Core State Standards in Mathematics Pre-Test
QUESTION 4: RIGOR (1e)
In the CCSSM, rigor is defined in the major topics as the pursuit conceptual understanding, procedural
skill and fluency, and application. Certain standards complement procedural skills, conceptual
understandings of specific content, or expect students to apply and extend a concept to a real world or
mathematical problem solving scenario. The “verbs” within each standard provide cues indicating the
aspect of rigor being addressed. For example, the standard below addresses a procedural skill by which
students should support their calculations using various representations: equations, rectangular arrays,
and/or area models.
CCSS.Math.Content.4.NBT.B.5 Multiply a whole number of up to four digits by a one-digit whole
number, and multiply two two-digit numbers, using strategies based on place value and the properties
of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area
models.

In tasks assessing Conceptual Understanding
- the assessed concept should be central to the task
- the necessary computational skill should be fairly low
- the task should be easy to solve if the student understands the concept
- the tasks do not reside in an application

In tasks assessing Procedural Skill & Fluency
- students are explicitly required to perform a calculation by manipulating numbers and operations
to derive an answer (with precision and accuracy)

In tasks assessing Application
- students are expected to use appropriate concepts and skills
- opportunities should exist for students to apply math concepts to real world & mathematical
problems
Indicate whether each of the standards explicitly call for conceptual understanding, procedural skill
and fluency, or application.

CCSS.Math.Content.4.NF.B.3a Understand addition and subtraction of fractions as joining and
separating parts referring to the same whole.

CCSS.Math.Content.4.NF.B.3c Add and subtract mixed numbers with like denominators, e.g.,
by replacing each mixed number with an equivalent fraction, and/or by using properties of
operations and the relationship between addition and subtraction.

CCSS.Math.Content.4.NF.B.3d Solve word problems involving addition and subtraction of
fractions referring to the same whole and having like denominators, e.g., by using visual
fraction models and equations to represent the problem.
5
Common Core State Standards in Mathematics Pre-Test
QUESTION 5: MULTIPLE REPRESENTATIONS (2a)
Promoting a culture of high expectations for all students is a fundamental goal of the Common Core
State Standards. In order to participate with success in the general curriculum, students with
disabilities, as appropriate, may be provided additional supports based on the principles of Universal
Design for Learning (UDL) - which foster student engagement by presenting information in multiple
ways and allowing for diverse avenues of action and expression.
Multiple Representations allow teachers to present content in multiple modalities to support flexible
thinking. These frameworks go beyond concrete representation (e.g. manipulatives) to promote the
realistic representation of concepts addressed in multiple settings.
Below is an example of a Multiple Representations Framework that supports CCSS Math Cluster 7.NS.A
Apply and extend previous understandings of addition and subtraction to add and subtract rational
numbers; represent addition and subtraction on a horizontal or vertical number line diagram.
How does each representation (Concrete, Pictorial, and Abstract) support the cluster?
6
Common Core State Standards in Mathematics Pre-Test
CONCRETE REPRESENTATIONS

2-color coin counters to represent negatives
and positives

Number Lines

Thermometers and other equally
partitioned tools
PICTORIAL REPRESENTATIONS

Number Lines (Horizontal)

Number Lines (Vertical)

Distance / Vector Model
Adding Integers
Addition is modeled as putting a second vector’s tail at the first
vector’s head and finding where the second vector’s head extends
to.
3 + -4 = -1
Subtracting Integers
Subtraction can be thought of as comparing the two vectors p, and
q, by putting both tails together (starting each from zero) and
asking the question: “How would one extend a vector from the head
of p to the head of q?” The length and direction of that vector would
be the result of the subtraction.
3 - -4 = 7

ABSTRACT REPRESENTATIONS
p – q = p + (-q)
Applying Properties of Numbers
p - -q = p + q
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Common Core State Standards in Mathematics Pre-Test
QUESTION 6: TASK ANALYSIS (2b, 2d)
Common Core-aligned assessment items should align to the meaning, depth and breadth of the
standard.
Explain whether or not following assessments items 1-4 assess at the level explicitly called for
within the language of the standard.
5.NBT.A.2: Explain patterns in the number of zeros of the product when multiplying a number by
powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied
or divided by a power of 10. Use whole-number exponents to denote powers of 10.
1. 42.5 x 104 =
2. Explain a rule for multiplying by 10; 100; 1000; any whole number power of 10.
3. A student says that she multiplies by 10 by moving the decimal point one place to the right.
Why does she do this?
4. The arrow above points to the number 44 on a number line numbered from 0 to 100.
Determine the number will the arrow point when the endpoint of 100 is changed to the ranges
below. Describe any patterns that you notice.
44
0
10
20
30
40
50
a. 1,000?
b. 100,000?
c. 1,000,000?
d. 10?
e. 1?
f.
.01?
8
60
70
80
90
100
Common Core State Standards in Mathematics Pre-Test
QUESTION 7: MATHEMATICAL PRACTICES (2b)
The Common Core State Standards for Mathematics articulates practice standards in addition to the
content standards. The Mathematical Practices work together with the content standards; connecting
in meaningful ways.
Mathematical Practices
MP.1 Make sense of problems and persevere in solving them
MP.2 Reason abstractly and quantitatively
MP.3 Construct viable arguments and critique the reasoning of others
MP.4 Model with mathematics
MP.5 Use appropriate tools strategically
MP.6 Attend to precision
MP.7 Look for and make use of structure
MP.8 Look for and express regularity in repeated reasoning
Characteristics of Mathematical Practice 3 [MP.3], include
o
o
o
Students justifying a conclusion
Students proving a statement
Students explaining the mathematics/reasoning used
Where students may be asked to
o use diagrams, words, and/or equations in their work
o “repair” a flawed argument
o use problem solving as a form of argument where the solution is to be laid out as a
series of logical, well-motivated steps using precise language
Describe how the following task connects MP.3 with standard 7.EE.A.2 Understand that rewriting an
expression in different forms in a problem context can shed light on the problem and how the quantities in
it are related.
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Common Core State Standards in Mathematics Pre-Test
Ms. Esposito asked her students to generate an expression that describes the number of
each term as an expression of the stage (s).
’s in
Four students each provided Ms. Esposito with a different expression to describe the number of
’s in a stage.
Student A: 5 + 4(s – 1)
Student B: 1 + 4(s)
Student C: s + s + 2s + 1
Student D: 6s - 1



Which of the students provided the correct expression?
Which expressions are equivalent?
Select the expression provided by one student and explain how it relates to the pattern.
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Common Core State Standards in Mathematics Pre-Test
QUESTION 8: INSTRUCTION and PLANNING (1e, 1f)
Although the CCSSM do not explicitly reference instructional delivery, certain practices, such as
strategic planning, are essential to ensure that instruction is well aligned to the intended objective(s).
Key planning strategies encourage teachers to

Study the standard to highlight key elements

Look at aligned assessment items to help plan with the end in mind (i.e. mastery level
problems, problems embedding the practices, problems assessing all aspects of rigor)

Select tasks that align well to the day’s objective (focusing more so on conceptual
understandings initially )

Solve the problems in advance

Consider various entry points to determine which will inform the day’s objective

Consider potential student errors and misconceptions
Which of the 6 planning strategies, if any, are evident in the lesson outlined on the next page?
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Common Core State Standards in Mathematics Pre-Test
A Grade 6 teacher asks students to determine the number of shaded squares border this 10 by 10 figure,
(without counting). Her goal is to have students ‘apply and extend their previous understandings of
arithmetic to algebraic expressions’.
The teacher takes a moment, in a whole group format, to review some of the anticipated and likely
errors that students would make. She then captures expressions that students used to generate the
correct answer. As an extension to the activity, the teacher asks students to use the captured student
responses (below) to generate numerical expressions for a 6 by 6 figure, then an algebraic expression
for an n by n figure.
STUDENT RESPONSES
SHARMEEN
4 X 10 – 4
COLIN
10 + 9 + 9 + 8
JOSEPH
10 + 10 + 8 + 8
MELISSA
102 – 82
TINA
9X4
ZACHARY
4X8+4
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Common Core State Standards in Mathematics Pre-Test
QUESTION 9: INSTRUCTION and LESSON DESIGN (1f, 2a, 2b, 2d, 4e)
Although the CCSSM do not explicitly reference instructional delivery, certain lesson components are
essential in order to ensure that instruction is well aligned to the intended objective(s).

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


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
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


planned objective
planned sequence (timed)
essential question (the big picture question)
evidence of backwards planning via relevant, thought-provoking questions, problems and tasks
that stimulate interest and elicit mathematical thinking
resources/ materials
academic vocabulary/terminology
focus questions
diagnostic checks for prior knowledge
launch/introductory activity
multiple representations (e.g., pictures, symbols, expressions, equations, graphics, models)
guided practice
activities structured around the gradual release of responsibility ( whole → collaborative →
independent)
address of misconceptions
reflective questions
homework
scaffolding, differentiation, intervention and support for a broad range of learners
demonstrations of learning
appropriate technology, tools, and media to support teaching and learning
extra supports for students working below grade level
extensions for students working at or above grade level
The tables below outline a Ratios and Proportions Framework for Grades 6 & 7. How can this type
of “framing” be used to ensure that teachers are embedding many of the essential components in
their lesson design?
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Common Core State Standards in Mathematics Pre-Test
RATIOS & PROPORTIONAL REASONING FRAMEWORK: TEACHER NOTES
UNDERSTANDING RATIOS
UNIT RATE
REPRESENTATIONS
A ratio is a pair of non-negative numbers, A : B, which are not
both 0 associating two (or more) quantities.
A units
B units
Some authors distinguish ratios from rates, using the term
“ratio” when units are the same and “rate” when units are
different; others use ratio to encompass both kinds of
situations. Standards use ratio in the second sense, applying
it to situations in which units are the same as well as to
situations in which units are different. Relationships of two
quantities in such situations may be described in terms of
ratios, rates, percents, or proportional relationships.
Ratios have associated rates, expressed as the unit
rate.
A
B
The unit rate is the numerical part of the rate; the
“unit” in “unit rate” is often used to highlight the 1 in
“for each 1” or “for every 1.”








Equivalent ratios arise by multiplying each measurement in a
ratio pair by the same positive number.
Proportional relationships involve collections of pairs of
measurements in equivalent ratios.
A proportion is an equation stating that two ratios are
equivalent.
Equivalent ratios have the same unit rate.
cA = A
cB
B
A proportional relationship is described by an equation
of the form y= kx, where k is a positive constant, often
called a constant of proportionality. The constant of
𝐵
proportionality, k, is equal to the value . The graph of
𝐴
a proportional relationship lies on a ray with endpoint
at the origin.
ITERATING RELATED QUANITITIES
TABLE OF EQUIVALENT RATIOS
MODELS
TAPE DIAGRAM/BAR MODEL
GRAPHING ON COOR PLANE
DOUBLE NUMBERLINES
EQUATIONS
EQUATIONS IN TWO VARIABLES; y=cx
where c is the constant of proportionality
RATIO NOTATION
Ratios can be indicated in words as
 3 to 2
 3 for every 2; e.g., 3 cups of flour for
every 2 eggs; 3 meters in 2 seconds
 3 out of every 5
 3 parts to 2 parts
Notation for ratios can include the
 Use of a colon, as in 3 : 2
 Use of quotient, as in 3/2
NOTE: Although it is traditional to move students quickly to solving proportions by setting up an equation, the Standards do not require this method in
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Grade 6. There are a number of strategies for solving problems that involve ratios.
Common Core State Standards in Mathematics Pre-Test
RATIOS & PROPORTIONAL REASONING FRAMEWORK: STRUCTURES OF PROBLEMS
TOPICS & CONTEXTS





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




Ratios
Rates
Similarity
Scale
Probability
Percents
Percent inc/dec
Linear Equations
Linear patterns and
relationships
Slope
Frequency Distributions
ITEM TYPES





RELATIONSHIPS
Missing Value
Rate comparisons
Ratio
Ratio comparisons,
equivalence
Determining proportional
and non-proportional
relationships

W/IN RATIOS

BTW/RATIOS
COMPLEXITY OF #’S

All positive whole and
rational numbers
REPRESENTATIONS





Word Problems
Graphs
Tables
Models
Equations
RATIOS & PROPORTIONAL REASONING FRAMEWORK EVIDENCE IN STUDENTS WORK
EVIDENCE OF TRANSITIONING
EVIDENCE OF PROPORTIONAL REASONING
STRATEGIES (not yet seeing &
STRATEGIES (seeing & making use of structure)
making use of structure)
 Relies on additive reasoning to
build up and build down
 Finds equivalent fractions/ratios
 Defaults to proportions/cross
multiplication w/out evidence of
multiplicative reasoning







Flexibly applies unit rate (A/B or B/A) where practical
Applies multiplicative relationships w/in or between ratios
Uses strategies/models effectively: tape diagram, tables, graphs,
number lines, equations
Students recognize the roles of “for every,” “for each,” and
“per”
Recognizes the proportional relationships have a constant of
proportionality and that the line of a proportional relationship
always passes through the origin
Recognize the constant of proportionality in tables, graphs,
equations, diagrams, and verbal descriptions of proportional
relationships
Establishes the unit rate triangle when graphing proportional
relationships
15
ERRORS & MISCONCEPTIONS



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


Uses non-proportional situations
Uses additive structures
Uses whole number reasoning
Using fractional reasoning
Misinterprets meaning of quantities
Misinterprets ‘remainders’
Mislabeling of units
Misplacement of numbers/units
Computational errors
Misinterpretation of terms
Common Core State Standards in Mathematics Pre-Test
QUESTION 10: INSTRUCTION and LESSON IMPLEMENTATION (1f)
Assess the Go Math Lesson or the Math in Focus lesson provided using the attached eQuip rubric.
16
Common Core State Standards in Mathematics Pre-Test
QUESTION 11: FLUENCIES (4d)
Generally, fluencies mark the end of a progression. Fluency expectations should build throughout the course of
the year and should be assessed in a timed setting (K through grade 6). Identify the mental fluencies with a 
and paper-pencil fluencies with a .
+/-
K
Fluently
add and
subtract
within 5
1
Fluently
add and
subtract
within 10
2
Fluently add
and subtract
within 20;
From memory
all sums of 2
one-digit
numbers
Fluently add
and subtract
within 100
3
Fluently
add and
subtract
within
1000 (using
number
strategies and
algorithms
based on place
value;
properties,
relationship
btw + / -)
4
Fluently
add and
subtract
multi-digit
whole
numbers
less than or
equal to
1,000,000
5
6
Fluently
add and
subtract
multi-digit
decimals
(using the
standard
algorithm)
(using the
standard
algorithm )
Mentally
add/subtract
10 or 100
to/from any
number 100900
x/÷
Fluently
multiply
and divide
within 100
Know from
memory
all
products
of two
one-digit
numbers.
17
Fluently
multiply
and divide
multi-digit
whole
numbers
using
standard
algorithm
(using
strategies
based on place
value and
properties,
relationship
between
multiplication /
division)
Fluently
divide
multi-digit
whole
numbers
(using the
standard
algorithm)
Fluently
multiply
and divide
multi-digit
decimals
(using the
standard
algorithm)
Common Core State Standards in Mathematics Pre-Test
QUESTION 12: COHERENCE
Certain topics and themes are woven throughout the standards. These themes progress from K-8 and
reflect the coherent nature of the standards across grades and within grades. Use the wording of the
standards on each card to organize the standards by grade (K-8), then determine the theme.
How do the fluencies…
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Common Core State Standards in Mathematics Pre-Test
Cool Down: Which of the following resources should be a part of a teacher’s Professional Learning Toolkit?
 The Common Core State Standards
 PARCC Model Frameworks
 PARCC Blueprints ,Test Specifications, and PLDs
 Progressions Documents
 Publishers’ Criteria (K-8)
 Illustrativemathematics.org
 Achievethecore.org
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