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Math Standards Document
Eighth Grade
Final Draft
Hawaii School Districts
Prepared by Marzano and Associates
Grade 8/Pre-Algebra Summary
Strand
Standard
Numbers and Operations
Numbers and Operations
Numbers and Operations
Numbers and Operations
Measurement/Geometry
and Spatial Sense
1
2
3
3
4/7
Measurement
Measurement
Geometry and Spatial
Sense
Geometry and Spatial
Sense
Patterns, Functions, and
Algebra
Patterns, Functions, and
Algebra
Patterns, Functions, and
Algebra
Patterns, Functions, and
Algebra
Data Analysis, Statistics,
and Probability
Data Analysis, Statistics,
and Probability
Data Analysis, Statistics,
and Probability
Data Analysis, Statistics,
and Probability
Total
4
4
5
Topic
Number of
Elements
2
2
1
2
3
9
Numbers and Number Systems
Operations
Computational Fluency
Estimation
Measurement Units and
Formulas/Visualization and Spatial
Reasoning
Measurement and Ratios
Measurement Tools and Techniques
Geometric Shapes and Their
Properties and Relationships
Transformation/Coordinate
Geometry
Patterns
9
Function
2
10
3
10
Numeric and Algebraic
Representations
Rates of Change
11
Data Collection and Representation
2
12
Data Interpretation
2
13
Predictions and Inferences
1
14
Probability
3
6
2
1
2
4
1
1
34
Topic: Numbers and Number Systems
Strand: Numbers and Operations
Standard 1: NUMBER SENSE: Understand numbers, ways of representing
numbers, relationships among numbers, and number systems.
Grade 8/Pre-Algebra
Level 4.0
In addition to Level 3.0, in-depth inferences and applications that go beyond
what was taught such as:
 describes why certain situations involve square roots and cube roots
 identifies and justifies whether or not given numbers (including square
roots) are rational numbers
Level 3.5
Level 3.0
While involved in tasks related to numbers and number systems the student
will:
 (MA.8.1.2/MA.PA.1.2) identify situations represented by square roots and
cube roots (e.g., provide examples of situations that use square roots and cube
roots. i.e., explains that the length of the hypotenuse of a right triangle may be
represented by square root)
 (MA.8.1.1/MA.PA.1.1) compare and order rational numbers and square roots
(e.g., orders a set of rational numbers and square roots on a number line)
The student exhibits no major errors or omissions.
Level 2.5
Level 2.0
In addition to Level 3.0 performance, in-depth inferences and applications with
partial success.
No major errors or omissions regarding the simpler details and process and partial
knowledge of the more complex ideas and processes.
There are no major errors or omissions regarding the simpler details and
processes as the student:
 recognizes or recalls specific terminology such as:
o rational, irrational numbers
o ratio, proportion
o square root
o perfect square
 performs basic processes such as:
o finding a square root and a cube root
o writing square roots as rational numbers or estimating square roots
between rational numbers
However, the student exhibits major errors or omissions regarding the more
complex ideas and processes.
Level 1.5
Level 1.0
Level 0.0
Partial knowledge of the simpler details and processes but major errors or omissions
regarding the more complex ideas and procedures.
With help, a partial understanding of some of the simpler details and processes and some of the
more complex ideas and processes.
Level 0.5
With help, a partial understanding of some of the simpler details and processes but
not the more complex ideas and processes.
Even with help, no understanding or skill demonstrated.
Sample Tasks for Levels 4.0, 3.0, & 2.0
Level 4.0
 Ask students to explain why certain situations involve square roots and cube
roots.
 Ask students to explain why given numbers (including square roots) are rational
or irrational numbers.
Level 3.0
 Ask students to provide examples of situations that use square roots and cube
roots (i.e., explain that the length of the hypotenuse of a right triangle may be
represented by a square root).
 Ask students to order a set of rational numbers and square roots on a number line.
Level 2.0
 Ask students to simplify square roots and cube roots.
 Ask students estimate square roots to the nearest integer.
 Ask students to identify a number as rational or irrational.
 Ask students to identify or produce definitions to the following terms: rational,
irrational numbers, ratio, proportion, square root, perfect square.
Topic: Operations
Strand: Numbers and Operations
Standard 2: OPERATION SENSE: Understand the meaning of operations and how
they relate to each other.
Grade 8/Pre-Algebra
Level 4.0
In addition to Level 3.0, in-depth inferences and applications that go beyond
what was taught such as:
 applies the order of operations to more complex problems involving
rational numbers (i.e., those including a radical sign and a fraction bar)
 describes/creates situations that would demonstrate the inverse
relationship between squares and square roots, and cubes and cube roots
Level 3.5
Level 3.0
While involved in tasks related to operations the student will:
 (MA.8.2.1/MA.PA.2.1) apply the order of operations when calculating with
rational numbers (e.g., expands his or her use of the order of operations to
include exponents and applies the order of operations in the correct sequence
when simplifying numeric expressions that involve rational numbers. i.e.,
fractions, decimals, integers)
 (MA.8.2.2/MA.PA.2.2) demonstrate the inverse relationship between square
numbers and square roots, and cubes and cubed roots (e.g., finds the area of a
square whose side length is a square root)
The student exhibits no major errors or omissions.
Level 2.5
Level 2.0
In addition to Level 3.0 performance, in-depth inferences and applications with
partial success.
No major errors or omissions regarding the simpler details and process and partial
knowledge of the more complex ideas and processes.
There are no major errors or omissions regarding the simpler details and
processes as the student:
 recognizes or recalls specific terminology such as:
o grouping symbols: radical sign and fraction bar
o exponent
 performs basic processes such as:
o describing the order of operations
o calculating a square and square root separately and a cube and
cubed root separately
However, the student exhibits major errors or omissions regarding the more
complex ideas and processes.
Level 1.5
Level 1.0
Level 0.0
Partial knowledge of the simpler details and processes but major errors or omissions
regarding the more complex ideas and procedures.
With help, a partial understanding of some of the simpler details and processes and some of the
more complex ideas and processes.
Level 0.5
With help, a partial understanding of some of the simpler details and processes but
not the more complex ideas and processes.
Even with help, no understanding or skill demonstrated.
Sample Tasks for Levels 4.0, 3.0, & 2.0
Level 4.0
 Ask students to apply the order of operations to problems involving rational
numbers that include a radical sign and a fraction bar.
 Ask students to describe or create a situation that would demonstrate the inverse
relationship between squares and square roots, and cubes and cube roots.
Level 3.0
 Ask students to apply the order of operations when simplifying numeric
expressions.
 Ask students to find the area of a square whose side length is a square root.
Level 2.0
 Ask students to describe the order of operations.
 Ask students to calculate a square and square root separately and a cube and cube
root separately.
 Ask students to identify or produce definitions to the following terms: grouping
symbols: radical sign and fraction bar, exponent.
Topic: Computational Fluency
Strand: Numbers and Operations
Standard 3: COMPUTATION STRATEGIES: Use computational tools and
strategies fluently and, when appropriate, use estimation.
Grade 8/Pre-Algebra
Level 4.0
In addition to Level 3.0, in-depth inferences and applications that go beyond
what was taught such as:
 adds, subtracts, multiplies and divides numbers with zero and negative
exponents
Level 3.5
Level 3.0
While involved in tasks related to computational fluency the student will:
 (MA.8.3.1/MA.PA.3.1) add, subtract, multiply, and divide numbers with
whole number exponents (e.g., uses arithmetic properties (associative,
commutative, distributive, identity properties) and the Law of Exponents
to calculate numbers that have whole number exponents and when
appropriate use the properties to make it easier to perform the
calculations. i.e., instead of multiplying 3 x 3 x 3 x 3 to calculate 34, the
student represents 34 as 32 x 32 which becomes 9 x 9 which can easily be
computed using mental math)
The student exhibits no major errors or omissions.
Level 2.5
Level 2.0
In addition to Level 3.0 performance, in-depth inferences and applications with
partial success.
No major errors or omissions regarding the simpler details and process and partial
knowledge of the more complex ideas and processes.
There are no major errors or omissions regarding the simpler details and
processes as the student:
 recognizes or recalls specific terminology such as:
o exponent
o factor
o associative, commutative, distributive, identity properties
 performs basic processes such as:
o calculating a number raised to an exponent; evaluating a one-term
expression that contains an exponent
However, the student exhibits major errors or omissions regarding the more
complex ideas and processes.
Level 1.5
Level 1.0
Level 0.0
Partial knowledge of the simpler details and processes but major errors or omissions
regarding the more complex ideas and procedures.
With help, a partial understanding of some of the simpler details and processes and some of the
more complex ideas and processes.
Level 0.5
With help, a partial understanding of some of the simpler details and processes but
not the more complex ideas and processes.
Even with help, no understanding or skill demonstrated.
Sample Tasks for Levels 4.0, 3.0, & 2.0
Level 4.0
 Ask students to simplify an expression with zero and negative exponents.
Level 3.0
 Ask students to calculate numbers that have whole number exponents and when
appropriate use properties of exponents to make it easier to perform the
calculations.
Level 2.0
 Ask students to simplify an expression with whole number exponents.
 Ask students to identify the property demonstrated by a given mathematical
statement (i.e., a(b+c) = ab+ac).
 Ask students to identify or produce definitions to the following terms: exponent,
factor, associative, commutative, distributive, identity properties.
Topic: Estimation
Strand: Numbers and Operations:
Standard 3: COMPUTATION STRATEGIES: Use computational tools and
strategies fluently and, when appropriate, use estimation.
Grade 8/Pre-Algebra
Level 4.0
In addition to Level 3.0, in-depth inferences and applications that go beyond
what was taught such as:
 determines a reasonable estimate using the upper and lower limits for the
solution of a problem
 creates or describes a real-world situation in which rounded answers
would lead to major problems
Level 3.5
Level 3.0
While engaged in tasks involving estimation the student will:
 (MA.8.3.2/MA.PA.3.2) estimate a reasonable upper and lower limit for the
solution to a problem (e.g., use appropriate estimation strategies to state
upper and lower bounds of the estimated answer)
 (MA.PA.3.3) explain that rounding answers in real-world situations may lead
to major problems (e.g., describes problems that could occur in real-world
situations if answers were rounded. i.e., rocket missing the moon, a bridge
collapsing, automobile brakes not working)
The student exhibits no major errors or omissions.
Level 2.5
Level 2.0
In addition to Level 3.0 performance, in-depth inferences and applications with
partial success.
No major errors or omissions regarding the simpler details and process and partial
knowledge of the more complex ideas and processes.
There are no major errors or omissions regarding the simpler details and
processes as the student:
 recognizes or recalls specific terminology such as:
o upper and lower limits
 performs basic processes such as:
o estimating one possible solution to a problem
o providing an example of an inappropriately rounded-off number
in a real-world situation
However, the student exhibits major errors or omissions regarding the more
complex ideas and processes.
Level 1.5
Level 1.0
Level 0.0
Partial knowledge of the simpler details and processes but major errors or omissions
regarding the more complex ideas and procedures.
With help, a partial understanding of some of the simpler details and processes and some of the
more complex ideas and processes.
Level 0.5
With help, a partial understanding of some of the simpler details and processes but
not the more complex ideas and processes.
Even with help, no understanding or skill demonstrated.
Sample Tasks for Levels 4.0, 3.0, & 2.0
Level 4.0
 Ask students to determine a reasonable estimate using the upper and lower bounds
for the solution of a problem.
 Ask students to create or describe a real-world situation in which rounded answers
would lead to major problems.
Level 3.0
 Ask students to use appropriate estimation strategies to state upper and lower
bounds of an estimated answer.
 Ask students to describe a problem that could occur in real-world situations if
answers were rounded.
Level 2.0
 Ask students to estimate one possible solution to a problem.
 Ask students to provide an example of an inappropriately rounded-off number in a
real-world situation.
 Ask students to identify or produce definitions to the following terms: upper and
lower limits.
Topic: Measurement Units and Formulas/Visualization and Spatial Reasoning
Strand: Measurement
Standard 4: FLUENCY WITH MEASUREMENT: Understand attributes, units,
and systems of units in measurement and develop and use techniques, tools, and
formulas for measuring.
Grade 8/Pre-Algebra
Level 4.0
In addition to Level 3.0, in-depth inferences and applications that go beyond
what was taught such as:
 explains why the selected unit(s) is/are appropriate for measuring the surface
area and volume of solids
 explains the components of the formulas used to determine the surface area
and volume of selected prisms, cylinders, and pyramids
 uses two-dimensional representations of pyramids, prisms, and cylinders to
solve complex problems using these figures (e.g., finds surface area of
compound figures)
Level 3.5
Level 3.0
While involved in tasks regarding measurement attributes and units the
student will:
 (MA.8.4.1/MA.PA.4.1) select and use appropriate units to measure the surface
area and volume of solids (e.g., selecs a unit based on the desired level of
precision, and explais why that unit was chosen rather than a different unit)
 (MA.8.4.4/MA.PA.4.4) uses formulas to determine the surface area and
volume of selected prisms, cylinders, and pyramids (e.g., apply the formula
for the volume of prisms (or cylinders) when the necessary measurements are
given; decomposes a prism (or cylinder or pyramid) into its different
faces/bases, and apply strategies or formulas to determine their areas)
 (MA.8.7.1/MA.PA.7.1) use two-dimensional representations of pyramids,
prisms, and cylinders to solve problems using these figures (e.g., use the twodimensional net of a cylinder to determine its surface area)
The student exhibits no major errors or omissions.
Level 2.5
Level 2.0
In addition to Level 3.0 performance, in-depth inferences and applications with
partial success.
No major errors or omissions regarding the simpler details and process and partial
knowledge of the more complex ideas and processes.
There are no major errors or omissions regarding the simpler details and
processes as the student:
 recognizes or recalls specific terminology such as:
o surface area
o volume
o net
o slant height (of pyramids and cones)
 performs basic processes such as:
o finding the surface area and volume of a solid and label answer
with appropriate units
o defining surface area and volume
o drawing and labeling nets of various three-dimensional figures
However, the student exhibits major errors or omissions regarding the more
complex ideas and processes.
Level 1.5
Level 1.0
Level 0.0
Partial knowledge of the simpler details and processes but major errors or omissions
regarding the more complex ideas and procedures.
With help, a partial understanding of some of the simpler details and processes and some of the
more complex ideas and processes.
Level 0.5
With help, a partial understanding of some of the simpler details and processes but
not the more complex ideas and processes.
Even with help, no understanding or skill demonstrated.
Sample Tasks for Levels 4.0, 3.0, & 2.0
Level 4.0
 Ask students to explain why selected units are appropriate for measuring the
surface area and volume of solids.
 Ask students to explain the components of the formula used to determine the
surface area and volume of selected prisms, cylinders, and pyramids.
 Ask students to use a net to find the surface area of compound figures involving
pyramids, prisms, and cylinders.
Level 3.0
 Ask students to select a unit based on the desired level of precision to find surface
area and volume and explain why the unit was chosen.
 Ask students to apply a formula to find the surface area and volume of prisms,
cylinders, and pyramids when given measurements.
 Ask students to use a two-dimensional net of a cylinder to determine its surface
area.
Level 2.0
 Ask students to find the surface area and volume of a solid and label the answer
with appropriate units.
 Ask students to define surface area and volume.
 Ask students to draw and label nets of various three-dimensional figures.
 Ask students to identify or produce definitions to the following terms: surface
area, volume, net, slant height (of pyramids and cones).
Topic: Measurement and Ratios
Strand: Measurement
Standard 4: FLUENCY WITH MEASUREMENT: Understand attributes, units,
and systems of units in measurement and develop and use techniques, tools, and
formulas for measuring.
Grade 8/Pre-Algebra
Level 4.0
In addition to Level 3.0, in-depth inferences and applications that go beyond
what was taught such as:
 uses ratios and proportions to solve complex measurement problems (e.g.,
solving for x when x represents part of a length)
 uses the right triangle relationships to solve complex problems involving
trigonometric ratios
Level 3.5
Level 3.0
While involved in tasks regarding measurement attributes and units the
student will:
 (MA.8.4.3/MA.PA.4.3) use ratios and proportions to solve measurement
problems (e.g., use ratios and proportions to determine an unknown
measurement when given known measurements. i.e., a student uses her height,
the length of her shadow, and the length of a flagpole’s shadow to determine
the flagpole’s height)
 (MA.PA.4.5) use the right triangle relationships (trigonometric ratios, sine,
cosine, and tangent) to solve problems (e.g., use an angle measurement in a
right triangle (other than the right triangle) and the length of one of the sides
to determine the lengths of the other two sides)
The student exhibits no major errors or omissions.
Level 2.5
Level 2.0
In addition to Level 3.0 performance, in-depth inferences and applications with
partial success.
No major errors or omissions regarding the simpler details and process and partial
knowledge of the more complex ideas and processes.
There are no major errors or omissions regarding the simpler details and
processes as the student:
 recognizes or recalls specific terminology such as:
o ratio, proportion
o similar figures
 performs basic processes such as:
o simplifying a ratio
o solving a proportion
o defining trigonometric ratios: sine, cosine, and tangent
However, the student exhibits major errors or omissions regarding the more
complex ideas and processes.
Level 1.5
Level 1.0
Level 0.0
Partial knowledge of the simpler details and processes but major errors or omissions
regarding the more complex ideas and procedures.
With help, a partial understanding of some of the simpler details and processes and some of the
more complex ideas and processes.
Level 0.5
With help, a partial understanding of some of the simpler details and processes but
not the more complex ideas and processes.
Even with help, no understanding or skill demonstrated.
Sample Tasks for Levels 4.0, 3.0, & 2.0
Level 4.0
 Ask students to use ratios and proportions to solve for x when x represents part of
a length of a figure (as opposed to the entire missing side length).
 Ask students to use the right triangle relationships to solve complex problems
involving trigonometric ratios.
Level 3.0
 Ask students to use ratios and proportions to determine an unknown measurement
when given known measurements.
 Ask students to use an angle measurement (other than the right angle) in a right
triangle and the length of one of the sides to determine the length of the other two
sides.
Level 2.0
 Ask students to write a ratio as a fraction in simplest form.
 Ask students to solve a given proportion.
 Ask students to define the trigonometric ratios: sine, cosine, and tangent.
 Ask students to identify or produce definitions to the following terms: ratio,
proportion, similar figures.
Topic: Measurement Tools and Techniques
Strand: Measurement
Standard 4: FLUENCY WITH MEASUREMENT: Understand attributes, units,
and systems of units in measurement and develop and use techniques, tools, and
formulas for measuring.
Grade 8/Pre-Algebra
Level 4.0
In addition to Level 3.0, in-depth inferences and applications that go beyond
what was taught such as:
 converts from one rate of change to another and use derived rate to solve
problems
Level 3.5
Level 3.0
While involved in tasks regarding measurement tools and techniques the
student will:
 (MA.8.4.2/MA.PA.4.2) express rates of change as a ratio of two different
measures, where units are included in the ratio, and use the derived rate to
solve problems (e.g., measure two quantities that are related (the capacity of
water that comes out of a water fountain in 10 seconds), express the quantities
as a ratio (rate), and use it to solve a problem. i.e., “How long would it take to
fill a gallon of water from a water fountain?”)
The student exhibits no major errors or omissions.
Level 2.5
Level 2.0
In addition to Level 3.0 performance, in-depth inferences and applications with
partial success.
No major errors or omissions regarding the simpler details and process and partial
knowledge of the more complex ideas and processes.
There are no major errors or omissions regarding the simpler details and
processes as the student:
 recognizes or recalls specific terminology such as:
o rate, rate of change
o unit rate
o ratio
 performs basic processes such as:
o writing a ratio comparing two different measures using
appropriate units
However, the student exhibits major errors or omissions regarding the more
complex ideas and processes.
Level 1.5
Level 1.0
Level 0.0
Partial knowledge of the simpler details and processes but major errors or omissions
regarding the more complex ideas and procedures.
With help, a partial understanding of some of the simpler details and processes and some of the
more complex ideas and processes.
Level 0.5
With help, a partial understanding of some of the simpler details and processes but
not the more complex ideas and processes.
Even with help, no understanding or skill demonstrated.
Sample Tasks for Levels 4.0, 3.0, & 2.0
Level 4.0
 Ask students to convert from one rate of change to another (i.e., miles per hour to
miles per minute) and used the derived rate to solve problems.
Level 3.0
 Ask students to express two related quantities in different units (i.e., 60 miles in 2
hours) as a ratio or rate, and use it to solve problems.
Level 2.0
 Ask students to write a ratio comparing two different measures using appropriate
units.
 Ask students to identify or produce definitions to the following terms: rate, rate of
change, unit rate, ratio.
Topic: Geometric Shapes and Their Properties and Relationships
Strand: Geometry and Spatial Sense
Standard 5: PROPERTIES AND RELATIONSHIPS: Analyze properties of objects
and relationships among the properties.
Grade 8/Pre-Algebra
Level 4.0
In addition to Level 3.0, in-depth inferences and applications that go beyond
what was taught such as:
 uses the Pythagorean theorem to determine whether a triangle with given side
lengths is a right triangle
 explains why conjectures about classes of two- and three-dimensional
shapes/objects are true
Level 3.5
Level 3.0
While engaged in tasks regarding geometric shapes and their properties and
relationships the student will:
 (MA.8.5.1/MA.PA.5.1) apply the Pythagorean theorem to solve problems
involving right triangles (e.g., uses the Pythagorean theorem to find an
unknown length in a problem involving a right triangle. i.e., finds the height of
the ladder needed to wash a window that is 25 feet above the ground if the
ladder is placed 4 feet from the side of the house)
 (MA.PA.5.2) evaluate conjectures about classes of two- and three-dimensional
shapes/objects (e.g., provide examples and logical reasons that support a
conjecture that was made about two- and three-dimensional shapes/objects
(all quadrilaterals with perpendicular diagonals are squares) or provide
counterexamples that refute the conjecture)
The student exhibits no major errors or omissions.
Level 2.5
Level 2.0
In addition to Level 3.0 performance, in-depth inferences and applications with
partial success.
No major errors or omissions regarding the simpler details and process and partial
knowledge of the more complex ideas and processes.
There are no major errors or omissions regarding the simpler details and
processes as the student:
 recognizes or recalls specific terminology such as:
o leg
o hypotenuse
 performs basic processes such as:
o naming the parts of a right triangle
o stating the Pythagorean theorem
o recognizing or recalling examples of characteristics of two- and
three-dimensional shapes
However, the student exhibits major errors or omissions regarding the more
complex ideas and processes.
Level 1.5
Level 1.0
Level 0.0
Partial knowledge of the simpler details and processes but major errors or omissions
regarding the more complex ideas and procedures.
With help, a partial understanding of some of the simpler details and processes and some of the
more complex ideas and processes.
Level 0.5
With help, a partial understanding of some of the simpler details and processes but
not the more complex ideas and processes.
Even with help, no understanding or skill demonstrated.
Sample Tasks for Levels 4.0, 3.0, & 2.0
Level 4.0
 Ask students to use the Pythagorean Theorem to determine whether a triangle
with given side lengths is a right triangle.
 Ask students to explain why conjectures made about classes of two- and threedimensional shapes and objects are true.
Level 3.0
 Ask students to use the Pythagorean Theorem to find an unknown length in a
problem involving a right triangle.
 Ask students to provide examples and logical reasons that support a conjecture
that was made about two- and three-dimensional shapes and objects.
Level 2.0
 Ask students to name the parts of a right triangle.
 Ask students to state the Pythagorean Theorem.
 Ask students to recognize or recall examples of characteristics of two- and threedimensional shapes.
 Ask students to identify or produce definitions to the following terms: leg,
hypotenuse.
Topic: Transformations/Coordinate Geometry
Strand: Geometry and Spatial Sense
Standard 6: TRANSFORMATIONS AND SYMMETRY: Use transformations and
symmetry to analyze mathematical situations.
Grade 8/Pre-Algebra
Level 4.0
In addition to Level 3.0, in-depth inferences and applications that go beyond
what was taught such as:
 performs two or more transformations on the same figure
 describes the size, position, and orientation of the original shape that has been
transformed multiple times and in multiple ways
 identifies the two-dimensional shape that was rotated about an axis to create a
given three-dimensional shape
 given coordinates of an image and the rule(s) for the transformation(s)
performs, determines the coordinates of the original figure
Level 3.5
Level 3.0
While engaged in tasks involving transformation the student will:
 (MA.8.6.1/MA.PA.6.1) perform a transformation (reflection, rotation,
translation) when given a figure and necessary parameters (e.g., reflects a
given figure over a given line of symmetry; rotates a given figure by a given
angle around a given center of rotation; translates a given figure in a given
direction by a given distance)
 (MA.8.6.2/MA.PA.6.2) describe the size, position, and orientation of shapes
under transformations and compositions of transformations (e.g., describes the
size, position, and orientation of a given shape after it has been reflected over
one line of reflection, and the resulting image has been reflected over a second
line of reflection)
 (MA.PA.6.3) describe three-dimensional shapes that are formed by rotating
two-dimensional shapes about an axis (e.g., illustrate/show/name the image
that is formed when a two-dimensional figure is rotated (spun) quickly around
an axis, and describes how features of the image correspond to the original
figure)
 (MA.8.8.1/MA.PA.8.1) use coordinate geometry to represent transformations
in the coordinate plane (e.g., determines the coordinates of a figure after it has
been transformed. i.e., uses the coordinates of a given figure and its distance
from the line of symmetry to locate the coordinates of its reflection)
The student exhibits no major errors or omissions.
Level 2.5
Level 2.0
In addition to Level 3.0 performance, in-depth inferences and applications with
partial success.
No major errors or omissions regarding the simpler details and process and partial
knowledge of the more complex ideas and processes.
There are no major errors or omissions regarding the simpler details and
processes as the student:
 recognizes or recalls specific terminology such as:
o transformation
o reflection
o rotation
o translation
o image
 performs basic processes such as:
o defining/recognizing or recalling types of transformations
o describing resulting images of a transformation
o simulating a rotation of a figure about an axis
o drawing the image of a figure on a coordinate grid after a
transformation has been performed
However, the student exhibits major errors or omissions regarding the more
complex ideas and processes.
Level 1.5
Partial knowledge of the simpler details and processes but major errors or omissions
Level 1.0
Level 0.0
regarding the more complex ideas and procedures.
With help, a partial understanding of some of the simpler details and processes and some of the
more complex ideas and processes.
Level 0.5
With help, a partial understanding of some of the simpler details and processes but
not the more complex ideas and processes.
Even with help, no understanding or skill demonstrated.
Sample Tasks for Levels 4.0, 3.0, & 2.0
Level 4.0
 Ask students to perform two or more transformations on the same figure.
 Ask students to describe the size, position, and orientation of a shape that has
been transformed multiple times and in multiple ways.
 Ask students to identify the two-dimensional shape that was rotated about an axis
to create a given three-dimensional shape.
 Ask students to determine the coordinates of the original figure when given the
coordinates of the image and rule(s) for the transformation(s) performed.
Level 3.0
 Ask students to reflect a figure over a given line of symmetry, rotate a figure by a
given angle about a given center of rotation, and translate a figure in a given
direction by a given distance.
 Ask students to describe the size, position, and orientation of a given shape after it
has been reflected over one line of reflection, and the resulting image has been
reflected over a second line of reflection.
 Ask students to illustrate, show, or name the image that is formed when a twodimensional shape is rotated quickly around an axis.
 Ask students to determine the coordinates of a figure after it has been
transformed.
Level 2.0
 Ask students to identify types of transformations.
 Ask students to describe the size, position, and orientation of a given shape after
one transformation has been performed.
 Ask students to simulate a rotation of a figure about an axis.
 Ask students to draw the image of a figure on a coordinate grid after one
transformation has been performed.
 Ask students to identify or produce definitions to the following terms:
transformation, reflection, rotation, translation, image.
Topic: Patterns
Strand: Patterns, Functions, and Algebra
Standard 9: PATTERNS AND FUNCTIONAL RELATIONSHIPS: Understand
various types of patterns and functional relationships.
Grade 8/Pre-Algebra
Level 4.0
In addition to Level 3.0, in-depth inferences and applications that go beyond
what was taught such as:
 creates a variety of patterns and represents them with tables, graphs, words,
and when possible, symbolic rules
Level 3.5
Level 3.0
While engaged in tasks involving patterns the student will:
 (MA.8.9.1/MA.PA.9.1) represent a variety of patterns (including recursive
patterns) with tables, graphs (including graphing technology when available),
words, and when possible, symbolic rules (e.g., identifies the rule that
generates a recursive sequence, describes the pattern in words, and gives the
next four numbers in the sequence. i.e., [1, 1, 2, 3, 5, 8, 13, __, __, __, __])
The student exhibits no major errors or omissions.
Level 2.5
Level 2.0
In addition to Level 3.0 performance, in-depth inferences and applications with
partial success.
No major errors or omissions regarding the simpler details and process and partial
knowledge of the more complex ideas and processes.
There are no major errors or omissions regarding the simpler details and
processes as the student:
 recognizes or recalls specific terminology such as:
o sequence (arithmetic, geometric)
o term
o common difference
o common ratio
 performs basic processes such as:
o describing the pattern in words, given a set of data
However, the student exhibits major errors or omissions regarding the more
complex ideas and processes.
Level 1.5
Level 1.0
Level 0.0
Partial knowledge of the simpler details and processes but major errors or omissions
regarding the more complex ideas and procedures.
With help, a partial understanding of some of the simpler details and processes and some of the
more complex ideas and processes.
Level 0.5
With help, a partial understanding of some of the simpler details and processes but
not the more complex ideas and processes.
Even with help, no understanding or skill demonstrated.
Sample Tasks for Levels 4.0, 3.0, & 2.0
Level 4.0
 Ask students to create a variety of patterns and represent them with tables, graphs,
words, and, when possible, symbolic rules.
Level 3.0
 Ask students to identify the rule that generates a sequence, describe the pattern in
words, and give the next four numbers in the sequence.
Level 2.0
 Ask students to identify the type of sequence (arithmetic, geometric, or neither).
 Ask students to identify or produce definitions to the following terms: sequence
(arithmetic, geometric), term, common difference, common ratio.
Topic: Numeric and Algebraic Representations
Strand: Patterns, Functions, and Algebra
Standard 10: SYMBOLIC REPRESENTATION: Use symbolic forms to represent,
model, and analyze mathematical situations.
Grade 8/Pre-Algebra
Level 4.0
In addition to Level 3.0, in-depth inferences and applications that go beyond
what was taught such as:
 translates among tables, graphs, and equations involving non-linear
relationships
 solves multi-step linear equations and inequalities with one variable using
algebraic methods
 describes a situation involving the comparison of linear relationships using
tables and graphs
Level 3.5
Level 3.0
While engaged in tasks involving numeric and algebraic representations the
student will:
 (MA.8.10.1/MA.PA.10.1) translate among tables, graphs (including graphing
technology when available), and equations involving linear relationships (e.g.,
use the information in a table to make a graph and equation; use a linear
equation to make a table and graph)
 (MA.8.10.2/MA.PA.10.2) solve linear equations and inequalities with one
variable using algebraic methods (e.g., solve a given equation or inequality for
the unknown values and show/explain how he/she determined the unknown
values)
 (MA.8.10.3/MA.PA.10.3) use tables and graphs to represent and compare
linear relationships (e.g., translate the information from a problem or equation
into tables and graphs and compares the tables and graphs of each
relationship, paying particular attention to the point of intersection and the
values leading up to the point of intersection. i.e., determine which it the better
video rental plan if Plan A is represented by the equation C = 5V and Plan B
is represented by the equation C = 2V + 20 where C is the cost in dollars and
V is the number of videos rented.)
The student exhibits no major errors or omissions.
Level 2.5
Level 2.0
In addition to Level 3.0 performance, in-depth inferences and applications with
partial success.
No major errors or omissions regarding the simpler details and process and partial
knowledge of the more complex ideas and processes.
There are no major errors or omissions regarding the simpler details and
processes as the student:
 recognizes or recalls specific terminology such as:
o linear, non-linear functions
 performs basic processes such as:
o drawing a graph using a table or equation separately
However, the student exhibits major errors or omissions regarding the more
complex ideas and processes.
Level 1.5
Level 1.0
Level 0.0
Partial knowledge of the simpler details and processes but major errors or omissions
regarding the more complex ideas and procedures.
With help, a partial understanding of some of the simpler details and processes and some of the
more complex ideas and processes.
Level 0.5
With help, a partial understanding of some of the simpler details and processes but
not the more complex ideas and processes.
Even with help, no understanding or skill demonstrated.
Sample Tasks for Levels 4.0, 3.0, & 2.0
Level 4.0
 Ask students to translate among tables, graphs, and equations involving non-linear
relationships.
 Ask students to solve multi-step linear equations and inequalities with one
variable using algebraic methods.
 Ask students to describe a situation involving the comparison of linear
relationships using tables and graphs.
Level 3.0
 Ask students to use the information in a table to make a graph and equation.
 Ask students to use a linear equation to make a table and graph.
 Ask students to solve a given equation or inequality for unknown values, and
show or explain how he/she determined the unknown values.
 Ask students to compare the tables and graphs of two linear relationships, paying
particular attention to the point of intersection and the values leading up to the
point of intersection.
Level 2.0
 Ask students to draw a graph using a table or equation separately.
 Ask students to identify or produce definitions to the following terms: linear, nonlinear functions.
Topic: Rates of Change
Strand: Patterns, Functions, and Algebra
Standard 10: SYMBOLIC REPRESENTATION: Use symbolic forms to represent,
model, and analyze mathematical situations.
Grade 8/Pre-Algebra
Level 4.0
In addition to Level 3.0, in-depth inferences and applications that go beyond
what was taught such as:
 compares different slopes to determine which rate of change is greater, and
justifies comparison
Level 3.5
Level 3.0
While engaged in tasks involving rates of change the student will:
 (MA.8.10.4/MA.PA.10.4) use the slope of a line to describe a constant
rate of change (e.g., determines the slope of a line and uses that
information to indicate the rate of change. i.e., finds the constant speed of
a train by determining the slope of a distance-time graph of the train’s
movement)
The student exhibits no major errors or omissions.
Level 2.5
Level 2.0
In addition to Level 3.0 performance, in-depth inferences and applications with
partial success.
No major errors or omissions regarding the simpler details and process and partial
knowledge of the more complex ideas and processes.
There are no major errors or omissions regarding the simpler details and
processes as the student:
 recognizes or recalls specific terminology such as:
o rate of change
o slope
 performs basic processes such as:
o finding the slope of a line
However, the student exhibits major errors or omissions regarding the more
complex ideas and processes.
Level 1.5
Level 1.0
Level 0.0
Partial knowledge of the simpler details and processes but major errors or omissions
regarding the more complex ideas and procedures.
With help, a partial understanding of some of the simpler details and processes and some of the
more complex ideas and processes.
Level 0.5
With help, a partial understanding of some of the simpler details and processes but
not the more complex ideas and processes.
Even with help, no understanding or skill demonstrated.
Sample Tasks for Levels 4.0, 3.0, & 2.0
Level 4.0
 Ask students to compare different slopes of lines to determine which rate of
change is greater, and justify the comparison.
Level 3.0
 Ask students to determine the slope of a line and use that information to describe
a rate of change.
Level 2.0
 Ask students to define slope as the ratio of a graph’s vertical change to its
horizontal change (i.e., rise/run).
 Ask students to find the slope of the line.
 Ask students to identify or produce definitions to the following terms: slope, rate
of change
Topic: Functions
Strand: Patterns, Functions, and Algebra
Standard 9: PATTERNS AND FUNCTIONAL RELATIONSHIPS: Understand
various types of patterns and functional relationships.
Grade 8/Pre-Algebra
Level 4.0
In addition to Level 3.0, in-depth inferences and applications that go beyond
what was taught such as:
 uses linear relationships with two variables to solve complex problems
 describes situations that can be defined as linear or non-linear
Level 3.5
Level 3.0
While engaged in tasks involving functions the student will:
 (MA.8.9.2/MA.PA.9.2) use linear relationships with two variables to solve
problems (e.g., organizes data about the two variables into a table and/or
graph, and uses the pattern or rule that defines the linear relationship to make
predictions about data not in the original set)
 (MA.8.9.3/MA.PA.9.3) identify functions as linear or non-linear and contrast
their properties from tables, graphs (including graphing technology when
available), or equations (e.g., uses the data in a table to determine if the data
represents a linear or non-linear function, and justifies the decision)
The student exhibits no major errors or omissions.
Level 2.5
Level 2.0
In addition to Level 3.0 performance, in-depth inferences and applications with
partial success.
No major errors or omissions regarding the simpler details and process and partial
knowledge of the more complex ideas and processes.
There are no major errors or omissions regarding the simpler details and
processes as the student:
 recognizes or recalls specific terminology such as:
o linear, nonlinear
o function
 performs basic processes such as:
o recognizing or recalling examples of linear relationships
o describing the difference between linear and non-linear functions
However, the student exhibits major errors or omissions regarding the more
complex ideas and processes.
Level 1.5
Level 1.0
Level 0.0
Partial knowledge of the simpler details and processes but major errors or omissions
regarding the more complex ideas and procedures.
With help, a partial understanding of some of the simpler details and processes and some of the
more complex ideas and processes.
Level 0.5
With help, a partial understanding of some of the simpler details and processes but
not the more complex ideas and processes.
Even with help, no understanding or skill demonstrated.
Sample Tasks for Levels 4.0, 3.0, & 2.0
Level 4.0
 Ask students to use linear relationships with two variables to solve complex
problems.
 Ask students to describe situations that can be defined as linear or non-linear.
Level 3.0
 Ask students to organize data about the two variables into a table and/or graph,
and uses the pattern or rule that defines the linear relationship to make predictions
about data not in the original set.
 Ask students to use data in a table or on a graph to determine if the data represents
a linear or non-linear function, and justify the decision.
Level 2.0
 Ask students to recognize or recall examples of linear relationships.
 Ask students to describe the difference between linear and non-linear functions.
 Ask students to identify or produce definitions to the following terms: linear,
nonlinear, function.
Topic: Data Collection and Representation
Strand: Data Analysis, Statistics, and Probability
Standard 11: FLUENCY WITH DATA: Pose questions and collect, organize, and
represent data to answer those questions.
Grade 8/Pre-Algebra
Level 4.0
In addition to Level 3.0, in-depth inferences and applications that go beyond
what was taught such as:
 explains why the chosen data representation is appropriate for the study
 suggests how to modify an experiment to make the data more valid
Level 3.5
Level 3.0
While involved in tasks regarding data collection and representation the
student will:
 (MA.8.11.1/MA.PA.11.1) design a study that compares two samples, collect
data, and select the appropriate representation double bar graph, back-to-back
stem-and-leaf plot, parallel box-and-whisker plots, scatter plot) (e.g., select a
representation that supports the desired purpose of the study and shows a
visual comparison of the data sets. i.e., in studying the relationship between
an 8th grader’s height and arm span, the student chooses to represent the data
in a scatter plot since scatter plots are designed to determine if two variables
are correlated)
 (MA.8.11.2/MA.PA.11.2) judge the validity of data based on the data
collection method (e.g., explains that the results of an experiment or survey
may be questionable because the data collection method or the way the sample
was chosen is questionable)
The student exhibits no major errors or omissions.
Level 2.5
Level 2.0
In addition to Level 3.0 performance, in-depth inferences and applications with
partial success.
No major errors or omissions regarding the simpler details and process and partial
knowledge of the more complex ideas and processes.
There are no major errors or omissions regarding the simpler details and
processes as the student:
 recognizes or recalls specific terminology such as:
o bar graph
o stem-and-leaf plot
o box-and-whisker plot
o scatter plot
 performs basic processes such as:
o recognizing or recalling examples of features of various forms of
representation
o making conclusions about a study; defining valid data
However, the student exhibits major errors or omissions regarding the more
complex ideas and processes.
Level 1.5
Level 1.0
Level 0.0
Partial knowledge of the simpler details and processes but major errors or omissions
regarding the more complex ideas and procedures.
With help, a partial understanding of some of the simpler details and processes and some of the
more complex ideas and processes.
Level 0.5
With help, a partial understanding of some of the simpler details and processes but
not the more complex ideas and processes.
Even with help, no understanding or skill demonstrated.
Sample Tasks for Levels 4.0, 3.0, & 2.0
Level 4.0
 Ask students to explain why the chosen data representation is appropriate for the
study.
 Ask students to suggest how to modify an experiment to make the data more
valid.
Level 3.0
 Ask students to select a representation that supports the desired purpose of the
study and shows a visual comparison of the data sets.
 Ask students to explain that the results of an experiment or survey may be
questionable because the data collection method or the way the sample was
chosen is questionable.
Level 2.0
 Ask students to recognize or recall examples of features of various forms of
representation.
 Ask students to make conclusions about a study.
 Ask students to define valid data.
 Ask students to identify or produce definitions to the following terms: bar graph,
stem-and-leaf plot, box-and-whisker plot, scatter plot.
Topic: Data Interpretation
Strand: Data Analysis, Statistics, and Probability
Standard 12: STATISTICS: Interpret data using methods of exploratory data
analysis.
Grade 8/Pre-Algebra
Level 4.0
In addition to Level 3.0, in-depth inferences and applications that go beyond
what was taught such as:
 describes why a situation is appropriate for scatter plots
 creates a data representation that would lead/mislead a person to a particular
conclusion about the data
Level 3.5
Level 3.0
While engaged in tasks involving data interpretation the student will:
 (MA.8.12.1/MA.PA.12.1) recognize situations appropriate for scatter plots
(e.g., choose to use a scatter plot when determining if a correlation between
two variables exists. i.e., compare the height of students’ to their arm span)
 (MA.8.12.2/MA.PA.12.2) analyze different representations of the same data to
describe how representations can be used to skew a person’s interpretation of
the data (e.g., adjust the interval or scale on a graph to change the
appearance of the graph and describe how the changes that were made affect
a person’s interpretation of the data)
The student exhibits no major errors or omissions.
Level 2.5
Level 2.0
In addition to Level 3.0 performance, in-depth inferences and applications with
partial success.
No major errors or omissions regarding the simpler details and process and partial
knowledge of the more complex ideas and processes.
There are no major errors or omissions regarding the simpler details and
processes as the student:
 recognizes or recalls specific terminology such as:
o scatter plot
o correlation (positive, negative, no correlation)
 performs basic processes such as:
o defining the purpose of a scatter plot
o describing how a graph can be misleading; recognizing whether or
not a graph is misleading; describing characteristics of a graph
that is misleading
However, the student exhibits major errors or omissions regarding the more
complex ideas and processes.
Level 1.5
Level 1.0
Level 0.0
Partial knowledge of the simpler details and processes but major errors or omissions
regarding the more complex ideas and procedures.
With help, a partial understanding of some of the simpler details and processes and some of the
more complex ideas and processes.
Level 0.5
With help, a partial understanding of some of the simpler details and processes but
not the more complex ideas and processes.
Even with help, no understanding or skill demonstrated.
Sample Tasks for Levels 4.0, 3.0, & 2.0
Level 4.0
 Ask students to explain why a scatter plot appropriately represents a given
situation.
 Ask students to create a graph that can skew a person’s interpretation of the data
represented.
Level 3.0
 Ask students to determine when to use a scatter plot to represent data.
 Ask students to adjust the scale or spacing along axes to emphasize (or deemphasize) how changes in data are related.
Level 2.0
 Ask students to define the purpose of a scatter plot.
 Ask students to describe how a graph can be misleading or characteristics of
misleading graphs.
 Ask students to identify whether or not a graph is misleading.
 Ask students to identify or produce definitions to the following terms: scatter plot,
correlation (positive, negative, no correlation).
Topic: Predictions and Inferences
Strand: Data Analysis, Statistics, and Probability
Standard 13: DATA ANALYSIS: Develop and evaluate inferences, predictions, and
arguments that are based on data.
Grade 8/Pre-Algebra
Level 4.0
In addition to Level 3.0, in-depth inferences and applications that go beyond
what was taught such as:
 uses conjectures to make predictions about data
Level 3.5
Level 3.0
While engaged in tasks involving predictions and inferences the student will:
 (MA.8.13.1/MA.PA.13.1) make conjectures about possible relationships
between two characteristics of a sample based on interpretations of scatter
plots (e.g., analyze a scatter plot and make a conjecture based on the presence
or absence of an approximate line of best fit)
The student exhibits no major errors or omissions.
Level 2.5
Level 2.0
In addition to Level 3.0 performance, in-depth inferences and applications with
partial success.
No major errors or omissions regarding the simpler details and process and partial
knowledge of the more complex ideas and processes.
There are no major errors or omissions regarding the simpler details and
processes as the student:
 recognizes or recalls specific terminology such as:
o conjecture
o scatter plot
 performs basic processes such as:
o describing the relationship between two quantities represented in a
scatter plot; stating any correlation
However, the student exhibits major errors or omissions regarding the more
complex ideas and processes.
Level 1.5
Level 1.0
Level 0.0
Partial knowledge of the simpler details and processes but major errors or omissions
regarding the more complex ideas and procedures.
With help, a partial understanding of some of the simpler details and processes and some of the
more complex ideas and processes.
Level 0.5
With help, a partial understanding of some of the simpler details and processes but
not the more complex ideas and processes.
Even with help, no understanding or skill demonstrated.
Sample Tasks for Levels 4.0, 3.0, & 2.0
Level 4.0
 Ask students to use a conjecture made about a scatter plot to make predictions
about the data.
Level 3.0
 Ask students to make a conjecture about a scatter plot based on whether or not a
line of best fit can represent the data.
Level 2.0
 Ask students to describe the relationship between two quantities represented in a
scatter plot (i.e., identify positive, negative, or no correlation).
 Ask students to identify or produce definitions to the following terms: scatter plot,
conjecture.
Topic: Probability
Strand: Data Analysis, Statistics, and Probability
Standard 14: PROBABILITY: Understand and apply basic notions of chance and
probability.
Grade 8/Pre-Algebra
Level 4.0
In addition to Level 3.0, in-depth inferences and applications that go beyond
what was taught such as:
 describes an alternative experiment/situation
 creates a situation that requires the calculation of probabilities for simple
events under different relationships
 uses the Fundamental Counting Principle to determine probabilities for
various events
Level 3.5
Level 3.0
While engaged in tasks involving probability the student will:
 (MA.8.14.1/MA.PA.14.1) judge the validity of conjectures that are based on
experiments or simulations (e.g., refer to the theoretical probability, sample
size, and data collection techniques to support the validity of a conjecture that
is based on experiments or simulations with predicable outcomes)
 (MA.8.14.2/MA.PA.14.2) calculate probabilities for simple events under
different relationships (e.g., inclusion, disjoint, complementary, independent,
dependent, with replacement, without replacement) (e.g., calculate the
probability of an event (or combinations of events) and show/explain how the
probability was determined)
 (MA.PA.14.3) use the Fundamental Counting Principle to calculate
combinations and permutations (e.g., show/describe/determine all the possible
outcomes (or permutations) of events. i.e., uses and organized list, a tree
diagram, chart, illustration)
The student exhibits no major errors or omissions.
Level 2.5
Level 2.0
In addition to Level 3.0 performance, in-depth inferences and applications with
partial success.
No major errors or omissions regarding the simpler details and process and partial
knowledge of the more complex ideas and processes.
There are no major errors or omissions regarding the simpler details and
processes as the student:
 recognizes or recalls specific terminology such as:
o probability (theoretical vs. experimental)
o combination, permutation
o independent & dependent events
 performs basic processes such as:
o making a conclusion based on the results of an experiment
o describing the different types of event relationships; calculate
probability
o defining the Fundamental Counting Principle
However, the student exhibits major errors or omissions regarding the more
complex ideas and processes.
Level 1.5
Level 1.0
Level 0.0
Partial knowledge of the simpler details and processes but major errors or omissions
regarding the more complex ideas and procedures.
With help, a partial understanding of some of the simpler details and processes and some of the
more complex ideas and processes.
Level 0.5
With help, a partial understanding of some of the simpler details and processes but
not the more complex ideas and processes.
Even with help, no understanding or skill demonstrated.
36
Sample Tasks for Levels 4.0, 3.0, & 2.0
Level 4.0
 Ask students to describe an alternative experiment/situation.
 Ask students to create a situation that requires the calculation of probabilities for
simple events under different relationships.
 Ask students to use Fundamental Counting Principle to determine probabilities for
various events.
Level 3.0
 Ask students to refer to the theoretical probability, sample size, and data
collection techniques to support the validity of a conjecture that is based on
experiments or simulations with predicable outcomes.
 Ask students to calculate the probability of an event or combinations of events
and show how the probability was determined.
 Ask students to use the Fundamental Counting Principle to determine all the
possible outcomes (or permutations) of events.
Level 2.0
 Ask students to make a conclusion based on the results of an experiment.
 Ask students to describe the different types of event relationships.
 Ask students to define the Fundamental Counting Principle.
 Ask students to identify or produce definitions to the following terms: probability
(theoretical vs. experimental), combination, permutation, independent &
dependent events.
37