Math Standards Document
Eighth Grade
Final Draft
Hawaii School Districts
Prepared by Marzano and Associates
Grade 8/Pre-Algebra Summary
Strand / Standard / Topic / Number of ElementsNumbers and Operations / 1 / Numbers and Number Systems / 2
Numbers and Operations / 2 / Operations / 2
Numbers and Operations / 3 / Computational Fluency / 1
Numbers and Operations / 3 / Estimation / 2
Measurement/Geometry and Spatial Sense / 4/7 / Measurement Units and Formulas/Visualization and Spatial Reasoning / 3
Measurement / 4 / Measurement and Ratios / 2
Measurement / 4 / Measurement Tools and Techniques / 1
Geometry and Spatial Sense / 5 / Geometric Shapes and Their Properties and Relationships / 2
Geometry and Spatial Sense / 6 / Transformation/Coordinate Geometry / 4
Patterns, Functions, and Algebra / 9 / Patterns / 1
Patterns, Functions, and Algebra / 9 / Function / 2
Patterns, Functions, and Algebra / 10 / Numeric and Algebraic Representations / 3
Patterns, Functions, and Algebra / 10 / Rates of Change / 1
Data Analysis, Statistics, and Probability / 11 / Data Collection and Representation / 2
Data Analysis, Statistics, and Probability / 12 / Data Interpretation / 2
Data Analysis, Statistics, and Probability / 13 / Predictions and Inferences / 1
Data Analysis, Statistics, and Probability / 14 / Probability / 3
Total / 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 / In addition to Level 3.0 performance, in-depth inferences and applications with partial success.
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 / No major errors or omissions regarding the simpler details and process and partial knowledge of the more complex ideas and processes.
Level 2.0 / 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 / Partial knowledge of the simpler details and processes but major errors or omissions regarding the more complex ideas and procedures.
Level 1.0 / 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.
Level 0.0 / 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: OperationsStrand: 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 / In addition to Level 3.0 performance, in-depth inferences and applications with partial success.
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 / No major errors or omissions regarding the simpler details and process and partial knowledge of the more complex ideas and processes.
Level 2.0 / 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 / Partial knowledge of the simpler details and processes but major errors or omissions regarding the more complex ideas and procedures.
Level 1.0 / 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.
Level 0.0 / 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 FluencyStrand: 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 / In addition to Level 3.0 performance, in-depth inferences and applications with partial success.
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 / No major errors or omissions regarding the simpler details and process and partial knowledge of the more complex ideas and processes.
Level 2.0 / 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 / Partial knowledge of the simpler details and processes but major errors or omissions regarding the more complex ideas and procedures.
Level 1.0 / 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.
Level 0.0 / 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: EstimationStrand: 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 / In addition to Level 3.0 performance, in-depth inferences and applications with partial success.
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 / No major errors or omissions regarding the simpler details and process and partial knowledge of the more complex ideas and processes.
Level 2.0 / 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 / Partial knowledge of the simpler details and processes but major errors or omissions regarding the more complex ideas and procedures.
Level 1.0 / 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.
Level 0.0 / 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 ReasoningStrand: 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 / In addition to Level 3.0 performance, in-depth inferences and applications with partial success.
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 two-dimensional net of a cylinder to determine its surface area)
The student exhibits no major errors or omissions.
Level 2.5 / No major errors or omissions regarding the simpler details and process and partial knowledge of the more complex ideas and processes.
Level 2.0 / 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 / Partial knowledge of the simpler details and processes but major errors or omissions regarding the more complex ideas and procedures.
Level 1.0 / 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.
Level 0.0 / Even with help, no understanding or skill demonstrated.
Sample Tasks for Levels 4.0, 3.0, & 2.0