Designing Quality Mathematics Assessments to PARCC · 2014-07-14 · Designing Quality Mathematics...

Designing Quality Mathematics Assessments Aligned to PARCC (6–12)

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The Latest and Greatest From PARCC:

Claims Cognitive complexities

Blueprints Evidence tables

EOY versus PBA PLDs

Individual Reflection on the Four Pursuits: What will be your commitments for 2013–2014?

A thorough review of your current local assessments on a unit‐by‐unit basis

High‐quality common assessments and the accurate scoring of those assessments

A robust formative assessment process for students and adults using each assessment instrument

Instruction that provides evidence of student understanding via the mathematical practices

Unless footers note otherwise, all pages are copyrighted to © Mona Toncheff 2013 and are REPRODUCIBLE. • solution-tree.com 1

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Designing Quality Mathematics Assessments Aligned to PARCC (6–12)

Mona Toncheff [email protected] http://puhsdmath.blogspot.com

Paradigm Shifts• Professional Development

– On-going collaborative team learning• Instruction

– Teaching for conceptual understanding as well as procedural fluency

• Content– Focus, coherence, rigor; conceptual

understanding and procedural fluency• Assessment

– Multifaceted process; emphasis on formative assessment

• Intervention– Required, not invitational

Today’s Learning Targets

• I can describe the work of collaborative team during the teaching-assessing-learning cycle.

• I can examine tools and criteria for effective assessment design on a unit-by-unit basis.

• I can become familiar with the new dynamic assessments from PARCC.

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Three Big Ideas

1. Focus on student learning

2. Focus on collaboration

3. Focus on results

--DuFour, DuFour, Eaker, & Many, Learning by Doing (2010)

Four PLC Questions1. What do we expect students to

learn?

2. How will we know students learned it?

3. What will we do when students do not learn?

4. What will we do when students do learn?

--DuFour, DuFour, Eaker, & Many, Learning by Doing (2010)

High-Leverage Unit-By-Unit Actions of Mathematics Collaborative Teams

• Teaching and learning

• Assessment instruments and tools

• Formative assessment feedback

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High-Leverage Unit-By-Unit Actions of Mathematics Collaborative Teams

First Turn-Last Turn • Read silently .

• First team member chooses an action to discuss and starts the conversation.

• Each member shares one comment while everyone listens (no cross-talk) and the originator has the last turn.

—Kanold & Larson, Common Core Mathematics in a PLC at Work, Leader’s Guide (2012)

What Is a Common Assessment?

“Common assessment means student learning will be assessed using the same instrument or process and according to the same criteria.”

—DuFour, DuFour, Eaker, & Many, Learning by Doing (2010), p. 63

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R E PRO DUCI B LE

Common Core Mathematics in a PLC at WorkTM, Leader’s Guide © 2012 Solution Tree Press • solution-tree.comVisit go.solution-tree.com/commoncore to download this page.

Figure 1.5: High-Leverage Unit-By-Unit Actions of

Mathematics Collaborative Teams

Teaching and Learning1. The team designs and implements agreed-on prior knowledge skills to be assessed and taught during

each lesson of the unit. The collaborative teacher team reaches agreement for teaching and learning in the lessons and unit.

2. The team designs and implements agreed-on lesson-design elements that ensure active student engagement with the mathematics. Students experience some aspect of the CCSS Mathematical Practices, such as Construct viable arguments and critique the reasoning of others or Attend to precision, within the daily lessons of every unit or chapter.

3. The team designs and implements agreed-on lesson-design elements that allow for student-led summaries and demonstrations of learning the daily lesson.

4. The team designs and implements agreed-on lesson-design elements that include the strategic use of tools—including technology—for developing student understanding.

Assessment Instruments and Tools1. The team designs and implements agreed-on common assessment instruments based on high-

quality exam designs. The collaborative team designs all unit exams, unit quizzes, final exams, writing assignments, and projects for the course.

2. The team designs and implements agreed-on common assessment instrument scoring rubrics for each assessment in advance of the exam.

3. The team designs and implements agreed-on common scoring and grading feedback (level of specificity to the feedback) of the assessment instruments to students.

Formative Assessment Feedback1. The team designs and implements agreed-on adjustments to instruction and intentional student

support based on both the results of daily formative classroom assessments and the results of student performance on unit or chapter assessment instruments such as quizzes and tests.

2. The team designs and implements agreed-on levels of rigor for daily in-class prompts and common high-cognitive-demand tasks used to assess student understanding of various mathematical concepts and skills. This also applies to variance in rigor and task selection for homework assignments and expectations for make-up work. This applies to depth, quality, and timeliness of teacher descriptive formative feedback on all student work.

3. The team designs and implements agreed-on methods to teach students to self-assess and set goals. Self-assessment includes students using teacher feedback, feedback from other students, or their own self-assessments to identify what they need to work on and to set goals for future learning.

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http://go.solution-tree.com/commoncore

R E PRO DUCI B LE

Common Core Mathematics in a PLC at WorkTM, High School © 2012 Solution Tree Press • solution-tree.comVisit go.solution-tree.com/commoncore to download this page.

Figure 4.1: The PLC Teaching-Assessing-Learning Cycle

Step One

Collaborative teams identify learning targets and design common unit tasks and assessment instruments.

Step Two

Teachers implement formative assessment classroom strategies.

Step Three

Students take action on in-class formative assessment feedback.

Step Five

Collaborative teams use ongoing assessment feedback to improve instruction.

Step Four

Students use assess-ment instruments from step one for motivation, reflection, and action.

The PLC Teaching- Assessing-Learning Cycle

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Think Back: CCSS Sustained Implementation Begins Here1. How would you know if your unit tests

are coherent, aligned and rigorous—really really good?

2. How would you know if they are being used for the right purposes?

Step 4: Teaching–Assessing–Learning Cycle

Students reflect on successes and focus next steps based on evidence of areas of weakness, during and after the unit of study as the assessment instrument is used for formative student learning…

How can this be done in your school or on your team for each unit cycle?

1. Create a student goal setting reflection process to identify errors and use the assessment results to form a plan.

2. Create a process for students to act on their plan and take action (allow it to improve their grade)


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Mathematics faculty reflect on successes and focus next steps based on evidence of areas of weakness, during and after the unit of study as the assessment instrument is used for formative adult learning.


Sustained implementation of the CCSS requires four pursuits:

1. A thorough review of your current local assessments on a unit-by-unit basis

2. High quality common assessments and the accurate scoring of those assessments

3. A robust formative assessment process for students and adults using each assessment instrument

4. Instruction that provides evidence of student understanding via the mathematical practices

Collaborative Team Actions

• Grade-7 ratios and proportions unit assessment

• Algebra I exponential unit assessment

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Questions About the TargetsWhat questions should teams consider when reviewing targets?

Plan: Assess What and How?• How important is this topic? Is this a CCSS

grade-level area for critical focus?

• What is the breadth and depth of learning targets for the topic?

• Are learning targets—skill level and understanding level—clear to everyone on the collaborative team?

• How will they be made clear to the students?

• What role will the CCSS Mathematical Practices have in the assessment?

Develop the Assessment • Determine sample questions and tasks for

the assessment.

Select, create, or modify assessment items or tasks and scoring rubrics to meet student needs.

• What format and methods will be used for student demonstrations of proficiency?

• Do tasks assess both the CCSS content standards and Mathematical Practices?

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R E PRO DUCI B LE

Common Core Mathematics in a PLC at WorkTM, Grades K–2 © 2012 Solution Tree Press • solution-tree.comVisit go.solution-tree.com/commoncore to download this page.

Figure 4.5: Common Assessment Planning Process

1. Plan: Assess what and how. How important is this topic? Is this one of the CCSS grade-level areas for critical focus? What is the breadth and depth of the learning targets for the topic? Are the learning targets—skill level and understanding level—clear to everyone on the collaborative team? How will they be made clear to the students? What role will the CCSS Mathematical Practices have in the assessment?

2. Develop: Determine the sample questions and tasks for the assessment. Select, create, or modify assessment items or tasks and scoring rubrics as needed to meet student needs. What will be the format and methods used for student demonstrations of proficiency? Are there tasks that assess both the CCSS content standards and Mathematical Practices?

3. Critique: Evaluate the assessment for quality. How does the collaborative team know it has written a high-quality assessment? Does the school have well-defined and understood criteria for high-quality assessment development?

4. Administer and score: A unit assessment is given to the students and immediately scored using the collaboratively developed scoring rubric, and students receive timely descriptive feedback concerning their performance. Ideally, grade-level collaborative teams grade unit assessments together to improve the accuracy of feedback students receive. Students receive results immediately—ideally, the next day, but at most within two class days (Reeves, 2011).

5. Revise: Evaluate assessment quality based on results, and revise as needed for the following year. The results should also be used to identify learning targets and assessment questions that may need to be repeated as part of the next unit of study to build student retention—for example, areas identified in the CCSS Frameworks for a more critical focus and emphasis.

Source: Adapted from Stiggins et al., 2006, pp. 106–117.

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Critique: Evaluate the Assessment for Quality• How does the collaborative team know

it has written a high-quality assessment?

• Does the school have well-defined and understood criteria for high-quality assessment development?

Critique: Evaluate the Assessment for Quality

Administer and Score“A unit assessment is given to the students and immediately scored using the collaboratively developed scoring rubric, and students receive timely descriptive feedback concerning their performance.

“Ideally, grade-level collaborative teams grade unit assessments together to improve the accuracy of feedback students receive. Students receive results immediately—ideally, the next day, but at most within two class days.”

—Kanold (Ed.), Common Core Mathematics in a PLC at Work™, Grades 3–5 (2012)

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Revision Evaluate assessment quality based on results, and revise as needed for the following year.

Results also should be used to identify learning targets and assessment questions that may need to be repeated as part of the next unit of study to build student retention.

Example: areas identified in the CCSS frameworks for a more critical focus and emphasis

Improving the Assessment• With your team, decide upon an action

step to improve the features of the assessment that were not rated a 4.

• Discuss and agree upon proposed improvements.

What Tasks Form Learning?

What decides the cognitive demand of a task?

It is decided not by whether it is a hard problem, but rather by the complexity of reasoning required by the student.

(Kanold, Briars, & Fennel, What Principals Need to Know About Teaching and Learning Mathematics (2011)

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Promoting Proficiency in the Standards for Mathematical Practice

“Not all tasks are created equal, and different tasks will provoke different levels and kinds of student thinking.”

—Stein, Smith, Henningsen, & Silver, Implementing Standards-Based Mathematics Instruction (2000)

“The level and kind of thinking in which students engage determines what they will learn.”

—Hiebert, et al., Making Sense: Teaching and Learning Mathematics With Understanding(1997)

Definition of Rigor

A balance between procedural fluency and conceptual understanding

(William G. McCallum, University of Arizona)

Our 1st Outcome for Today!

Revisit the engagement of students in high cognitive demand tasks— or DOK levels 3 and 4

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R E PRO DUCI B LE

Common Core Mathematics in a PLC at WorkTM, High School © 2012 Solution Tree Press • solution-tree.comVisit go.solution-tree.com/commoncore to download this page.

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How Do We Develop This Complexity of Understanding? • Emphasis on the use of student thinking

within instruction and assessment

• Incorporation of mathematical explanations

• Use of multiple representations (technology)

• Learning opportunities and assessments that include inquiry and exploration

• Explicit attention to the development of mathematical concepts and skills over multiple grades (learning trajectories)

Why Focus on Mathematical Tasks?

• Tasks form the basis for students’ opportunities to learn what mathematics is and how one does it.

• Tasks influence learners by directing their attention to particular aspects of content and by specifying ways to process information.

• The level and kind of thinking required by mathematical instructional tasks influences what students learn.

• Differences in the level and kind of thinking that teachers, schools, and districts use is a major source of inequity in students’ opportunities to learn mathematics.

Lower-Level Tasks• Memorization

– What are the decimal equivalents for the fractions ½ and ¼?

• Procedures without connections

– Convert the fraction 3/8 to a decimal.

(Stein & Smith, “Mathematical Tasks as Framework for ReflMathematics Teaching in the Middle School, January 199

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Higher-Level Tasks

• Procedures with connections• Using a 10x10 grid, identify the decimal and

percent equivalents of 3/5.

• Doing mathematicso Shade 6 small squares in a 4x10 rectangle.

Using the rectangle, explain how to determine: The decimal part of area that is shaded;

The fractional part of area that is shaded.

(Stein & Smith, “Mathematical Tasks as Framework for ReflMathematics Teaching in the Middle School, January 199

What Are Great Mathematical Tasks?

• Center on an interesting problem, offering several methods of solution

• Are directed at essential mathematical content as specified in the standards

• Require examination and perseverance (challenging to students)

• Beg for discussion, offering rich discourse on mathematics involved

• Build student understanding, following a clear set of learning expectations

• Warrant a summary look back with reflection and extension opportunities(www.mathedleadership.org/ccss/greattasks.html)

What Tasks Are Your Students Engaged in?

TASK: Review your assessment and sort the questions into the four categories.

Lower-Level Tasks Higher-Level TasksMemorization Procedures with

connections

Procedures without connections

Doing mathematics

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Levels of Demand

(Source: This page provides a summary of M. K. Stein & M. S. Smith’s 1998 article, “Mathematical Tasks as a Framework for Reflection: From Research to Practice” in Mathematics Teaching in the Middle School, 3, 268–275.)

Lower-Level Demands Memorization

a. Involve either reproducing previously learned facts, rules, formulas, or definitions or committing facts, rules, formulas or definitions to memory.

b. Cannot be solved using procedures because a procedure does not exist or because the time frame in which the task is being completed is too short to use a procedure.

c. Are not ambiguous. Such tasks involve the exact reproduction of previously seen material, and what is to be reproduced is clearly and directly stated.

d. Have no connection to the concepts or meaning that underlies the facts, rules, formulas, or definitions being learned or reproduced.

Procedures Without Connections

a. Are algorithmic. Use of the procedure either is specifically called for or is evident from prior instruction, experience, or placement of the task.

b. Require limited cognitive demand for successful completion. Little ambiguity exists about what needs to be done and how to do it.

c. Have no connection to the concepts or meaning that underlies the procedures being used. d. Are focused on producing correct answers versus developing mathematical understanding. e. Require no explanations or explanations that focus solely on describing the procedure that was used.

Higher-Level Demands Procedures With Connections

a. Focus students’ attention on the use of procedures for the purpose of developing deeper levels of understanding of mathematical concepts and ideas.

b. Suggest explicitly or implicitly pathways to follow that are broad general procedures that have close connections to underlying conceptual ideas as opposed to narrow algorithms that are opaque with respect to underlying concepts.

c. Usually are represented in multiple ways, such as visual diagrams, manipulatives, symbols and problem situations. Making connections among multiple representations helps develop meaning.

d. Requires some degree of cognitive effort. Although general procedures may be followed, they cannot be followed mindlessly. Students need to engage with conceptual ideas that underlie the procedures to complete the task successfully and that develop understanding.

Doing Mathematics a. Require complex and nonalgorithmic thinking—a predictable, well-rehearsed approach or pathway is not

explicitly suggested by the task, task instructions, or a worked-out example. b. Require students to explore and understand the nature of mathematical concepts, processes, or

relationships. c. Demand self-monitoring or self-regulation of one’s own cognitive processes. d. Require students to access relevant knowledge and experiences and make appropriate use of them in

working through the task. e. Require students to analyze the task and actively examine task constraints that may limit possible solution

strategies and solutions. f. Require considerable cognitive effort and may involve some level of anxiety for the student because of the

unpredictable nature of the solution process required.

© National Council of Teachers of Mathematics 1998. Used with permission.Do not duplicate.

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How Are Tasks Formative?

I use?

Which coupon should I use?

Opportunities for All Students to Engage in Challenging Tasks?• Examine tasks in your instructional

materials:– Higher cognitive demand?– Lower cognitive demand?

• Where are the challenging tasks?• Do all students have the opportunity to

grapple with challenging tasks?• Examine the tasks in your assessments:

– Higher cognitive demand?– Lower cognitive demand?

Cognitive Complexities From PARCC

1. Mathematical content

2. Mathematical practices

3. Stimulus material

4. Response mode

5. Processing demand

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Master Claim: On-Track for college and career readiness. The degree to which a student is college and career ready (or “on-track” to being ready) in mathematics. The student solves grade-level /course-level problems in

mathematics as set forth in the Standards for Mathematical Content with connections to the Standards for Mathematical Practice.

Sub-Claim A: Major Content1 with Connections to Practices

The student solves problems involving the Major Content1 for her

grade/course with connections to the Standards for Mathematical

Practice.

Sub-Claim B: Additional & Supporting Content2 with Connections to

Practices

The student solves problems involving the Additional and Supporting

Content2 for her grade/course with connections to the Standards for

Mathematical Practice.

Sub-Claim E: Fluency in applicable grades (3-6)

The student demonstrates fluency as set forth in the Standards for Mathematical

Content in her grade.

Claims Structure: Mathematics

Sub-Claim C: Highlighted Practices MP.3,6 with Connections to Content3

(expressing mathematical reasoning)

The student expresses grade/course-level appropriate mathematical reasoning by constructing viable

arguments, critiquing the reasoning of others, and/or attending to precision

when making mathematical statements.

Sub-Claim D: Highlighted Practice MP.4 with Connections to Content (modeling/application)

The student solves real-world problems with a degree of difficulty appropriate to the grade/course by applying knowledge and skills articulated in the standards for the

current grade/course (or for more complex problems, knowledge and skills articulated in the standards for previous grades/courses), engaging particularly in the Modeling

practice, and where helpful making sense of problems and persevering to solve them (MP. 1),reasoning abstractly and quantitatively (MP. 2), using appropriate tools

strategically (MP.5), looking for and making use of structure (MP.7), and/or looking for and expressing regularity in repeated reasoning (MP.8).

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PARCC Blueprints

Evidence Statement Tables

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April 2013

New Approach to Designing Performance Level Descriptors: PARCC Mathematics Summative Assessment

In October 2012 PARCC established five performance levels.

• Level 5: Students performing at this level demonstrate a distinguished command of the knowledge, skills, and practices embodied by the Common Core State Standards assessed at their grade level.

• Level 4: Students performing at this level demonstrate a strong command of the knowledge, skills, and practices embodied by the Common Core State Standards assessed at their grade level.

• Level 3: Students performing at this level demonstrate a moderate command of the knowledge, skills, and practices embodied by the Common Core State Standards assessed at their grade level.

• Level 2: Students performing at this level demonstrate a partial command of the knowledge, skills, and practices embodied by the Common Core State Standards assessed at their grade level.

• Level 1: Students performing at this level demonstrate a minimal command of the knowledge, skills, and practices embodied by the Common Core State Standards assessed at their grade level.

Claims Driving Design: Students Are “On Track” to College and Career Readiness

The Master Claim driving the design of the PARCC assessments is “Students are ‘on track’ or ready for college and careers.” This Master Claim reflects the overall goal of the Common Core State Standards and PARCC Model Content Frameworks—to prepare students for college and careers, and specifically to ensure that students have the skills and understandings required for success. The measure of progress toward this essential goal will be reflected by a student’s overall performance on the summative components (both the Performance-Based Assessment and End-of-Year Assessment components) of the PARCC assessment system. While students will receive one summative mathematics score and performance level, they will also receive information about their performance in the following areas:

• Sub-claim A: Students solve problems involving the major content1 for their grade level with connections to practices.

1 Major and additional and supporting content at each grade level are defined in the PARCC Model Content

Frameworks, available at http://www.parcconline.org/parcc-model-content-frameworks.

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http://www.parcconline.org/parcc-model-content-frameworks

April 2013

• Sub-Claim B: Students solve problems involving the additional and supporting content for their grade level with connections to practices.

• Sub-claim C: Students express mathematical reasoning by constructing mathematical arguments and critiques.

• Sub-Claim D: Students solve real world problems engaging particularly in the modeling practice.

• Sub-Claim E: Students demonstrate fluency in areas set forth in the Standards for Content in grades 3-6.

PARCC’s Process in Developing Performance Level Descriptors:

1. The PLD process began with scheduled week-long PLD grade band meetings where

state representatives developed the language for descriptors of the standards/

evidence statements that students are expected to demonstrate at each of the levels.

2. Following these meetings, the information for each grade level PLD was gathered

and formulated into a grade-level synthesis document that captured the essence of

the PLD work. These were reviewed by the Center for Assessment, Educational

Testing System, College Board and the PARCC Technical Advisory Committee.

3. Using this feedback, the Math Operational Working Group (Math OWG) met to:

add more specificity to the PLDs;

ensure a common language for descriptors was apparent across the

grade level;

ensure alignment to the standards; and

ensure grade-to-grade consistency and differentiation were evident.

4. PARCC K-12 State Leads and Higher Education Leadership Team (HELT) provided

feedback on the PLDs. Math OWG members reviewed and incorporated the State

Lead and HELT feedback.

5. The Draft Math PLDs were presented to the PARCC Executive Committee and

Advisory Committee on College Readiness Steering Committee for approval.

22

April 2013

Factors Contributing to Cognitive Complexity: Early in the design phase PARCC determined that development of a new complexity framework and a different process and structure for performance level descriptors was needed.

1. Mathematical Content At each grade level, there is a range in the level of demand in the content standards--from low to moderate to high complexity. Within Mathematical Content, complexity is affected by:

• Numbers: Whole numbers vs. fractions • Expressions and Equations: The types of numbers or operations in an

expression or equation ( 3/7, √ ) • Diagrams, graphs, or other concrete representations: may contribute to

greater overall complexity than simpler graphs such as scatterplots. • Problem structures: Word problems with underlying algebraic structures vs.

word problems with underlying arithmetic structures.

Cognitive Complexity

Mathematical Content

Mathematical Practices

Stimulus Material

Response Mode

Processing Demand

23

April 2013

2. Mathematical Practices (MPs) MPs involve what students are asked to do with mathematical content, such as engage in application and analysis of the content. The actions that students perform on mathematical objects also contribute to Mathematical Practices complexity.

• Low Complexity items primarily involve recalling or recognizing concepts or procedures specified in the Standards.

• High Complexity items make heavy demands on students, because students are expected to use reasoning, planning, synthesis, analysis, judgment, and creative thought. They may be expected to justify mathematical statements or construct a formal mathematical argument.

3. Stimulus Material This dimension of cognitive complexity accounts for the number of different pieces of stimulus material in an item, as well as the role of technology tools in the item.

• Low Complexity involves a single piece of (or no) stimulus material (e.g., table, graph, figure, etc.) OR a single online tool (generally, incremental technology).

• High Complexity involves two pieces of stimulus material with online tool(s) OR three pieces of stimulus material with or without online tools.

4. Response Mode The way in which examinees are required to complete assessment activities influences an item’s cognitive complexity.

• Low complexity response modes in mathematics involve primarily selecting responses and producing short responses, rather than generating more extended responses.

• High Complexity response modes require students to construct extended written responses that may also incorporate the use of online tools such as an equation editor, graphing tool, or other online feature that is essential to responding.

5. Processing Demand Reading load and linguistic demands in item stems, instructions for responding to an item, and response options contribute to the cognitive complexity of items.

24

PLDs: Performance Level Descriptors (In Draft)

PARCC Performance-Level Descriptors

PARCC Prototype

25

Sustained Implementation of CCSS requires four pursuits:

1. A thorough review of your current local assessments on a unit-by-unit basis

2. High-quality common assessments and the accurate scoring of those assessments

3. A robust formative assessment process for students and adults, using each assessment Instrument

4. Instruction that provides evidence of student understanding via the mathematical practices

Today’s Learning Targets

• I can describe the work of collaborative team during the teaching-assessing-learning cycle.

• I can examine tools and criteria for effective assessment design on a unit-by-unit basis.

• I can become familiar with the new dynamic assessments from PARCC.

Reflection: What will be your commitments for assessment in 2013–2014?

Mona Toncheff [email protected] http://puhsdmath.blogspot.com

26

Unit Assessment NRatios and Proportions

Learning arget Standard Testuestions

Pointswarded

Percentorrect

I can compute unit rates associated withratios of fractions.For full credit be sure to include:

ProcessSolution

7.RP.1 1 3

/6

I can decide whether two quantities are in aproportional relationship.For full credit be sure to include:

ProcessSolutionJustification

7. RP.2a 4 6

/10

I can identify unit rate from multiplerepresentations.For full credit be sure to include:

ProcessSolution

7.RP.2b 7 11

/12

I can define characteristics of the graph ofproportional relationships.For full credit be sure to include:

ProcessSolution

7.RP.2d 12 13

/5

I can use proportional relationships to solvemultistep ratio and percent problems.For full credit be sure to include:

ProcessSolution

7.RP.3 14 17

/11

Unit erformance task 7.RP 18 /6

REPRODUCIBLE

© Toncheff 2013 • solution-tree.comReproducible.

27

Unit AssessmentRatios and Proportions

Page 2 of 5

7.RP.1 I can compute unit rates associated withratios of fractions.

1. If it takes Carlos 15 weeks to make 3birdhouses, how long will it take him tomake 11 birdhouses?

2. Joanie rode her bike at a constant speed of

10 miles per hour. At the same speed,

how many miles will she ride in 6 hours?

3. You can get 640 calories from eating 8apples. How many calories can you getfrom eating 1 apple?

7.RP.2a I can decide whether two quantities arein a proportional relationship.

4. The graph shows the number of miles adriver travels between Phoenix and Tucson,AZ over a number of hours. About how fastdoes she drive per hour?

5. Temperatures were taken after every 10minutes for 4 experiments.

Which of the experiments’ temperatureshave a proportional relationship with time?Explain your reasoning.

(2pt)

(2pt)

(2pt)

(2pt)

(5pt)

REPRODUCIBLE


28


Page 4 of 5

7.RP.2d I can define characteristics of the graphof proportional relationships.

12. Darryl is reading a book at the rate of 4.5pages per minute. What ordered pair on agraph of his reading rate would representthe number of minutes it would take him toread 90 pages?

A.B.C.D.

13. The ordered pairsappear on a graph showing the total distancebicycled after a certain number of hours.

A) What ordered pair on that graph showsthe unit rate?

B) What does the unit rate represent?

7.RP.3 I can use proportional relationships tosolve multi-step ratio and percent problems.

14. Jake sold a total of $8,400 worth of clothinglast week at his store. If his commission is12% of sales, how much commission did heearn?

15. If the sales tax is 8.5%, about how much willan $89.95 pair of sneakers cost, includingsales tax?

16. The regular price of a pair of sneakers is$65. They are on sale for 20% off. What isthe sale price?

17. A balloon holds 4.2 cubic feet of air. Theballoon is blown up larger to hold 5.6 cubicfeet of air. What is the percent of changefor the volume of air inside the balloon?

(1pt)

(4pt)

(2pt)

(3pt)

(3pt)

(3pt)

REPRODUCIBLE


30


Page 5 of 5

18. Amy and her family were traveling during their vacation he looked at her watch at 1 point inthe diagram below, and then again at Point 2 in the diagram below. Her mom told her how farthey traveled in that time, noted below.

A. Based on this information, what is the unit rate of the car? Explain in words what thatunit rate means in the context of the problem.

B. Amy’s dad said that the entire trip was 1200 mile ow many hours will ittake to complete the trip xplain your reasoning in words.

REPRODUCIBLE

2© 2011 University of Pittsburgh, Institute for Learning.

Available for download as part of “Common Core-Aligned Task With Instructional Support: Mathematics, Grade 7 Math: Proportional Reasoning,” page 5, at goo.gl/gDKCC 31

Learning Target Test

Questions Score Percent

F-BF.1a: Identify explicit and recursive patterns within a situation. Write an exponential function given a situation. 1−2 /4

F-BF.3: Infer how the change of parameters a,b,h and k of x hy ab k−= + transform the graph. 3−4 /4

F-LE.1a: Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.

5−6 /5

F-LE.1b: Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.

7−8 /4

F-LE.1c: Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

9−10 /6

F-LE.2: Write an exponential function given a pattern, a set of ordered pairs, a graph or a description of an exponential situation.

11−12 /4

F-LE.3: Compare exponential growth to linear growth using graphs and tables. 13−14 /4

F-LE.5: Identify common ratio (b) and initial value (a) of xy ab= from a given context. 15−16 /5

REPRODUCIBLE

0© 2012 Phoenix Union High School District, Algebra Honors Team. Used with permission.

Reproducible.32

I can identify explicit and recursive patterns within a situation. I can write an exponential function given a situation.

t

v⎛ ⎞= ⎜ ⎟⎝ ⎠

t

v⎛ ⎞= ⎜ ⎟⎝ ⎠

v t= − +

v t= − +

a

at =2(a0)t

ta t

t a

at = ao(t)

2

at = ao(2)t

at =2 ao

REPRODUCIBLE

© Toncheff 2013 • solution-tree.comReproducible.206

33

I can infer how the change in parameters a,b,h and k of an exponential function transform the graph.

∙2

∙2

xy = − +

I can prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.

xh x

REPRODUCIBLE

© Toncheff 2013 • solution-tree.comReproducible. 207

34

xy =

x y change

I can recognize situations in which one quantity changes at a constant rate per unit interval relative to another.

s

o?

I can recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

Justify your response.

REPRODUCIBLE


35

yx

x y

I can write an

exponential function given a pattern, a set of ordered pairs, a graph, or a description of an exponential situation.

xy =

xy =

x

y⎛ ⎞= ⎜ ⎟⎝ ⎠

x

y⎛ ⎞= ⎜ ⎟⎝ ⎠

xf x ab=

I can compare exponential growth to linear growth using graphs and tables.

a b c d

x

a

bc

d

x

y

REPRODUCIBLE

© Toncheff 2013 • solution-tree.comReproducible. 209

36

x

y

x

y

x

y

x

xs x =

q x x=

r x x=

t x x=

I can identify common ratio (b) and initial value (a) of

xy ab= from a given context.

• •

••

REPRODUCIBLE


37

Performance Level Descriptors – Grade 6 Mathematics

April 2013 Page 56 of 189

Grade 6 Math : Sub-Claim A

The student solves problems involving the Major Content for grade/course with connections to the Standards for


Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command

Level 5: Distinguished

Command

Multiplying and

Dividing with

Fractions

Applies and extends

previous understandings

of multiplication and

division to divide

fractions with common

denominators and to solve

word problems with

prompting (e.g., four-

fifths of a candy bar is

divided into one fifth

pieces; how many pieces

are there?) embedded

within the problem.

Applies and extends



division to divide

fractions and solve word

problems with prompting

embedded within the

problem.

Applies and extends



division to solve word

problems involving

division of fractions by

fractions.

Applies and extends



division to create and

solve word problems

involving division of

fractions by fractions.

Ratios Uses ratio and rate

reasoning to solve

mathematical problems,

including ratio, unit rate,

percent of a number and

simple unit conversion

problems.

Uses a limited variety of

representations and

strategies to solve these

problems.

Finds missing values in

Uses ratio and rate

reasoning to solve real-

world and mathematical

problems, including ratio,

unit rate, percent and unit

conversion problems.

Uses a limited variety of

representations and


problems.


tables and plots values on

Uses ratio and rate




unit rate, percent and unit


Uses different

representations and


problems.



Uses ratio and rate




unit rate percent and unit


Uses and connects a

variety of representations

and strategies to solve

these problems.



38






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command


the coordinate plane.

the coordinate plane. the coordinate plane. the coordinate plane.

Rational Numbers Understands that positive

and negative numbers

describe mathematical or

real-world quantities

which have opposite

values or directions and

can be represented on a

number line.

Determines the absolute

value of a rational

number.

Plots ordered pairs on a

coordinate plane to solve

mathematical problems.

Understands that positive




which have opposite



number line and

compared with or without

the use of a number line.

Understands the absolute

value of a rational

number.



real-world and






which have opposite



number line and



Understands (or

recognizes) that when two

ordered pairs differ only

by signs, the locations of

the points are related by

reflections across one or

both axes.

Understands and

interprets the absolute

value of a rational

number.







which have opposite



number line and



Understands (or

recognizes) that when two

ordered pairs differ only

by signs, the locations of

the points are related by

reflections across one or

both axes.

Understands and

interprets the absolute

value of a rational

number.



39






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

real-world and


Distinguishes

comparisons of absolute

value from statements

about order.

real-world and


Distinguishes

comparisons of absolute

value from statements

about order.

Recognizes patterns and

makes generalizations

about characteristics of

positive and negative

numbers.

Expressions and

Inequalities

Evaluates numerical and

algebraic expressions

including those that

contain whole number

exponents.

Identifies parts of an

algebraic or numerical

expression using

mathematical terms.

Reads and evaluates

numerical and algebraic

expressions, including

those that contain whole

number exponents.

Writes numerical

expressions and some

algebraic expressions,



exponents.



Writes, reads and

evaluates numerical and




exponents.



expression using

mathematical terms.

Identifies equivalent

expressions using

properties of operations.

Writes, reads and

evaluates numerical and




exponents.



expression using

mathematical terms and

views one or more parts

of an expression as a

single entity.

40






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

expression using

mathematical terms.


expressions using



expressions using


Uses variables to

represent numbers and

writes expressions and

single-step equations to

solve mathematical

problems.

Relates tables and graphs

to the equations.

Graphs inequalities to

represent a constraint or

condition in a

mathematical problem.

Uses variables to




solve real-world or



to the equations.

Writes and graphs

inequalities to represent a

constraint or condition in

a real-world or


Uses variables to




solve real-world and

mathematical problems

and understand their

solutions.


to equations.

Writes and graphs



a real-world or


Understands that there are

an infinite number of

solutions for an

inequality.

Uses variables to


write expressions and


solve real-world and


and understand their

solutions.

Analyzes the relationship

between dependent and

independent variables and

relates tables and graphs

to equations.

Writes and graphs



a real-world or


Understands that there are

41






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

an infinite number of

solutions for an

inequality.

42



Grade 6 Math: Sub-Claim B

The student solves problems involving the Additional and Supporting Content for her grade/course with

connections to the Standards for Mathematical Practice.

Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

Factors and

Multiples

Finds greatest common

factors and least common

multiples.



multiples.

Uses the distributive

property to rewrite the

sum of two whole

numbers with a common

factor as a multiple of a

sum of two whole

numbers with a common

factor.



multiples.

In most cases, uses the

distributive property to

rewrite the sum of two

whole numbers with a

common factor as a

multiple of a sum of two

whole numbers with no

common factor.



multiples.

Consistently uses the

distributive property to

rewrite the sum of two

whole numbers with a

common factor as a

multiple of a sum of two

whole numbers with no

common factor.

Geometry Solves mathematical

problems involving area

of polygons by either

composing into rectangles

or decomposing into

triangles and other shapes.

Determines measurements

of polygons in the

coordinate plane.

Uses nets of three-

dimensional figures to

find surface area.

Finds volume of right

Solves real-world and


involving area of

polygons by either

composing into rectangles

or decomposing into



of polygons in the

coordinate plane.

Determines and uses nets

of three-dimensional

figures to find surface

area.



involving area of

polygons by composing

into rectangles or

decomposing into



of polygons in the

coordinate plane.




area.



involving area of

polygons by composing

into rectangles or

decomposing into



of polygons in the

coordinate plane.




area.

43






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

rectangular prisms with

fractional edge lengths by

packing them with unit

cubes and using formulas.











Uses volume formulas to

find unknown

measurements.







find unknown

measurements.

Understands the concepts

of area and volume in

order to solve

unstructured and/or

complex problems.

Statistics and

Probability

Recognizes a statistical

question and understand

that a set of collected data

has a distribution which

can be described by its

center, spread and overall

shape.

Understands the purpose

of center and that it can be

summarized with a single

number.


question and understands





shape.


of center and that it can be


number.


question and understands





shape.


of center and variability

and that each can be


Recognizes and describes

a statistical question, and

understands that a set of

collected data has a

distribution which can be

described by its center,

spread and overall shape.


of center and variability

and that each can be


44






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

Displays numerical data

in plots on a number line,

including dot plots and

histograms.

Summarizes numerical

data sets in relation to

their context, such as by

reporting the number of

observations and using

measures of center.



including dot plots,

histograms and box plots.





observations, describing

the nature of the attributes

under investigation and

using measures of center

and variability.

number.




histograms and box plots.









and variability.

Determines which

measures of center and

variability are the most

appropriate for a set of

data.

number.




histograms and box plots,

and determines which

display is the most

appropriate.









and variability.

Determines which

measures of center and

variability are the most

appropriate for a set of

data.

45



Grade 6: Sub-Claim C

The student expresses grade/course-level appropriate mathematical reasoning by constructing viable arguments,

critiquing the reasoning of others and/or attending to precision when making mathematical statements.

Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

Properties of

Operations

Constructs and

communicates an

incomplete response

based on the properties of

operations and the

relationship between

addition and subtraction

or between multiplication

and division, which may

include:

an approach based on

a conjecture and/or

stated or faulty

assumptions

an incomplete or

illogical progression

of steps

major calculation

errors

limited use of grade-

level vocabulary,

symbols and labels

partial justification of

a conclusion

Constructs and

communicates a response


operations and the




and division, including:

a logical approach

based on a conjecture

and/or stated

assumptions

a logical, but

incomplete,

progression of steps

minor calculation

errors

some use of grade-

level vocabulary,

symbols and labels


a conclusion

Evaluates the validity of

other’s approaches and

conclusions.

Clearly constructs and

communicates a complete

response based on the

properties of operations

and the relationship

between addition and

subtraction or between

multiplication and

division, including:

a logical approach


and/or stated

assumptions

a logical progression

of steps

precision of

calculation

correct use of grade-

level vocabulary,

symbols and labels

justification of a

conclusion

Evaluates, interprets and

critiques the validity of

other’s responses,

approaches and








multiplication and


a logical approach


and/or stated

assumptions


of steps

precision of

calculation


level vocabulary,

symbols and labels

justification of a

conclusion

generalization of an

argument or

conclusion


46






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

reasoning. critiques the validity and

efficiency of other’s

responses, approaches

and reasoning, and

provides counter-

examples where

applicable.

Concrete

Referents

and Diagrams

Constructs and

communicates an

incomplete response

based on concrete

referents provided in the

prompt such as: diagrams,

number line diagrams or

coordinate plane

diagrams, which may

include:


a conjecture and/or

stated or faulty

assumptions

an incomplete or


of steps

major calculation

errors


Constructs and


based on concrete


prompt or in simple cases,

constructed by the student

such as: diagrams that are

connected to a written

(symbolic) method,


coordinate plane

diagrams, including:

a logical approach


and/or stated

assumptions

a logical, but

incomplete,


minor calculation



response based on

concrete referents

provided in the prompt or




(symbolic) method,


coordinate plane


a logical approach


and/or stated

assumptions


of steps

precision of

calculation



response based on

concrete referents



such as diagrams that are


(symbolic) method,


coordinate plane


a logical approach


and/or stated

assumptions


of steps

precision of

calculation

47






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

level vocabulary,

symbols and labels


a conclusion

errors

some use of grade-

level vocabulary,

symbols and labels


a conclusion



conclusions.


level vocabulary,

symbols and labels

justification of a

conclusion




approaches and

reasoning.


level vocabulary,

symbols and labels

justification of a

conclusion


argument or

conclusion


critiques the validity and



and reasoning, and

provides a counter-

example where

applicable.

Distinguish

Correct

Explanation/

Reasoning from

that which is

Flawed

Constructs and


to a given equation,

multi-step problem,

proposition or conjecture,

including:


a conjecture and/or

stated or faulty

assumptions

Constructs and


response to a given

equation, multi-step

problem, proposition or

conjecture, including:

a logical approach


and/or stated

assumptions



response to a given




a logical approach


and/or stated

assumptions



response to a given




a logical approach


and/or stated

assumptions

48






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

an incomplete or


of steps

major calculation

errors


level vocabulary,

symbols and labels


a conclusion

a logical, but

incomplete,


minor calculation

errors

some use of grade-

level vocabulary,

symbols and labels


a conclusion



conclusion.

Identifies and describes

errors in solutions.


of steps

precision of

calculation


level vocabulary,

symbols and labels

justification of a

conclusion




approaches and

reasoning.


errors in solutions and

presents correct solutions.


of steps

precision of

calculation


level vocabulary,

symbols and labels

justification of a

conclusion


argument or

conclusion





and reasoning, and

provides a counter-

example where

applicable.




Distinguishes correct

explanation/reasoning

49






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

from that which is flawed.

If there is a flaw, presents

correct reasoning.

50



Grade 6 : Sub-Claim D

The student solves real-world problems with a degree of difficulty appropriate to the grade/course by applying

knowledge and skills articulated in the standards for the current grade/course (or for more complex problems,

knowledge and skills articulated in the standards for previous grades/courses), engaging particularly in the

Modeling practice, and where helpful making sense of problems and persevering to solve them, reasoning

abstractly, and quantitatively, using appropriate tools strategically, looking for the making use of structure

and/or looking for and expressing regularity in repeated reasoning.

Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

Modeling Devises a plan to apply

mathematics in solving

problems arising in

everyday life, society and

the workplace by:

using stated

assumptions and

approximations to

simplify a real-world

situation

identifying important

quantities

using provided tools

to create models

analyzing

relationships

mathematically to

draw conclusions

writing an algebraic

expression or equation

to describe a situation

applying proportional

reasoning

Devises a plan to apply


problems arising in


the workplace by:

using stated

assumptions and

approximations to


situation

illustrating

relationships between

important quantities

by using provided

tools to create models

analyzing

relationships

mathematically

between important

quantities to draw

conclusions

interpreting

mathematical results



problems arising in


the workplace by:

using stated

assumptions and

making assumptions

and approximations to


situation

mapping relationships

between important

quantities by selecting

appropriate tools to

create models

analyzing

relationships

mathematically

between important

quantities to draw

conclusions

interpreting



problems arising in


the workplace by:

using stated

assumptions and

making assumptions



situation

analyzing and/or

creating constraints,

relationships and

goals


between important



create models

analyzing

relationships

mathematically

51










Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

using geometry

using functions to

describe how one

quantity of interest

depends on another

using estimates of

known quantities in a

chain of reasoning

that yields an estimate

of an unknown

quantity

in a simplified context

reflecting on whether

the results make sense

modifying the model

if it has not served its

purpose





reasoning

using geometry

writing/using

functions to describe

how one quantity of

interest depends on

another

using reasonable

estimates of known

quantities in a chain of

reasoning that yields

an estimate of an

unknown quantity


in the context of the

situation



improving the model


purpose

writing a complete,

clear and correct

algebraic expression

or equation to

describe a situation


reasoning

using geometry

writing/using


how one quantity of

interest depends on

another

using reasonable

estimates of known

between important

quantities to draw

conclusions

justifying and

defending models

which lead to a

conclusion

interpreting



situation



improving the model


purpose

writing a complete,

clear and correct


or equation to



reasoning

52










Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command



an estimate of an

unknown quantity

using geometry

writing/using


how one quantity of

interest depends on

another

using reasonable

estimates of known



an estimate of an

unknown quantity

53



Grade 6 : Sub-Claim E

The student demonstrates fluency in areas set forth in the Standards for Content in grades 3-6.

Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

Fluency Divides multi-digit

numbers and adds,

subtracts, multiplies and

divides multi-digit

decimals using the

standard algorithm with

some level of accuracy.

Accurately divides multi-

digit numbers and adds,

subtracts, multiplies and

divides multi-digit

decimals using the

standard algorithm.

Fluently (accurately in a

timely manner) divides

multi-digit numbers and

adds, subtracts, multiplies

and divides multi-digit

decimals using the

standard algorithm.

Fluently (accurately and

quickly) divides multi-

digit whole numbers and

adds, subtracts, multiplies

and divides multi-digit

decimals using the

standard algorithm and

assesses reasonableness of

the result.

54






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

Proportional

Relationships

Uses proportional

relationships to solve real-


problems, including

simple ratio/percent

problems.

Computes unit rates of

quantities associated with

ratios of fractions.

Decides whether two

quantities are in a

proportional relationship

and identifies the constant

of proportionality (unit

rate) in tables, equations,

diagrams, verbal

descriptions and graphs.

Uses equations

representing a


to solve simple

mathematical and real-

world problems, including

simple ratio and percent

problems.

Analyzes and uses

proportional relationships

to solve real-world and


including simple

ratio/percent problems.




Decides whether two

quantities are in a





diagrams, verbal


Interprets a point (x, y) on

the graph of a


in terms of the situation,

with special attention to

the points (0, 0) and (1, r)

where r is the unit rate.

Analyzes and uses




including multi-step





Decides whether two

quantities are in a





diagrams, verbal



the graph of a






Analyzes and uses




including multi-step





Decides whether two

quantities are in a





diagrams, verbal



the graph of a






55






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

Represents proportional

relationships by equations

and uses them to solve



simple ratio and percent

problems.






multi-step ratio and

percent problems.






multi-step ratio and

percent problems.

Compares proportional

relationships given in

different forms (tables,

equations, diagrams,

verbal, graphs).

Determines when it is

appropriate to use unit

rate and understands

when it has its limitations.

Operations with

Fractions

Performs operations on


rational numbers in

simple mathematical and

real-world problems.

Represents addition and

subtraction on a

horizontal or vertical

number line and



rational numbers in multi-

step mathematical and


Determines

reasonableness of a

solution.



rational numbers in multi-



Determines

reasonableness of a

solution and interprets

solutions in real-world



rational numbers in


world problems.

Determines

reasonableness of a

solution and interprets

solutions in real-world

56






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

recognizes situations in

which opposite quantities

combine to make zero.


subtraction on a


number line and




contexts.


subtraction on a


number line and




contexts.


subtraction on a


number line and




Using the properties of

operations, justifies the

steps taken to solve multi-


real-world problems

involving rational

numbers.

Expressions,

Equations and

Inequalities

Applies properties of

operations as strategies to

add, subtract and expand

linear expressions.

Solves two-step linear

equations with rational

coefficients.

In a mathematical context,

uses variables to represent



add, subtract, factor and

expand linear expressions.

Solves two-step linear

equations with rational

coefficients.

In a mathematical or real-

world context, uses





Rewrites an expression in

different forms.

Fluently solves multi-step

linear equations with

rational coefficients.





Describes the relationship

between equivalent

quantities that are

expressed algebraically in

different forms in a

problem context and

57






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

quantities, construct and

solve simple equations

and inequalities, and

graph solution sets.

variables to represent




graph and solution sets.

In mathematical or real-

world contexts, uses





graph and interpret

solution sets.

explains their equivalence

in light of the context of

the problem.

Fluently solves multi-step

linear equations with


In a mathematical or real-

world contexts, uses





graph and interpret

solution sets.

58






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

Representing

Geometric

Figures

Draws geometric figures

– freehand, with a ruler

and protractor, or with

technology – and describe

some of their attributes.

Constructs triangles with

given angle and side

conditions.


– with a ruler and

protractor or with

technology – and

describes their attributes.



conditions.

Describes the two-

dimensional figures that

result from slicing three-

dimensional figures by a

plane parallel or

perpendicular to a base or

face.


– with a ruler and

protractor or with

technology – and




conditions and notices

when those conditions

determine a unique

triangle, more than one

triangle or no triangle.

Describes two-



dimensional figures.

Draws, with precision,

geometric figures – with a

ruler and protractor or

with technology – and




conditions and notices

when those conditions

determine a unique

triangle, more than one

triangle or no triangle.

Describes two-



dimensional figures.

Drawings and

Measurement

Solves mathematical

problems involving

circumference, area,

surface area and volume

of two- and three-

dimensional objects.

Solves problems

involving scale drawings

of geometric figures.

Solves mathematical and

real-world problems

involving circumference,

area, surface area and

volume of two- and three-


Solves problems


of geometric figures,


real-world problems




dimensional objects,

including composite

objects.

Solves problems


real-world problems




dimensional objects,

including composite

objects.

Identifies or produces a

59






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

Uses facts about angle

relationships to determine

the measure of unknown

angles.

including reproducing a

scale drawing at a

different scale.

Represents angle

relationships using

equations to solve for

unknown angles.




scale drawing at a

different scale.

Represents angle

relationships using


unknown angles.

logical conclusion about

the relationship between

the circumference and

area of a circle.

Solves problems




scale drawing at a

different scale.

Represents angle

relationships using


unknown angles.

Random

Sampling and

Comparative

Inferences

Draws inferences about a

population from a table or

graph of random samples.

Draws simple informal

comparative inferences

about two populations.

Understands and uses

random sampling to draw

inferences about a

population.

Draws informal


about two populations.



inferences about a

population.

Draws informal


about two populations,

including assessing the

degree of visual overlap

of two numerical data

distributions with similar



inferences about a

population.

Analyzes whether a

sample is representative

of a population.

Draws informal


about two populations,

60






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

variabilities.

Generates multiple

samples of the same size

to gauge the variation in

estimates or predictions.

including assessing the

degree of visual overlap

of two numerical data

distributions with similar

variabilities.

Generates multiple

samples of the same size

to gauge the variation in

estimates or predictions.

Chance Processes

and Probability

Models

Understands that the

probability of a chance

event is a number

between 0 and 1 that

expresses the likelihood

of the event occurring.

Finds probabilities when

given sample spaces for

simple events using

methods such as

organized lists and tables.



event is a number




Develops a model to

approximate the


event and predicts

approximate frequencies

when given the

probability or by

observing frequencies in

data generated from the

process.



event is a number




Develops a model to

approximate the


event and predicts


when given the

probability or by



process.



event is a number




Develops a model to

approximate the


event, predicts


when given the

probability or by



process, and compares

probabilities from a

61






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

Finds probabilities when

given sample spaces for

simple and compound

events using methods

such as organized lists,

tables and tree diagrams.

Generates a sample space

to determine the

probability of simple or

compound events using

methods such as

organized lists, tables,

tree diagrams or

simulations.

Uses a simulation to

estimate the probability of

a compound event.

model to observed

frequencies.

Generates a sample space

to determine the

probability of simple or

compound events using

methods such as

organized lists, tables or

tree diagrams.

Designs and uses a

simulation to estimate the

probability of a

compound event.

62






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

Properties of

Operations

Constructs and

communicates an

incomplete response


operations and the





include:


a conjecture and/or

stated or faulty

assumptions

an incomplete or


of steps

major calculation

errors


level vocabulary,

symbols and labels


a conclusion

Constructs and



operations and the





a logical approach


and/or stated

assumptions

a logical, but

incomplete,


minor calculation

errors

some use of grade-

level vocabulary,

symbols and labels


a conclusion



conclusion.








multiplication and


a logical approach


and/or stated

assumptions


of steps

precision of

calculation


level vocabulary,

symbols and labels

justification of a

conclusion




approaches and reasoning.








multiplication and


a logical approach


and/or stated

assumptions


of steps

precision of

calculation


level vocabulary,

symbols and labels

justification of a

conclusion


argument or

conclusion


63






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

critique the validity and


responses, approaches and

reasoning, and provides

counter-examples where

applicable.

Concrete

Referents

and Diagrams

Constructs and

communicates an

incomplete response

based on concrete




coordinate plane

diagrams, which may

include:


a conjecture and/or

stated or faulty

assumptions

an incomplete or


of steps

major calculation

errors


level vocabulary,

Constructs and


based on concrete






(symbolic) method,


coordinate plane


a logical approach


and/or stated

assumptions

a logical, but

incomplete,


minor calculation

errors



response based on

concrete referents





(symbolic) method,


coordinate plane


a logical approach


and/or stated

assumptions


of steps

precision of

calculation




response based on

concrete referents





(symbolic) method,


coordinate plane


a logical approach


and/or stated

assumptions


of steps

precision of

calculation


64






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

symbols and labels


a conclusion

some use of grade-

level vocabulary,

symbols and labels


a conclusion



conclusions.

level vocabulary,

symbols and labels

justification of a

conclusion





level vocabulary,

symbols and labels

justification of a

conclusion


argument or

conclusion





reasoning, and provides a

counter-example where

applicable.

Distinguish

Correct

Explanation/

Reasoning from

that which is

Flawed

Constructs and


to a given equation, multi-

step problem, proposition

or conjecture, including:


a conjecture and/or

stated or faulty

assumptions

an incomplete or


of steps

Constructs and


response to a given




a logical approach


and/or stated

assumptions

a logical, but

incomplete,



response to a given




a logical approach


and/or stated

assumptions


of steps



response to a given




a logical approach


and/or stated

assumptions


of steps

65






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

major calculation

errors


level vocabulary,

symbols and labels


a conclusion


minor calculation

errors

some use of grade-

level vocabulary,

symbols and labels


a conclusion



conclusion.



precision of

calculation


level vocabulary,

symbols and labels

justification of a

conclusion








precision of

calculation


level vocabulary,

symbols and labels

justification of a

conclusion


argument or

conclusion







applicable.








correct reasoning.

66










Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command



problems arising in


the workplace by:

using stated

assumptions and

approximations to


situation


quantities


to create models

analyzing

relationships

mathematically to

draw conclusions







problems arising in


the workplace by:

using stated

assumptions and

approximations to


situation

illustrating



by using provided


analyzing

relationships

mathematically

between important

quantities to draw

conclusions

interpreting



problems arising in


the workplace by:

using stated

assumptions and

making assumptions



situation


between important



create models

analyzing

relationships

mathematically

between important

quantities to draw

conclusions



problems arising in


the workplace by:

using stated

assumptions and

making assumptions



situation

analyzing and/or


relationships and

goals


between important



create models

analyzing

relationships

67










Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

reasoning

using geometry

using functions to

describe how one


depends on another

using estimates of


chain of reasoning


of an unknown

quantity





modifying the model


purpose





reasoning

using geometry

writing/using


how one quantity of

interest depends on

another

using reasonable

estimates of known



an estimate of an

unknown quantity

interpreting



situation



improving the model


purpose

writing a complete,

clear and correct


or equation to



reasoning

using geometry

writing/using


how one quantity of

interest depends on

another

using reasonable

mathematically

between important

quantities to draw

conclusions

justifying and

defending models

which lead to a

conclusion

interpreting



situation



improving the model


purpose

writing a complete,

clear and correct


or equation to



reasoning

68










Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

estimates of known



an estimate of an

unknown quantity

using geometry

writing/using


how one quantity of

interest depends on

another

using reasonable

estimates of known



an estimate of an

unknown quantity

69






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

Expressions and

Equations

Evaluates simple

numerical expressions

using properties of integer

exponents.

Partially solves equations

of the form x2 = p, where

p is a perfect square, by

representing the positive

solution of the equation.

Evaluates and generates

equivalent numerical

expressions applying

properties of integer

exponents.

Solves equations of the

form x2 = p, where p is a

perfect square and solves

equations of the form x3 =

p, where p is a perfect

cube.





exponents.

Demonstrates a general

understanding of the

structure of these

properties within a real-

world context.


form x2 = p and x

3 = p,

representing solutions

using √ symbols.





exponents.

Demonstrates a solid

understanding of the

structure of these

properties within a real-

world context.


form x2 = p and x

3 = p,

representing solutions

using √ symbols.

Scientific

Notation

Using scientific notation,

estimates very large

quantities.

Performs operations with

numbers expressed in

scientific notation,

without technology.


estimates very large and

very small quantities.




without technology.



very small quantities and

determines how many

times as large one number

is in relation to another.




without technology.



very small quantities and

determines how many

times as large one number

is in relation to another.



scientific notation, with

and without technology.

70






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

Interprets scientific

notation that has been

generated by technology.

Interprets scientific

notation that has been

generated by technology.

Proportional

Relationships and

Linear Equations

Graphs linear

relationships, in the form

y= mx+b, including

proportional relationships.

Interprets the unit rate as

the slope of the graph of a

proportional relationship.

Makes some comparisons

between two different


represented in different

ways.

Graphs linear

relationships, in the form

y=mx+b, including





and applies these concepts

to solve real-world

problems.

Compares two different



ways.

Graphs linear

relationships in the form

y=mx+b, including






to solve real-world

problems.




ways.

Interprets y=mx+b as

defining a linear function.

Graphs linear

relationships in the form

y=mx+b, including






to solve real-world

problems.

Uses similar triangles to

show that the slope is the

same between any two

distinct points on a non-

vertical line in the

coordinate plane.




ways.

Interprets y=mx+b as

71






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

defining a linear function.

Solving Linear

Equations

Solves linear equations in

one variable, with rational

number coefficients,


require combining like

terms.

Solves linear equations in




require use of the

distributive property or

combining like terms.


real-world linear

equations in one variable,

with rational number

coefficients, including

those that require use of

the distributive property

or combining of like

terms.

Solves complex


world linear equations in




require use of the

distributive property or

combining of like terms.

Simultaneous

Linear Equations

Solves mathematical

problems leading to pairs

of simultaneous linear

equations graphically or

by inspection.

Analyzes and solves


leading to pairs of

simultaneous linear

equations graphically and

algebraically.

Analyzes and solves


world problems leading to

pairs of simultaneous

linear equations

algebraically, graphically

and by inspection.

Understands the

relationship between the

graphic representation and

the algebraic solution to

the system.

Analyzes and solves


world problems leading to

pairs of simultaneous

linear equations

algebraically, graphically

and by inspection.

Understands the

relationship between the

graphic representation and

algebraic solution to the

system.

Verifies a solution

utilizing multiple methods

to prove accuracy.

72






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

Functions

Understands that a

function is a rule that

assigns to each input

exactly one output and

can be graphed as a set of

ordered pairs.

Understands that a

function is a rule that

assigns to each input

exactly one output and


ordered pairs.

Compares some of the

properties of two

functions represented in

different ways.

Understands that a

function is a rule

assigning to each input

exactly one output, which


ordered pairs.

Compares properties of

two functions represented

in different ways.

Identifies functions that

are non-linear.

Understands that a

function is a rule

assigning to each input

exactly one output which


ordered pairs.



in different ways.

Identifies and proves

functions as non-linear.

Congruence and

Similarity

Describes the effect of

translations, rotations and

reflections on two-

dimensional figures

without coordinates and

determines whether two

given figures are

congruent.


dilations, translations,

rotations and reflections

on two-dimensional

figures with and without

coordinates, and

determines whether two

given figures are

congruent or similar

through one or more

transformations.




on two-dimensional


coordinates, determines

whether two given figures

are congruent or similar

through one or more

transformations and

describes the sequence of

transformations to justify

congruence or similarity

of two figures.




on two-dimensional


coordinates, determines

whether two given figures

are congruent or similar

through one or more

transformations and

describes multiple

sequences of

transformations to justify

congruence or similarity

of two figures.

73






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

Pythagorean

Theorem

Applies the Pythagorean

Theorem in a simple

planar case.


Theorem in a simple

planar case and to find the

distance between two

points in a coordinate

system.


Theorem in a simple

planar case and to find the

distance between two

points in a coordinate

system and in a simple

three-dimensional case in

both mathematical and



Theorem in a planar case

and to find the distance

between two points in a

coordinate system and in

a three-dimensional case

in both mathematical and

real-world multi-step

problems.

Recognizes situations to

apply the Pythagorean

Theorem in multi-step

problems.

74






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

Rational

Numbers

Distinguishes between

rational and irrational

numbers and understands

that these numbers have

decimal expansions and

can locate them

approximately on a

number line.



numbers, understands that

these numbers have


can locate them

approximately on a

number line, and converts

between terminating

decimals or repeating

decimals of the form

(0.aaa…) and fractional

representations of rational

numbers.




these numbers have


can locate them

approximately on a


between terminating


decimals and fractional


numbers.




these numbers have


can locate them

approximately on a


between terminating


decimals and fractional


numbers.

Analyzes and generalizes

patterns and structures of

repeating decimals.

Modeling with

Functions

Constructs a function to

model a linear

relationship between two

quantities in a table or a

graph.

Determines the rate of

change and initial value

of the function from a

table or graph that

contains the initial value.


model a linear


quantities without a

context.

Given two (x,y) values in

a table of values or a

graph, determines the rate

of change and initial

value of the function.


model a linear


quantities described with

or without a context.

Given a description of a

relationship or two (x,y)

values in a table of values

or a graph, determines the

rate of change and initial


model a linear


quantities described

within or without a

context.

Given a description of a

relationship or two (x,y)

values in a table of values

or a graph, determines the

75






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

Analyzes the graph of a

linear function to describe

the functional relationship

between two quantities.

Analyzes the graph of a

linear function to describe



Sketches the graph of a

function when given a

written description.


Analyzes and describes



Sketches a graph of a



rate of change and initial


Analyzes, describes and

contextualizes the

functional relationship


Sketches a graph of a



Volume Knows the formulas for

the volume of cones,

cylinders and spheres, and

uses them to find the

volume of solids in


Knows the formulas for




volume of solids in


world problems.





volume or dimensions of

solids in mathematical

and real-world problems.

Applies these formulas to

multiple composite

mathematical solids.





volume or dimensions of

composite solids in


world problems.

Applies these formulas to

multiple composite

mathematical solids and

utilize these formulas

within a novel context.

Bivariate Data Analyzes and describes

the patterns of association

that can be seen in



that can be seen in



that can be seen in

Justifies the patterns of

association that can be

seen in bivariate data by

76






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

bivariate data by

constructing, displaying

and interpreting scatter

plots and two-way tables.

Uses a given equation of a

linear model to solve

problems in context.

Informally fits a straight

line to a scatter plot that

suggests a linear

association.

bivariate data by




Uses the equation of a





suggests a linear

association.

bivariate data by









suggests a linear

association and assesses

the model fit.









suggests a linear

association and assesses

the model fit.

Compares linear models

used to fit the same set of

data to determine which

better fits are.

77





critiquing the reasoning of others and/or attending to precision when making mathematical statements. Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

Properties of

Operations

Constructs and

communicates an

incomplete response


operations and the





include:


a conjecture and/or

stated or faulty

assumptions

an incomplete or


of steps

major calculation

errors


level vocabulary,

symbols and labels


a conclusion

Constructs and



operations and the





a logical approach


and/or stated

assumptions

a logical, but

incomplete,


minor calculation

errors

some use of grade-

level vocabulary,

symbols and labels


a conclusion



conclusion.








multiplication and


a logical approach


and/or stated

assumptions


of steps

precision of

calculation


level vocabulary,

symbols and labels

justification of a

conclusion












multiplication and


a logical approach


and/or stated

assumptions


of steps

precision of

calculation


level vocabulary,

symbols and labels

justification of a

conclusion


argument or

conclusion


78






Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command




reasoning, and provides

counter-examples where

applicable.

Concrete

Referents

and Diagrams

Constructs and

communicates an

incomplete response

based on concrete




coordinate plane

diagrams, which may

include:


a conjecture and/or

stated or faulty

assumptions

an incomplete or


of steps

major calculation

errors


level vocabulary,

Constructs and


based on concrete






(symbolic) method,


coordinate plane


a logical approach


and/or stated

assumptions

a logical, but

incomplete,


minor calculation

errors



response based on

concrete referents





(symbolic) method,


coordinate plane


a logical approach


and/or stated

assumptions


of steps

precision of

calculation




response based on

concrete referents





(symbolic) method,


coordinate plane


a logical approach


and/or stated

assumptions


of steps

precision of

calculation


79






Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

symbols and labels


a conclusion.

some use of grade-

level vocabulary,

symbols and labels


a conclusion



conclusions.

level vocabulary,

symbols and labels

justification of a

conclusion





level vocabulary,

symbols and labels

justification of a

conclusion


argument or

conclusion







applicable.

Distinguish

Correct

Explanation/

Reasoning from

that which is

Flawed

Constructs and


to a given equation, multi-

step problem, proposition

or conjecture, including:


a conjecture and/or

stated or faulty

assumptions

an incomplete or


of steps

Constructs and


response to a given




a logical approach


and/or stated

assumptions

a logical, but

incomplete,



response to a given




a logical approach


and/or stated

assumptions


of steps



response to a given




a logical approach


and/or stated

assumptions


of steps

80






Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

major calculation

errors


level vocabulary,

symbols and labels


a conclusion


minor calculation

errors

some use of grade-

level vocabulary,

symbols and labels


a conclusion



conclusion.



precision of

calculation


level vocabulary,

symbols and labels

justification of a

conclusion








precision of

calculation


level vocabulary,

symbols and labels

justification of a

conclusion


argument or

conclusion







applicable.



presents correct solutions





correct reasoning.

81



Grade 8: Sub-Claim D






and/or looking for and expressing regularity in repeated reasoning. Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command



problems arising in


the workplace by:

using stated

assumptions and

approximations to


situation


quantities


to create models

analyzing

relationships

mathematically to

draw conclusions







problems arising in


the workplace by:

using stated

assumptions and

approximations to


situation

illustrating



by using provided


analyzing

relationships

mathematically

between important

quantities to draw

conclusions

interpreting



problems arising in


the workplace by:

using stated

assumptions and

making assumptions



situation


between important



create models

analyzing

relationships

mathematically

between important

quantities to draw

conclusions



problems arising in


the workplace by:

using stated

assumptions and

making assumptions



situation

analyzing and/or


relationships and

goals


between important



create models

analyzing

relationships

82










Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

reasoning

using geometry

using functions to

describe how one


depends on another

using estimates of


chain of reasoning


of an unknown

quantity





modifying the model


purpose





reasoning

using geometry

writing/using


how one quantity of

interest depends on

another

using reasonable

estimates of known



an estimate of an

interpreting



situation



improving the model


purpose

writing a complete,

clear and correct


or equation to



reasoning

using geometry

writing/using


how one quantity of

interest depends on

another

using reasonable

mathematically

between important

quantities to draw

conclusions

justifying and

defending models

which lead to a

conclusion

interpreting



situation



improving the model


purpose

writing a complete,

clear and correct


or equation to



reasoning

83










Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

unknown quantity

estimates of known



an estimate of an

unknown quantity

using geometry

writing/using


how one quantity of

interest depends on

another

using reasonable

estimates of known



an estimate of an

unknown quantity

84

Performance Level Descriptors – Algebra I


Algebra I: Sub-Claim A

The student solves problems involving the Major Content for her grade/course with connections to the Standards

for Mathematical Practice.

Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

Expressions

Writes equivalent

numerical and polynomial

expressions in one

variable, using addition,

subtraction and

multiplication.

Identifies components of

exponential and quadratic

expressions.

Writes equivalent


expressions in one


subtraction, multiplication

and factoring.

Interprets parts of


expressions that represent

a quantity in terms of its

context.

Writes equivalent


expressions in one



and factoring, including

multi-step problems.

Interprets parts of

complicated exponential

and quadratic expressions

that represent a quantity

in terms of its context.

Writes equivalent


expressions in one



and factoring, including

multi-step problems in

mathematical and

contextual situations.

Interprets parts of

complicated exponential

and quadratic expressions

that represent a quantity

in terms of its context.

Evaluates expressions for

accuracy and justifies the

results.

Interpreting

Functions

Determines if a given

relation is a function.

Uses and evaluates with

function notation.

Given a context, writes a

linear function.



Within a context, uses and

evaluates with function

notation.


linear function.



Within a context, uses,

interprets and evaluates

with function notation.


linear or quadratic






Given a context, writes

and analyzes a linear or

85






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

For linear or quadratic

functions that model

relationships within a

context, determines key

features.

Determines the domain of

linear and quadratic

functions.




context, determines key

features and graphs the

function.

Determines the domain

and relates it to the

quantitative relationship it

describes for linear,

quadratic and exponential

(limited to domains in the

integers) functions.

function.




context, determines and

interprets key features,

graphs the function and

solves problems.




describes for a linear,

quadratic, exponential


integers), square root and

absolute value functions.

quadratic function.




context, determines and

interprets key features,

graphs the function, and

solves problems.





quadratic, exponential


integers), square root,

cube root, piece-wise,

step and absolute value

functions.

Rate of Change

Calculates the average

rate of change of a linear,


function (presented

symbolically or as a table)

over a specified interval.


rate of change of a linear,


function (presented


over a specified interval

or estimate the rate of

change from a graph.

Calculates and interprets

the average rate of change

of a linear, exponential,

quadratic, square root,

cube root and piece-wise-

defined function

(presented symbolically

or as a table) over a




quadratic, square root,


defined function



86






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

specified interval, or

estimates the rate of





Compares rate of change

associated with different

intervals.

Solving

Algebraically

Algebraically solves

linear equations, linear

inequalities and

quadratics in one variable

(at complexity appropriate

to the course).



inequalities and



to the course), including

those with variable

coefficients and literal

equations.



inequalities and




those with variable


equations.

Utilizes structure and

rewriting as strategies for

solving.



inequalities and




those with variable


equations, and identifies

and corrects errors in a

given solution.

Utilizes structure and

rewriting as strategies for

solving.

Solving

Graphically

Graphs the solution set of

equations or linear

inequalities.

Uses technology to graph

or make tables to find the


equations, linear

inequalities or system of

linear inequalities.



equations, linear



Writes a system of linear

Graphs and analyzes the

solution set of equations,

linear inequalities or

system of linear

inequalities.

87






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

intersection(s) of two

polynomial functions.




inequalities given a

context.

Uses technology to graph,

make tables or find

successive

approximations to find the





context.


make tables or find

successive




88



Algebra I: Sub-Claim B



Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

Equivalent

Expressions and

Functions

Identifies an equivalent

form of a quadratic

expression or function to

identify its zeros and

symmetry.

Determines an equivalent

form of a quadratic

expression or function to

reveal and explains its

zeros, extreme values and

symmetry.

Determines an equivalent

form of a quadratic or

exponential (with integer

domain) expression or

function to reveal and

explain its properties.

Given a scenario,

determines the most

appropriate form of a

quadratic or exponential

(with integer domain)

function.

Interpreting

Graphs of

Functions

Graphs linear and

quadratic functions,

showing key features.

Graphs linear, quadratic

and cubic (in which linear

and quadratic factors are

available) functions,


Graphs linear, quadratic,

cubic (in which linear and

quadratic factors are

available), square root,


defined functions,


Determines a quadratic,

cubic (in which linear and

quadratic factors are

available), square root,


defined function, given a

graph with key features

identified.

Function

Transformations

Identifies the effects of a

single transformation on

graphs of linear and


limited to f(x)+k and

kf(x).



graphs of linear and


including f(x)+k, kf(x),

f(kx) and f(x+k), and finds

the value of k given a

transformed graph.

Identifies the effects of

multiple transformations

on graphs of linear and

quadratic functions and

finds the value of k given

a transformed graph.

Experiments with cases

using technology.

Given the equation of a

transformed linear or

quadratic function, creates

an appropriate graph.

Experiments with cases

using technology.

Multiple

Representations

of Functions

Writes systems of linear

equations in multi-step

contextual problems.

Given a symbolic

Writes systems of linear



Represents linear and

Writes and analyzes

systems of linear



Writes and analyzes

systems of linear



89






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

representation, real-life

scenario, graph, verbal

description, sequence or

input-and output pairs for

linear and exponential

functions (with domains

in the integers), solves

problems.

Compares the properties

of two linear and/or

quadratic functions


ways.

exponential (with domain

in the integers) functions

symbolically, graphically

and with input-output

pairs to solve



of two linear, exponential


integers) and/or quadratic


different ways.




symbolically, in real-life

scenarios, graphically,

with a verbal description,

as a sequence and with

input-output pairs to solve

mathematical and





integers), quadratic,

square root and/or

absolute value functions

represented in multiple

ways.









mathematical and





integers), quadratic,

square root, absolute

value, cube root, piece-

wise and/or step functions

represented in multiple

ways.

Summarizing

Representing and

Interpreting Data

Given an appropriate

representation of

categorical or quantitative

data, summarizes the data

and characteristics of the

representation(s).

Determines an appropriate

representation of


data, summarizing the

data and characteristics of

the representation(s).


representation of


data, summarizing and

interpreting the data and

characteristics of the

representation(s).


representation of


data, summarizing and

interpreting the data and


representation(s).

90






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

Describes possible

associations and trends in

the data.

Describes and interprets

possible associations and

trends in the data.

91



Algebra I: Sub-Claim C

The student expresses course-level appropriate mathematical reasoning by constructing viable arguments,


Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

Reasoning

Constructs and

communicates an

incomplete response

based on:

the principle that a

graph of an equation

in two variables is the

set of all its solutions

reasoning about linear

and exponential

growth

properties of rational

numbers or irrational

numbers

transformations of

functions

a chain of reasoning to

justify or refute

algebraic, function-

related or linear

equation propositions

or conjectures

a given equation or

system of equations

by :

Constructs and


based on:






and exponential

growth



numbers

transformations of

functions

a chain of reasoning

to justify or refute


related, or linear


or conjectures

a given equation or

system of equations

by:

using a logical



response based on:






and exponential

growth



numbers

transformations of

functions




related, or linear


or conjectures

a given equation or

system of equations

by:

using a logical



response based on:






and exponential

growth



numbers

transformations of

functions




related, or linear


or conjectures

a given equation or

system of equations

by:

using a logical

92






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

using an approach


and/or stated or faulty

assumptions

providing an

incomplete or illogical

progression of steps or

chain of reasoning

making an intrusive

calculation error

using limited grade-

level vocabulary,

symbols and labels

providing a partial

justification of a

conclusion based on

own calculations

approach based on a

conjecture and/or

stated assumptions

providing a logical,

but incomplete,


chain of reasoning

performing minor

calculation errors

using some grade-

level vocabulary,

symbols and labels

providing a partial

justification of a

conclusion based on

own calculations

evaluating the validity

of others’ approaches

and conclusions

approach based on a

conjecture and/or

stated assumptions,

utilizing mathematical

connections (when

appropriate)

providing a logical


chain of reasoning

with appropriate

justification

showing precision of

calculations

using correct grade-

level vocabulary,

symbols and labels

providing a

justification of a

conclusion

evaluating,

interpreting and

critiquing the validity

of others’ responses,

approaches – utilizing

mathematical

connections (when

appropriate) – and

reasoning

approach based on a

conjecture and/or

stated assumptions,


connections (when

appropriate)

providing an efficient

and logical


chain of reasoning

with appropriate

justification


calculation


level vocabulary,

symbols and labels

providing a

justification of a

conclusion

determining whether

an argument or

conclusion is

generalizable.

evaluating,

interpreting and


and efficiency of

93






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

others’ responses,

approaches and

reasoning – utilizing

mathematical

connections (when


providing a counter-

example where

applicable

94



Algebra I: Sub-Claim D







Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command



problems arising in


the workplace by:

using stated

assumptions and

approximations to


situation


quantities


to create models

analyzing

relationships

mathematically to

draw conclusions





reasoning and

Devises and enacts a plan

to apply mathematics in

solving problems arising

in everyday life, society

and the workplace by:

using stated

assumptions and

approximations to


situation

illustrating




to create models

analyzing

relationships

mathematically

between important

quantities to draw

conclusions

interpreting







using stated

assumptions and

making assumptions



situation (includes

micro-models)


between important

quantities

selecting appropriate


analyzing

relationships

mathematically

between important

quantities to draw

conclusions






using stated

assumptions and

making assumptions



situation (includes

micro-models)


between important

quantities



analyzing

relationships

mathematically

between important

quantities to draw

conclusions

95










Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

percentages

applying common

geometric principles

and theorems

using functions to

describe how one


depends on another

using statistics

using estimates of


chain of reasoning


of an unknown

quantity




modifying the model


purpose





reasoning and

percentages

applying geometric

principles and

theorems

writing and using


how one quantity of

interest depends on

another

using statistics

using reasonable

estimates of known


interpreting



situation



improving the model


purpose

writing a complete,

clear and correct


or equation to



reasoning and

percentages

applying geometric

principles and

theorems

writing and using

functions in any form

to describe how one


analyzing and/or


relationships and

goals

interpreting



situation



improving the model


purpose

writing a complete,

clear and correct


or equation to



reasoning and

percentages justifying

and defending models

which lead to a

conclusion

96










Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command


an estimate of an

unknown quantity

depends on another

using statistics

using reasonable

estimates of known



an estimate of an

unknown quantity

applying geometric

principles and

theorems

writing and using


to describe how one


depends on another

using statistics

using reasonable

estimates of known



an estimate of an

unknown quantity

97

Performance Level Descriptors – Geometry


Geometry: Sub-Claim A



Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

Congruence

Transformations

Uses given geometric

theorems and properties

of rigid motions, lines,

angles, triangles and

parallelograms to solve

routine problems and

reason about angle

measurement, triangles,

distance, line properties

and/or congruence.







prove statements about

angle measurement,

triangles, distance, line

properties and/or

congruence.

Determines and uses

appropriate geometric







angle measurement,


properties and/or

congruence.

Determines and uses






non-routine problems and


angle measurement,


properties and/or

congruence.

Similarity

Identifies transformation

relationships in geometric

figures.

Uses transformations to

determine relationships

among geometric figures

and to solve problems.

Uses transformations and

congruence and similarity

criteria for triangles to

prove relationships among

geometric figures and to

solve problems.



criteria for triangles and

to prove relationships

among composite


solve multi-step

problems.

Similarity in

Trigonometry

Uses trigonometric ratios

and the Pythagorean

Theorem to determine the

missing sides and missing

angles of a right triangle.

Uses trigonometric ratios,

the Pythagorean Theorem


between sine and cosine

to solve right triangles in

applied problems.






applied problems.

Uses similarity






applied non-routine

problems.

98



Geometry: Sub-Claim A



Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

transformations with right

triangles to define

trigonometric ratios for

acute angles.

Uses similarity


triangles to define


acute angles.

Modeling and

Applying

Uses provided geometric

relationships in the



and perimeter.

Applies geometric

concepts to describe,

model and solve applied

problems related to the

Pythagorean theorem,

geometric shapes, their

measures and properties.

Uses geometric



problems involving area,

perimeter and ratios of

lengths.

Applies geometric


model and solve applied

problems related to the




Uses geometric





lengths.

Applies geometric

concepts and

trigonometric ratios to

describe, model and solve

applied problems related

to the Pythagorean

theorem, density,



Uses geometric





lengths.

Applies geometric

concepts and

trigonometric ratios to

describe, model and solve

applied problems

(including design

problems) related to the


density, geometric shapes,

their measures and

properties.

99



Geometry: Sub-Claim B



Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

Transformations

Given a figure and a

transformation, draws the

transformed figure.



transformed figure.

Specifies a sequence of

transformations that will

carry a figure onto

another.



transformed figure.

Uses precise geometric

terminology to specify a

sequence of


carry a figure onto itself

or another.


sequence of

transformations, draws

the transformed figure.


terminology to specify

more than one sequence

of transformations that

will carry a figure onto

itself or another.

Geometric

Constructions

Makes basic geometric

constructions: copying a

segment, copying an

angle, bisecting an angle,

bisecting a segment,

including the

perpendicular bisector of

a line segment.

Makes geometric


segment, copying an



including the


a line segment.

Given a line and a point

not on the line, constructs

perpendicular and parallel

lines.

Makes geometric


segment, copying an



including the


a line segment.


not on the line and with a

a variety of tools and

methods, constructs


lines, as well as an

Makes geometric


segment, copying an



including the


a line segment.




lines, an equilateral

triangle, a square and a

regular hexagon inscribed

100






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

equilateral triangle, a

square and a regular

hexagon inscribed in a

circle.

in a circle with a variety

of tools and methods with


methods to prove

geometric theorems.

Applying

Geometric

Properties and

Theorems

Applies provided

properties and theorems

of angles, segments and

arcs in circles to solve

problems.

Applies properties and

theorems of angles,

segments and arcs in

circles to solve problems.

Completes the square to

find the center and radius

of a circle given by an

equation.


theorems of angles,


circles to solve problems

and model relationships.




equation.


theorems of angles,


circles to solve problems,

model relationships and

formulate generalizations.

Complete the square to



equation.

Geometric

Formulas

Using formulas,

determines the volume of

cylinders, pyramids,

cones and spheres.

Identifies the shapes of

two-dimensional cross-

sections of three-


Using formulas,



cones and spheres.

Gives an informal

argument for the formula

for the circumference of a

circle and area of a circle

including dissection

arguments.


solve mathematical and

contextual problems that

involve cylinders,

pyramids, cones and

spheres.

Gives an informal



circle, area of a circle and

volume of a cylinder




involve cylinders,

pyramids, cones and

spheres.

Uses dissection

arguments, Cavalieri’s

principle and informal

limit arguments to support

the formula for the

101






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command



sections of three-



arguments.



sections of three-

dimensional objects and

identifies three-

dimensional objects

generated by rotations of

two-dimensional objects.

circumference of a circle,

area of a circle, volume of

a cylinder, pyramid and

cone.



sections of three-


identifies three-

dimensional objects



102



Geometry: Sub-Claim C



Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

Reasoning Constructs and

communicates:



algebraic and/or

geometric

propositions or

conjectures

geometric reasoning

in a coordinate

setting, OR

a response to a multi-

step problem,

by :

using an approach



assumptions

providing an


o chain of

reasoning, or

o progression of

steps

Constructs and

communicates:



algebraic and/or

geometric, and/or

function or number

system related

propositions or

conjectures

geometric reasoning

in a coordinate

setting, OR

a response to a multi-

step problem,

by:

using a logical

approach based on a

conjecture and/or

stated assumptions


but incomplete,

o chain of

reasoning, or


communicates:



algebraic and/or

geometric, and/or

function or number

system related

propositions or

conjectures

geometric reasoning

in a coordinate

setting, OR

a complete response

to a multi-step

problem,

by:

using a logical

approach based on a

conjecture and/or

stated assumptions,


connections (when

appropriate)


communicates:



algebraic and/or

geometric, and/or

function or number

system related

propositions or

conjectures

geometric reasoning

in a coordinate

setting, OR

a complete response

to a multi-step

problem,

by:

using a logical

approach based on a

conjecture and/or

stated assumptions,


connections (when

appropriate)

103






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

making an intrusive

calculation error


level vocabulary,

symbols and labels

providing a partial

justification of a

conclusion based on

own calculations

o progression of

steps

performing minor

calculation errors

using some grade-

level vocabulary,

symbols and labels

providing a partial

justification of a

conclusion based on

own calculations



and conclusions

providing a logical

o chain of

reasoning, or

o progression of

steps

with appropriate

justification

performing precision

of calculation


level vocabulary,

symbols and labels

providing a

justification of a

conclusion

evaluating,

interpreting, and




mathematical

connections (when


reasoning


and logical

o chain of

reasoning, or

o progression of

steps

with appropriate

justification


of calculation


level vocabulary,

symbols and labels

providing a

justification of a

conclusion

determining whether

an argument or

conclusion is

generalizable

evaluating,

interpreting, and


and efficiency of


approaches and


104






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

mathematical

connections (when



example where

applicable

105



Geometry: Sub-Claim D







Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command



problems arising in


the workplace by:

using stated

assumptions and

approximations to


situation


quantities


to create models

analyzing

relationships

mathematically to

draw conclusions





reasoning and

Devises and enact a plan





using stated

assumptions and

approximations to


situation

illustrating




to create models

analyzing

relationships

mathematically

between important

quantities to draw

conclusions

interpreting







using stated

assumptions and

making assumptions



situation (includes

micro-models)


between important

quantities



analyzing

relationships

mathematically

between important

quantities to draw

conclusions






using stated

assumptions and

making assumptions



situation (includes

micro-models)


between important

quantities



analyzing

relationships

mathematically

between important

quantities to draw

conclusions

106










Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

percentages

applying common


and theorems

using functions to

describe how one


depends on another

using statistics

using estimates of


chain of reasoning


of an unknown

quantity




modifying the model


purpose





reasoning and

percentages

applying geometric

principles and

theorems

writing and using


how one quantity of

interest depends on

another

using statistics

using reasonable

estimates of known


interpreting



situation



improving the model


purpose

writing a complete,

clear and correct


or equation to



reasoning and

percentages

applying geometric

principles and

theorems

writing and using


to describe how one


analyzing and/or


relationships and

goals

interpreting



situation



improving the model


purpose

writing a complete,

clear and correct


or equation to



reasoning and



which lead to a

conclusion

107










Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command


an estimate of an

unknown quantity

depends on another

using statistics

using reasonable

estimates of known



an estimate of an

unknown quantity

applying geometric

principles and

theorems

writing and using


to describe how one


depends on another

using statistics

using reasonable

estimates of known



an estimate of an

unknown quantity

108

Performance Level Descriptors – Algebra II


Algebra II: Sub-Claim A



Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

Equivalent

Expressions

Uses provided

mathematical properties

and structure of

polynomial and

exponential expressions to

create equivalent

expressions.

Uses mathematical

properties and structure of

polynomial, exponential

and rational expressions

to create equivalent

expressions.

Rewrites exponential

expressions to reveal

quantities of interest that

may be useful.

Uses mathematical


polynomial, exponential,

rational and radical

expressions to create

equivalent expressions

that aid in solving

mathematical and

contextual problems with

two steps required.




may be useful.

Uses mathematical


polynomial, exponential,

rational and radical

expressions to create

equivalent expressions

that aid in solving

mathematical and

contextual problems with

three or more steps

required.




may be useful.

Interpreting

Functions

Uses provided


and relationships to reveal

key features of a

polynomial or exponential

function, using them to

sketch the graph.

Interprets key features of

graphs and tables, and

uses mathematical

properties and

relationships to reveal key

features a polynomial,

exponential or rational

function, using them to

sketch the graph.

Uses mathematical

properties and


features a polynomial,

exponential, rational,

trigonometric or

logarithmic function,

using them to sketch the

graph and identify



quantities, and applying

Uses mathematical

properties and


features of a polynomial,

exponential, rational,

trigonometric or

logarithmic function,

using them to sketch the

graph and identify



quantities, and applying

109






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

the remainder theorem

where appropriate.

the remainder theorem

where appropriate.

Identifies how changing

the parameters of the

function impacts key

features of the graph.

Rate of Change


rate of change of a


function (presented




rate of change of a


function (presented


over a specified interval,

or estimates the rate of




of a polynomial,

exponential, logarithmic

or trigonometric function








of a polynomial,


or trigonometric function








intervals.

Modeling

Builds a function that

models mathematical or

contextual situations,

limited to those requiring

arithmetic and geometric

sequences, and uses the

models to solve or

Builds a function that

models mathematical or

contextual situations

including those requiring

trigonometric functions,

sequences and

combinations of these and

Models mathematical and

contextual situations with

functions, including those

requiring trigonometric

functions, sequences and


other functions, and uses

Models mathematical and

contextual situations with

functions, including those

requiring multiple

trigonometric functions,

sequences and


110






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

interpret problems. other functions, and uses

the models to solve or

interpret problems.

the models to solve,

interpret or generalize

about problems.

other functions, and uses

the models to solve,

interpret or generalize

about problems.

Statistics &

Probability

Determines whether a

sample survey,

experiment or

observational study is

most appropriate.

Determines why a sample

survey, experiment or


most appropriate.

Given an inappropriate

choice of a sample


observational study,

identifies and supports the

appropriate choice.


choice of a sample



determines how to change

the scenario to make the

choice appropriate.

111



Algebra II: Sub-Claim B



Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

Interpreting

Functions

Given a function

represented algebraically,

graphically, numerically

or by verbal description,

writes an equivalent

version of the function,

and identify key features.

Given a function




writes multiple equivalent

versions of the function

and identifies key

features.

Given multiple functions

in different forms

(algebraically,


or by verbal description),


versions of the function,

and identifies and

compares key features.


in different forms

(algebraically,





and identifies and


Determines how the

change of a parameter in

each function impacts

their other

representations.

Equivalent

Expressions

Uses commutative and

associative properties to

perform operations with

complex numbers

Uses commutative,

associative and

distributive properties to


complex numbers.

Rewrites simple rational

expressions using

inspection.

Uses commutative,

associative and



complex numbers.


expressions using

inspection or long

division.

Uses commutative,

associative and



complex numbers.


expressions using

inspection or long

division, and determines

how one form is more

useful than the others.

112






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

Function

Transformations



graphs of polynomial,


and trigonometric

functions – limited to

f(x)+k and kf(x) – and

determines if the resulting

function is even or odd.





and trigonometric

functions – including

f(x)+k, kf(x), f(kx), and

f(x+k) – and determines if

the resulting function is

even or odd.



on graphs of polynomial,


and trigonometric

functions, and determines

if the resulting function is

even or odd.

Given a context that infers

particular transformations,

identifies the effects on



and trigonometric



even or odd.

Trigonometry

Given a trigonometric

value for an angle in

degrees, and its quadrant,

utilizes the structure and

relationships of

trigonometry to identify

other trigonometric values

for that angle.



radians, and its quadrant,


relationships of

trigonometry, including

relationships in the unit

circle, to identify other

trigonometric values for

that angle.



radians and its quadrant,


relationships of





that angle, and describes


the radian measure and

the subtended arc in the

circle.





relationships of





that angle, and in

contextual situation,

describes the relationship

between the radian

measure and the

subtended arc in the

circle.

Solving Equations

and Systems

Solves problems

involving linear,

Solves problems

involving linear,

Solves multi-step

contextual word problems

Finds similarities and/or

differences between

113






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command


(with real solutions)

equations and systems of

equations, using inverses

where appropriate.

exponential, quadratic

(with real or complex

solutions) and

trigonometric equations

and systems of equations,

and – where appropriate –

using inverses and

constructing linear and/or

exponential models.

involving linear,



solutions) and




using inverses and

constructing linear and/or

exponential models.

solution approaches of

multiple contextual word

problems involving linear,



solutions) and



using inverses and –

where appropriate – using

inverses and constructing

linear and/or exponential

models.

Data – Univariate

and Bivariate

Uses the mean and

standard deviation of a

data set to fit it to a

normal distribution.

Uses a fitted exponential

function to solve a multi-

step contextual problem.

Uses the mean and




Fits an exponential

function in order to solve

a multi-step contextual

problem.

Uses the mean and




Fits an exponential or a

trigonometric function in

order to solve a multi-step

contextual problem.

Uses the mean and




Fits an exponential or a

trigonometric function in

order to solve a multi-step

contextual problem.

Identifies when these

procedures are not

appropriate.

Inference

Identifies when sample

data can be used to make

Uses sample data to make

inferences about the


inferences and justify

Uses sample data to

critique inferences and

114






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command


corresponding population.


conclusions about the


Decides if a specified

model is consistent with

results from a given data-

generating process.






generating process.

Probability

Recognizes and calculates

conditional probability or

independence in a

contextual problem.

Recognizes, calculates

and uses conditional

probability or

independence in a

contextual problem, using

appropriate set language

and appropriate

representations including

two-way frequency tables.



probability and

independence in a multi-

step contextual problem,

using appropriate set

language and appropriate



Applies the addition rule

of probability.



probability and


step contextual problem,






of probability and

interprets the answers in

terms of the model.

115



Algebra II: Sub-Claim C



Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command


communicates:

a response to a given

equation or system of

equations


justify or refute

algebraic, function or

number system related

propositions or

conjectures

a response based on

data

a response based on

the graph of an

equation in two

variables, the

principle that a graph

is a solution set or the


zeros and factors of

polynomials

a response based on

trigonometric

functions and the unit

circle

a response based on

Constructs and

communicates:



equations


justify or refute



propositions or

conjectures

a response based on

data

a response based on

the graph of an

equation in two

variables, the





polynomials

a response based on

trigonometric


circle

a response based on

Constructs and

communicates:



equations


justify or refute



propositions or

conjectures,

a response based on

data

a response based on

the graph of an

equation in two

variables, the





polynomials

a response based on

trigonometric


circle

a response based on

Constructs and

communicates:



equations


justify or refute



propositions or

conjectures

a response based on

data

a response based on

the graph of an

equation in two

variables, the





polynomials

a response based on

trigonometric


circle

a response based on

116






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

transformations of

functions

OR

a response based on

properties of

exponents

by :

using an approach



assumptions

providing an


o chain of reasoning

or

o progression of

steps

making an intrusive

calculation error


level vocabulary,

symbols and labels

providing a partial

justification of a

conclusion based on

own calculations

transformations of

functions

OR

a response based on

properties of

exponents

by :

using a logical

approach based on a

conjecture and/or

stated assumptions


but incomplete,


or

o progression of

steps

performing minor

calculation errors

using some grade-

level vocabulary,

symbols and labels

providing a partial

justification of a

conclusion based on

own calculations

transformations of

functions

OR

a response based on

properties of

exponents

by :

using a logical

approach based on a

conjecture and/or

stated assumptions,


connections (when

appropriate)

providing a logical


with appropriate

justification

or

o progression of

steps


of calculation


level vocabulary,

symbols and labels

transformations of

functions

OR

a response based on

properties of

exponents

by :

using a logical

approach based on a

conjecture and/or

stated assumptions,


connections (when

appropriate)


and logical


or

o progression of

steps

with appropriate

justification


of calculation


level vocabulary,

117






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command



and conclusions

providing a

justification of a

conclusion

evaluating,

interpreting and




mathematical

connections (when


reasoning

symbols and labels

providing a

justification of a

conclusion

determining whether

an argument or

conclusion is

generalizable

evaluating,

interpreting and


and efficiency of


approaches and


mathematical

connections (when



example where

applicable

118



Algebra II: Sub-Claim D


knowledge and skills articulated in the standards for the current grade/course (or for more complex problems, knowledge

and skills articulated in the standards for previous grades/courses), engaging particularly in the Modeling practice, and

where helpful making sense of problems and persevering to solve them, reasoning abstractly, and quantitatively, using

appropriate tools strategically, looking for the making use of structure and/or looking for and expressing regularity in

repeated reasoning.

Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

Solve Multi-step

Contextual Word

Problems



problems arising in

everyday life, society


using stated

assumptions and

approximations to


situation


given quantities


to create a model

analyzing

relationships

mathematically to

draw conclusions

writing an

expression, equation

or function to




problems arising in



using stated

assumptions and

approximations to


situation

illustrating



by using provided

tools to create an

appropriate, but

inaccurate model

analyzing

relationships

mathematically

between important

given quantities to



problems arising in



using stated

assumptions and

approximations to


situation

mapping



by selecting


create the appropriate

model

analyzing

relationships

mathematically

between important

quantities (either



problems arising in



using stated

assumptions and

approximations to


situation

mapping



by selecting



model

analyzing

relationships

mathematically

between important

quantities (either

119









repeated reasoning.

Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

draw conclusions

interpreting


in a simplified

context


the results make

sense

modifying the model


purpose

writing an


or function to


using geometry to

solve design

problems

given or created) to

draw conclusions

interpreting



situation


the results make

sense

improving the model


purpose

writing a complete,

clear and correct


or function to


using geometry to

solve design

problems


draw conclusions

analyzing and/or


relationships and

goals

justifying and

defending models

which lead to a

conclusion

interpreting



situation


the results make

sense

improving the model


purpose

writing a complete,

clear and correct


120









repeated reasoning.

Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

or function to


using geometry to

solve design

problems

Full Models Devises a plan to apply


problems arising in



using stated

assumptions and

approximations to


situation


given quantities


to create a models

analyzing

relationships



problems arising in



using stated

assumptions and

approximations to


situation

illustrating



by using provided

tools to create an

appropriate but



problems arising in



using stated

assumptions and

approximations to


situation

mapping



by selecting





problems arising in



using stated

assumptions and

approximations to


situation

mapping



by selecting



121









repeated reasoning.

Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

mathematically to

draw conclusions

writing an


or function to


using securely held

content incompletely

reporting a

conclusion, with

some inaccuracy

within the reporting

inaccurate models

analyzing

relationships

mathematically

between important

given quantities to

draw conclusions

interpreting


in a simplified

context


the results make

sense

modifying the model


purpose

writing an


or function to


using geometry to

solve design

models

analyzing

relationships

mathematically

between important

quantities (either


draw conclusions

interpreting



situation


the results make

sense

improving the model


purpose

writing a complete,

clear and correct


or function to


models

analyzing

relationships

mathematically

between important

quantities (either


draw conclusions

interpreting



situation


the results make

sense

improving the model


purpose

writing a complete,

clear and correct


or function to


122









repeated reasoning.

Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

problems

using securely held


reporting a

conclusion

using geometry to

solve design

problems

using securely held

content

briefly reporting the

conclusion

accurately reporting

the conclusion

modifying or

changing the model if

it has not served its

purpose

analyzing and/or


relationships and

goals

justifying and

defending models

which lead to a

conclusion

using geometry to

solve design

problems

using securely held

content

briefly reporting and

justifying the

conclusion

accurately reporting

and justifying the

conclusion

modifying or



purpose

123









repeated reasoning.

Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

Decisions from Data Devises a plan to apply


problems arising in



using stated

assumptions and

approximations to


situation


quantities


to create models

analyzing

relationships

mathematically to

draw conclusions

writing an


or function to




problems arising in



using stated

assumptions and

approximations to


situation

illustrating



by using provided


analyzing

relationships

mathematically

between important

quantities to draw

conclusions

interpreting



problems arising in



using stated

assumptions and

making assumptions

and approximations

to simplify a real-

world situation

mapping



by selecting


create models

analyzing

relationships

mathematically

between important

quantities to draw



problems arising in



using stated

assumptions and

making assumptions

and approximations

to simplify a real-

world situation

mapping



by selecting


create models

analyzing

relationships

mathematically

between important

quantities to draw

124









repeated reasoning.

Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

indiscriminately

using data from a

data source


in a simplified

context


the results make

sense

modifying the model


purpose

writing an


or function to


selecting and using

some relevant data

from a data source

making an evaluation

or recommendation

conclusions

interpreting



situation


the results make

sense

improving the model


purpose

writing a complete,

clear and correct


or function to


identifying and using

relevant data from a

data source

making an

appropriate

evaluation or

recommendation

conclusions

interpreting



situation


the results make

sense

improving the model


purpose

writing a complete,

clear and correct


or function to


analyzing and/or


relationships and

goals

justifying and

defending models

which lead to a

125









repeated reasoning.

Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

conclusion



data source

making an

appropriate

evaluation or

recommendation

126

Performance Level Descriptors – Math I


Math I: Sub-Claim A



Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

Expressions and

Equations

Given a specified quantity

of interest, manipulates

linear formulas and

equations to solve for a

given variable requiring

one-step.

In contextual cases,

identifies components of

exponential expressions

and linear formulas.

Manipulates linear

formulas and equations

for a given variable in

context.

Identifies parts of

contextual exponential

expressions and solves

equations that require

seeing structure.

Manipulates linear

formulas and equations to

highlight a quantity of

interest in context.

Interprets parts of




seeing structure.

Manipulates complicated

linear formulas and

equations to highlight a

quantity of interest in

context.

Interprets parts of




seeing structure.

Rate of Change


rate of change of a linear

and exponential function



specified interval.


rate of change of a linear

and exponential function









square root, cube root and

piece-wise-defined

function (presented








square root, cube root and

piece-wise-defined

function (presented







intervals.

Interpreting

Functions









127



Math I: Sub-Claim A



Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

Uses and evaluate with

function notation.

Writes arithmetic

sequences.

For linear functions that

model relationships

within a context,

determines key features.


linear functions.

Within a context, uses and

evaluates with function

notation.

Writes arithmetic and

geometric sequences.


model relationships

within a context,

determines key features

and graphs the function.




describes for linear and

exponential (limited to

domains in the integers)

functions.


interprets and evaluate


Writes and uses


sequences to model

situations.


model relationships

within a context,

determines and interprets

key features, graphs the

function and solves

problems.






domains in the integers),

square root and absolute

value functions.




Writes and uses


sequences to model

situations.


model relationships

within a context,

determines and interprets

key features, graphs the

function and solves

problems.







square root, cube root,

piece-wise, step and

absolute value functions.

128



Math I: Sub-Claim A



Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

Graphing

Solutions


equations or linear

inequalities.






equations, linear








equations, linear





context.


make tables or find

successive




Graphs and analyzes the

solution set of equations,

linear inequalities or

system of linear

inequalities.



context.


make tables or find

successive




Congruence







reason about angle

measurement, triangles,

distance, line properties

and/or congruence.








angle measurement,


properties and/or

congruence.

Determines and uses








angle measurement,


properties and/or

congruence.

Determines and uses






non-routine problems and


angle measurement,


properties and/or

congruence.

129



Math I: Sub-Claim B



Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

Statistics

Given an appropriate

representation of


data, summarizes the data

and characteristics of the

representation(s).


representation of


data, and summarizes the

data and characteristics of

the representation(s).


representation of


data, and summarizes and

interprets the data and


representation(s).

Describes possible

associations and trends in

the data.


representation of


data, and summarizes and

interprets the data and


representation(s).

Describes and interprets

possible associations and

trends in the data.

Transformations



transformed figure.



transformed figure.

Specifies a sequence of


carry a figure onto

another.



transformed figure.


terminology to specify a

sequence of


carry a figure onto itself

or another.


sequence of

transformations, draws the

transformed figure.


terminology to specify

more than one sequence

of transformations that

will carry a figure onto

itself or another.

Solving Systems

Given a system of linear

equations, solves

contextual problems

exactly and approximately

Given a system of linear

equations, solves

contextual problems

exactly and approximately

Solves multi-step


require writing and

analyzing systems of

Solves multi-step


require writing and

analyzing systems of

130



Math I: Sub-Claim B



Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

with two equations and

two unknowns with

integer coefficients and

solutions.

with two equations and

two unknowns with

rational coefficients and

solutions.

linear equations exactly

and approximately (e.g.,

with graphs), focusing on

pairs of linear equations

with rational coefficients

and solutions.

linear equations exactly

and approximately (e.g.,

with graphs), focusing on

pairs of linear equations

with real coefficients and

solutions.

Solves a given system of

three linear equations and

three unknowns with


Contextual

Problems

Functions

Given a symbolic

representation, real-life

scenario, graph, verbal

description, sequence or

input-and output pairs for

linear and exponential

functions (with domains

in the integers), solves

problems.


of two linear functions


ways.




symbolically, graphically

and with input-output

pairs to solve



of two linear and/or




different ways.









mathematical and





integers), square root

and/or absolute value










mathematical and





integers), square root,

absolute value, cube root,

piece-wise and/or step

131



Math I: Sub-Claim B



Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

multiple ways. functions represented in

multiple ways.

132



Math I: Sub-Claim C



Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command


communicates an

incomplete response

based on:






and exponential

growth



numbers

transformations of

functions


justify or refute


related, or linear


or conjectures

a given equation or

system of equations

by :

Constructs and


based on:






and exponential

growth



numbers

transformations of

functions


justify or refute


related, or linear


or conjectures

a given equation or

system of equations

by:

using a logical



response based on:






and exponential

growth



numbers

transformations of

functions


justify or refute


related, or linear


or conjectures

a given equation or

system of equations

by:

using a logical



response based on:






and exponential

growth



numbers

transformations of

functions


justify or refute


related, or linear


or conjectures

a given equation or

system of equations

by:

using a logical

133



Math I: Sub-Claim C



Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

using an approach



assumptions

providing an



chain of reasoning

making an intrusive

calculation error


level vocabulary,

symbols and labels

providing a partial

justification of a

conclusion based on

own calculations

approach based on a

conjecture and/or

stated assumptions


but incomplete,


chain of reasoning

performing minor

calculation errors

using some grade-

level vocabulary,

symbols and labels

providing a partial

justification of a

conclusion based on

own calculations



and conclusions

approach based on a

conjecture and/or

stated assumptions,


connections (when

appropriate)

providing a logical


chain of reasoning

with appropriate

justification


calculations


level vocabulary,

symbols and labels

providing a

justification of a

conclusion

evaluating,

interpreting and




mathematical

connections (when


reasoning

approach based on a

conjecture and/or

stated assumptions,


connections (when

appropriate)


and logical


chain of reasoning

with appropriate

justification


calculation


level vocabulary,

symbols and labels

providing a

justification of a

conclusion

determining whether

an argument or

conclusion is

generalizable

o evaluating,

interpreting and


and efficiency of

134



Math I: Sub-Claim C



Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command


approaches and


mathematical

connections (when



example where

applicable

135



Math I: Sub-Claim D







Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command



problems arising in


the workplace by:

using stated

assumptions and

approximations to


situation


quantities


to create models

analyzing

relationships

mathematically to

draw conclusions





reasoning and






using stated

assumptions and

approximations to


situation

illustrating




to create models

analyzing

relationships

mathematically

between important

quantities to draw

conclusions

interpreting







using stated

assumptions and

making assumptions



situation (includes

micro-models)


between important

quantities



analyzing

relationships

mathematically

between important

quantities to draw

conclusions






using stated

assumptions and

making assumptions



situation (includes

micro-models)


between important

quantities



analyzing

relationships

mathematically

between important

quantities to draw

conclusions

136



Math I: Sub-Claim D







Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

percentages

applying common


and theorems

using functions to

describe how one


depends on another

using statistics

using estimates of


chain of reasoning


of an unknown

quantity




modifying the model


purpose





reasoning and

percentages

applying geometric

principles and

theorems

writing and using


how one quantity of

interest depends on

another

using statistics

using reasonable

estimates of known

interpreting



situation



improving the model


purpose

writing a complete,

clear and correct


or equation to



reasoning and

percentages

applying geometric

principles and

theorems

writing and using


to describe how one

analyzing and/or


relationships and

goals

interpreting



situation



improving the model


purpose

writing a complete,

clear and correct


or equation to



reasoning and



which lead to a

conclusion

137



Math I: Sub-Claim D







Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command



an estimate of an

unknown quantity


depends on another

using statistics

using reasonable

estimates of known



an estimate of an

unknown quantity

applying geometric

principles and

theorems

writing and using


to describe how one


depends on another

using statistics

using reasonable

estimates of known



an estimate of an

unknown quantity

138

Performance Level Descriptors – Math II


Math II: Sub-Claim A



Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

Quadratics and

Exponential

Expressions



expressions with integer

exponents.

Interprets the structure of


expressions with rational

exponents and to reveal

information by viewing at

least one of their parts as

a single entity.



expressions that contain

real exponents.

In cases where two steps

are required, writes

equivalent expressions to

reveal information by

viewing one or more of

their parts as a single

entity, including factoring

and/or completing the

square for quadratics.




real exponents.

In cases where three or

more steps are required,

writes equivalent


information by viewing

one or more of their parts

as a single entity,

including factoring and/or

completing the square for

quadratics.

Quadratic

Equations

Identifies solutions to

quadratic equations in one

variable with integer or

rational number

coefficients.

Solves quadratic

equations in one variable

with rational number

coefficients, using

methods such as

completing the square,

inspection, taking square

roots, the quadratic

formula or factoring.

Solves quadratic


with real number

coefficients, using

methods appropriate to

the initial form, including




formula and factoring.

Solves quadratic


with real number

coefficients, using

methods appropriate to

the initial form, including




formula and factoring.

Recognizes when the

quadratic formula gives

139






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

complex solutions.

Graphing

Exponential and

Quadratic

Functions

Identifies the key features

of a quadratic or

exponential function.

For quadratic or

exponential functions that

model relationships

within a context,

determines key features

and sketches a graph of

the function.


a quadratic function.

Given a context, writes an

exponential function.

For quadratic or

exponential functions that

model relationships

within a context,

determines key features,

where appropriate, graph

the function and solves

problems.




describes for a quadratic

function.



function, determines key

features, where

appropriate, graph the

function and solves

problems.




describes for a quadratic

function.

Rate of Change


rate of change of a


function (presented




rate of change of a


function (presented







of a quadratic or

exponential function








of a quadratic and







140






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command



intervals.

Polynomial,

Rational and

Radical

Expressions


expressions when adding,

subtracting and

multiplying polynomials,

or expressions that

contain integer exponents.

Adds, subtracts and

multiplies polynomials.


exponents, rewrites


rational exponents.

Adds, subtracts and

multiplies polynomials.


exponents, rewrites


radicals or rational

exponents.

Adds, subtracts and

multiplies polynomials in

multi-step problems.


exponents, rewrites


radicals and rational

exponents.

Similarity

Identifies transformation

relationships in geometric

figures.

Uses transformations to

determine relationships

among geometric figures

and solve problems.



criteria for triangles to

prove relationships among


solve problems.



criteria for triangles and

to prove relationships

among composite


solve multi-step

problems.

Similarity in

Trigonometry

Uses trigonometric ratios

and the Pythagorean

Theorem to determine the

missing sides and missing

angles of a right triangle.






applied problems.






applied problems.

Uses similarity






applied non-routine

problems.

141






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command


triangles to define


acute angles.

Uses similarity


triangles to define


acute angles.

142



Math II: Sub-Claim B



Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

Probability

Recognizes and calculates

conditional probability or

independence in a

contextual problem.



probability or

independence in a

contextual problem using

appropriate set language

and appropriate





probability and


step contextual problem






of probability.



probability and


step contextual problem






of probability and

interprets the answers in

terms of the model.

Statistics

Represents data on a

scatter plot.

Informally, determines

whether a quadratic

model is a good fit.


scatter plot and describes

how the variables are

related.

Informally, determine

whether a quadratic

model is a good fit.

Fits a quadratic function

to data to solve problems

in the context of the data.




related.


to data to solve problems

in the context of the data

and informally assesses

the fit of the function by

plotting and analyzing

residuals.




related.


to data to solve multi-step

problems in the context of

the data and informally

assesses the fit of the

function by plotting and

analyzing residuals.

Geometric

Formulas

Using formulas,


Using formulas,






143






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command


cones and spheres.


cones and spheres.

Gives an informal



circle and area of a circle


arguments.


involve cylinders,

pyramids, cones and

spheres.

Gives an informal



circle, area of a circle and

volume of a cylinder


arguments.


involve cylinders,

pyramids, cones and

spheres.

Uses dissection

arguments, Cavalieri’s

principle and informal

limit arguments to support

the formula for the

circumference of a circle,

area of a circle, volume of

a cylinder, pyramid, and

cone.

Graphs

From a graph, identifies

intercepts, maxima and

minima, end behavior and

zeros – where appropriate.

Identifies the effect on a

linear and quadratic graph

of replacing f(x) by one of

the following f(x)+k,

kf(x), f(kx), and f(x+k)

for specific values of k.

Graphs exponential or


identifying intercepts,

maxima and minima, end

behavior and zeros –

where appropriate.

Identifies and illustrates

the effect on a linear and

quadratic graph of

replacing f(x) by one of

the following: f(x)+k,


for specific values of k;

Graphs and compares

exponential, quadratic,

piece-wise-defined

functions, including step

functions and absolute

value functions,



behavior and zeros –

where appropriate.

Identifies and illustrates

the effect on a linear and

quadratic graph of

Graphs and compares

exponential, quadratic,

square root, cube root,

piece-wise-defined

functions, including step

functions and absolute

value functions,



behavior and zeros, –

where appropriate.

Identifies, illustrates and

interprets the effect on a

144






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

finds the values of k given

the graphs.

replacing f(x) by f(x)+k,


for specific values of k;


the graphs.

linear and quadratic graph

of replacing f(x) by


f(x+k) for specific values

of k given the graphs;


the graphs.

Multiple

Representations

of Functions

Given equivalent

expressions, identifies

features of a quadratic or

exponential function,

including – where

appropriate – zeros,

extreme values and

percent rate of change.


two functions within the

same representation.

Writes a quadratic or


defined by an expression

in different but equivalent

forms to reveal and

explain different

properties of the function,

including – where


extreme values, symmetry

and percent rate of

change.

Within a routine context,

compares properties of


in different ways

(algebraically,


or verbally).





forms to reveal and

explain different


including – where



and percent rate of

change.

Combines standard

functions using an

arithmetic operation.

Within a context,

compares properties of


in different ways





forms to reveal and

explain different


including – where



and percent rate of

change.

Combines standard

functions using multiple

arithmetic operations.

Within a non-routine

context, compares

properties of two


145






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

Given a graph, solve a

system of a linear and

quadratic equation.

(algebraically,


or verbally).

Solves a simple system of

a linear and quadratic

equation algebraically or

graphically.

different ways

(algebraically,


or verbally).

Solves a simple system of

a linear and quadratic

equation algebraically and

graphically.

Number Systems

Classifies rational,

irrational and complex

numbers.

Uses commutative and

associative properties to

perform operation with

complex numbers.



numbers.

Uses commutative,

associative and



complex numbers.



numbers.

Uses commutative,

associative and



complex numbers

Determines if the sum of

two rational and/or

irrational numbers is

rational or irrational.



numbers.

Uses commutative,

associative and



complex numbers

Determines if the sum or

product of two rational

and/or irrational numbers

is rational or irrational.

146



Math II: Sub-Claim C



Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command


communicates an

incomplete response

based on:






and exponential

growth



numbers

transformations of

functions


justify or refute


related, or linear


or conjectures

a given equation or

system of equations

by :

Constructs and


based on:






and exponential

growth



numbers

transformations of

functions


justify or refute


related, or linear


or conjectures

a given equation or

system of equations

by:

using a logical



response based on:






and exponential

growth



numbers

transformations of

functions


justify or refute


related, or linear


or conjectures

a given equation or

system of equations

by:

using a logical



response based on:






and exponential

growth



numbers

transformations of

functions


justify or refute


related, or linear


or conjectures

a given equation or

system of equations

by:

using a logical

147






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

using an approach



assumptions

providing an



chain of reasoning

making an intrusive

calculation error


level vocabulary,

symbols and labels

providing a partial

justification of a

conclusion based on

own calculations

approach based on a

conjecture and/or

stated assumptions


but incomplete,


chain of reasoning

performing minor

calculation errors

using some grade-

level vocabulary,

symbols and labels

providing a partial

justification of a

conclusion based on

own calculations



and conclusions

approach based on a

conjecture and/or

stated assumptions,


connections (when

appropriate)

providing a logical


chain of reasoning

with appropriate

justification


calculations


level vocabulary,

symbols and labels

providing a

justification of a

conclusion

evaluating,

interpreting and


of others’ responses


mathematical

connections (when


reasoning

approach based on a

conjecture and/or

stated assumptions,


connections (when

appropriate)


and logical


chain of reasoning

with appropriate

justification


calculation


level vocabulary,

symbols and labels

providing a

justification of a

conclusion

determining whether

an argument or

conclusion is

generalizable

evaluating,

interpreting, and


and efficiency of

148






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command


approaches and


mathematical

connections (when

appropriate) and


example where

applicable

149



Math II: Sub-Claim D







Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command



problems arising in


the workplace by:

using stated

assumptions and

approximations to


situation


quantities


to create models

analyzing

relationships

mathematically to

draw conclusions





reasoning and






using stated

assumptions and

approximations to


situation

illustrating




to create models

analyzing

relationships

mathematically

between important

quantities to draw

conclusions

interpreting







using stated

assumptions and

making assumptions



situation (includes

micro-models)


between important

quantities



analyzing

relationships

mathematically

between important

quantities to draw

conclusions






using stated

assumptions and

making assumptions



situation (includes

micro-models)


between important

quantities



analyzing

relationships

mathematically

between important

quantities to draw

conclusions

150










Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

percentages

applying common


and theorems

using functions to

describe how one


depends on another

using statistics

using estimates of


chain of reasoning


of an unknown

quantity




modifying the model


purpose





reasoning and

percentages

applying geometric

principles and

theorems

writing and using


how one quantity of

interest depends on

another

using statistics

using reasonable

estimates of known

interpreting



situation



improving the model


purpose

writing a complete,

clear and correct


or equation to



reasoning and

percentages

applying geometric

principles and

theorems

writing and using


to describe how one


analyzing and/or


relationships and

goals

interpreting



situation



improving the model


purpose

writing a complete,

clear and correct


or equation to



reasoning and



which lead to a

conclusion

151










Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command



an estimate of an

unknown quantity

depends on another

using statistics

using reasonable

estimates of known



an estimate of an

unknown quantity

applying geometric

principles and

theorems

writing and using


to describe how one


depends on another

using statistics

using reasonable

estimates of known



an estimate of an

unknown quantity

152

Performance Level Descriptors – Math III


Math III: Sub-Claim A



Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

Equivalent

Expressions

Uses the structure of

polynomial and

exponential expressions to

create equivalent

expressions.





expressions.




to create equivalent that

aid in solving


with two steps required.





expressions in solving


with three or more steps

required.

Interpreting

Functions

Uses provided


and relationships to reveal

key features a

polynomial function to

sketch a graph.

Identifies zeros of easily

factorable quadratics and

cubics.

Interprets key features of

graphs and tables, and

uses mathematical

properties and


features of a polynomial

or rational function to

sketch a graph.

Identifies zeros and

sketches graphs of easily

factorable quadratics and

cubics.

Uses mathematical

properties and



rational, trigonometric or

logarithmic to sketch a

graph and identify



quantities.


sketches graphs of

quadratics and cubics,

applying the remainder

theorem where

appropriate.

Uses mathematical

properties and



rational, trigonometric or

logarithmic function to

sketch a graph and

identify characteristics of


two quantities.

Identifies how changing

the parameters of the

function impacts key

features of the graph.


sketches graphs of

quadratics and cubics,

153






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

applying the remainder

theorem where

appropriate.

Rate of Change


rate of change of a

polynomial function



specified interval.


rate of change of a

polynomial function








of a polynomial,

logarithmic or

trigonometric function








of a polynomial,

logarithmic or

trigonometric function








intervals.

Solving Equations

Solves mathematical

equations directly and

indirectly using structure,

technology, graphs,

formulas, tables of values

and/or successive

approximations.

Solves mathematical



technology, graphs,


and/or successive

approximations, and

identifies extraneous

solutions.

Solves mathematical



technology, graphs,


and/or successive

approximations, and gives

examples of how

extraneous solutions may

arise.

Solves mathematical


strategies based on inference

directly and indirectly, using

structure, technology, graphs,


and/or successive

approximations, and gives

examples of how extraneous

solutions may arise.

154






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

Modeling with

Geometry

Uses provided geometric




and perimeter.

Applies geometric


model and solve

contextual problems

related to geometric

shapes, their measures

and properties.

Uses geometric





lengths.

Applies geometric


model and solve

contextual problems

related to geometric

shapes, their measures

and properties.

Uses geometric





lengths.

Applies geometric


model and solve

contextual problems

related to density,



Uses geometric





lengths.

Applies geometric


model and solve

contextual problems

(including design

problems) related to

density, geometric shapes,

their measures and

properties.

Statistics &

Probability

Determines whether a

sample survey,

experiment or


most appropriate.

Determines why a sample



most appropriate.


choice of a sample



identifies and supports the

appropriate choice.


choice of a sample



determines how to change

the scenario to make the

choice appropriate.

155



Math III: Sub-Claim B



Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

Interpreting

Functions

Given a function




writes an equivalent

version of the function

and identifies key

features.

Given a function






and identifies key

features.


in different forms

(algebraically,





and identifies and

compare key features.


in different forms

(algebraically,





and identifies and


Determines how the

change of a parameter in

each function impacts

their other

representations.

Expressions and

Equations

Solves problems

involving polynomial

equations, using inverses

where appropriate.

Solves problems

involving polynomial and

trigonometric equations,


using inverses and

constructing linear,

quadratic and/or

exponential models


expressions using

inspection.

Solves multi-step

contextual word problems

involving polynomial and



using inverses and


quadratic and/or

exponential models.


expressions using

Finds similarities and/or

differences between

solution approaches of

multiple contextual word

problems involving

polynomial and



using inverses and


quadratic and/or

exponential models.

156






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

inspection or long

division.


expressions using

inspection or long

division, and determines

how one form is more

useful than the others.

Function

Transformations





and trigonometric

functions, limited to

f(x)+k and kf(x), and

determines if the resulting

function is even or odd.





and trigonometric

functions including


f(x+k), and determines if

the resulting function is

even or odd.



on graphs of polynomial,


and trigonometric



even or odd.

Given a context that infers

particular transformations,

identifies the effects on



and trigonometric



even or odd.

Trigonometry



degrees and its quadrant,


relationships of

trigonometry to identify

other trigonometric values

for that angle.





relationships of





that angle in radians.





relationships of





that angle in radians and

describe the relationship





relationships of





that angle in radians and –

in contextual situations –

157






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

between the radian

measure and the


circle.

describe the relationship

between the radian

measure and the


circle.

Data – Univariate

and Bivariate

Uses the mean and




Uses the mean and




Uses a fitted

trigonometric function to

solve a multi-step

contextual problem.

Uses the mean and




Fits a trigonometric



problem.

Uses the mean and




Fits a trigonometric



problem.

Identifies when these

procedures are not

appropriate.

Inference

Identifies when sample

data can be used to make







inferences and justify





results from a given data

generating processes.

Uses sample data to

critique inferences and






generating processes.

158






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

Properties and

Theorems

Applies provided

properties and theorems

of angles, segments and

arcs in circles to solve

problems.



sections of three-



theorems of angles,


circles to solve problems.




equation.



sections of three-



theorems of angles,


circles to solve problems

and model relationships.




equation.



sections of three-


identifies three-

dimensional objects




theorems of angles,


circles to solve problems,

model relationships and

formulate generalizations.




equation.



sections of three-


identifies three-

dimensional objects



Geometric

Constructions

Makes basic geometric


segment, copying an



including the


a line segment.

Makes geometric


segment, copying an



including the


a line segment.

Makes geometric


segment, copying an



including the


a line segment.

Makes geometric


segment, copying an



including the


a line segment.

159






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command




lines.


not on the line and by

using a variety of tools

and methods, constructs


lines, and an equilateral



in a circle.




lines, and an equilateral



in a circle with a variety

of tools and methods with


methods to prove

geometric theorems.

160



Math III: Sub-Claim C



Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command


communicates an

incomplete response

based on:

a given equation or

system of equations


justify or refute

algebraic, function, or


propositions or

conjectures

data

the graph of an

equation in two

variables, the





polynomials

trigonometric


circle

transformations of

functions, OR

properties of

Constructs and


based on:

a given equation or

system of equations


justify or refute



propositions or

conjectures

data

the graph of an

equation in two

variables, the





polynomials

trigonometric


circle

transformations of

functions, OR

properties of

exponents



response based on:

a given equation or

system of equations


justify or refute



propositions or

conjectures

data

the graph of an

equation in two

variables, the





polynomials

trigonometric


circle

transformations of

functions, OR

properties of

exponents



response based on:

a given equation or

system of equations


justify or refute



propositions or

conjectures,

data

the graph of an

equation in two

variables, the





polynomials

trigonometric


circle

transformations of

functions, OR

properties of

exponents

161






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

exponents

by :

using an approach



assumptions

providing an



chain of reasoning

making an intrusive

calculation error


level vocabulary,

symbols and labels

providing a partial

justification of a

conclusion based on

own calculations

by:

using a logical

approach based on a

conjecture and/or

stated assumptions


but incomplete,


chain of reasoning

performing minor

calculation errors

using some grade-

level vocabulary,

symbols and labels

providing a partial

justification of a

conclusion based on

own calculations



and conclusions

by:

using a logical

approach based on a

conjecture and/or

stated assumptions,


connections (when

appropriate)

providing a logical


chain of reasoning

with appropriate

justification


calculations


level vocabulary,

symbols and labels

providing a

justification of a

conclusion

evaluating,

interpreting and




by:

using a logical

approach based on a

conjecture and/or

stated assumptions,


connections (when

appropriate)


and logical


chain of reasoning

with appropriate

justification


calculation


level vocabulary,

symbols and labels

providing a

justification of a

conclusion

determining whether

an argument or

conclusion is

generalizable

162






Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

mathematical

connections (when


reasoning

evaluating,

interpreting and


and efficiency of


approaches and


mathematical

connections (when

appropriate) and


example where

applicable

163



Math III : Sub-Claim D







Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command



problems arising in


the workplace by:

using stated

assumptions and

approximations to


situation


quantities


to create models

analyzing

relationships

mathematically to

draw conclusions









using stated

assumptions and

approximations to


situation

illustrating




to create models

analyzing

relationships

mathematically

between important

quantities to draw

conclusions

interpreting







using stated

assumptions and

approximations to


situation


between important

quantities



analyzing

relationships

mathematically

between important

quantities (either


draw conclusions

interpreting






using stated

assumptions and

approximations to


situation


between important

quantities



analyzing

relationships

mathematically

between important

quantities (either


draw conclusions

analyzing and/or

164










Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command




modifying the model


purpose






situation



improving the model


purpose

writing a complete,

clear and correct


or equation to



relationships and

goals

interpreting



situation



improving the model


purpose

writing a complete,

clear and correct


or equation to


using geometry to

solve design problems

165










Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command

Full Models Devises a plan to apply


problems arising in


the workplace by:

using stated

assumptions and

approximations to


situation


given quantities


to create inaccurate

models

analyzing

relationships

mathematically to

draw conclusions

writing an expression,

equation or function


using securely held



problems arising in


the workplace by:

using stated

assumptions and

approximations to


situation

illustrating




to create appropriate

but inaccurate models

analyzing

relationships

mathematically

between important

given quantities to

draw conclusions

interpreting



problems arising in


the workplace by:

using stated

assumptions and

approximations to


situation


between important

quantities


tools to create the

appropriate models

analyzing

relationships

mathematically

between important

quantities (either


draw conclusions



problems arising in


the workplace by:

using stated

assumptions and

approximations to


situation


between important

quantities


tools to create the

appropriate models

analyzing

relationships

mathematically

between important

quantities (either


draw conclusions

166










Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command


reporting a

conclusion, with some

inaccuracy within the

reporting





modifying the model


purpose




using securely held


reporting a conclusion

interpreting



situation



improving the model


purpose

writing a complete,

clear and correct


or function to describe

a situation

using securely held

content briefly

reporting the

conclusion, accurately

reporting the

conclusion

modifying or



purpose

interpreting



situation



improving the model


purpose

writing a complete,

clear and correct



a situation

analyzing and/or


relationships and

goals

justifying and

defending models

which lead to a

conclusion

using geometry to

167










Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command


using securely held

content briefly

reporting and

justifying the


reporting and

justifying the

conclusion

using securely held

content

modifying or



purpose

Decisions from

Data



problems arising in


the workplace by:

using stated

assumptions and

approximations to



problems arising in


the workplace by:

using stated

assumptions and

approximations to



problems arising in


the workplace by:

using stated

assumptions and

approximations to



problems arising in


the workplace by:

using stated

assumptions and

approximations to

168










Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command


situation


quantities


to create models

analyzing

relationships

mathematically to

draw conclusions




indiscriminately using

data from a data

source


situation

illustrating




to create an

appropriate but

inaccurate models

analyzing

relationships

mathematically

between important

given quantities to

draw conclusions

interpreting





modifying the model


purpose


situation


between important

quantities


tools to create the

appropriate models

analyzing

relationships

mathematically

between important

quantities (either


draw conclusions

interpreting



situation



improving the model


purpose


situation


between important

quantities


tools to create the

appropriate models

analyzing

relationships

mathematically

between important

quantities (either


draw conclusions

interpreting



situation



improving the model


purpose

169










Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command




selecting and using

some relevant data

from a data source

making an evaluation

or recommendation

writing a complete,

clear and correct



a situation



data source

making an appropriate

evaluation or

recommendation

writing a complete,

clear and correct



a situation

analyzing and/or


relationships and

goals

justifying and

defending models

which lead to a

conclusion

using geometry to


using securely held

content briefly

reporting and

justifying the


reporting and

justifying the

conclusion


170










Level 2: Partial

Command

Level 3: Moderate

Command

Level 4: Strong

Command


Command


data source

making an appropriate

evaluation or

recommendation

171

Designing Quality Mathematics Assessments to PARCC · 2014-07-14 · Designing Quality Mathematics...

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