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COMMONWEALTH OF PENNSYLVANIA Mark Schweiker, Governor DEPARTMENT OF EDUCATION Charles B. Zogby, Secretary OFFICE OF ELEMENTARY AND SECONDARY EDUCATION Thomas P. Carey, Deputy Secretary BUREAU OF CURRICULUM AND ACADEMIC SERVICES Michael J. Kozup, Director DIVISION OF CURRICULUM & INSTRUCTION Nancy Neil, Chief DIVISION OF FEDERAL PROGRAMS Jim M. Sheffer, Chief DIVISION OF EVALUATION & REPORTS Lee Plempel, Chief DIVISION OF SCHOOL BASED IMPROVEMENT Marian Sutter, Chief BUREAU OF SPECIAL EDUCATION Frances James-Warkomski, Director John Tommasini, Assistant Director
PENNSYLVANIA DEPARTMENTOF EDUCATION
333 Market StreetHarrisburg, PA 17126-0333
The Pennsylvania Department of Education (PDE) will not discriminate in its educational programs, activities or employment practices, based on race, color, national origin, sex, sexual orientation, disability, age, religion, ancestry, union membership, or any other legally protected category. Announcement of this policy is in accordance with State law including the Pennsylvania Human Relations Act and with Federal law, including Title VI of the Civil Rights Act of 1964, Title IX of the Education Amendments of 1972, Section 504 of the Rehabilitation Act of 1973, the Age Discrimination in Employment Act of 1967, and the Americans with Disabilities Act of 1990. If you have any questions about this publication, or for additional copies, contact: Bureau of Curriculum and Academic Services, Department of Education, 333 Market Street, Harrisburg, PA 17126-0333, Telephone: 717 787-8913. Any complaint of harassment or discrimination pertaining to education should be directed to the Equal Employment Opportunity Manager, Department of Education, 333 Market Street, Harrisburg, PA 17126-0333, Voice Telephone: 717-787-4417, Text Telephone TTY: 717-783-8445, Fax: 717-783-9348. For further information on accommodations for persons with disabilities, contact the ADA Coordinator, Department of Education at the same address, Voice Telephone: 717-783-9791, Fax: 717-772-2317, at the same Text Telephone TTY.
i
Mathematics Instructional Rubrics Mathematics Instructional Rubrics Mathematics Instructional Rubrics Mathematics Instructional Rubrics
This resource was developed through the Pennsylvania Academic Educational Excellence Network (PEEN): A Partnership between the Pennsylvania Department of Education and Pennsylvania Association of Intermediate Units
Primary Development Team
Grace Cisek Berks County Intermediate Unit Janet Lorant Consultant Frank Marburger Pennsylvania Department of Education Sheila Simyak Berks County Intermediate Unit Melody Wilt Berks County Intermediate Unit
Core Team Members
Judy Battista Tulpehocken Area School District Jennifer Cooper South Western School District Lorraine Felker Kutztown Area School District Kris Gushue Wilson School District Becky Kercher Berks County Intermediate Unit Shelley Noel Grove City Area School District Rebecca Shaffer West York Area School District Donna Spatz Antietam School District Deborah White Tulpehocken Area School District Julie Yoder Governor Mifflin School District
ii
Pilot Team Members
Thomas Baraniak Meyersdale Area School District Karen Baum Spring Grove Area School District Maureen Bilik Mount Pleasant Area School District Jacqueline Burton Philadelphia City School District Blair Caboot Abington Heights School District Alice Coleman Scranton City School District Charlene Collins Philadelphia City School District Pat Crist General McLane School District Susan Dougherty Reading School District Diane Eskin Reading School District Susan Estep Central Cambria School District Heather Godine Central York School District Kathy Hohenadel Northeastern York School District Tracey Karlie Meyersdale Area School District Mike Lacey Baldwin-Whitehall School District Noreen Lynott Scranton City School District Barbara MacDonald North Allegheny School District Susan Mace South Western School District Deborah Matthews Philadelphia City School District Virginia Merkel Reading School District Debra Migden Philadelphia City School District Joyce Minnis Conneaut School District Lou Ann Nudi Monaca School District Christina Pagnotto North Allegheny School District Tom Parker General McLane School District Kristy Paterson Northwest Tri-County IU 5 Virginia Perfetto Scranton City School District Vince Pricci Abington Heights School District Debra Printz Reading School District Juliann Ranieri Beaver Area School District Jill Ruch Philadelphia City School District Debra Schwenk Reading School District Ilene Silverman Philadelphia City School District Marva Stacey Altoona Area School District Darlene Stone Rochester Area School District Patti Trotz Central Cambria School District Anna Walker Philadelphia City School District Robert Weber General McLane School District Deb Wirth Dover Area School District
iii
TTTTABLE OF ABLE OF ABLE OF ABLE OF CCCCONTENTSONTENTSONTENTSONTENTS Acknowledgements .............................................................................................. i Table of Contents................................................................................................. iii
MATHEMATICS INSTRUCTIONAL RUBRICS
Introduction and Overview................................................................................... 1 Mathematics Instructional Rubrics for all Organizational Levels ......................................................................................... 10
Analytic……………………………………………………………………………......10 24 36
Holistic…………………………………………………………………………….......Combination………………………………………………………………………......
Organizational Level Rubrics, Prompts, and Student Anchor Papers ............................................................................... 40
Communication Analytic…………………………………………………………..... 41 Reasoning Analytic………………………………………………………………...... 56 Representation Analytic…………………………………………………………...... 71 Communication Holistic……………………………………………………………...84 Reasoning Holistic…………………………………………………………………....95 Representation Holistic……………………………………………………………..108 Combination………………………………………………………………………....123
Resources ......................................................................................................... 153
Websites…………………………………………………………………………..... 153 References………………………………………………………………………….. 155 Glossary…………………………………………………………………..…….…... 157
Mathematics Instructional Rubrics – Section One 1
INTRODUCTIONINTRODUCTIONINTRODUCTIONINTRODUCTION The Mathematics Instructional Rubrics manual contains scoring guides for students to evaluate and improve their mathematical problem solving skills. It also contains many teacher resources to help students learn how to use the rubrics. Pennsylvania educators developed these materials through a project conducted by Partnership for Educational Excellence Network. Every rubric in the manual has two components: the criteria for judging a student performance and descriptions of student performances ranging from a low of a one-point score to a high of a four-point score. Accompanying each rubric are student anchor papers illustrating student work representative of each score point in the rubric, explanations of the score points, teacher suggestions on how to introduce the rubric to a class of students, and a prompt. Purpose The purpose of the Mathematics Instructional Rubrics is to help students increase their proficiency in solving open-ended mathematical problems such as those found on the Pennsylvania System of School Assessment (PSSA). They are designed in conjunction with the Mathematics General Rubric found in the Pennsylvania System of School Assessment Handbook. The rubrics in this document are designed specifically for student use and are written in student appropriate language. The developers of the Mathematics Instructional Rubrics hoped to create tools that would identify and clarify specific performance expectations and provide goals for student achievement. As rubrics define what performances students should do to demonstrate mastery of the mathematics standards, students learn the criteria for achievement levels through their use. Rubrics help learners to look at themselves in positive ways while continuing to take steps toward improving performance. Each student is provided feedback on his or her performance in a developmental continuum. In the continual improvement model, teachers need to supply quality feedback about what students are learning, what they can do to improve, and whether their mastery level is in line with expectations. Rubrics are tools to achieve this end. Teachers can determine mathematical error patterns and student strengths and specify next steps in the continuum of improvement of skills. With practice, students are eventually able to self-assess what is required to move from their present scoring levels to greater proficiency.
“Rubrics are a fantastic way to develop mathematical metacognition.”
A rubric is a scoring guide for evaluation of student work that includes specific performance criteria in a continuum of leveled descriptions from low to high.
Mathematics Instructional Rubrics – Section One 2
One of the most significant changes in a standards-based classroom is the continual effort to revise and improve student work. The goal is mastery of the standards, not coverage of material. Each student requires coaching on his or her progress along the path to mastery. Students and teachers must be able to check the individual’s work against a scoring guide and determine what modifications are required. The process of checking and determining what steps are needed for further improvement is a unique and critical diagnostic interaction between the student and the teacher with the aim of mastery for all. The Process of Development The development of the Mathematics Instructional Rubrics was a collaborative process between teachers, intermediate unit personnel, and members of the Pennsylvania Department of Education who worked with national mathematics consultants to produce the end product. Initially, a core group of mathematics teachers working in primary, intermediate, middle, and high school teams developed a set of draft rubrics. A four-point rubric was selected, rather than the six data points in the state rubric. The rationale for this change came from teachers who reported that the four points represented what worked best in their classrooms. They also thought that the zero point tended to discourage students from working toward improvement. Working through intermediate units, teachers from across the state joined with the core group of teachers and met during the 2000-2001 school year to discuss appropriate mathematical problems and the process of field-testing the rubrics in their classrooms. The student rubrics were then piloted in PA schools in urban, suburban, and rural settings. Participating teachers maintained logs of their experiences and provided the feedback needed to further refine the rubrics. The rubrics were checked for coherence across grade levels and finalized. Field-test teachers also provided their anecdotal logs, general lesson plan outlines, mathematical problems that accompany each rubric, and sample student papers illustrating various score points of each rubric. Examples of these have been included in this document with each rubric. While the Mathematics Instructional Rubrics is a complete set of scoring tools that has been field-tested and revised by teachers across the state, district personnel should not hesitate to further revise the rubrics to meet their district and classroom needs.
“Student responsesimprove dramatically with the use of rubrics.”
“Practice is the key to success with students.”
Mathematics Instructional Rubrics – Section One 3
A Description of the Rubrics The rubrics are written in appropriate language for instruction at four developmental levels: primary, intermediate, middle, and high school. They are not written at specific grade levels so that district educators can adapt and select the rubric that best fits their students. Each developmental level is encased in a separate binder. Within the developmental levels, there are rubrics written for the purpose of increasing students’ problem solving abilities in three skill areas: communication, reasoning, and representation. Communication is the part of a problem solving skill where a student is required to explain “how” they solve a problem using the language of mathematics. Reasoning is the skill area of explaining and defending “why” a student chooses to solve a problem using a selected strategy. Representation is the skill area of correctly “showing one’s work.” This last skill might include creating and using graphs, models, and drawings. To develop these skill areas there are three types of rubrics: analytic, holistic, and combination. Analytic rubrics include discrete criteria and specific traits. There is a score for each trait. Analytic rubrics lend themselves to formative evaluations of student skill development. The holistic rubrics include clusters of descriptors for a concluding or cumulative performance for a summative evaluation. The combination rubrics include all of the criteria from the holistic communication, reasoning, and representation skill areas in one rubric. The following paragraphs further describe the characteristics of each type of rubric. In the analytic rubric there are separate and discrete descriptive criteria for a skill development area, such as communication, across a range of score points and there are specific characteristics to be evaluated in separate columns. The four levels represent a range of performances from strongest to weakest. Separating the descriptors allows the teacher or student to look at the individual characteristics to be evaluated. The responses are viewed in a part-to-whole relationship, hence the term, “analytic.” The teacher and student must select the relevant score point that best matches the characteristic of the response. The benefits of this format are that it permits an evaluation of each characteristic of the response and asks the evaluator to analyze the response for strengths and weaknesses. This type of rubric is best suited to an analysis of the student’s work. As each component is considered separately, the analytic rubric format lends itself to formative evaluations of student work that can be used to guide the student to improve specific aspects of his or her work.
“As a first year teacher I learned some very important lessons while working with the rubrics. I learned that my students have done very little writing in mathematics.”
“It helps to developa chart in your room which explains ‘How to get a 4’ on a rubric.”
Mathematics Instructional Rubrics – Section One 4
The holistic rubric clusters descriptive criteria for one skill development area, such as communication, around a range of score points, four through one, representing a range of performances from strongest to weakest. Clustering the descriptors enables the user to look at a response as a whole, hence the term holistic. The teacher or student using this rubric must select the score point that completely matches the overall response. All criteria of a given score point must be met to earn that score point. This allows the evaluator to look at the total or integrated response. This type of rubric is best suited to offering a single score response to a student’s work. As all criteria are considered as a group, this format lends itself to a summative assessment of student work on a given task. The combined rubric organizes descriptive criteria for all the skill development areas, communication, representation, and reasoning. It does this collectively in one rubric across a range of score points and combines the content included in the three holistic rubrics. By including the descriptors for all three of the skill development areas, this format enables the user to look at a student’s response as it relates to communication, representation, and reasoning. The responses are evaluated for each skill area as a part-to-whole relationship. The teacher and student must select the score point that generally best describes each aspect of the response. The benefit of the combination rubric is that it provides an evaluation of all three-skill development areas in one tool. Recommended Sequence for Introducing Rubrics to Students Figure One is designed to represent the types and difficulty levels of Mathematics Instructional Rubrics that can be used to assist students in responding to open-ended questions. When solving a problem, students’ responses need to include communication of “how” a problem was solved, “why” a specific strategy was selected and representation that supports the conclusion. This manual includes rubrics or tools to assist students in creating responses to an open- ended problem that includes communication, reasoning and representation. Figure One is designed as a flow chart to illustrate the recommended sequence for introducing the rubrics to students. Since analytic rubrics are generally considered to be the easiest of rubrics for students to use, they are shown in the first section of the flow chart. This section is shaded in yellow to remind teachers to introduce the rubrics with caution and to teach students how to use these tools. Notice that the analytic section includes rubrics to assist students in creating responses that communicate, reason, and represent their answers. These tools are usually used to diagnose student errors in solving problems and are formative in design. If students struggle with all aspects of solving open-ended problems then it is recommended that
“Students initially have a high resistance to the requirements of rubrics. However, with consistent use of rubrics they show improvement in their written explanations.”
Mathematics Instructional Rubrics – Section One 5
Figure One
RECOMMENDED SEQUENCE OF RUBRIC INTRODUCTION
Analytic 1. Communication 2. Reasoning
3. Representation
Holistic 1. Communication 2. Reasoning 3. Representation
Combination
Mathematics Instructional Rubrics – Section One 6
they be instructed in the separate components of a response, beginning with the communication rubric. If students are adept at communicating and representing their responses, but struggle with reasoning, then it is appropriate to use the reasoning rubric to diagnose their error patterns and instruct them on the criteria necessary to successfully explain “why” they chose a specific strategy. When students are comfortable with the analytic rubrics or if they have had experience in the use of rubrics it is then appropriate to use the holistic form. These rubrics are represented in Figure One in the blue section, since blue is a comfort color. This is to signify that when students are comfortable with the criteria that are specified in levels three and four of the analytic rubric it is time to assess them holistically on their responses. The holistic section of the flow chart includes rubrics for students to use when working on responses to communicate, reason and represent their answers. These tools are designed as summative tools to assess the total student response. They are generally more difficult to use and hold student responses to a higher level. Students must meet all the descriptive criteria of the response point in order to earn the score at that level. If students score at the lower levels of the rubric (one or two) the teacher may use the analytic form of the rubric to diagnose the error pattern and assist them in refining their skills. The combined rubric organizes the descriptive criteria for all of the skill development areas of communication, reasoning and representation into one rubric across a range of score points. These rubrics are represented on Figure One in the green section since green signifies the ability to move ahead. When students are comfortable with this rubric they are ready to “go” and respond to open ended questions with all of the required components of communication, reasoning and representation. This rubric is an excellent tool when students are comfortable in all of the skill areas for problem solving, however, it would be overwhelming to a student that has not had previous experience with rubrics. While this flow chart is a recommended sequence for the introduction of rubrics to students, this sequence is totally dependent upon students’ experiences with rubrics and their developmental level. Primary and intermediate students may spend much more time using the analytic and holistic rubrics until they are developmentally ready to use the combined rubric or until they have had experience with analytic and holistic. Once they are ready to use the combined rubric, it can always be folded so that only one or more of the sections are being used with the appropriate concepts of solving open-ended problems. Secondary students may use the combined rubric more easily as they have had more experience in solving open-ended problems as part of the curriculum and on the PSSA. If, however, they are experiencing difficulty in one aspect of
“Choose simple problems to begin introducing rubrics.”
Mathematics Instructional Rubrics – Section One 7
problem solving (reasoning or representation) it would be appropriate to use the analytic rubric for diagnostic purposes. The ultimate goal of using the rubrics in this manual is to enhance students’ abilities to create responses to open-ended problems that include communication of “how” a problem was solved; “why” a specific strategy was selected; and representation that supports the conclusion. The Mathematics Instructional Rubrics will assist students and teachers in facilitating that process. The Mathematics Instructional Rubrics Format The Mathematics Instructional Rubrics are arranged by the four organizational levels - primary, intermediate, middle, and high school. Each document is divided into four sections.
• Section One includes the introduction and overview of the project; explanations of the development of the mathematics rubrics; and general guidelines to introduce rubrics to students.
• Section Two provides a set of all the rubrics, primary through high school, a copy of the PSSA Mathematics rubric, and an explanation of the difference between the PSSA six-point rubric and four-point instructional students rubrics.
• Section Three contains student rubrics for each organizational level (primary, intermediate, middle, or high school); teachers’ comments of how to use the rubric in their classroom; samples of student problems; and student anchor papers representing score points of the rubric.
• Section Four contains resources that would be helpful in learning more about the use and development of rubrics with students. These resources include: additional prompts, website addresses that provide more information on rubrics, and a glossary.
In Figure Two an outline is provided to guide the teacher in how to work with students through the initial stages of using rubrics to independent use of rubrics. The authors caution that these are only general suggestions and are not intended to take the place of individual teacher’s lesson designs. Additionally, there are teacher suggestions with each rubric in Section Three of this guide that have been provided by the classroom teachers who field-tested the rubrics in their classrooms.
“As a teacher with 35 years of experience, using these rubrics is most stimulating. It demands time and expertise and is rewarding.”
Mathematics Instructional Rubrics – Section One 8
Figure Two
GENERAL GUIDELINES TO INTRODUCE STUDENTS TO RUBRICS
1. Introduce the concept of rubrics to students emphasizing that rubrics provide a guide of what is expected in a problem solving response. Use an analogy to help explain how rubrics are used such as the straw building activity. (A detailed version of a straw building lesson plan is located in section four, the resource section, of this manual.) Explain the rationale of how the rubrics can be used as a target for continual improvement, and that they can serve as a checklist to proof student work.
2. Using the overhead master of a rubric, present the rubric, reading each item
aloud and underlining key vocabulary words. Discuss vocabulary words in each line making sure students understand their meanings.
3. Explain the differences between each level on the rubric. Emphasize the use
of words such as, “all, most, some, few or none” or “completely, mostly, and partly” and their relationship to a 4, 3, 2, 1 score.
4. Present a practice problem and brainstorm together how to solve it. Once
students have chosen a solution, work through it with them emphasizing the use of the scoring guide to help write the solution. Check the student work and guide the responses. Place key mathematical terms on the board and tell students to use them in their written explanations.
5. Present another practice problem and have students read their responses
listing all the math terms they use. Create a list of reminders of what to do based on the discussions of the student responses. Create a list of problem solving procedures from this discussion. An example follows: • Read the problem and decide how to solve it. • Read the problem a second time rethinking the solution and what will be
needed to solve it. • Organize what needs to be done using the rubric as a checklist. • Solve the problem. • Use the rubric to check the response.
Mathematics Instructional Rubrics – Section One 9
6. Have the students score their own responses using the rubric. Have students discuss how they could improve their response or score.
7. Present several more practice problems having students work in cooperative
groups using the rubric to guide their answers. Using the rubric have the groups exchange papers to score another group’s work. Discuss the rationale for their scores.
8. Utilize student papers from the anchor sets to illustrate the differences
between a 4, 3, 2, or 1 score.* 9. Have students individually solve a problem using the rubric. Give each
student feedback on their answer using the rubric. Guide students in what their next steps should be to improve their scores.
10. Continue to practice solving open-ended problems using rubrics. Encourage
students to improve; they will increase their skills with practice. Suggested Accommodations for Special Needs Students Allow extra time to complete the actual problem. Read the rubric and the problem or prompt orally to the student. Use the help of an aide to keep the student focused on the task. Provide a calculator or manipulative. Read the problem several times to the student emphasizing key words. Utilize dark-lined composition or graph paper for visually impaired students. Highlight important information and key words on the rubric. Add lines to the response pages. Use “post-it” notes to show students where to write parts of their work.
*Note: Teachers using student papers as examples should adhere to the Federal Educational Rights and Privacy Act (FERPA) guidelines.
Mathematics Instructional Rubrics – Section Two 10
HOW DO MATHEMATICS IHOW DO MATHEMATICS IHOW DO MATHEMATICS IHOW DO MATHEMATICS INSTRUCTIONAL RUBRICSNSTRUCTIONAL RUBRICSNSTRUCTIONAL RUBRICSNSTRUCTIONAL RUBRICS COMPARE TO THE PSSA COMPARE TO THE PSSA COMPARE TO THE PSSA COMPARE TO THE PSSA RUBRIC?RUBRIC?RUBRIC?RUBRIC?
The goal of the Mathematics Instructional Rubrics project was to develop a set of assessment tools students could use to improve their responses to open-ended problems both in the classroom and on the state assessment (the PSSA). While creating the rubrics, the core development team consistently referenced the expectations for student responses to the expectations for a level five on the PSSA; however, they chose to create four-point rubrics for two reasons. First, the team unanimously agreed that they did not want a zero on the instructional rubric because not completing a problem or being off task is unacceptable in daily classroom instruction. Second, as it is important for students to learn to evaluate their own work, there was some concern that an odd number on the rubrics (five rather than the state’s current six) might cause some students to lean toward the central tendency of a “3”. Given both of these considerations and teacher experience with four-point rubrics, the team elected to design a four-point rubric. According to the PA Mathematics Assessment Handbook, (p. 8, 2000) in order for a student to score at the level five on the PSSA rubric a student response must have the following characteristics:
“Correct answer with correct procedures/correct calculations shown or described and a written explanation that supports the work shown. The explanation tells what was done in the solution process and explains why the steps were done (or the reason(s) for the steps to be taken). No blemishes, that is, everything is correct. May have a minor omission in calculation or explanation where the omitted step or explanation may be of the level of 2 + 2 = 4 (something that is usually done mentally and considered trivial and understood).”
To achieve a five on the state rubric, the student response must have thoroughly addressed an explanation of their answer, a representation of the answer, and the reasoning for solving the problem. These three characteristics of the highest score on the state’s rubric were the impetus for the development of rubrics based on these three separate characteristics of quality mathematic performance (communication, representation, and reasoning). The Mathematics Instructional Rubrics were developed to be classroom tools and to be utilized by students so they were developed using “student appropriate language” to explain “how” and “why” a student solved a problem and chose to represent the answer. A comparison between the state’s scoring guide and the instructional rubrics is as follows: • A level four on the Mathematics Instructional Rubrics (MIR) is consistent with the PSSA
level five. • A level three on the MIR is consistent to a combination of the three and four on the PSSA
Rubric. • A level two on the MIR is comparable to a level two on the PSSA Rubric. The MIR level two
varies slightly from the PSSA level two in that the state rubric does allow for a correct answer at level two.
• A level one on the MIR is consistent with the criteria established on the PSSA rubric when levels one and zero are combined.
Mathematics Instructional Rubrics – Section Two 11
PSSA
Mathematics Instructional Rubrics – Section Two 12
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Mathematics Instructional Rubrics – Section Two 13
Mat
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de a
n e
xpla
nat
ion
of
som
e of
th
e st
eps
nee
ded
to s
olve
th
e pr
oble
m.
I u
se s
ome
of t
he
prop
er
labe
ls, d
etai
ls, s
ymbo
ls, a
nd
corr
ect
mat
h t
erm
s in
my
expl
anat
ion
.
1 M
y w
ork
is n
ot o
rgan
ized
, doe
s n
ot
mak
e se
nse
, an
d is
ver
y co
nfu
sin
g.
I in
clu
de a
n e
xpla
nat
ion
of
few
or
non
e of
th
e st
eps
nee
ded
to s
olve
th
e pr
oble
m.
I d
o n
ot u
se p
rope
r la
bels
, de
tail
s, s
ymbo
ls, a
nd
corr
ect
mat
h t
erm
s in
my
expl
anat
ion
.
Mathematics Instructional Rubrics – Section Two 14
Mat
hem
atic
al C
omm
un
icat
ion
- A
nal
ytic
Ru
bri
c -
Mid
dle
Sch
ool
Lev
el
Th
is is
how
I s
olve
th
e pr
oble
m.
O
rgan
izat
ion
Is
my
resp
onse
org
aniz
ed t
o sh
ow
how
I s
olve
th
e pr
oble
m?
E
xpla
nat
ion
D
oes
my
expl
anat
ion
incl
ude
how
I
solv
e th
e pr
oble
m?
U
se o
f M
ath
emat
ical
Ter
ms
Doe
s m
y re
spon
se in
clu
de p
rope
r co
nce
pts,
labe
ls, d
etai
ls, s
ymbo
ls,
and
corr
ect
mat
h t
erm
s?
4
My
resp
onse
is w
ell
orga
niz
ed
and
is lo
gica
l.
I in
clu
de a
n a
ccu
rate
, com
plet
e,
and
thor
ough
exp
lan
atio
n o
f al
l of
th
e st
eps
nee
ded
to c
orre
ctly
sol
ve
the
prob
lem
.
I u
se a
ll o
f th
e pr
oper
con
cept
s,
labe
ls, d
etai
ls, s
ymbo
ls, a
nd
corr
ect
mat
h t
erm
s in
my
expl
anat
ion
.
3 M
y re
spon
se is
mos
tly
orga
niz
ed
and
is lo
gica
l.
I in
clu
de a
n a
ccu
rate
exp
lan
atio
n o
f m
ost
of t
he
step
s n
eede
d to
co
rrec
tly
solv
e th
e pr
oble
m.
I u
se m
ost
of t
he
prop
er c
once
pts,
la
bels
, det
ails
, sym
bols
, an
d co
rrec
t m
ath
ter
ms
in m
y ex
plan
atio
n.
2 M
y re
spon
se is
poo
rly
orga
niz
ed
or is
illo
gica
l.
I in
clu
de a
n e
xpla
nat
ion
of
som
e of
th
e st
eps
nee
ded
to s
olve
th
e pr
oble
m.
I u
se s
ome
of t
he
prop
er c
once
pts,
la
bels
, det
ails
, sym
bols
, an
d co
rrec
t m
ath
ter
ms
in m
y ex
plan
atio
n.
1 M
y re
spon
se is
not
org
aniz
ed
and
is il
logi
cal.
I in
clu
de a
n e
xpla
nat
ion
of
few
or
non
e of
th
e st
eps
nee
ded
to s
olve
th
e pr
oble
m.
I u
se f
ew o
r n
one
of t
he
prop
er
con
cept
s, la
bels
, det
ails
, sym
bols
, an
d co
rrec
t m
ath
ter
ms
in m
y ex
plan
atio
n.
Mathematics Instructional Rubrics – Section Two 15
Mat
hem
atic
al C
omm
un
icat
ion
- A
nal
ytic
Ru
bri
c -
Hig
h S
choo
l L
evel
T
his
is h
ow I
sol
ve t
he
prob
lem
.
O
rgan
izat
ion
Is
my
resp
onse
org
aniz
ed t
o sh
ow h
ow I
so
lve
the
prob
lem
?
E
xpla
nat
ion
D
oes
my
expl
anat
ion
incl
ude
how
I s
olve
th
e pr
oble
m?
U
se o
f M
ath
emat
ical
Ter
ms
Doe
s m
y re
spon
se in
clu
de a
ccu
rate
labe
ls,
sym
bols
, con
cept
s, t
erm
inol
ogy,
an
d re
pres
enta
tion
s?
4
My
resp
onse
is t
hor
ough
, wel
l or
gan
ized
an
d lo
gica
l.
I in
clu
de a
com
plet
e an
d co
rrec
t ex
plan
atio
n u
sin
g al
l of
th
e re
leva
nt
and
spec
ific
det
ails
fro
m t
he
prob
lem
wh
en
desc
ribi
ng
the
proc
edu
res
use
d to
arr
ive
at t
he
corr
ect
solu
tion
.
I u
se a
ll o
f th
e la
bels
, sym
bols
, con
cept
s,
term
inol
ogy,
an
d re
pres
enta
tion
s ac
cura
tely
, cle
arly
, an
d su
ccin
ctly
in m
y re
spon
se.
3 M
y re
spon
se is
mos
tly
orga
niz
ed a
nd
logi
cal.
I in
clu
de a
cor
rect
exp
lan
atio
n u
sin
g m
ost
of t
he
rele
van
t an
d sp
ecif
ic d
etai
ls
from
th
e pr
oble
m w
hen
des
crib
ing
the
proc
edu
res
use
d to
arr
ive
at t
he
corr
ect
solu
tion
.
I u
se m
ost
of t
he
labe
ls, s
ymbo
ls,
con
cept
s, t
erm
inol
ogy,
an
d re
pres
enta
tion
s ac
cura
tely
, cle
arly
, an
d su
ccin
ctly
in m
y re
spon
se.
2 M
y re
spon
se is
poo
rly
orga
niz
ed o
r is
il
logi
cal.
I in
clu
de a
n e
xpla
nat
ion
usi
ng
som
e of
th
e re
leva
nt
and
spec
ific
det
ails
fro
m t
he
prob
lem
wh
en d
escr
ibin
g th
e pr
oced
ure
s u
sed
to a
rriv
e at
a s
olu
tion
.
I u
se s
ome
of t
he
labe
ls, s
ymbo
ls,
con
cept
s, t
erm
inol
ogy,
an
d re
pres
enta
tion
s ac
cura
tely
in m
y re
spon
se.
1 M
y re
spon
se is
not
org
aniz
ed a
nd
is
illo
gica
l.
I in
clu
de a
n e
xpla
nat
ion
usi
ng
few
or
non
e of
th
e re
leva
nt
or s
peci
fic
deta
ils
from
th
e pr
oble
m w
hen
des
crib
ing
the
proc
edu
res.
I u
se f
ew o
r n
one
of t
he
labe
ls, s
ymbo
ls,
con
cept
s, t
erm
inol
ogy,
or
repr
esen
tati
ons
in m
y re
spon
se.
Mathematics Instructional Rubrics – Section Two 16
Mat
hem
atic
al R
easo
nin
g -
An
alyt
ic R
ub
ric
– P
rim
ary
Lev
el
Th
is is
wh
y I
solv
e th
e pr
oble
m t
his
way
.
E
xpla
nat
ion
of
the
Ste
ps
Doe
s m
y ex
plan
atio
n in
clu
de
com
plet
e an
d or
gan
ized
in
form
atio
n?
R
easo
n f
or t
he
Ch
oice
of
Str
ateg
y D
oes
my
expl
anat
ion
tel
l wh
y I
chos
e th
is s
trat
egy?
U
se o
f M
ath
Ter
ms
Do
I u
se c
orre
ct la
bels
, det
ails
, sy
mbo
ls, a
nd
mat
h t
erm
s to
ex
plai
n m
y w
ork?
4 M
y ex
plan
atio
n in
clu
des
all
of
the
step
s, is
wel
l org
aniz
ed, a
nd
mak
es s
ense
.
I ex
plai
n w
hy
I ch
ose
this
str
ateg
y to
cor
rect
ly s
olve
th
e pr
oble
m
incl
udi
ng
all
the
info
rmat
ion
n
eede
d.
I u
se a
ll o
f th
e co
rrec
t la
bels
, de
tail
s, s
ymbo
ls, a
nd
mat
h t
erm
s to
exp
lain
my
wor
k.
3 M
y ex
plan
atio
n in
clu
des
mos
t of
th
e st
eps,
is o
rgan
ized
, an
d m
akes
sen
se.
I ex
plai
n w
hy
I ch
ose
this
str
ateg
y to
cor
rect
ly s
olve
th
e pr
oble
m a
nd
incl
ude
mos
t of
th
e in
form
atio
n
nee
ded.
I u
se m
ost
of t
he
corr
ect
labe
ls,
deta
ils,
sym
bols
, an
d m
ath
ter
ms
to e
xpla
in m
y w
ork.
2 M
y ex
plan
atio
n in
clu
des
som
e of
th
e st
eps,
is n
ot o
rgan
ized
, bu
t m
akes
som
e se
nse
.
I ex
plai
n w
hy
I ch
ose
this
str
ateg
y to
sol
ve t
he
prob
lem
an
d in
clu
de
som
e of
th
e in
form
atio
n n
eede
d.
I u
se s
ome
labe
ls, d
etai
ls,
sym
bols
, an
d m
ath
ter
ms
to
expl
ain
my
wor
k.
1 M
y ex
plan
atio
n in
clu
des
few
or
non
e of
th
e st
eps,
is n
ot
orga
niz
ed, a
nd
does
not
mak
e se
nse
.
I do
not
exp
lain
wh
y I
chos
e th
is
stra
tegy
to
solv
e th
e pr
oble
m a
nd
I in
clu
de li
ttle
or
no
info
rmat
ion
n
eede
d to
su
ppor
t m
y st
rate
gy.
I d
o n
ot u
se la
bels
, det
ails
, sy
mbo
ls, o
r m
ath
ter
ms
to e
xpla
in
my
wor
k.
Mathematics Instructional Rubrics – Section Two 17
M
ath
emat
ical
Rea
son
ing
- A
nal
ytic
Ru
bri
c –
Inte
rmed
iate
Lev
el
Th
is is
wh
y I
solv
e th
e pr
oble
m t
his
way
.
Exp
lan
atio
n o
f th
e S
trat
egy
Doe
s m
y ex
plan
atio
n in
clu
de
orga
niz
ed in
form
atio
n?
R
easo
n f
or t
he
Ch
oice
of
Str
ateg
y D
oes
my
expl
anat
ion
rev
eal w
hy
I ch
ose
this
str
ateg
y?
U
se o
f M
ath
Ter
ms
Doe
s m
y re
spon
se in
clu
de p
rope
r la
bels
, det
ails
, sym
bols
, an
d th
e u
se o
f th
e co
rrec
t m
ath
ter
ms?
4 M
y ex
plan
atio
n d
escr
ibes
my
stra
tegy
in a
hig
hly
sk
illf
ul
way
th
at is
cle
ar, o
rgan
ized
, an
d m
akes
se
nse
.
I co
mp
lete
ly e
xpla
in w
hy
I ch
ose
this
str
ateg
y to
cor
rect
ly s
olve
th
e pr
oble
m.
I u
se a
ll o
f th
e pr
oper
labe
ls,
deta
ils,
sym
bols
, an
d co
rrec
t m
ath
te
rms
in m
y ex
plan
atio
n.
3 M
y ex
plan
atio
n d
escr
ibes
my
stra
tegy
in a
mos
tly
skil
lfu
l w
ay
that
is o
rgan
ized
an
d m
akes
sen
se.
I m
ostl
y ex
plai
n w
hy
I ch
ose
this
st
rate
gy t
o co
rrec
tly
solv
e th
e pr
oble
m.
I u
se m
ost
of t
he
prop
er la
bels
, de
tail
s, s
ymbo
ls, a
nd
corr
ect
mat
h
term
s in
my
expl
anat
ion
.
2 M
y ex
plan
atio
n d
escr
ibes
my
stra
tegy
in a
som
ewh
at s
kil
lfu
l w
ay t
hat
is n
ot o
rgan
ized
bu
t m
akes
som
e se
nse
.
I p
arti
ally
exp
lain
wh
y I
chos
e th
is
stra
tegy
to
solv
e th
e pr
oble
m.
I u
se s
ome
of t
he
prop
er la
bels
, de
tail
s, s
ymbo
ls, a
nd
corr
ect
mat
h
term
s in
my
expl
anat
ion
.
1 M
y ex
plan
atio
n d
oes
not
des
crib
e m
y st
rate
gy in
a s
kill
ful w
ay, i
s n
ot o
rgan
ized
, an
d do
es n
ot m
ake
sen
se.
I d
o n
ot e
xpla
in w
hy
I ch
ose
this
st
rate
gy t
o so
lve
the
prob
lem
.
I d
o n
ot u
se p
rope
r la
bels
, det
ails
, sy
mbo
ls, a
nd
corr
ect
mat
h t
erm
s in
my
expl
anat
ion
.
Mathematics Instructional Rubrics – Section Two 18
M
ath
emat
ical
Rea
son
ing
- A
nal
ytic
Ru
bri
c –
Mid
dle
Sch
ool
Lev
el
Th
is is
wh
y I
solv
e th
e pr
oble
m u
sin
g a
sele
ct s
trat
egy.
Exp
lan
atio
n o
f th
e S
trat
egy
Doe
s m
y ex
plan
atio
n in
clu
de
orga
niz
ed in
form
atio
n?
R
atio
nal
e fo
r th
e S
trat
egy
Ch
oice
Doe
s m
y ex
plan
atio
n r
evea
l wh
y I
chos
e th
is s
trat
egy?
U
se o
f M
ath
emat
ical
Ter
ms
Doe
s m
y ex
plan
atio
n in
clu
de
prop
er c
once
pts,
labe
ls, d
etai
ls,
sym
bols
, an
d co
rrec
t m
ath
ter
ms?
4 M
y ex
plan
atio
n o
f th
e st
rate
gy
incl
ude
s al
l de
tail
s an
d is
wel
l or
gan
ized
.
I fu
lly
expl
ain
wh
y I
chos
e th
is
effe
ctiv
e an
d ef
fici
ent
stra
tegy
th
at
lead
s to
a c
omp
lete
ly c
orre
ct
con
clu
sion
or
solu
tion
.
I u
se a
ll o
f th
e pr
oper
con
cept
s,
labe
ls, d
etai
ls, s
ymbo
ls, a
nd
corr
ect
mat
h t
erm
s in
my
expl
anat
ion
.
3 M
y ex
plan
atio
n o
f th
e st
rate
gy
incl
ude
s m
ost
deta
ils
and
is
orga
niz
ed.
I m
ostl
y ex
plai
n w
hy
I ch
ose
this
ef
fect
ive
stra
tegy
th
at le
ads
to a
co
mp
lete
ly c
orre
ct c
oncl
usi
on o
r so
luti
on.
I u
se m
ost
of t
he
prop
er c
once
pts,
la
bels
, det
ails
, sym
bols
, an
d co
rrec
t m
ath
ter
ms
in m
y ex
plan
atio
n.
2 M
y ex
plan
atio
n o
f th
e st
rate
gy
incl
ude
s so
me
deta
ils
and
is
diso
rgan
ized
.
I pa
rtia
lly
expl
ain
wh
y I
chos
e th
is
stra
tegy
th
at le
ads
to a
par
tial
ly
corr
ect
con
clu
sion
or
solu
tion
.
I u
se s
ome
of t
he
prop
er c
once
pts,
la
bels
, det
ails
, sym
bols
, an
d co
rrec
t m
ath
ter
ms
in m
y ex
plan
atio
n.
1 M
y ex
plan
atio
n o
f th
e st
rate
gy
incl
ude
s fe
w o
r n
o de
tail
s an
d is
n
ot o
rgan
ized
.
I do
not
exp
lain
wh
y I
chos
e th
is
stra
tegy
th
at le
ads
to a
n i
nco
rrec
t co
ncl
usi
on o
r so
luti
on.
I u
se f
ew o
r n
one
of t
he
prop
er
con
cept
s, la
bels
, det
ails
, sym
bols
, an
d co
rrec
t m
ath
ter
ms
in m
y ex
plan
atio
n.
Mathematics Instructional Rubrics – Section Two 19
Mat
hem
atic
al R
easo
nin
g -
An
alyt
ic R
ub
ric
– H
igh
Sch
ool
My
impl
emen
tati
on in
clu
des
orga
niz
ed m
ath
emat
ical
ste
ps/p
roce
dure
s an
d re
leva
nt
deta
ils.
Im
ple
men
tati
on o
f th
e S
trat
egy
Doe
s my
impl
emen
tatio
n in
clud
e or
gani
zed
step
s, pr
oced
ures
, and
re
leva
nt d
etai
ls?
R
atio
nal
e fo
r th
e S
trat
egy
Ch
oice
D
oes m
y ex
plan
atio
n re
veal
why
I se
lect
th
e co
ncep
ts p
rese
nted
in m
y so
lutio
n?
U
se o
f M
ath
emat
ical
Ter
ms
Doe
s my
resp
onse
incl
ude
accu
rate
labe
ls,
sym
bols
, con
cept
s, te
rmin
olog
y, a
nd
repr
esen
tatio
ns?
4 M
y im
plem
enta
tion
is fu
lly
deve
lope
d an
d is
com
plet
ely
supp
orte
d by
rele
vant
and
spec
ific
deta
ils fr
om th
e pr
oble
m.
I inc
lude
a c
orre
ct, t
horo
ugh,
and
wel
l or
gani
zed
expl
anat
ion
com
plet
ely
base
d up
on m
athe
mat
ical
trut
hs o
f why
I se
lect
th
e co
ncep
ts a
nd re
pres
enta
tions
pre
sent
ed
in m
y co
rrec
t sol
utio
n.
I use
all
of th
e la
bels
, sym
bols
, con
cept
s, te
rmin
olog
y, a
nd re
pres
enta
tions
ac
cura
tely
, cle
arly
, and
succ
inct
ly in
my
resp
onse
.
3 M
y im
plem
enta
tion
is m
ostly
de
velo
ped
and
supp
orte
d by
rele
vant
an
d sp
ecifi
c de
tails
from
the
prob
lem
.
I inc
lude
a c
orre
ct e
xpla
natio
n ba
sed
on
mat
hem
atic
al tr
uths
of w
hy I
sele
ct th
e co
ncep
ts a
nd re
pres
enta
tions
pre
sent
ed in
m
y co
rrec
t sol
utio
n.
I use
mos
t of t
he la
bels
, sym
bols
, co
ncep
ts, t
erm
inol
ogy,
and
re
pres
enta
tions
acc
urat
ely,
cle
arly
, and
su
ccin
ctly
in m
y re
spon
se.
2 M
y im
plem
enta
tion
is so
mew
hat
deve
lope
d an
d su
ppor
ted
by re
leva
nt
and
spec
ific
deta
ils fr
om th
e pr
oble
m.
I inc
lude
a p
artia
lly c
orre
ct a
nd so
mew
hat
orga
nize
d ex
plan
atio
n of
why
I se
lect
the
conc
epts
and
repr
esen
tatio
ns p
rese
nted
in
my
solu
tion.
I use
som
e of
the
labe
ls, s
ymbo
ls,
conc
epts
, ter
min
olog
y, a
nd
repr
esen
tatio
ns a
ccur
atel
y in
my
resp
onse
.
1 M
y im
plem
enta
tion
is r
arel
y su
ppor
ted
by re
leva
nt a
nd sp
ecifi
c de
tails
from
the
prob
lem
.
I inc
lude
an
inco
rrec
t, di
sorg
aniz
ed, o
r m
inim
al e
xpla
natio
n of
why
I se
lect
the
conc
epts
and
repr
esen
tatio
ns p
rese
nted
in
my
solu
tion.
I use
few
or
none
of t
he la
bels
, sym
bols
, co
ncep
ts, t
erm
inol
ogy,
or r
epre
sent
atio
ns
in m
y re
spon
se.
Mathematics Instructional Rubrics – Section Two 20
Mat
hem
atic
al R
epre
sen
tati
on -
An
alyt
ic R
ub
ric
– P
rim
ary
Lev
el
Th
is is
how
I s
how
my
info
rmat
ion
in a
gra
ph o
r pi
ctu
re.
A
ccu
racy
D
oes
my
grap
h o
r pi
ctu
re
corr
ectl
y sh
ow t
he
info
rmat
ion
?
F
orm
at
Do
I co
rrec
tly
show
all
th
e pa
rts
I n
eed
in t
he
grap
h o
r pi
ctu
re?
C
oncl
usi
on
Do
I su
ppor
t m
y an
swer
wit
h t
he
grap
h o
r pi
ctu
re?
4
M
y gr
aph
or
pict
ure
an
d in
form
atio
n a
re
com
ple
tely
cor
rect
.
I
put
all
of t
he
part
s I
nee
d in
my
grap
h o
r pi
ctu
re.
I
com
ple
tely
su
ppor
t m
y co
rrec
t an
swer
wit
h t
he
grap
h
or p
ictu
re.
3
M
y gr
aph
or
pict
ure
an
d in
form
atio
n a
re m
ostl
y co
rrec
t.
I pu
t m
ost
of t
he
part
s I
nee
d in
my
grap
h o
r pi
ctu
re.
I
mos
tly
supp
ort
my
corr
ect
answ
er w
ith
th
e gr
aph
or
pict
ure
.
2
My
grap
h o
r pi
ctu
re o
r in
form
atio
n is
par
tly
corr
ect.
I pu
t so
me
of t
he
part
s I
nee
d in
my
grap
h o
r pi
ctu
re.
I
par
tly
supp
ort
my
answ
er
wit
h t
he
grap
h o
r pi
ctu
re.
1
My
grap
h o
r pi
ctu
re o
r in
form
atio
n is
not
cor
rect
.
I pu
t fe
w o
r n
one
of t
he
part
s I
nee
d in
my
grap
h
or p
ictu
re.
I
do n
ot s
upp
ort
my
answ
er
wit
h t
he
grap
h o
r pi
ctu
re.
Mathematics Instructional Rubrics – Section Two 21
Mat
hem
atic
al R
epre
sen
tati
on -
An
alyt
ic R
ub
ric
– In
term
edia
te L
evel
T
his
is h
ow I
rep
rese
nt
my
data
an
d an
swer
.
Acc
ura
cy
Doe
s m
y re
pres
enta
tion
cor
rect
ly
disp
lay
the
data
?
F
orm
at
Do
I re
pre
sen
t al
l of
the
labe
ls,
oper
atio
n s
ymbo
ls, t
itle
s, a
nd/
or
keys
for
the
type
of
disp
lay
or
grap
h I
ch
ose?
C
oncl
usi
on
Do
I st
ate
my
con
clu
sion
wit
h
supp
ort
from
my
repr
esen
tati
on?
4 M
y re
pres
enta
tion
is c
omp
lete
lyco
rrec
t in
dis
play
ing
all
of t
he
data
.
I in
clu
de a
ll o
f th
e la
bels
, op
erat
ion
sym
bols
, tit
les,
an
d/or
ke
ys fo
r th
e ty
pe o
f di
spla
y or
gr
aph
I c
hos
e.
I st
ate
my
corr
ect
con
clu
sion
co
mp
lete
lysu
ppor
ted
from
my
repr
esen
tati
on.
3 M
y re
pres
enta
tion
is m
ostl
y co
rrec
t in
dis
play
ing
mos
t of
th
e da
ta.
I in
clu
de m
ost
of t
he
labe
ls,
oper
atio
n s
ymbo
ls, t
itle
s, a
nd/
or
keys
for
th
e ty
pe o
f di
spla
y or
gr
aph
I c
hos
e.
I st
ate
my
corr
ect
con
clu
sion
m
ostl
y su
ppor
ted
from
my
repr
esen
tati
on.
2 M
y re
pres
enta
tion
is p
artl
y co
rrec
t in
dis
play
ing
som
e of
th
e da
ta.
I in
clu
de s
ome
of t
he
labe
ls,
oper
atio
n s
ymbo
ls, t
itle
s, a
nd/
or
keys
for
th
e ty
pe o
f di
spla
y or
gr
aph
I c
hos
e.
I st
ate
my
con
clu
sion
par
tly
supp
orte
d fr
om m
y re
pres
enta
tion
.
1
My
repr
esen
tati
on is
not
cor
rect
in
dis
play
ing
the
data
.
I in
clu
de f
ew o
r n
one
of t
he
labe
ls, o
pera
tion
sym
bols
, tit
les,
an
d/or
key
s fo
r th
e ty
pe o
f di
spla
y or
gra
ph I
ch
ose.
I st
ate
my
con
clu
sion
wit
hou
t su
ppor
t fr
om m
y re
pres
enta
tion
.
Mathematics Instructional Rubrics – Section Two 22
Mat
hem
atic
al R
epre
sen
tati
on -
An
alyt
ic R
ub
ric
– M
idd
le S
choo
l L
evel
T
his
is h
ow I
rep
rese
nt
the
prob
lem
an
d so
luti
on.
A
ccu
racy
D
oes
my
repr
esen
tati
on a
ccu
rate
ly
disp
lay
the
prob
lem
?
F
orm
at
Do
I in
clu
de a
ll o
f th
e la
bels
, ope
rati
on
sym
bols
, tit
les,
an
d/or
key
s fo
r th
e ty
pe o
f re
pres
enta
tion
I c
hos
e?
C
oncl
usi
on
Do
I st
ate
my
con
clu
sion
wit
h s
upp
ort
from
my
repr
esen
tati
on?
4
My
repr
esen
tati
on is
com
ple
tely
ac
cura
te a
nd
incl
ude
s al
l of
th
e da
ta a
nd
all
of t
he
nec
essa
ry
rela
tion
ship
s in
th
e pr
oble
m.
I in
clu
de a
ll o
f th
e la
bels
, ope
rati
on
sym
bols
, tit
les,
an
d/or
key
s fo
r th
e ty
pe
of r
epre
sen
tati
on I
ch
ose.
I st
ate
my
corr
ect
con
clu
sion
co
mp
lete
ly s
upp
orte
d fr
om m
y re
pres
enta
tion
.
3
My
repr
esen
tati
on is
mos
tly
accu
rate
an
d in
clu
des
mos
t of
th
e da
ta a
nd
mos
t of
th
e n
eces
sary
re
lati
onsh
ips
in t
he
prob
lem
.
I in
clu
de m
ost
of t
he
labe
ls, o
pera
tion
sy
mbo
ls, t
itle
s, a
nd/
or k
eys
for
the
type
of
rep
rese
nta
tion
I c
hos
e.
I st
ate
my
corr
ect
con
clu
sion
mos
tly
supp
orte
d fr
om m
y re
pres
enta
tion
.
2 M
y re
pres
enta
tion
is p
arti
ally
ac
cura
te a
nd
incl
ude
s so
me
of t
he
data
an
d so
me
of t
he
nec
essa
ry
rela
tion
ship
s in
th
e pr
oble
m.
I in
clu
de s
ome
of t
he
labe
ls, o
pera
tion
sy
mbo
ls, t
itle
s, a
nd/
or k
eys
for
the
type
of
rep
rese
nta
tion
I c
hos
e.
I st
ate
my
con
clu
sion
par
tly
supp
orte
d fr
om m
y re
pres
enta
tion
or
my
repr
esen
tati
on le
ads
to a
n
un
stat
ed c
oncl
usi
on.
1 M
y re
pres
enta
tion
is n
ot a
ccu
rate
an
d in
clu
des
few
or
non
e of
th
e da
ta o
r n
eces
sary
rel
atio
nsh
ips
in
the
prob
lem
.
I in
clu
de f
ew o
r n
one
of t
he
labe
ls,
oper
atio
n s
ymbo
ls, t
itle
s, a
nd/
or k
eys
for
the
type
of
repr
esen
tati
on I
ch
ose.
I st
ate
my
con
clu
sion
wit
hou
t su
ppor
t fr
om m
y re
pres
enta
tion
or
I do
not
sta
te a
con
clu
sion
.
Mathematics Instructional Rubrics – Section Two 23
Mat
hem
atic
al R
epre
sen
tati
on -
An
alyt
ic R
ub
ric
– H
igh
Sch
ool
Lev
el
Th
is is
how
I r
epre
sen
t th
e pr
oble
m a
nd
solu
tion
.
Acc
ura
cy
Do
I ac
cura
tely
rep
rese
nt
all o
f m
y da
ta?
F
orm
at
Do
I co
rrec
tly
rep
rese
nt
all n
eces
sary
re
lati
onsh
ips?
C
oncl
usi
on
Do
I st
ate
my
con
clu
sion
an
d/or
ge
ner
aliz
atio
n w
ith
su
ppor
t fr
om m
y re
pres
enta
tion
?
4
My
repr
esen
tati
on(s
) is
wel
l pr
esen
ted
wit
h a
ll d
etai
ls w
ell
exec
ute
d in
th
at it
is c
ompl
ete,
ac
cura
te, c
lear
, cor
rect
, an
d ea
sy
to in
terp
ret.
I re
pres
ent
the
elem
ents
in a
n
insi
ghtf
ul s
elec
tion
/for
mat
th
at
illu
stra
tes
all
of t
he
nec
essa
ry
rela
tion
ship
s in
th
e pr
oble
m.
I st
ate
my
corr
ect
con
clu
sion
an
d/or
gen
eral
izat
ion
co
mp
lete
ly s
upp
orte
d fr
om m
y re
pres
enta
tion
.
3
My
repr
esen
tati
on(s
) is
ap
prop
riat
e in
th
at it
is m
ostl
y co
mpl
ete,
acc
ura
te, c
lear
, cor
rect
, an
d ea
sy t
o in
terp
ret.
I re
pres
ent
the
elem
ents
in a
n
appr
opri
ate
sele
ctio
n/f
orm
at t
hat
il
lust
rate
s m
ost
of t
he
nec
essa
ry
rela
tion
ship
s in
th
e pr
oble
m.
I st
ate
my
corr
ect
con
clu
sion
an
d/or
gen
eral
izat
ion
mos
tly
supp
orte
d fr
om m
y re
pres
enta
tion
.
2 M
y re
pres
enta
tion
(s)
is
som
ewh
at c
ompl
ete,
acc
ura
te
and/
or p
arti
ally
cor
rect
, an
d so
me
part
s ar
e ea
sy t
o in
terp
ret.
I pa
rtia
lly
repr
esen
t th
e el
emen
ts
in a
sel
ecti
on/f
orm
at t
hat
il
lust
rate
s so
me
of t
he
nec
essa
ry
rela
tion
ship
s in
th
e pr
oble
m.
I st
ate
my
con
clu
sion
an
d/or
ge
ner
aliz
atio
n p
artl
y su
ppor
ted
from
my
repr
esen
tati
on.
1 M
y re
pres
enta
tion
(s)
is
inco
mpl
ete,
inac
cura
te, i
nco
rrec
t,
and
is d
iffi
cult
to
inte
rpre
t.
I re
pres
ent
the
elem
ents
in a
se
lect
ion
/for
mat
th
at il
lust
rate
few
or
non
e of
th
e n
eces
sary
re
lati
onsh
ips
in t
he
prob
lem
.
I m
ay o
r m
ay n
ot s
tate
my
con
clu
sion
an
d/or
gen
eral
izat
ion
w
ith
out
supp
ort
from
my
repr
esen
tati
on.
Mathematics Instructional Rubrics – Section Two 24
Mat
hem
atic
al C
omm
unic
atio
n H
olis
tic
Rub
ric
- Pri
mar
y Le
vel
This
is h
ow I
solv
e th
e pr
oble
m.
4
• M
y w
ork
is w
ell o
rgan
ized
and
eas
y to
und
erst
and.
•
I exp
lain
all
of th
e st
eps t
hat I
use
to g
et a
corr
ect a
nsw
er.
• I u
se a
ll of
the
corr
ect l
abel
s, de
tails
, sym
bols
, and
mat
h te
rms t
o ex
plai
n m
y w
ork.
3 •
My
wor
k is
mos
tly
orga
nize
d an
d ea
sy to
und
erst
and.
•
I exp
lain
mos
t of t
he st
eps t
hat I
use
to g
et a
corr
ect a
nsw
er.
• I u
se m
ost o
f the
corr
ect l
abel
s, de
tails
, sym
bols
, and
mat
h te
rms t
o ex
plai
n m
y w
ork.
2 •
My
wor
k is
poo
rly
orga
nize
d an
d is
har
d to
und
erst
and.
•
I exp
lain
som
e of
the
step
s tha
t I u
se to
get
an
answ
er.
• I u
se s
ome
labe
ls, d
etai
ls, s
ymbo
ls, a
nd m
ath
term
s to
expl
ain
my
wor
k.
1
• M
y w
ork
is n
ot o
rgan
ized
to sh
ow h
ow I
solv
e th
e pr
oble
m.
• I d
o no
t exp
lain
the
step
s tha
t I u
se to
get
an
answ
er.
• I d
o no
t use
labe
ls, d
etai
ls, s
ymbo
ls, o
r mat
h te
rms t
o ex
plai
n m
y w
ork.
Mathematics Instructional Rubrics – Section Two 25
M
athe
mat
ical
Com
mun
icat
ion
- Hol
isti
c R
ubri
c - I
nter
med
iate
Lev
el
This
is h
ow I
solv
e th
e pr
oble
m.
4
• M
y w
ork
is w
ell o
rgan
ized
, mak
es se
nse,
and
is e
asy
to u
nder
stan
d.
• I i
nclu
de a
n ac
cura
te e
xpla
natio
n of
all
of th
e st
eps n
eede
d to
corr
ectly
solv
e th
e pr
oble
m.
• I u
se a
ll of
the
prop
er la
bels
, det
ails
, sym
bols
, and
corr
ect m
ath
term
s in
my
expl
anat
ion.
3 •
My
wor
k is
mos
tly
orga
nize
d, m
akes
sens
e, a
nd is
eas
y to
und
erst
and.
•
I inc
lude
an
accu
rate
exp
lana
tion
of m
ost o
f the
step
s nee
ded
to co
rrec
tly so
lve
the
prob
lem
. •
I use
mos
t of t
he p
rope
r lab
els,
deta
ils, s
ymbo
ls, a
nd co
rrec
t mat
h te
rms i
n m
y ex
plan
atio
n.
2
• M
y w
ork
is p
oorl
y or
gani
zed,
har
d to
und
erst
and,
and
is co
nfus
ing.
•
I inc
lude
an
expl
anat
ion
of s
ome
of th
e st
eps n
eede
d to
solv
e th
e pr
oble
m.
• I u
se s
ome
of th
e pr
oper
labe
ls, d
etai
ls, s
ymbo
ls, a
nd co
rrec
t mat
h te
rms i
n m
y ex
plan
atio
n.
1
• M
y w
ork
is n
ot o
rgan
ized
, doe
s not
mak
e se
nse,
and
is v
ery
conf
usin
g.
• I i
nclu
de a
n ex
plan
atio
n of
few
or
none
of t
he st
eps n
eede
d to
solv
e th
e pr
oble
m.
• I d
o no
t use
pro
per l
abel
s, de
tails
, sym
bols
, and
corr
ect m
ath
term
s in
my
expl
anat
ion.
Mathematics Instructional Rubrics – Section Two 26
M
athe
mat
ical
Com
mun
icat
ion
- Hol
isti
c R
ubri
c - M
iddl
e Sc
hool
Lev
el
This
is h
ow I
solv
e th
e pr
oble
m.
4
• M
y re
spon
se is
wel
l org
aniz
ed a
nd is
logi
cal.
• I i
nclu
de a
n ac
cura
te, c
ompl
ete,
and
thor
ough
exp
lana
tion
of a
ll of
the
step
s nee
ded
to
corr
ectly
solv
e th
e pr
oble
m.
• I u
se a
ll of
the
prop
er co
ncep
ts, l
abel
s, de
tails
, sym
bols
, and
corr
ect m
ath
term
s in
my
expl
anat
ion.
3 •
My
resp
onse
is m
ostl
y or
gani
zed
and
is lo
gica
l. •
I inc
lude
an
accu
rate
exp
lana
tion
of m
ost o
f the
step
s nee
ded
to co
rrec
tly so
lve
the
prob
lem
. •
I use
mos
t of t
he p
rope
r con
cept
s, la
bels
, det
ails
, sym
bols
, and
corr
ect m
ath
term
s in
my
expl
anat
ion.
2 •
My
resp
onse
is p
oorl
y or
gani
zed
or is
illo
gica
l. •
I inc
lude
an
expl
anat
ion
of s
ome
of th
e st
eps n
eede
d to
solv
e th
e pr
oble
m.
• I u
se s
ome
of th
e pr
oper
conc
epts
, lab
els,
deta
ils, s
ymbo
ls, a
nd co
rrec
t mat
h te
rms i
n m
y ex
plan
atio
n.
1
• M
y re
spon
se is
not
org
aniz
ed a
nd is
illo
gica
l. •
I inc
lude
an
expl
anat
ion
of fe
w o
r no
ne o
f the
step
s nee
ded
to so
lve
the
prob
lem
. •
I use
few
or
none
of t
he p
rope
r con
cept
s, la
bels
, det
ails
, sym
bols
, and
corr
ect m
ath
term
s in
my
expl
anat
ion.
Mathematics Instructional Rubrics – Section Two 27
M
athe
mat
ical
Com
mun
icat
ion
- Hol
isti
c R
ubri
c - H
igh
Scho
ol L
evel
Th
is is
how
I so
lve
the
prob
lem
.
4 •
My
resp
onse
is th
orou
gh, w
ell o
rgan
ized
, and
logi
cal.
• I i
nclu
de a
com
plet
e an
d co
rrec
t exp
lana
tion
usin
g al
l of t
he re
leva
nt a
nd sp
ecifi
c det
ails
from
the
prob
lem
w
hen
desc
ribi
ng th
e pr
oced
ures
use
d to
arr
ive
at th
e cor
rect
solu
tion.
•
I use
all
of th
e la
bels
, sym
bols
, con
cept
s, te
rmin
olog
y, a
nd re
pres
enta
tions
acc
urat
ely,
clea
rly
and
succ
inct
ly in
my
resp
onse
.
3 •
My
resp
onse
is m
ostl
y or
gani
zed
and
logi
cal.
• I i
nclu
de a
corr
ect e
xpla
natio
n us
ing
mos
t of t
he re
leva
nt a
nd sp
ecifi
c det
ails
from
the
prob
lem
whe
n de
scri
bing
the
proc
edur
es u
sed
to a
rriv
e at
the
corr
ect s
olut
ion.
•
I use
mos
t of t
he la
bels
, sym
bols
, con
cept
s, te
rmin
olog
y, a
nd re
pres
enta
tions
acc
urat
ely,
clea
rly,
and
su
ccin
ctly
in m
y re
spon
se.
2
• M
y re
spon
se is
poo
rly
orga
nize
d or
is il
logi
cal.
• I i
nclu
de a
n ex
plan
atio
n us
ing
som
e of
the
rele
vant
and
spec
ific d
etai
ls fr
om th
e pr
oble
m w
hen
desc
ribi
ng
the
proc
edur
es u
sed
to a
rriv
e at
a so
lutio
n.
• I u
se s
ome
of th
e la
bels
, sym
bols
, con
cept
s, te
rmin
olog
y, a
nd re
pres
enta
tions
acc
urat
ely
in m
y re
spon
se.
1
• M
y re
spon
se is
not
org
aniz
ed a
nd is
illo
gica
l. •
I inc
lude
an
expl
anat
ion
usin
g fe
w o
r no
ne o
f the
rele
vant
or s
peci
fic d
etai
ls fr
om th
e pr
oble
m w
hen
desc
ribi
ng th
e pr
oced
ures
. •
I use
few
or
none
of t
he la
bels
, sym
bols
, con
cept
s, te
rmin
olog
y, o
r rep
rese
ntat
ions
in m
y re
spon
se.
Mathematics Instructional Rubrics – Section Two 28
Mat
hem
atic
al R
easo
ning
- H
olis
tic
Rub
ric
- Pri
mar
y Le
vel
This
is w
hy I
solv
e th
e pr
oble
m th
is w
ay.
4
• M
y ex
plan
atio
n in
clud
es a
ll of
the
step
s, is
wel
l org
aniz
ed, a
nd m
akes
sens
e.
• I e
xpla
in w
hy I
chos
e th
is st
rate
gy to
corr
ectly
solv
e th
e pr
oble
m in
clud
ing
all t
he
info
rmat
ion
need
ed.
• I u
se a
ll of
the
corr
ect l
abel
s, de
tails
, sym
bols
, and
mat
h te
rms t
o ex
plai
n m
y w
ork.
3 •
My
expl
anat
ion
incl
udes
mos
t of t
he st
eps,
is o
rgan
ized
, and
mak
es se
nse.
•
I exp
lain
why
I ch
ose
this
stra
tegy
to co
rrec
tly so
lve
the
prob
lem
and
incl
ude
mos
t of t
he
info
rmat
ion
need
ed.
• I u
se m
ost o
f the
corr
ect l
abel
s, de
tails
, sym
bols
, and
mat
h te
rms t
o ex
plai
n m
y w
ork.
2 •
My
expl
anat
ion
incl
udes
som
e of
the
step
s, is
not
org
aniz
ed, b
ut m
akes
som
e se
nse.
•
I exp
lain
why
I ch
ose
this
stra
tegy
to so
lve
the
prob
lem
and
incl
ude
som
e of
the
info
rmat
ion
need
ed.
• I u
se s
ome
labe
ls, d
etai
ls, s
ymbo
ls, a
nd m
ath
term
s to
expl
ain
my
wor
k.
1
• M
y ex
plan
atio
n in
clud
es fe
w o
r no
ne o
f the
step
s, is
not
org
aniz
ed, a
nd d
oes n
ot m
ake
sens
e.
• I d
o no
t exp
lain
why
I ch
ose
this
stra
tegy
to so
lve
the
prob
lem
and
I in
clud
e lit
tle o
r no
info
rmat
ion
need
ed to
supp
ort m
y st
rate
gy.
• I d
o no
t use
labe
ls, d
etai
ls, s
ymbo
ls, o
r mat
h te
rms t
o ex
plai
n m
y w
ork.
Mathematics Instructional Rubrics – Section Two 29
Mat
hem
atic
al R
easo
nin
g -
Hol
isti
c R
ub
ric
- In
term
edia
te L
evel
T
his
is w
hy
I so
lve
the
prob
lem
th
is w
ay.
4
• M
y ex
plan
atio
n d
escr
ibes
my
stra
tegy
in a
hig
hly
sk
illf
ul
way
th
at is
cle
ar, o
rgan
ized
, an
d m
akes
sen
se.
•
I co
mp
lete
ly e
xpla
in w
hy
I ch
ose
this
cor
rect
str
ateg
y to
sol
ve t
he
prob
lem
. •
I u
se a
ll o
f th
e pr
oper
labe
ls, d
etai
ls, s
ymbo
ls, a
nd
corr
ect
mat
h t
erm
s in
my
expl
anat
ion
.
3 •
My
expl
anat
ion
des
crib
es m
y st
rate
gy in
a m
ostl
y sk
illf
ul
way
th
at is
org
aniz
ed a
nd
mak
es s
ense
. •
I m
ostl
y ex
plai
n w
hy
I ch
ose
this
cor
rect
str
ateg
y to
sol
ve t
he
prob
lem
. •
I u
se m
ost
of t
he
prop
er la
bels
, det
ails
, sym
bols
, an
d co
rrec
t m
ath
ter
ms
in m
y ex
plan
atio
n.
2
• M
y ex
plan
atio
n d
escr
ibes
my
stra
tegy
in a
som
ewh
at s
kil
lfu
l w
ay t
hat
is n
ot o
rgan
ized
bu
t m
akes
som
e se
nse
. •
I p
arti
ally
exp
lain
wh
y I
chos
e th
is s
trat
egy
to s
olve
th
e pr
oble
m.
• I
use
som
e of
th
e pr
oper
labe
ls, d
etai
ls, s
ymbo
ls, a
nd
corr
ect
mat
h t
erm
s in
my
expl
anat
ion
.
1 •
My
expl
anat
ion
doe
s n
ot d
escr
ibe
my
stra
tegy
in a
ski
llfu
l way
, is
not
org
aniz
ed, a
nd
does
n
ot m
ake
sen
se.
• I
do
not
exp
lain
wh
y I
chos
e th
is s
trat
egy
to s
olve
th
e pr
oble
m.
• I
do
not
use
pro
per
labe
ls, d
etai
ls, s
ymbo
ls, a
nd
corr
ect
mat
h t
erm
s in
my
expl
anat
ion
.
Mathematics Instructional Rubrics – Section Two 30
Mat
hem
atic
al R
easo
nin
g -
Hol
isti
c R
ub
ric
- M
idd
le S
choo
l L
evel
T
his
is w
hy
I so
lve
the
prob
lem
usi
ng
a se
lect
str
ateg
y.
4
• M
y ex
plan
atio
n o
f th
e st
rate
gy in
clu
des
all
deta
ils
and
is w
ell o
rgan
ized
. •
I fu
lly
expl
ain
wh
y I
chos
e th
is e
ffec
tive
an
d ef
fici
ent
stra
tegy
th
at le
ads
to a
com
ple
tely
co
rrec
t co
ncl
usi
on o
r so
luti
on.
• I
use
all
of
the
prop
er c
once
pts,
labe
ls, d
etai
ls, s
ymbo
ls, a
nd
corr
ect
mat
h t
erm
s in
my
expl
anat
ion
.
3 •
My
expl
anat
ion
of
the
stra
tegy
incl
ude
s m
ost
deta
ils
and
is o
rgan
ized
. •
I m
ostl
y ex
plai
n w
hy
I ch
ose
this
eff
ecti
ve s
trat
egy
that
lead
s to
a c
omp
lete
ly c
orre
ct
con
clu
sion
or
solu
tion
. •
I u
se m
ost
of t
he
prop
er c
once
pts,
labe
ls, d
etai
ls, s
ymbo
ls, a
nd
corr
ect
mat
h t
erm
s in
my
expl
anat
ion
.
2 •
My
expl
anat
ion
of
the
stra
tegy
incl
ude
s so
me
deta
ils
and
is d
isor
gan
ized
. •
I pa
rtia
lly
expl
ain
wh
y I
chos
e th
is s
trat
egy
that
lead
s to
a p
arti
ally
cor
rect
con
clu
sion
or
solu
tion
. •
I u
se s
ome
of t
he
prop
er c
once
pts,
labe
ls, d
etai
ls, s
ymbo
ls, a
nd
corr
ect
mat
h t
erm
s in
my
expl
anat
ion
.
1 •
My
expl
anat
ion
of
the
stra
tegy
incl
ude
s fe
w o
r n
o de
tail
s an
d is
not
org
aniz
ed.
• I
do n
ot e
xpla
in w
hy
I ch
ose
this
str
ateg
y th
at le
ads
to a
n i
nco
rrec
t co
ncl
usi
on o
r so
luti
on.
• I
use
few
or
non
e of
th
e pr
oper
con
cept
s, la
bels
, det
ails
, sym
bols
, an
d co
rrec
t m
ath
ter
ms
in m
y ex
plan
atio
n.
Mathematics Instructional Rubrics – Section Two 31
Mat
hem
atic
al R
easo
nin
g -
Hol
isti
c R
ub
ric
- H
igh
Sch
ool
Lev
el
My
impl
emen
tati
on in
clu
des
orga
niz
ed m
ath
emat
ical
ste
ps/p
roce
dure
s an
d re
leva
nt
deta
ils.
4 •
My
impl
emen
tati
on is
fu
lly
deve
lope
d an
d co
mpl
etel
y su
ppor
ted
by r
elev
ant
and
spec
ific
de
tail
s fr
om t
he
prob
lem
. •
I in
clu
de a
cor
rect
, th
orou
gh, a
nd
wel
l org
aniz
ed e
xpla
nat
ion
bas
ed o
n m
ath
emat
ical
tru
ths
of
wh
y I
sele
ct t
he
con
cept
s an
d re
pres
enta
tion
s pr
esen
ted
in m
y co
rrec
t so
luti
on.
• I
use
all
of
the
labe
ls, s
ymbo
ls, c
once
pts,
ter
min
olog
y, a
nd
repr
esen
tati
ons
accu
rate
ly, c
lear
ly,
and
succ
inct
ly in
my
resp
onse
.
3 •
My
impl
emen
tati
on is
mos
tly
deve
lope
d an
d su
ppor
ted
by r
elev
ant
and
spec
ific
det
ails
fro
m
the
prob
lem
. •
I in
clu
de a
cor
rect
exp
lan
atio
n b
ased
on
mat
hem
atic
al t
ruth
s of
wh
y I
sele
ct t
he
con
cept
s an
d re
pres
enta
tion
s pr
esen
ted
in m
y co
rrec
t so
luti
on.
• I
use
mos
t of
th
e la
bels
, sym
bols
, con
cept
s, t
erm
inol
ogy,
an
d re
pres
enta
tion
s ac
cura
tely
, cl
earl
y, a
nd
succ
inct
ly in
my
resp
onse
.
2 •
My
impl
emen
tati
on is
som
ewh
at d
evel
oped
an
d su
ppor
ted
by r
elev
ant
and
spec
ific
det
ails
fr
om t
he
prob
lem
. •
I in
clu
de a
par
tial
ly c
orre
ct a
nd
som
ewh
at o
rgan
ized
exp
lan
atio
n o
f w
hy
I se
lect
th
e co
nce
pts
and
repr
esen
tati
ons
pres
ente
d in
my
solu
tion
. •
I u
se s
ome
of t
he
labe
ls, s
ymbo
ls, c
once
pts,
ter
min
olog
y, a
nd
repr
esen
tati
ons
accu
rate
ly in
my
resp
onse
.
1 •
My
impl
emen
tati
on is
rar
ely
supp
orte
d by
rel
evan
t an
d sp
ecif
ic d
etai
ls f
rom
th
e pr
oble
m.
• I
incl
ude
an
inco
rrec
t, d
isor
gan
ized
, or
min
imal
exp
lan
atio
n o
f w
hy
I se
lect
th
e co
nce
pts
and
repr
esen
tati
ons
pres
ente
d in
my
solu
tion
. •
I u
se f
ew o
r n
one
of t
he
labe
ls, s
ymbo
ls, c
once
pts,
ter
min
olog
y, o
r re
pres
enta
tion
s in
my
resp
onse
.
Mathematics Instructional Rubrics – Section Two 32
Mat
hem
atic
al R
epre
sen
tati
on -
Hol
isti
c R
ub
ric
- P
rim
ary
Lev
el
Th
is is
how
I s
how
my
info
rmat
ion
in a
gra
ph o
r pi
ctu
re.
4
•
My
grap
h o
r pi
ctu
re a
nd
info
rmat
ion
are
com
ple
tely
cor
rect
. •
I pu
t al
l of
th
e pa
rts
I n
eed
in m
y gr
aph
or
pict
ure
. •
I co
mp
lete
ly s
upp
ort
my
corr
ect
answ
er w
ith
th
e gr
aph
or
pict
ure
.
3 •
My
grap
h o
r pi
ctu
re a
nd
info
rmat
ion
are
mos
tly
corr
ect.
•
I pu
t m
ost
of t
he
part
s I
nee
d in
my
grap
h o
r pi
ctu
re.
• I
mos
tly
supp
ort
my
corr
ect
answ
er w
ith
th
e gr
aph
or
pict
ure
.
2 •
My
grap
h o
r pi
ctu
re o
r in
form
atio
n is
par
tly
corr
ect.
•
I pu
t so
me
of t
he
part
s I
nee
d in
my
grap
h o
r pi
ctu
re.
• I
par
tly
supp
ort
my
answ
er w
ith
th
e gr
aph
or
pict
ure
.
1 •
My
grap
h o
r pi
ctu
re o
r in
form
atio
n is
not
cor
rect
. •
I pu
t fe
w o
r n
one
of t
he
part
s I
nee
d in
my
grap
h o
r pi
ctu
re.
• I
do n
ot s
upp
ort
my
answ
er w
ith
th
e gr
aph
or
pict
ure
.
Mathematics Instructional Rubrics – Section Two 33
Mat
hem
atic
al R
epre
sen
tati
on -
Hol
isti
c R
ub
ric
- In
term
edia
te L
evel
T
his
is h
ow I
rep
rese
nt
my
data
an
d an
swer
.
4 •
My
repr
esen
tatio
n is
com
plet
ely
corr
ect i
n di
spla
ying
all
of th
e da
ta.
• I i
nclu
de a
ll of
the
labe
ls, o
pera
tion
sym
bols
, titl
es, a
nd/o
r key
s for
the
type
of d
ispl
ay o
r gra
ph I
chos
e.
• I s
tate
my
corr
ect c
oncl
usio
n co
mpl
etel
y su
ppor
ted
from
my
repr
esen
tatio
n.
3
• M
y re
pres
enta
tion
is m
ostly
cor
rect
in d
ispl
ayin
g m
ost o
f the
dat
a.
• I i
nclu
de m
ost o
f the
labe
ls, o
pera
tion
sym
bols
, titl
es, a
nd/o
r key
s for
the
type
of d
ispl
ay o
r gra
ph I
chos
e.
• I s
tate
my
corr
ect c
oncl
usio
n m
ostly
supp
orte
d fr
om m
y re
pres
enta
tion.
2 •
My
repr
esen
tatio
n is
par
tly c
orre
ct in
dis
play
ing
som
e of
the
data
. •
I inc
lude
som
e of
the
labe
ls, o
pera
tion
sym
bols
, titl
es, a
nd/o
r key
s for
the
type
of d
ispl
ay o
r gra
ph I
chos
e.
• I s
tate
my
conc
lusi
on p
artly
supp
orte
d fr
om m
y re
pres
enta
tion.
1 •
My
repr
esen
tatio
n is
not
cor
rect
in d
ispl
ayin
g th
e da
ta.
• I i
nclu
de fe
w o
r no
ne o
f the
labe
ls, o
pera
tion
sym
bols
, titl
es, a
nd/o
r key
s for
the
type
of d
ispl
ay o
r gr
aph
I cho
se.
• I s
tate
my
conc
lusi
on w
ithou
t sup
port
from
my
repr
esen
tatio
n.
Mathematics Instructional Rubrics – Section Two 34
Mat
hem
atic
al R
epre
sen
tati
on -
Hol
isti
c R
ub
ric
- M
idd
le S
choo
l L
evel
T
his
is h
ow I
rep
rese
nt
the
prob
lem
an
d so
luti
on.
4
• M
y re
pres
enta
tion
is c
omp
lete
ly a
ccu
rate
an
d in
clu
des
all
of t
he
data
an
d al
l of
th
e n
eces
sary
rel
atio
nsh
ips
in t
he
prob
lem
. •
I in
clu
de a
ll o
f th
e la
bels
, ope
rati
on s
ymbo
ls, t
itle
s, a
nd/
or k
eys
for
the
type
of
repr
esen
tati
on I
ch
ose.
•
I st
ate
my
corr
ect
con
clu
sion
com
ple
tely
su
ppor
ted
from
my
repr
esen
tati
on.
3
• M
y re
pres
enta
tion
is m
ostl
y ac
cura
te a
nd
incl
ude
s m
ost
of t
he
data
an
d m
ost
of t
he
nec
essa
ry r
elat
ion
ship
s in
th
e pr
oble
m.
• I
incl
ude
mos
t of
th
e la
bels
, ope
rati
on s
ymbo
ls, t
itle
s, a
nd/
or k
eys
for
the
type
of
repr
esen
tati
on I
ch
ose.
•
I st
ate
my
corr
ect
con
clu
sion
mos
tly
supp
orte
d fr
om m
y re
pres
enta
tion
.
2 •
My
repr
esen
tati
on is
par
tial
ly a
ccu
rate
an
d in
clu
des
som
e of
th
e da
ta a
nd
som
e of
th
e n
eces
sary
rel
atio
nsh
ips
in t
he
prob
lem
. •
I in
clu
de s
ome
of t
he
labe
ls, o
pera
tion
sym
bols
, tit
les,
an
d/or
key
s fo
r th
e ty
pe o
f re
pres
enta
tion
I c
hos
e.
• I
stat
e m
y co
ncl
usi
on p
artl
y su
ppor
ted
from
my
repr
esen
tati
on o
r m
y re
pres
enta
tion
lead
s to
an
un
stat
ed c
oncl
usi
on.
1
• M
y re
pres
enta
tion
is n
ot a
ccu
rate
an
d in
clu
des
few
or
non
e of
th
e da
ta o
r n
eces
sary
re
lati
onsh
ips
in t
he
prob
lem
. •
I in
clu
de f
ew o
r n
one
of t
he
labe
ls, o
pera
tion
sym
bols
, tit
les,
an
d/or
key
s fo
r th
e ty
pe o
f re
pres
enta
tion
I c
hos
e.
• I
stat
e m
y co
ncl
usi
on w
ith
out
supp
ort
from
my
repr
esen
tati
on o
r I
do n
ot s
tate
a c
oncl
usi
on.
Mathematics Instructional Rubrics – Section Two 35
Mat
hem
atic
al R
epre
sen
tati
on -
Hol
isti
c R
ub
ric
- H
igh
Sch
ool
Lev
el
Th
is is
how
I r
epre
sen
t th
e pr
oble
m a
nd
solu
tion
.
4 •
My
repr
esen
tati
on(s
) is
wel
l pre
sen
ted
wit
h a
ll d
etai
ls w
ell e
xecu
ted
in t
hat
it is
com
plet
e,
accu
rate
, cle
ar, c
orre
ct, a
nd
easy
to
inte
rpre
t.
• I
repr
esen
t th
e el
emen
ts in
an
insi
ghtf
ul s
elec
tion
/for
mat
th
at il
lust
rate
s al
l of
th
e n
eces
sary
re
lati
onsh
ips
in t
he
prob
lem
. •
I st
ate
my
corr
ect
con
clu
sion
an
d/or
gen
eral
izat
ion
com
ple
tely
su
ppor
ted
from
my
repr
esen
tati
on.
3
• M
y re
pres
enta
tion
(s)
is a
ppro
pria
te in
th
at it
is m
ostl
y co
mpl
ete,
acc
ura
te, c
lear
, cor
rect
, an
d ea
sy t
o in
terp
ret.
•
I re
pres
ent
the
elem
ents
in a
n a
ppro
pria
te s
elec
tion
/for
mat
th
at il
lust
rate
s m
ost
of t
he
nec
essa
ry
rela
tion
ship
s in
th
e pr
oble
m.
• I
stat
e m
y co
rrec
t co
ncl
usi
on a
nd/
or g
ener
aliz
atio
n m
ostl
y su
ppor
ted
from
my
repr
esen
tati
on.
2
• M
y re
pres
enta
tion
(s)
is s
omew
hat
com
plet
e, a
ccu
rate
, an
d/or
par
tial
ly c
orre
ct a
nd
som
e pa
rts
are
easy
to
inte
rpre
t.
• I
part
iall
y re
pres
ent
the
elem
ents
in a
sel
ecti
on/f
orm
at t
hat
illu
stra
tes
som
e of
th
e n
eces
sary
re
lati
onsh
ips
in t
he
prob
lem
. •
I st
ate
my
con
clu
sion
an
d/or
gen
eral
izat
ion
par
tly
supp
orte
d fr
om m
y re
pres
enta
tion
.
1 •
My
repr
esen
tati
on(s
) is
inco
mpl
ete,
inac
cura
te, i
nco
rrec
t, a
nd
is d
iffi
cult
to
inte
rpre
t •
I re
pres
ent
the
elem
ents
in a
sel
ecti
on/f
orm
at t
hat
illu
stra
te f
ew o
r n
one
of t
he
nec
essa
ry
rela
tion
ship
s in
th
e pr
oble
m.
• I
may
or
may
not
sta
te m
y co
ncl
usi
on a
nd/
or g
ener
aliz
atio
n w
ith
out
supp
ort
from
my
repr
esen
tati
on.
Mathematics Instructional Rubrics – Section Two 36
M
athe
mat
ics C
ombi
natio
n R
ubri
c –
Prim
ary
Lev
el
Com
mun
icat
ions
Th
is is
how
I so
lve
the
prob
lem
.
R
easo
ning
Th
is is
why
I so
lve
the
prob
lem
this
way
.
R
epre
sent
atio
n Th
is is
how
I sh
ow m
y in
form
atio
n in
a g
raph
or
pict
ure.
4
• M
y w
ork
is w
ell o
rgan
ized
and
eas
y to
un
ders
tand
. •
I exp
lain
all
of th
e st
eps t
hat I
use
to
get a
corr
ect a
nsw
er.
• M
y ex
plan
atio
n in
clud
es a
ll of
the
step
s, is
wel
l or
gani
zed,
and
mak
es se
nse.
•
I exp
lain
why
I ch
ose
this
stra
tegy
to co
rrec
tly
solv
e th
e pr
oble
m in
clud
ing
all t
he in
form
atio
n ne
eded
. •
I use
all
of th
e co
rrec
t lab
els,
deta
ils, s
ymbo
ls,
and
mat
h te
rms t
o ex
plai
n m
y w
ork.
•
My
grap
h or
pic
ture
and
info
rmat
ion
are
com
plet
ely
corr
ect.
• I p
ut a
ll of
the
part
s I n
eed
in m
y gr
aph
or
pict
ure.
•
I com
plet
ely
supp
ort m
y co
rrec
t ans
wer
w
ith th
e gr
aph
or p
ictu
re.
3
• M
y w
ork
is m
ostl
y or
gani
zed
and
easy
to
und
erst
and.
•
I exp
lain
mos
t of t
he st
eps t
hat I
use
to
get a
corr
ect a
nsw
er.
• M
y ex
plan
atio
n in
clud
es m
ost o
f the
step
s, is
or
gani
zed,
and
mak
es se
nse.
•
I exp
lain
why
I ch
ose
this
stra
tegy
to co
rrec
tly
solv
e th
e pr
oble
m a
nd in
clud
e m
ost o
f the
in
form
atio
n ne
eded
. •
I use
mos
t of t
he co
rrec
t lab
els,
deta
ils,
sym
bols
, and
mat
h te
rms t
o ex
plai
n m
y w
ork.
•
My
grap
h or
pic
ture
and
info
rmat
ion
are
mos
tly
corr
ect.
• I p
ut m
ost o
f the
par
ts I
need
in m
y gr
aph
or
pict
ure.
•
I mos
tly
supp
ort m
y co
rrec
t ans
wer
with
the
grap
h or
pic
ture
.
2
• M
y w
ork
is p
oorl
y or
gani
zed
and
is
hard
to u
nder
stan
d.
• I e
xpla
in s
ome
of th
e st
eps t
hat I
use
to
get
an
answ
er.
• M
y ex
plan
atio
n in
clud
es s
ome
of th
e st
eps,
is
not o
rgan
ized
, but
mak
es so
me
sens
e.
• I e
xpla
in w
hy I
chos
e th
is st
rate
gy to
solv
e th
e pr
oble
m a
nd in
clud
e so
me
of th
e in
form
atio
n ne
eded
. •
I use
som
e la
bels
, det
ails
, sym
bols
, and
mat
h te
rms t
o ex
plai
n m
y w
ork.
•
My
grap
h or
pic
ture
or i
nfor
mat
ion
is p
artl
y co
rrec
t. •
I put
som
e of
the
part
s I n
eed
in m
y gr
aph
or
pict
ure.
•
I par
tly
supp
ort m
y an
swer
with
the
grap
h or
pic
ture
.
1
• M
y w
ork
is n
ot o
rgan
ized
to sh
ow h
ow
I sol
ve th
e pr
oble
m.
• I d
o no
t exp
lain
the
step
s tha
t I u
se to
ge
t an
answ
er.
• M
y ex
plan
atio
n in
clud
es fe
w o
r no
ne o
f the
st
eps,
is n
ot o
rgan
ized
and
doe
s not
mak
e se
nse.
• I d
o no
t exp
lain
why
I ch
ose
this
stra
tegy
to
solv
e th
e pr
oble
m a
nd I
incl
ude
little
or n
o in
form
atio
n ne
eded
to su
ppor
t my
stra
tegy
. •
I do
not u
se la
bels
, det
ails
, sym
bols
, or m
ath
term
s to
expl
ain
my
wor
k.
•
My
grap
h or
pic
ture
or i
nfor
mat
ion
is n
ot
corr
ect.
• I p
ut fe
w o
r no
ne o
f the
par
ts I
need
in m
y gr
aph
or p
ictu
re.
• I d
o no
t sup
port
my
answ
er w
ith th
e gr
aph
or p
ictu
re.
W
hen
the
com
mun
icat
ion
and
reas
onin
g ru
bric
s are
use
d se
para
tely
, the
y bo
th a
sses
s a st
uden
ts’ a
bilit
y to
use
the
prop
er la
bels
, det
ails,
sym
bols,
and
mat
h te
rms i
n th
e ex
plan
atio
n. W
hen
the
indi
vidu
al ru
bric
s are
uni
ted
to
crea
te th
e co
mbi
natio
n ru
bric
the
stat
emen
t con
cern
ing
the
use
of la
bels,
etc
, bec
omes
redu
ndan
t. Th
eref
ore,
the
labe
ls st
atem
ent i
s writ
ten
only
in th
e re
ason
ing
colu
mn,
whe
re it
is m
ost a
ppro
pria
te.
Mathematics Instructional Rubrics – Section Two 37
M
athe
mat
ics C
ombi
natio
n R
ubri
c –
Inte
rmed
iate
Lev
el
Com
mun
icat
ion
This
is h
ow I
solv
e th
e pr
oble
m.
R
easo
ning
Th
is is
why
I so
lve
the
prob
lem
this
way
.
R
epre
sent
atio
n Th
is is
how
I re
pres
ent m
y da
ta a
nd a
nsw
er.
4 •
My
wor
k is
wel
l org
aniz
ed, m
akes
sens
e, a
nd is
ea
sy to
und
erst
and.
•
I inc
lude
an
accu
rate
exp
lana
tion
of a
ll of
the
step
s nee
ded
to c
orre
ctly
solv
e th
e pr
oble
m.
• M
y ex
plan
atio
n de
scrib
es m
y st
rate
gy in
a h
ighl
y sk
illfu
l way
that
is c
lear
, org
aniz
ed, a
nd m
akes
se
nse.
•
I com
plet
ely
expl
ain
why
I cho
se th
is c
orre
ct
stra
tegy
to so
lve
the
prob
lem
. •
I use
all
of th
e pr
oper
labe
ls, d
etai
ls, s
ymbo
ls,
and
corr
ect m
ath
term
s in
my
expl
anat
ion.
• M
y re
pres
enta
tion
is c
ompl
etel
y co
rrec
t in
disp
layi
ng a
ll of
the
data
. •
I inc
lude
all
of th
e la
bels
, ope
ratio
n sy
mbo
ls, t
itles
, and
/or k
eys f
or th
e ty
pe o
f di
spla
y or
gra
ph I
chos
e.
• I s
tate
my
corr
ect c
oncl
usio
n co
mpl
etel
y su
ppor
ted
from
my
repr
esen
tatio
n.
3
• M
y w
ork
is m
ostly
org
aniz
ed, m
akes
sens
e an
d is
ea
sy to
und
erst
and.
•
I inc
lude
an
accu
rate
exp
lana
tion
of m
ost o
f the
st
eps n
eede
d to
cor
rect
ly so
lve
the
prob
lem
.
• M
y ex
plan
atio
n de
scrib
es m
y st
rate
gy in
a
mos
tly sk
illfu
l way
that
is o
rgan
ized
and
mak
es
sens
e.
• I m
ostly
exp
lain
why
I ch
ose
this
cor
rect
stra
tegy
to
solv
e th
e pr
oble
m.
• I u
se m
ost o
f the
pro
per l
abel
s, de
tails
, sym
bols
, an
d co
rrec
t mat
h te
rms i
n m
y ex
plan
atio
n.
• M
y re
pres
enta
tion
is m
ostly
cor
rect
in
disp
layi
ng m
ost o
f the
dat
a.
• I i
nclu
de m
ost o
f the
labe
ls, o
pera
tion
sym
bols
, titl
es, a
nd/o
r key
s for
the
type
of
disp
lay
or g
raph
I ch
ose.
•
I sta
te m
y co
rrec
t con
clus
ion
mos
tly
supp
orte
d fr
om m
y re
pres
enta
tion.
2 •
My
wor
k is
poo
rly
orga
nize
d, h
ard
to u
nder
stan
d an
d is
con
fusi
ng.
• I i
nclu
de a
n ex
plan
atio
n of
som
e of
the
step
s ne
eded
to so
lve
the
prob
lem
.
• M
y ex
plan
atio
n de
scrib
es m
y st
rate
gy in
a
som
ewha
t ski
llful
way
that
is n
ot o
rgan
ized
but
m
akes
som
e se
nse.
•
I par
tially
exp
lain
why
I ch
ose
this
stra
tegy
to
solv
e th
e pr
oble
m a
nd in
clud
e so
me
of th
e in
form
atio
n ne
eded
to su
ppor
t my
stra
tegy
. •
I use
som
e of
the
prop
er la
bels
, det
ails
, sym
bols
, an
d co
rrec
t mat
h te
rms i
n m
y ex
plan
atio
n.
• M
y re
pres
enta
tion
is p
artly
cor
rect
in
disp
layi
ng so
me
of th
e da
ta.
• I i
nclu
de so
me
of th
e la
bels
, ope
ratio
n sy
mbo
ls, t
itles
, and
/or k
eys f
or th
e ty
pe o
f di
spla
y or
gra
ph I
chos
e.
• I s
tate
my
conc
lusi
on p
artly
supp
orte
d fr
om
my
repr
esen
tatio
n.
1
• M
y w
ork
is n
ot o
rgan
ized
, doe
s not
mak
e se
nse,
an
d is
ver
y co
nfus
ing.
•
I inc
lude
an
expl
anat
ion
of fe
w o
r no
ne o
f the
st
eps n
eede
d to
solv
e th
e pr
oble
m.
• M
y ex
plan
atio
n do
es n
ot d
escr
ibe
my
stra
tegy
in
a sk
illfu
l way
, is n
ot o
rgan
ized
, and
doe
s not
m
ake
sens
e.
• I d
o no
t exp
lain
why
I ch
ose
this
stra
tegy
to
solv
e th
e pr
oble
m a
nd I
incl
ude
little
or n
o in
form
atio
n ne
eded
to su
ppor
t my
stra
tegy
. •
I do
not u
se p
rope
r lab
els,
deta
ils, s
ymbo
ls, a
nd
corr
ect m
ath
term
s in
my
expl
anat
ion.
• M
y re
pres
enta
tion
is n
ot c
orre
ct in
di
spla
ying
the
data
. •
I inc
lude
few
or
none
of t
he la
bels
, op
erat
ion
sym
bols
, titl
es, a
nd/o
r key
s for
th
e ty
pe o
f dis
play
or g
raph
I ch
ose.
•
I sta
te m
y co
nclu
sion
with
out s
uppo
rt fr
om
my
repr
esen
tatio
n.
Whe
n th
e co
mm
unic
atio
n an
d re
ason
ing
rubr
ics a
re u
sed
sepa
rate
ly, t
hey
both
ass
ess a
stud
ents
’ abi
lity
to u
se th
e pr
oper
labe
ls, d
etai
ls, sy
mbo
ls, a
nd m
ath
term
s in
the
expl
anat
ion.
Whe
n th
e in
divi
dual
rubr
ics a
re u
nite
d to
cr
eate
the
com
bina
tion
rubr
ic th
e st
atem
ent c
once
rnin
g th
e us
e of
labe
ls, e
tc, b
ecom
es re
dund
ant.
Ther
efor
e, th
e la
bels
stat
emen
t is w
ritte
n on
ly in
the
reas
onin
g co
lum
n, w
here
it is
mos
t app
ropr
iate
.
Mathematics Instructional Rubrics – Section Two 38
Mat
hem
atic
s Com
bina
tion
Rub
ric
– M
iddl
e Sc
hool
Lev
el
Com
mun
icat
ions
Th
is is
how
I so
lve
the
prob
lem
.
Rea
soni
ng
This
is w
hy I
solv
e th
e pr
oble
m u
sing
a se
lect
stra
tegy
.
Rep
rese
ntat
ion
This
is h
ow I
repr
esen
t the
pro
blem
and
solu
tion.
4
• M
y re
spon
se is
wel
l org
aniz
ed a
nd is
lo
gica
l. •
I inc
lude
an
accu
rate
, com
plet
e, a
nd
thor
ough
exp
lana
tion
of a
ll of
the
step
s ne
eded
to c
orre
ctly
solv
e th
e pr
oble
m.
• M
y ex
plan
atio
n of
the
stra
tegy
incl
udes
all
deta
ils a
nd is
wel
l org
aniz
ed.
• I f
ully
exp
lain
why
I ch
ose
this
effe
ctiv
e an
d ef
ficie
nt st
rate
gy th
at le
ads t
o a
com
plet
ely
corr
ect c
oncl
usio
n or
solu
tion.
•
I use
all
of th
e pr
oper
con
cept
s, la
bels
, det
ails
, sy
mbo
ls, a
nd c
orre
ct m
ath
term
s in
my
expl
anat
ion.
• M
y re
pres
enta
tion
is c
ompl
etel
y ac
cura
te a
nd
incl
udes
all
of th
e da
ta a
nd a
ll of
the
nece
ssar
y re
latio
nshi
ps in
the
prob
lem
. •
I inc
lude
all
of th
e la
bels
, ope
ratio
n sy
mbo
ls,
title
s, an
d/or
key
s for
the
type
of r
epre
sent
atio
n I
chos
e.
• I s
tate
my
corr
ect c
oncl
usio
n co
mpl
etel
y su
ppor
ted
from
my
repr
esen
tatio
n.
3
• M
y re
spon
se is
mos
tly o
rgan
ized
or i
s lo
gica
l. •
I inc
lude
an
accu
rate
exp
lana
tion
of m
ost o
f th
e st
eps n
eede
d to
cor
rect
ly so
lve
the
prob
lem
.
• M
y ex
plan
atio
n of
the
stra
tegy
incl
udes
mos
t de
tails
and
is o
rgan
ized
. •
I mos
tly e
xpla
in w
hy I
chos
e th
is e
ffect
ive
stra
tegy
that
lead
s to
a co
mpl
etel
y co
rrec
t co
nclu
sion
or s
olut
ion.
•
I use
mos
t of t
he p
rope
r con
cept
s, la
bels
, de
tails
, sym
bols
, and
cor
rect
mat
h te
rms i
n m
y ex
plan
atio
n.
• M
y re
pres
enta
tion
is m
ostly
acc
urat
e an
d in
clud
es
mos
t of t
he d
ata
and
mos
t of t
he n
eces
sary
re
latio
nshi
ps in
the
prob
lem
. •
I inc
lude
mos
t of t
he la
bels
, ope
ratio
n sy
mbo
ls,
title
s, an
d/or
key
s for
the
type
of r
epre
sent
atio
n I
chos
e.
• I s
tate
my
corr
ect c
oncl
usio
n m
ostly
supp
orte
d fr
om m
y re
pres
enta
tion.
2
• M
y re
spon
se is
poo
rly
orga
nize
d or
is
illog
ical
. •
I inc
lude
an
expl
anat
ion
of so
me
of th
e st
eps n
eede
d to
solv
e th
e pr
oble
m.
• M
y ex
plan
atio
n of
the
stra
tegy
incl
udes
som
e de
tails
and
is d
isor
gani
zed.
•
I par
tially
exp
lain
why
I ch
ose
this
stra
tegy
that
le
ads t
o a
part
ially
cor
rect
con
clus
ion
or
solu
tion.
•
I use
som
e of
the
prop
er c
once
pts,
labe
ls,
deta
ils, s
ymbo
ls, a
nd c
orre
ct m
ath
term
s in
my
expl
anat
ion.
• M
y re
pres
enta
tion
is p
artia
lly a
ccur
ate
and
incl
udes
som
e of
the
data
and
som
e of
the
nece
ssar
y re
latio
nshi
ps in
the
prob
lem
. •
I inc
lude
som
e of
the
labe
ls, o
pera
tion
sym
bols
, tit
les,
and/
or k
eys f
or th
e ty
pe o
f rep
rese
ntat
ion
I ch
ose.
•
I sta
te m
y co
nclu
sion
par
tly su
ppor
ted
from
my
repr
esen
tatio
n or
my
repr
esen
tatio
n le
ads t
o an
un
stat
ed c
oncl
usio
n.
1
• M
y re
spon
se is
not
org
aniz
ed a
nd is
ill
ogic
al.
• I i
nclu
de a
n ex
plan
atio
n of
few
or
none
of
the
step
s nee
ded
to so
lve
the
prob
lem
.
• M
y ex
plan
atio
n of
the
stra
tegy
incl
udes
few
or
no d
etai
ls a
nd is
not
org
aniz
ed.
• I d
o no
t exp
lain
why
I ch
ose
this
stra
tegy
that
le
ads t
o an
inco
rrec
t con
clus
ion
or so
lutio
n.
• I u
se fe
w o
r no
ne o
f the
pro
per c
once
pts,
labe
ls, d
etai
ls, s
ymbo
ls, a
nd c
orre
ct m
ath
term
s in
my
expl
anat
ion.
• M
y re
pres
enta
tion
is n
ot a
ccur
ate
and
incl
udes
few
or
non
e of
the
data
or n
eces
sary
rela
tions
hips
in
the
prob
lem
. •
I inc
lude
few
or
none
of t
he la
bels
, ope
ratio
n sy
mbo
ls, t
itles
, and
/or k
eys f
or th
e ty
pe o
f re
pres
enta
tion
I cho
se.
• I s
tate
my
conc
lusi
on w
ithou
t sup
port
from
my
repr
esen
tatio
n or
I do
not
stat
e a
conc
lusi
on.
W
hen
the
com
mun
icat
ion
and
reas
onin
g ru
bric
s are
use
d se
para
tely
, the
y bo
th a
sses
s a st
uden
ts’ a
bilit
y to
use
the
prop
er la
bels
, det
ails,
sym
bols,
and
mat
h te
rms i
n th
e ex
plan
atio
n. W
hen
the
indi
vidu
al ru
bric
s are
uni
ted
to
crea
te th
e co
mbi
natio
n ru
bric
the
stat
emen
t con
cern
ing
the
use
of la
bels,
etc
, bec
omes
redu
ndan
t. Th
eref
ore,
the
labe
ls st
atem
ent i
s writ
ten
only
in th
e re
ason
ing
colu
mn,
whe
re it
is m
ost a
ppro
pria
te.
Mathematics Instructional Rubrics – Section Two 39
Whe
n th
e co
mm
unic
atio
n an
d re
ason
ing
rubr
ics a
re u
sed
sepa
rate
ly, t
hey
both
ass
ess a
stud
ents
’ abi
lity
to u
se th
e pr
oper
labe
ls, d
etai
ls, sy
mbo
ls, a
nd m
ath
term
s in
the
expl
anat
ion.
Whe
n th
e in
divi
dual
rubr
ics a
re u
nite
d to
cr
eate
the
com
bina
tion
rubr
ic th
e st
atem
ent c
once
rnin
g th
e us
e of
labe
ls, e
tc, b
ecom
es re
dund
ant.
Ther
efor
e, th
e st
atem
ent i
s lef
t in
the
reas
onin
g ru
bric
, whe
re it
is m
ost a
ppro
pria
te.
M
athe
mat
ics C
ombi
natio
n R
ubri
c –
Hig
h Sc
hool
Lev
el
Com
mun
icat
ion
This
is h
ow I
solv
e th
e pr
oble
m.
R
easo
ning
M
y im
plem
enta
tion
incl
udes
org
aniz
ed m
athe
mat
ical
st
eps/
proc
edur
es a
nd re
leva
nt d
etai
ls.
R
epre
sent
atio
n Th
is is
how
I re
pres
ent t
he p
robl
em a
nd so
lutio
n.
4 •
My
resp
onse
is th
orou
gh, w
ell o
rgan
ized
, and
lo
gica
l whe
n de
scrib
ing
all m
y pr
oced
ures
. •
I inc
lude
a c
ompl
ete
and
corr
ect e
xpla
natio
n us
ing
all o
f the
rele
vant
and
spec
ific
deta
ils
from
the
prob
lem
whe
n de
scrib
ing
the
proc
edur
es u
sed
to a
rriv
e at
the
corr
ect s
olut
ion.
• M
y im
plem
enta
tion
is fu
lly d
evel
oped
and
co
mpl
etel
y su
ppor
ted
by re
leva
nt a
nd sp
ecifi
c de
tails
from
the
prob
lem
. •
I inc
lude
a c
orre
ct e
xpla
natio
n ba
sed
on
mat
hem
atic
al tr
uths
of w
hy I
sele
ct th
e co
ncep
ts
and
repr
esen
tatio
ns p
rese
nted
in m
y co
rrec
t so
lutio
n.
• I u
se a
ll of
the
labe
ls, s
ymbo
ls, c
once
pts,
term
inol
ogy,
and
repr
esen
tatio
ns a
ccur
atel
y,
clea
rly, a
nd su
ccin
ctly
in m
y re
spon
se.
• M
y re
pres
enta
tion(
s) is
wel
l pre
sent
ed w
ith
all d
etai
ls w
ell e
xecu
ted
in th
at it
is c
ompl
ete,
ac
cura
te, c
lear
, cor
rect
, and
eas
y to
inte
rpre
t. •
I rep
rese
nt th
e el
emen
ts in
an
insi
ghtfu
l se
lect
ion/
form
at th
at il
lust
rate
s all
of th
e ne
cess
ary
rela
tions
hips
in th
e pr
oble
m.
• I s
tate
my
corr
ect c
oncl
usio
n an
d/or
ge
nera
lizat
ion
com
plet
ely
supp
orte
d fr
om
my
repr
esen
tatio
n.
3
• M
y re
spon
se is
mos
tly o
rgan
ized
and
logi
cal
whe
n de
scrib
ing
mos
t of m
y pr
oced
ures
. •
I inc
lude
a c
orre
ct e
xpla
natio
n us
ing
mos
t of t
he
rele
vant
and
spec
ific
deta
ils fr
om th
e pr
oble
m
whe
n de
scrib
ing
the
proc
edur
es u
sed
to a
rriv
e at
th
e co
rrec
t sol
utio
n.
• M
y im
plem
enta
tion
is m
ostly
dev
elop
ed a
nd
supp
orte
d by
rele
vant
and
spec
ific
deta
ils fr
om th
e pr
oble
m.
• I i
nclu
de a
cor
rect
, exp
lana
tion
base
d on
m
athe
mat
ical
trut
hs o
f why
I se
lect
the
conc
epts
an
d re
pres
enta
tions
pre
sent
ed in
my
corr
ect
solu
tion.
•
I use
mos
t of t
he la
bels
, sym
bols
, con
cept
s, te
rmin
olog
y, a
nd re
pres
enta
tions
acc
urat
ely,
cl
early
, and
succ
inct
ly in
my
resp
onse
.
• M
y re
pres
enta
tion(
s) is
app
ropr
iate
in th
at it
is
mos
tly c
ompl
ete,
acc
urat
e, c
lear
, cor
rect
, an
d ea
sy to
inte
rpre
t. •
I rep
rese
nt th
e el
emen
ts in
an
appr
opria
te
sele
ctio
n/fo
rmat
that
illu
stra
tes m
ost o
f the
ne
cess
ary
rela
tions
hips
in th
e pr
oble
m.
• I s
tate
my
corr
ect c
oncl
usio
n an
d/or
ge
nera
lizat
ion
mos
tly su
ppor
ted
from
my
repr
esen
tatio
n.
2
• M
y re
spon
se is
poo
rly
orga
nize
d or
is il
logi
cal
whe
n de
scrib
ing
som
e of
my
proc
edur
es.
• I i
nclu
de a
n ex
plan
atio
n us
ing
som
e of
the
rele
vant
and
spec
ific
deta
ils fr
om th
e pr
oble
m
whe
n de
scrib
ing
the
proc
edur
es u
sed
to a
rriv
e at
a
solu
tion.
• M
y im
plem
enta
tion
is so
mew
hat d
evel
oped
and
su
ppor
ted
by re
leva
nt a
nd sp
ecifi
c de
tails
from
the
prob
lem
. •
I inc
lude
a p
artia
lly c
orre
ct a
nd so
mew
hat
orga
nize
d ex
plan
atio
n of
why
I se
lect
the
conc
epts
an
d re
pres
enta
tions
pre
sent
ed in
my
solu
tion.
•
I use
som
e of
the
labe
ls, s
ymbo
ls, c
once
pts,
term
inol
ogy,
and
repr
esen
tatio
ns a
ccur
atel
y in
my
resp
onse
.
• M
y re
pres
enta
tion(
s) is
som
ewha
t com
plet
e,
accu
rate
, and
/or p
artia
lly c
orre
ct, a
nd so
me
parts
are
eas
y to
inte
rpre
t. •
I par
tially
repr
esen
t the
ele
men
ts in
a
sele
ctio
n/fo
rmat
that
illu
stra
tes s
ome
of th
e ne
cess
ary
rela
tions
hips
in th
e pr
oble
m.
• I s
tate
my
conc
lusi
on a
nd/o
r gen
eral
izat
ion
part
ly su
ppor
ted
from
my
repr
esen
tatio
n.
1
• M
y re
spon
se is
not
org
aniz
ed a
nd is
illo
gica
l w
hen
desc
ribin
g fe
w o
r no
ne o
f my
proc
edur
es.
• I i
nclu
de a
n ex
plan
atio
n us
ing
few
or
none
of
the
rele
vant
or s
peci
fic d
etai
ls fr
om th
e pr
oble
m
whe
n de
scrib
ing
the
proc
edur
es.
• M
y im
plem
enta
tion
is r
arel
y su
ppor
ted
by
rele
vant
and
spec
ific
deta
ils fr
om th
e pr
oble
m.
• I i
nclu
de a
n in
corr
ect,
diso
rgan
ized
, or m
inim
al
expl
anat
ion
of w
hy I
sele
ct th
e co
ncep
ts a
nd
repr
esen
tatio
ns p
rese
nted
in m
y so
lutio
n.
• I u
se fe
w o
r no
ne o
f the
labe
ls, s
ymbo
ls, c
once
pts,
term
inol
ogy,
or r
epre
sent
atio
ns in
my
resp
onse
.
• M
y re
pres
enta
tion(
s) is
inco
mpl
ete,
in
accu
rate
, inc
orre
ct, a
nd is
diff
icul
t to
inte
rpre
t. •
I rep
rese
nt th
e el
emen
ts in
a se
lect
ion/
form
at
that
illu
stra
te fe
w o
r no
ne o
f the
nec
essa
ry
rela
tions
hips
in th
e pr
oble
m.
• I s
tate
my
conc
lusi
on a
nd/o
r gen
eral
izat
ion
with
out s
uppo
rt fr
om m
y re
pres
enta
tion.
Mathematics Instructional Rubrics – Section Three 40
OVERVIEW OF THE RUBROVERVIEW OF THE RUBROVERVIEW OF THE RUBROVERVIEW OF THE RUBRIC, PROMPT, AND ANCHIC, PROMPT, AND ANCHIC, PROMPT, AND ANCHIC, PROMPT, AND ANCHOR OR OR OR
PAPER SECTIONPAPER SECTIONPAPER SECTIONPAPER SECTION
This section of the manual includes everything that is needed to get started using the Mathematics Instructional Rubrics (MIR) in the classroom. The following components are included in this section:
• Rubrics - scoring guides that can be used with students to instruct them on how to communicate, reason, and represent their answers to open-ended questions. Included with the rubrics are tips from teachers on how to introduce the rubrics and how to get students accustomed to using rubrics to improve their responses.
• Prompts - questions or problems that were used to pilot the rubrics with
students and orient them to the concept of scoring their own work. Please note that some prompts were used to pilot more than one rubric.
• Anchor Papers - examples of student papers that illustrate each level or
point on the rubric.* When beginning to use rubrics with students it is important to remember that they are designed to be student tools. Students must be taught how to use them and then be given feedback to encourage growth in their responses. It is through practice that students refine their responses and move to the proficient level. It is important to remember that students need to be comfortable with these tools. This comfort is gained through continual practice and feedback.
The anchor papers provide students with examples of how other students in the state solved the problem or prompt that is provided. Teachers are encouraged to create their own anchor papers and prompts to allow students to see what is considered an exemplary answer. Additional sample prompts are included in the resource section of this document, and others may also be found on the websites listed in Section Four. *Note: Teachers using student papers as examples in the classroom should adhere to the Federal Educational Rights and Privacy Act (FERPA) guidelines.
Mathematics Instructional Rubrics – Section Three 41
Rubric: Communication - Analytic – High School Level Problem: Take-Home Pay Pilot Teacher Tips to Introduce the Rubric: Present the Communications Analytic Rubric and explain that it is a four-point rubric. Help the students understand the analytic rubric for communication by examining each of the traits and the differences between levels. Emphasize to students that communication is the hallmark of success for employment in today’s world. Students’ ability to communicate mathematically truly demonstrates their depth of understanding of mathematical concepts. Additional Teacher Comments:
• As a teacher with 35 years experience, using these rubrics is most stimulating. It demands time and expertise, and is rewarding.
• Often students do not see the sense of having to do so much writing to substantiate a correct solution. They wonder if the rubrics are testing them on their mathematical ability or their language arts ability.
• Students need different levels of intensity and explicitness of instruction to master the use of the rubrics. Some students need only a brief explanation while others need consistent and explicit instruction, practice, and feedback.
• Some students need to be taught the vocabulary of the rubrics to understand them. The words “organized, logical, well, most, poorly, not,” etc., have to be taught.
• Some of the tasks need to be broken down into smaller units for some special education students.
Mathematics Instructional Rubrics – Section 42
Mat
hem
atic
al C
omm
unic
atio
n - A
naly
tic R
ubri
c - H
igh
Scho
ol L
evel
Th
is is
how
I so
lve
the
prob
lem
.
Org
aniz
atio
n Is
my
resp
onse
org
aniz
ed to
show
how
I so
lve
the
prob
lem
?
E
xpla
natio
n D
oes m
y ex
plan
atio
n in
clud
e ho
w I s
olve
the
prob
lem
?
U
se o
f Mat
hem
atic
al T
erm
s D
oes m
y re
spon
se in
clud
e ac
cura
te la
bels
, sy
mbo
ls, c
once
pts,
term
inol
ogy,
and
re
pres
enta
tions
?
4 M
y re
spon
se is
thor
ough
, wel
l org
aniz
ed a
nd
logi
cal.
I inc
lude
a c
ompl
ete
and
corr
ect e
xpla
natio
n us
ing
all o
f the
rele
vant
and
spec
ific
deta
ils
from
the
prob
lem
whe
n de
scrib
ing
the
proc
edur
es u
sed
to a
rriv
e at
the
corr
ect s
olut
ion.
I use
all
of th
e la
bels
, sym
bols
, con
cept
s, te
rmin
olog
y, a
nd re
pres
enta
tions
acc
urat
ely,
cl
early
, and
succ
inct
ly in
my
resp
onse
.
3 M
y re
spon
se is
mos
tly o
rgan
ized
and
logi
cal.
I inc
lude
a c
orre
ct e
xpla
natio
n us
ing
mos
t of t
he
rele
vant
and
spec
ific
deta
ils fr
om th
e pr
oble
m
whe
n de
scrib
ing
the
proc
edur
es u
sed
to a
rriv
e at
th
e co
rrec
t sol
utio
n.
I use
mos
t of t
he la
bels
, sym
bols
, con
cept
s, te
rmin
olog
y, a
nd re
pres
enta
tions
acc
urat
ely,
cl
early
, and
succ
inct
ly in
my
resp
onse
.
2 M
y re
spon
se is
poo
rly
orga
nize
d or
is
illog
ical
.
I inc
lude
an
expl
anat
ion
usin
g so
me
of th
e re
leva
nt a
nd sp
ecifi
c de
tails
from
the
prob
lem
w
hen
desc
ribin
g th
e pr
oced
ures
use
d to
arr
ive
at
a so
lutio
n.
I use
som
e of
the
labe
ls, s
ymbo
ls, c
once
pts,
term
inol
ogy,
and
repr
esen
tatio
ns a
ccur
atel
y in
m
y re
spon
se.
1 M
y re
spon
se is
not
org
aniz
ed a
nd is
illo
gica
l.
I inc
lude
an
expl
anat
ion
usin
g fe
w o
r no
ne o
f th
e re
leva
nt o
r spe
cific
det
ails
from
the
prob
lem
w
hen
desc
ribin
g th
e pr
oced
ures
.
I use
few
or
none
of t
he la
bels
, sym
bols
, co
ncep
ts, t
erm
inol
ogy,
or r
epre
sent
atio
ns in
my
resp
onse
.
Mathematics Instructional Rubrics – Section Three 43
Take-Home Pay Toni is paid $18 an hour as a plumber. For overtime (time beyond 40 hours) she earns 1 and 1/2 times her normal rate. The regular deductions from her pay are federal taxes (14%), state tax (3.1%), local tax (1%), social security tax (7.5%), and union dues (1.2%). Last week Toni worked 52 hours. What was her net (take-home) pay? Justify your answer by explaining all of the steps you used. Do all work for this problem on this page. Remember you must show all the steps you used to solve the problem even if you have used a calculator. To receive the highest score, all calculation steps must be shown and explained in writing. Numeric answers must always be labeled.
PromptCommunicationAnalytic Rubric
Mathematics Instructional Rubrics – Section Three 44
Mathematics Instructional Rubrics Explanation of Score Points for
Student Anchor Paper
Date: 3/10/01 Grade Level: High School
Title of Rubric:Title of Rubric:Title of Rubric:Title of Rubric: Communications - Analytic
Set of comments: Score Point isScore Point isScore Point isScore Point is 4 \ 4 \ 4 because:
The explanation is thorough, well organized, and logical throughout. A complete and correct explanation is provided, including specific details and computations. All labels and symbols ($, %, X, +, -) are provided. The paper includes the correct solution.
Mathematics Instructional Rubrics – Section Three 45
Student Anchor Paper Score Point 4/4/4
Mathematics Instructional Rubrics – Section Three 46
Mathematics Instructional Rubrics Explanation of Score Points for
Student Anchor Paper
Date: 3/10/01 Grade Level: High School
Title of Rubric:Title of Rubric:Title of Rubric:Title of Rubric: Communications - Analytic
Set of comments: Score Point isScore Point isScore Point isScore Point is 4 \ 3 \ 4 because:
The response is thorough, well organized and logical. Most of the relevant and specific details were included when describing the procedures. Student neglected to include specific information about how the $27 overtime rate was computed. All labels, symbols, concepts, terms, and representations were included.
Mathematics Instructional Rubrics – Section Three 47
Student Anchor Paper Score Point 4/3/4
Mathematics Instructional Rubrics – Section Three 48
Mathematics Instructional Rubrics Explanation of Score Points for
Student Anchor Paper
Date: 3/10/01 Grade Level: High School
Title of Rubric:Title of Rubric:Title of Rubric:Title of Rubric: Communications - Analytic
Set of comments: Score Point isScore Point isScore Point isScore Point is 4 \ 2 \ 4 because:
The explanation is thorough and well organized in describing all procedures. A complete and detailed explanation is given with one computational rounding error leading to an incorrect solution. All labels, symbols, and terminology (percent, multiplied, divided, deductions) are used clearly and accurately.
Mathematics Instructional Rubrics – Section Three 49
Student Anchor Paper Score Point 4/2/4
Mathematics Instructional Rubrics – Section Three 50
Mathematics Instructional Rubrics Explanation of Score Points for
Student Anchor Paper
Date: 3/10/01 Grade Level: High School
Title of Rubric:Title of Rubric:Title of Rubric:Title of Rubric: Communications - Analytic
Set of comments: Score Point isScore Point isScore Point isScore Point is 3 \ 1 \ 4 because:
The work is mostly organized mathematically and logically, however description of categories are not specifically given. An explanation is completely lacking. All mathematical terms are accurately denoted.
Mathematics Instructional Rubrics – Section Three 51
Student Anchor Paper Score Point 3/1/4
Mathematics Instructional Rubrics – Section Three 52
Mathematics Instructional Rubrics Explanation of Score Points for
Student Anchor Paper
Date: 3/10/01 Grade Level: High School
Title of Rubric:Title of Rubric:Title of Rubric:Title of Rubric: Communications - Analytic
Set of comments: Score Point isScore Point isScore Point isScore Point is 3 \ 2 \ 2 because:
Solution is mostly organized but difficult to follow. The procedure is correct, but the solution is incorrect. Labels are used, but are not clear.
Mathematics Instructional Rubrics – Section Three 53
Student Anchor Paper Score Point 3/2/2
Mathematics Instructional Rubrics – Section Three 54
Mathematics Instructional Rubrics Explanation of Score Points for
Student Anchor Paper
Date: 3/10/01 Grade Level: High School
Title of Rubric:Title of Rubric:Title of Rubric:Title of Rubric: Communications - Analytic
Set of comments: Score Point isScore Point isScore Point isScore Point is 1 \ 1 \ 1 because:
The response is not organized and does not show the procedure followed. The response contains few relevant or specific details, including incorrect computation. There are few symbols shown and labeling is very limited.
Mathematics Instructional Rubrics – Section Three 55
Student Anchor Paper Score Point 1/1/1
Mathematics Instructional Rubrics – Section Three 56
Rubric: Reasoning - Analytic – High School Level Problem: Theater Tickets Pilot Teacher Tips to Introduce the Rubric: Prior to having students use the rubrics, have them experience the same activity as part of a rubrics team. Put students into groups of four or five and give them the Straw Building problem and group materials.*
In the straw building activity, students ask questions like: “Do the three shapes have to be different?” “Can we cut or bend the straws?” “What is a rubric?” “Can we have more time?” “What does self-supporting mean?”
After students complete the activity, present the rubric. Some students express anger that they had not known that height mattered. Give them five additional minutes and some extra tape to modify their building. Then evaluate the buildings based upon the rubric. This activity is a segue to the rational for using rubrics. Additional Teacher Comments:
• The analytic rubric seems to be the most straightforward and least intimidating for the students.
• Chose simple problems to begin introducing rubrics. • The students think that with the analytical rubric they have a better chance to pick up points
because of the way it is broken into accuracy, format, and conclusion. • Many of students are used to rubrics since they are utilized in performance-based
assessments in all disciplines. • The hardest part of using rubrics is picking out good questions. Once these are selected,
the rest is smooth sailing. The students are very good and the results are basically what one would expect.
• The vocabulary of the rubric must be taught. • Feedback regarding what the student has done well and its relationship to the rubric and
what the student needs to improve needs to be given.
* A lesson plan for this activity can be found in the resource section of this manual.
Mathematics Instructional Rubrics – Section Three 57
Mat
hem
atic
al R
easo
ning
- A
naly
tic R
ubri
c –
Hig
h Sc
hool
D
oes m
y im
plem
enta
tion
incl
ude
orga
nize
d m
athe
mat
ical
step
s/pr
oced
ures
and
rele
vant
det
ails
?
Impl
emen
tatio
n of
the
Stra
tegy
D
oes m
y im
plem
enta
tion
incl
ude
orga
nize
d st
eps,
proc
edur
es, a
nd
rele
vant
det
ails
?
R
atio
nale
for
the
Stra
tegy
Cho
ice
Doe
s my
expl
anat
ion
reve
al w
hy I
sele
ct
the
conc
epts
pre
sent
ed in
my
solu
tion?
U
se o
f Mat
hem
atic
al T
erm
s D
oes m
y re
spon
se in
clud
e ac
cura
te
labe
ls, s
ymbo
ls, c
once
pts,
term
inol
ogy,
and
repr
esen
tatio
ns?
4
My
impl
emen
tatio
n is
fully
de
velo
ped
and
is c
ompl
etel
y su
ppor
ted
by re
leva
nt a
nd sp
ecifi
c de
tails
from
the
prob
lem
.
I inc
lude
a c
orre
ct, t
horo
ugh,
and
wel
l-or
gani
zed
expl
anat
ion
com
plet
ely
base
d up
on m
athe
mat
ical
trut
hs o
f why
I se
lect
th
e co
ncep
ts a
nd re
pres
enta
tions
pre
sent
ed
in m
y co
rrec
t sol
utio
n.
I use
all
of th
e la
bels
, sym
bols
, co
ncep
ts, t
erm
inol
ogy,
and
re
pres
enta
tions
acc
urat
ely,
cle
arly
, and
su
ccin
ctly
in m
y re
spon
se.
3 M
y im
plem
enta
tion
is m
ostly
de
velo
ped
and
supp
orte
d by
rele
vant
an
d sp
ecifi
c de
tails
from
the
prob
lem
.
I inc
lude
a c
orre
ct e
xpla
natio
n ba
sed
on
mat
hem
atic
al tr
uths
of w
hy I
sele
ct th
e co
ncep
ts a
nd re
pres
enta
tions
pre
sent
ed in
m
y co
rrec
t sol
utio
n.
I use
mos
t of t
he la
bels
, sym
bols
, co
ncep
ts, t
erm
inol
ogy,
and
re
pres
enta
tions
acc
urat
ely,
cle
arly
, and
su
ccin
ctly
in m
y re
spon
se.
2 M
y im
plem
enta
tion
is so
mew
hat
deve
lope
d an
d su
ppor
ted
by re
leva
nt
and
spec
ific
deta
ils fr
om th
e pr
oble
m.
I inc
lude
a p
artia
lly c
orre
ct a
nd so
mew
hat
orga
nize
d ex
plan
atio
n of
why
I se
lect
the
conc
epts
and
repr
esen
tatio
ns p
rese
nted
in
my
solu
tion.
I use
som
e of
the
labe
ls, s
ymbo
ls,
conc
epts
, ter
min
olog
y, a
nd
repr
esen
tatio
ns a
ccur
atel
y in
my
resp
onse
.
1 M
y im
plem
enta
tion
is r
arel
y su
ppor
ted
by re
leva
nt a
nd sp
ecifi
c de
tails
from
the
prob
lem
.
I inc
lude
an
inco
rrec
t, di
sorg
aniz
ed, o
r m
inim
al e
xpla
natio
n of
why
I se
lect
the
conc
epts
and
repr
esen
tatio
ns p
rese
nted
in
my
solu
tion.
I use
few
or
none
of t
he la
bels
, sy
mbo
ls, c
once
pts,
term
inol
ogy,
or
repr
esen
tatio
ns in
my
resp
onse
.
Mathematics Instructional Rubrics – Section Three 58
Theater Tickets
Movie theaters keep track of how many tickets they sell for each show. The Penn Theater sells adult’s tickets for $7.00 and child’s tickets for $4.00. For a recent show, the Penn sold 272 tickets and collected $1,694 in ticket sales. How many of each type of ticket were sold? For this problem, show all of your steps (even if you used a calculator) and explain why you did each step.
PromptReasoning
Analytic Rubric
Mathematics Instructional Rubrics – Section Three 59
Mathematics Instructional Rubrics Explanation of Score Points for
Student Anchor Paper
Date: 3/10/01 Grade Level: High School
Title of Rubric:Title of Rubric:Title of Rubric:Title of Rubric: Reasoning - Analytic
Set of comments: Score Point isScore Point isScore Point isScore Point is 4 / 4 / 4 because:
The implementation is thorough and contains specific detail (e.g., total tickets, money, identified variables). The explanation is well organized and includes reasons for selecting the representation (2 variables – 2 equations, solving by substitution and check). Labels, concepts, symbols, and terms are accurate and clear (variables defined and solve system of equation, substitution).
Mathematics Instructional Rubrics – Section Three 60
Student Anchor Paper Score Point 4/4/4
Mathematics Instructional Rubrics – Section Three 61
Mathematics Instructional Rubrics Explanation of Score Points for
Student Anchor Paper
Date: 3/10/01 Grade Level: High School
Title of Rubric:Title of Rubric:Title of Rubric:Title of Rubric: Reasoning - Analytic
Set of comments: Score Point isScore Point isScore Point isScore Point is 4 \ 3 \ 4 because:
The implementation is fully developed and contains specific details from the problem (identification of variables). The explanation is correct and organized, but not thorough. Reason for setting up 2 equations with 2 variables is missing. Correctly describes the process of elimination but never identifies it by name as elimination or linear combination. Labels, symbols, concepts, terms, and representations are accurate and clear (answers labeled, use of variables, and processes of substitution and linear combination).
Mathematics Instructional Rubrics – Section Three 62
Student Anchor Paper Score Point 4/3/4
Mathematics Instructional Rubrics – Section Three 63
Mathematics Instructional Rubrics Explanation of Score Points for
Student Anchor Paper
Date: 3/10/01 Grade Level: High School
Title of RuTitle of RuTitle of RuTitle of Rubric:bric:bric:bric: Reasoning - Analytic
Set of comments: Score Point isScore Point isScore Point isScore Point is 3 / 1 / 2 because:
Student has a thorough mathematical solution with some organization concerns. Student does not explain why they selected processes/procedures. Student labeled variables and solutions but did not use mathematical terminology.
Mathematics Instructional Rubrics – Section Three 64
Student Anchor Paper Score Point 3/1/2
Mathematics Instructional Rubrics – Section Three 65
Mathematics Instructional Rubrics Explanation of Score Points for
Student Anchor Paper
Date: 3/10/01 Grade Level: High School
Title of Rubric:Title of Rubric:Title of Rubric:Title of Rubric: Reasoning - Analytic
Set of comments: Score Point isScore Point isScore Point isScore Point is 2 / 3 / 2 because:
The student’s work is somewhat developed and the method contains errors. Correct, well-organized step-by-step process leads to the solution. Labels are incorrect and are missing.
Mathematics Instructional Rubrics – Section Three 66
Student Anchor Paper Score Point 2/3/2
Mathematics Instructional Rubrics – Section Three 67
Mathematics Instructional Rubrics Explanation of Score Points for
Student Anchor Paper
Date: 3/10/01 Grade Level: High School
Title of Rubric:Title of Rubric:Title of Rubric:Title of Rubric: Reasoning - Analytic
Set of comments: Score Point isScore Point isScore Point isScore Point is 2 / 2 / 2 because:
The implementation is sometimes supported (make an equation). Although the explanation is weak, it is not incorrect and it is not disorganized. Some label terms are used ($, answer labeled, system of equations, matrix).
Mathematics Instructional Rubrics – Section Three 68
Student Anchor Paper Score Point 2/2/2
Mathematics Instructional Rubrics – Section Three 69
Mathematics Instructional Rubrics Explanation of Score Points for
Student Anchor Paper
Date: 3/10/01 Grade Level: High School
Title of Rubric:Title of Rubric:Title of Rubric:Title of Rubric: Reasoning - Analytic
Set of comments: Score Point isScore Point isScore Point isScore Point is 1 / 1 / 1 because:
Implementation is limited with insufficient connections to the problem. No explanation on why the matrix equation was used or how it was solved. The variables were identified, but solutions were not explained.
Mathematics Instructional Rubrics – Section Three 70
Student Anchor Paper Score Point 1/1/1
Mathematics Instructional Rubrics – Section Three 71
Rubric: Representation - Analytic – High School Level Problem: National Honor Society Luncheon Pilot Teacher Tips to Introduce the Rubric: Begin the process by giving the students the open-ended question to solve. Do not give them a rubric to follow at first, nor answer any of their questions. Essentially use this activity as a pre-test. Then present the analytic rubric to the students. Use the rubric to grade the pre-tests. Students are generally surprised to see how many aspects of the rubric are missing from their solutions. Next, have the students work through a problem together using the rubric as a guide. Students comment on how much explanation is necessary to get a “4.” It is then time to give them a problem to solve on their own using the rubric. Additional Teacher Comments:
• The analytic rubric seems to be the most straightforward and least intimidating for the students.
• The students like the format that separates it into “Accuracy, Format, and Conclusion.” • Present the PSSA rubric to a class along with various examples from last year’s PSSA
exam. This helps to show the students the type of responses that are expected. • Students have done very little writing in mathematics in the past. Most students seem
very hesitant to clearly explain their work and would rather the work stand on its own merit.
• Students seem to be scared to “problem solve.” They would much rather be presented with an idea or topic, learn about it, and be quizzed or tested on it. They hesitate to put the knowledge they have previously learned to use and apply it to a current situation.
• The analytic rubrics are much easier to use and students would rather be graded using them because they offer more of an opportunity to score well since the students’ work is scored in more than one category.
Mathematics Instructional Rubrics – Section Three 72
Mat
hem
atic
al R
epre
sen
tati
on -
An
alyt
ic R
ub
ric
– Hig
h S
choo
l L
evel
T
his
is h
ow I
rep
rese
nt
the
prob
lem
an
d so
luti
on.
Acc
ura
cy
Do
I ac
cura
tely
rep
rese
nt
all o
f m
y da
ta?
F
orm
at
Do
I co
rrec
tly
rep
rese
nt
all n
eces
sary
re
lati
onsh
ips?
C
oncl
usi
on
Do
I st
ate
my
con
clu
sion
an
d/or
ge
ner
aliz
atio
n w
ith
su
ppor
t fr
om m
y re
pres
enta
tion
?
4
My
repr
esen
tati
on(s
) is
wel
l pr
esen
ted
wit
h a
ll d
etai
ls w
ell
exec
ute
d in
th
at it
is c
ompl
ete,
ac
cura
te, c
lear
, cor
rect
, an
d ea
sy
to in
terp
ret.
I re
pres
ent
the
elem
ents
in a
n
insi
ghtf
ul s
elec
tion
/for
mat
th
at
illu
stra
tes
all
of t
he
nec
essa
ry
rela
tion
ship
s in
th
e pr
oble
m.
I st
ate
my
corr
ect
con
clu
sion
an
d/or
gen
eral
izat
ion
co
mp
lete
ly s
upp
orte
d fr
om m
y re
pres
enta
tion
.
3
My
repr
esen
tati
on(s
) is
ap
prop
riat
e in
th
at it
is m
ostl
y co
mpl
ete,
acc
ura
te, c
lear
, cor
rect
, an
d ea
sy t
o in
terp
ret.
I re
pres
ent
the
elem
ents
in a
n
appr
opri
ate
sele
ctio
n/f
orm
at t
hat
il
lust
rate
s m
ost
of t
he
nec
essa
ry
rela
tion
ship
s in
th
e pr
oble
m.
I st
ate
my
corr
ect
con
clu
sion
an
d/or
gen
eral
izat
ion
mos
tly
supp
orte
d fr
om m
y re
pres
enta
tion
.
2 M
y re
pres
enta
tion
(s)
is
som
ewh
at c
ompl
ete,
acc
ura
te
and/
or p
arti
ally
cor
rect
, an
d so
me
part
s ar
e ea
sy t
o in
terp
ret.
I pa
rtia
lly
repr
esen
t th
e el
emen
ts
in a
sel
ecti
on/f
orm
at t
hat
il
lust
rate
s so
me
of t
he
nec
essa
ry
rela
tion
ship
s in
th
e pr
oble
m.
I st
ate
my
con
clu
sion
an
d/or
ge
ner
aliz
atio
n p
artl
y su
ppor
ted
from
my
repr
esen
tati
on.
1 M
y re
pres
enta
tion
(s)
is
inco
mpl
ete,
inac
cura
te, i
nco
rrec
t,
and
is d
iffi
cult
to
inte
rpre
t.
I re
pres
ent
the
elem
ents
in a
se
lect
ion
/for
mat
th
at il
lust
rate
few
or
non
e of
th
e n
eces
sary
re
lati
onsh
ips
in t
he
prob
lem
.
I m
ay o
r m
ay n
ot s
tate
my
con
clu
sion
an
d/or
gen
eral
izat
ion
w
ith
out
supp
ort
from
my
repr
esen
tati
on.
Mathematics Instructional Rubrics – Section Three 73
National Honor Society Luncheon The National Honor Society is planning their annual luncheon honoring new members and their families. Costs from two catering companies are being considered by the club. Delicious Catering Services charges $20 per person and Tastee Caterers charge a flat fee of $800 plus $10 per person. Write a system of equations to represent this situation and solve the system by graphing. Show all of your work. What advice would you give the National Honor Society to help them decide what caterer to use? Include an explanation of why you chose to give this advice.
PromptRepresentation
Analytic Rubric
Mathematics Instructional Rubrics – Section Three 74
Mathematics Instructional Rubrics Explanation of Score Points for
Student Anchor Paper
Date: 5/21/01 Grade Level: High School
Title of Rubric:Title of Rubric:Title of Rubric:Title of Rubric: Representation - Analytic
Set of comments: Score Point isScore Point isScore Point isScore Point is 4 \ 4 \ 4 because:
The table and graph are accurate with all details included. All elements are represented. The conclusion is completely supported by the representations.
Mathematics Instructional Rubrics – Section Three 75
Student Anchor Paper Score Point 4/4/4
Mathematics Instructional Rubrics – Section Three 76
Mathematics Instructional Rubrics Explanation of Score Points for
Student Anchor Paper
Date: 3/10/01 Grade Level: High School
Title of Rubric:Title of Rubric:Title of Rubric:Title of Rubric: Representation - Analytic
Set of comments: Score Point isScore Point isScore Point isScore Point is 4 \ 3 \ 4 because:
The table and graph are accurate with all details included (labels, numeric increments, etc.). All necessary relationships are identified with one exception. Although the 2 equations are correct they are never appropriately associated to the companies (Delicious and Tastee). The correct conclusion is completely supported from the representation (use of “according to the graph,” “the line is lower,” the “line climbs slower”).
Mathematics Instructional Rubrics – Section Three 77
Student Anchor Paper Score Point 4/3/4
Mathematics Instructional Rubrics – Section Three 78
Mathematics Instructional Rubrics Explanation of Score Points for
Student Anchor Paper
Date: 5/21/01 Grade Level: High School
Title of Rubric:Title of Rubric:Title of Rubric:Title of Rubric: Representation - Analytic
Set of comments: Score Point isScore Point isScore Point isScore Point is 3 \ 3 \ 2 because:
The graph is correct but sideways, making it difficult to interpret. The format selected is appropriate however the table does not show the point of intersection. The conclusion is partially supported. The point of intersection (80 people) is not mentioned.
Mathematics Instructional Rubrics – Section Three 79
Student Anchor Paper Score Point 3/3/2
Mathematics Instructional Rubrics – Section Three 80
Mathematics Instructional Rubrics Explanation of Score Points for
Student Anchor Paper
Date: 5/21/01 Grade Level: High School
Title of Rubric:Title of Rubric:Title of Rubric:Title of Rubric: Representation - Analytic
Set of comments: Score Point isScore Point isScore Point isScore Point is 2 \ 3 \ 1 because:
The graph’s scale and labels are missing. The table and equations are accurate. The graph poorly represents data from the table. Most elements are represented in an appropriate format. Incorrect conclusion is not supported.
Mathematics Instructional Rubrics – Section Three 81
Student Anchor Paper Score Point 2/3/1
Mathematics Instructional Rubrics – Section Three 82
Mathematics Instructional Rubrics Explanation of Score Points for
Student Anchor Paper
Date: 5/21/01 Grade Level: High School
Title of Rubric:Title of Rubric:Title of Rubric:Title of Rubric: Representation - Analytic
Set of comments: Score Point isScore Point isScore Point isScore Point is 1 \ 2 \ 1 because:
Variables are not identified. Graph is not labeled. The equations partially represent elements of the problem. The graph shows the intersection of two appropriate lines but it is not labeled. The conclusion is incorrect and is not supported by the representation.
Mathematics Instructional Rubrics – Section Three 83
Student Anchor Paper Score Point 1/2/1
Mathematics Instructional Rubrics – Section Three 84
Rubric: Communication - Holistic – High School Level Problem: Take-Home Pay Pilot Teacher Tips to Introduce the Rubric: It is helpful to introduce rubrics to students by writing a number of open-ended problems. Select several to use with your classes. Many students are already familiar with the PSSA Mathematics Test. Present the PSSA rubric to your classes, along with various released problems form last year’s PSSA exam. Present this rubric. Explain the differences between this rubric and the PSSA rubric. Then give this problem to solve using the holistic four-point scale. Additional Teacher Comments:
• After using the rubric, ask students what additional information can be put into the rubric to assist them in obtaining better scores. The big question seems to surround the difference between “how” a solution is computed and “why” that solution was selected. This emphasizes the need to have students continually explain why they do what they do.
• As a first year teacher, working with rubrics revealed some very important lessons. The first discovery is that students have done very little writing in mathematics in the past. Most seem very hesitant to clearly explain their work. On the other hand, the enthusiasm that is shown by most students in working with rubrics is very moving. They try very hard to do their best in completing the open-ended tasks. All in all, rubrics work very well to improve student learning.
Mathematics Instructional Rubrics – Section Three 85
Mat
hem
atic
al C
omm
unic
atio
n - H
olis
tic R
ubri
c - H
igh
Scho
ol L
evel
Th
is is
how
I so
lve
the
prob
lem
.
4 •
My
resp
onse
is th
orou
gh, w
ell o
rgan
ized
, and
logi
cal.
• I i
nclu
de a
com
plet
e an
d co
rrec
t exp
lana
tion
usin
g al
l of t
he re
leva
nt a
nd sp
ecifi
c det
ails
from
the
prob
lem
w
hen
desc
ribi
ng th
e pr
oced
ures
use
d to
arr
ive
at th
e cor
rect
solu
tion.
•
I use
all
of th
e la
bels
, sym
bols
, con
cept
s, te
rmin
olog
y, a
nd re
pres
enta
tions
acc
urat
ely,
clea
rly
and
succ
inct
ly in
my
resp
onse
.
3 •
My
resp
onse
is m
ostl
y or
gani
zed
and
logi
cal.
• I i
nclu
de a
corr
ect e
xpla
natio
n us
ing
mos
t of t
he re
leva
nt a
nd sp
ecifi
c det
ails
from
the
prob
lem
whe
n de
scri
bing
the
proc
edur
es u
sed
to a
rriv
e at
the
corr
ect s
olut
ion.
•
I use
mos
t of t
he la
bels
, sym
bols
, con
cept
s, te
rmin
olog
y, a
nd re
pres
enta
tions
acc
urat
ely,
clea
rly,
and
su
ccin
ctly
in m
y re
spon
se.
2
• M
y re
spon
se is
poo
rly
orga
nize
d or
is il
logi
cal.
• I i
nclu
de a
n ex
plan
atio
n us
ing
som
e of
the
rele
vant
and
spec
ific d
etai
ls fr
om th
e pr
oble
m w
hen
desc
ribi
ng
the
proc
edur
es u
sed
to a
rriv
e at
a so
lutio
n.
• I u
se s
ome
of th
e la
bels
, sym
bols
, con
cept
s, te
rmin
olog
y, a
nd re
pres
enta
tions
acc
urat
ely
in m
y re
spon
se.
1
• M
y re
spon
se is
not
org
aniz
ed a
nd is
illo
gica
l. •
I inc
lude
an
expl
anat
ion
usin
g fe
w o
r no
ne o
f the
rele
vant
or s
peci
fic d
etai
ls fr
om th
e pr
oble
m w
hen
desc
ribi
ng th
e pr
oced
ures
. •
I use
few
or
none
of t
he la
bels
, sym
bols
, con
cept
s, te
rmin
olog
y, o
r rep
rese
ntat
ions
in m
y re
spon
se.
Mathematics Instructional Rubrics – Section Three 86
Take-Home Pay Toni is paid $18 an hour as a plumber. For overtime (time beyond 40 hours) she earns 1 and 1/2 times her normal rate. The regular deductions from her pay are federal taxes (14%), state tax (3.1%), local tax (1%), social security tax (7.5%), and union dues (1.2%). Last week Toni worked 52 hours. What was her net (take-home) pay? Justify your answer by explaining all of the steps you used. Do all work for this problem on this page. Remember you must show all the steps you used to solve the problem even if you have used a calculator. To receive the highest score, all calculation steps must be shown and explained in writing. Numerical answers must always be labeled.
PromptCommunicationHolistic Rubric
Mathematics Instructional Rubrics – Section Three 87
Mathematics Instructional Rubrics Explanation of Score Points for
Student Anchor Paper
Date: 3/10/01 Grade Level: High School
Title of Rubric:Title of Rubric:Title of Rubric:Title of Rubric: Communication - Holistic
Set of comments: Score Point isScore Point isScore Point isScore Point is 4 because:
The explanation is well organized and describes all procedures logically and sequentially. All of the relevant and specific details from the problem are noted with correct terms (regular pay, overtime pay, percentages, decimals). The paper includes the correct solution.
Mathematics Instructional Rubrics – Section Three 88
Student Anchor Paper Score Point 4
Mathematics Instructional Rubrics – Section Three 89
Mathematics Instructional Rubrics Explanation of Score Points for
Student Anchor Paper
Date: 3/10/01 Grade Level: High School
Title of Rubric:Title of Rubric:Title of Rubric:Title of Rubric: Communication - Holistic
Set of comments: Score Point isScore Point isScore Point isScore Point is 3 because:
The explanation is mostly organized and describes most of the procedures logically. It lacks an explanation of how the overtime rate and number of hours was found. The explanation includes proper concepts and labels. Mathematical terms used are correct (percentages, money deducted, add, multiple, divide). The paper includes the correct solution.
Mathematics Instructional Rubrics – Section Three 90
Student Anchor Paper Score Point 3
Mathematics Instructional Rubrics – Section Three 91
Mathematics Instructional Rubrics Explanation of Score Points for
Student Anchor Paper
Date: 3/10/01 Grade Level: High School
Title of Title of Title of Title of Rubric:Rubric:Rubric:Rubric: Communication - Holistic
Set of comments: Score Point isScore Point isScore Point isScore Point is 2 because:
The computation is partially accurate. There are errors in computing tax deductions result in an incorrect solution (net pay). The explanations of how the tax deductions were computed are unclear. There is minimal use of labeling and poor use of mathematical terminology.
Mathematics Instructional Rubrics – Section Three 92
Student Anchor Paper Score Point 2
Mathematics Instructional Rubrics – Section Three 93
Mathematics Instructional Rubrics Explanation of Score Points for
Student Anchor Paper
Date: 3/10/01 Grade Level: High School
Title of Rubric:Title of Rubric:Title of Rubric:Title of Rubric: Communication - Holistic
Set of comments: Score Point isScore Point isScore Point isScore Point is 1 because:
Although the student gives the correct answer, the response is not organized with few procedures shown. Details are lacking in the computation of individual totals. There is minimal labeling. The representation of the process of finding total tax deductions is incorrect.
Mathematics Instructional Rubrics – Section Three 94
Student Anchor Paper Score Point 1
Mathematics Instructional Rubrics – Section Three 95
Rubric: Reasoning - Holistic – High School Level Problem: Theater Tickets Pilot Teacher Tips so Introduce the Rubric: Students need to understand what mathematical reasoning is. Reasoning is about “why I did what I did” and not about how the problem was solved. Teach your students to connect the steps to the solution with justifications. To prepare for the use of rubrics and this problem, discuss the Holistic Reasoning Rubric on an overhead. Have the students read it and discuss it while working through it. Hand out the Holistic Reasoning Rubric to the students, the formula page from the PSSA, and a blank piece of scrap paper. Give students adequate time to complete the problem (most need about 20 minutes). Additional Teacher Comments:
• Use the Holistic Reasoning Rubric with a geometry class. Provide the students with a copy of the rubric and stress to the class what a “4” represents.
• For additional practice with rubrics create a four-point rubric that students use for completing their homework assignments.
• The vocabulary of the rubrics must be taught. • Students need different amounts of practice and feedback with rubrics.
Mathematics Instructional Rubrics – Section Three 96
Mat
hem
atic
al R
easo
nin
g -
Hol
isti
c R
ub
ric
- H
igh
Sch
ool
Lev
el
My
impl
emen
tati
on in
clu
des
orga
niz
ed m
ath
emat
ical
ste
ps/p
roce
dure
s an
d re
leva
nt
deta
ils.
4 •
My
impl
emen
tati
on is
fu
lly
deve
lope
d an
d co
mpl
etel
y su
ppor
ted
by r
elev
ant
and
spec
ific
de
tail
s fr
om t
he
prob
lem
. •
I in
clu
de a
cor
rect
, th
orou
gh, a
nd
wel
l org
aniz
ed e
xpla
nat
ion
bas
ed o
n m
ath
emat
ical
tru
ths
of
wh
y I
sele
ct t
he
con
cept
s an
d re
pres
enta
tion
s pr
esen
ted
in m
y co
rrec
t so
luti
on.
• I
use
all
of
the
labe
ls, s
ymbo
ls, c
once
pts,
ter
min
olog
y, a
nd
repr
esen
tati
ons
accu
rate
ly, c
lear
ly,
and
succ
inct
ly in
my
resp
onse
.
3 •
My
impl
emen
tati
on is
mos
tly
deve
lope
d an
d su
ppor
ted
by r
elev
ant
and
spec
ific
det
ails
fro
m
the
prob
lem
. •
I in
clu
de a
cor
rect
exp
lan
atio
n b
ased
on
mat
hem
atic
al t
ruth
s of
wh
y I
sele
ct t
he
con
cept
s an
d re
pres
enta
tion
s pr
esen
ted
in m
y co
rrec
t so
luti
on.
• I
use
mos
t of
th
e la
bels
, sym
bols
, con
cept
s, t
erm
inol
ogy,
an
d re
pres
enta
tion
s ac
cura
tely
, cl
earl
y, a
nd
succ
inct
ly in
my
resp
onse
.
2 •
My
impl
emen
tati
on is
som
ewh
at d
evel
oped
an
d su
ppor
ted
by r
elev
ant
and
spec
ific
det
ails
fr
om t
he
prob
lem
. •
I in
clu
de a
par
tial
ly c
orre
ct a
nd
som
ewh
at o
rgan
ized
exp
lan
atio
n o
f w
hy
I se
lect
th
e co
nce
pts
and
repr
esen
tati
ons
pres
ente
d in
my
solu
tion
. •
I u
se s
ome
of t
he
labe
ls, s
ymbo
ls, c
once
pts,
ter
min
olog
y, a
nd
repr
esen
tati
ons
accu
rate
ly in
my
resp
onse
.
1 •
My
impl
emen
tati
on is
rar
ely
supp
orte
d by
rel
evan
t an
d sp
ecif
ic d
etai
ls f
rom
th
e pr
oble
m.
• I
incl
ude
an
inco
rrec
t, d
isor
gan
ized
, or
min
imal
exp
lan
atio
n o
f w
hy
I se
lect
th
e co
nce
pts
and
repr
esen
tati
ons
pres
ente
d in
my
solu
tion
. •
I u
se f
ew o
r n
one
of t
he
labe
ls, s
ymbo
ls, c
once
pts,
ter
min
olog
y, o
r re
pres
enta
tion
s in
my
resp
onse
.
Mathematics Instructional Rubrics – Section Three 97
Theater Tickets
Movie theaters keep track of how many tickets they sell for each show. The Penn Theater sells adult’s tickets for $7.00 and child’s tickets for $4.00. For a recent show, the Penn sold 272 tickets and collected $1694 in ticket sales. How many of each type of ticket were sold? For this problem, show all of your steps (even if you used a calculator) and explain why you did each step.
PromptReasoning
Holistic Rubric
Mathematics Instructional Rubrics – Section Three 98
Mathematics Instructional Rubrics Explanation of Score Points for
Student Anchor Paper
Date: 3/10/01 Grade Level: High School
Title of Rubric:Title of Rubric:Title of Rubric:Title of Rubric: Reasoning - Holistic
Set of comments: Score Point isScore Point isScore Point isScore Point is 4 because:
The implementation is thorough and contains specific detail (total tickets, money, identified adult and children’s tickets). The explanation is well organized and includes reasons for selecting the representation (two variables, two equations, solving by substitution, checking, correct solution). Labels, concepts, symbols, and terms are accurate and clear (variables defined, solve system of equation, substitution).
Mathematics Instructional Rubrics – Section Three 99
Student Anchor Paper Score Point 4
Mathematics Instructional Rubrics – Section Three 100
Mathematics Instructional Rubrics Explanation of Score Points for
Student Anchor Paper
Date: 3/10/01 Grade Level: High School
Title of Rubric:Title of Rubric:Title of Rubric:Title of Rubric: Reasoning - Holistic
Set of comments: Score Point isScore Point isScore Point isScore Point is 3 because:
The implementation is mostly supported by the problem (division by seven is unclear). It is correct and well organized and the solution is correct. Reasons for selecting operations and processes are missing. Most of the labels, concepts and terms are clear. The use of variables (a, c, x) is inconsistent and unclear.
Mathematics Instructional Rubrics – Section Three 101
Student Anchor Paper Score Point 3
Mathematics Instructional Rubrics – Section Three 102
Mathematics Instructional Rubrics Explanation of Score Points for
Student Anchor Paper
Date: 3/10/01 Grade Level: High School
Title of Rubric:Title of Rubric:Title of Rubric:Title of Rubric: Reasoning - Holistic
Set of comments: Score Point isScore Point isScore Point isScore Point is 3 because:
The implementation is mostly supported by details from the problem (“this means that”). A correct and organized explanation is included with a correct solution, however the explanation is not thorough. Most of the symbols and concepts are used accurately. Money is not labeled and there is some inconsistent use of dollars and cents (7.00, 1694).
Mathematics Instructional Rubrics – Section Three 103
Student Anchor Paper Score Point 3
Mathematics Instructional Rubrics – Section Three 104
Mathematics Instructional Rubrics Explanation of Score Points for
Student Anchor Paper
Date: 3/10/01 Grade Level: High School
Title of Rubric:Title of Rubric:Title of Rubric:Title of Rubric: Reasoning - Holistic
Set of comments: Score Point isScore Point isScore Point isScore Point is 2 because:
The implementation is sometimes supported by the problem (system of equations, matrix). A somewhat organized explanation of why the matrix representation was chosen is given. Although the solution is correct, the explanation of the reasoning is weak. Some concepts and representations are accurate.
Mathematics Instructional Rubrics – Section Three 105
Student Anchor Paper Score Point 2
Mathematics Instructional Rubrics – Section Three 106
Mathematics Instructional Rubrics Explanation of Score Points for
Student Anchor Paper
Date: 3/10/01 Grade Level: High School
Title of Rubric:Title of Rubric:Title of Rubric:Title of Rubric: Reasoning - Holistic
Set of comments: Score Point isScore Point isScore Point isScore Point is 1 because:
The implementation is rarely supported by detail (multiplied the number of adults by ....). The explanation is incorrect and disorganized. Few labels, concepts, or terms are used in the response.
Mathematics Instructional Rubrics – Section Three 107
Student Anchor Paper Score Point 1
Mathematics Instructional Rubrics – Section Three 108
Rubric: Representation - Holistic – High School Level Problem: National Honor Society Pilot Teacher Tips to Introduce the Rubric: Teach your students that representations take on many forms. It is important that students understand the essential characteristics of each form (e.g., graphs, symbols, diagrams, tables). Explain rubrics to students. Hand out the rubric and ask students to read it. Discuss how a rubric tells them what to do. When the students have no more questions, hand out a practice problem. Grade the practice problem as a group and discuss the criteria necessary to score at the level 4. Additional Teacher Comments:
• There are usually no problems using the rubrics if students are experienced at using them in mathematics.
• The hardest part of the experience is picking out good questions to use.
Mathematics Instructional Rubrics – Section Three 109
Mat
hem
atic
al R
epre
sen
tati
on -
Hol
isti
c R
ub
ric
– Hig
h S
choo
l L
evel
T
his
is h
ow I
rep
rese
nt
the
prob
lem
an
d so
luti
on.
4
• M
y re
pres
enta
tion
(s)
is w
ell p
rese
nte
d w
ith
all
det
ails
wel
l exe
cute
d in
th
at it
is c
ompl
ete,
ac
cura
te, c
lear
, cor
rect
, an
d ea
sy t
o in
terp
ret.
•
I re
pres
ent
the
elem
ents
in a
n in
sigh
tfu
l sel
ecti
on/f
orm
at t
hat
illu
stra
tes
all
of t
he
nec
essa
ry
rela
tion
ship
s in
th
e pr
oble
m.
• I
stat
e m
y co
rrec
t co
ncl
usi
on a
nd/
or g
ener
aliz
atio
n c
omp
lete
ly s
upp
orte
d fr
om m
y re
pres
enta
tion
.
3 •
My
repr
esen
tati
on(s
) is
app
ropr
iate
in t
hat
it is
mos
tly
com
plet
e, a
ccu
rate
, cle
ar, c
orre
ct, a
nd
easy
to
inte
rpre
t.
• I
repr
esen
t th
e el
emen
ts in
an
app
ropr
iate
sel
ecti
on/f
orm
at t
hat
illu
stra
tes
mos
t of
th
e n
eces
sary
re
lati
onsh
ips
in t
he
prob
lem
. •
I st
ate
my
corr
ect
con
clu
sion
an
d/or
gen
eral
izat
ion
mos
tly
supp
orte
d fr
om m
y re
pres
enta
tion
.
2 •
My
repr
esen
tati
on(s
) is
som
ewh
at c
ompl
ete,
acc
ura
te, a
nd/
or p
arti
ally
cor
rect
an
d so
me
part
s ar
e ea
sy t
o in
terp
ret.
•
I pa
rtia
lly
repr
esen
t th
e el
emen
ts in
a s
elec
tion
/for
mat
th
at il
lust
rate
s so
me
of t
he
nec
essa
ry
rela
tion
ship
s in
th
e pr
oble
m.
• I
stat
e m
y co
ncl
usi
on a
nd/
or g
ener
aliz
atio
n p
artl
y su
ppor
ted
from
my
repr
esen
tati
on.
1
• M
y re
pres
enta
tion
(s)
is in
com
plet
e, in
accu
rate
, in
corr
ect,
an
d is
dif
ficu
lt t
o in
terp
ret.
•
I re
pres
ent
the
elem
ents
in a
sel
ecti
on/f
orm
at t
hat
illu
stra
te f
ew o
r n
one
of t
he
nec
essa
ry
rela
tion
ship
s in
th
e pr
oble
m.
• I
may
or
may
not
sta
te m
y co
ncl
usi
on a
nd/
or g
ener
aliz
atio
n w
ith
out
supp
ort
from
my
repr
esen
tati
on.
Mathematics Instructional Rubrics – Section Three 110
National Honor Society Luncheon The National Honor Society is planning their annual luncheon honoring new members and their families. Costs from two catering companies are being considered by the club. Delicious Catering Services charges $20 per person and Tastee Caterers charge a flat fee of $800 plus $10 per person. Write a system of equations to represent this situation and solve the system by graphing. Show all of your work. What advice would you give the National Honor Society to help them decide what caterer to use? Include an explanation of why you chose to give this advice.
PromptRepresentationHolistic Rubric
Mathematics Instructional Rubrics – Section Three 111
Mathematics Instructional Rubrics Explanation of Score Points for
Student Anchor Paper
Date: 5/21/01 Grade Level: High School
Title Title Title Title of Rubric:of Rubric:of Rubric:of Rubric: Representation - Holistic
Set of comments: Score Point isScore Point isScore Point isScore Point is 4 because:
All details are well presented and represented. The correct conclusion is well supported by the representation.
Mathematics Instructional Rubrics – Section Three 112
Student Anchor Paper Score Point 4
Mathematics Instructional Rubrics – Section Three 113
Mathematics Instructional Rubrics Explanation of Score Points for
Student Anchor Paper
Date: 5/21/01 Grade Level: High School
Title of Rubric:Title of Rubric:Title of Rubric:Title of Rubric: Representation - Holistic
Set of comments: ScoScoScoScore Point isre Point isre Point isre Point is 4 because:
All details are well presented. All necessary relationships are represented. The conclusion is completely supported by the representation.
Mathematics Instructional Rubrics – Section Three 114
Student Anchor Paper Score Point 4
Mathematics Instructional Rubrics – Section Three 115
Mathematics Instructional Rubrics Explanation of Score Points for
Student Anchor Paper
Date: 5/21/01 Grade Level: High School
Title of Rubric:Title of Rubric:Title of Rubric:Title of Rubric: Representation - Holistic
Set of comments: Score Point isScore Point isScore Point isScore Point is 3 because:
The explanation is supported but the conclusion stating why the advice is given is unclear. The graph is not labeled.
Mathematics Instructional Rubrics – Section Three 116
Student Anchor Paper Score Point 3
Mathematics Instructional Rubrics – Section Three 117
Mathematics Instructional Rubrics Explanation of Score Points for
Student Anchor Paper
Date: 5/21/01 Grade Level: High School
Title of Rubric:Title of Rubric:Title of Rubric:Title of Rubric: Representation - Holistic
Set of comments: Score PScore PScore PScore Point isoint isoint isoint is 2 because:
The incorrect conclusion is supported by the inaccurately drawn graph, but is not supported by the correct table.
Mathematics Instructional Rubrics – Section Three 118
Student Anchor Paper Score Point 2
Mathematics Instructional Rubrics – Section Three 119
Mathematics Instructional Rubrics Explanation of Score Points for
Student Anchor Paper
Date: 5/21/01 Grade Level: High School
Title of Rubric:Title of Rubric:Title of Rubric:Title of Rubric: Representation - Holistic
Set of comments: Score Point isScore Point isScore Point isScore Point is 1 because:
The general conclusion is not specifically supported by the correct table and graph. The graph is not labeled.
Mathematics Instructional Rubrics – Section Three 120
Student Anchor Paper Score Point 1
Mathematics Instructional Rubrics – Section Three 121
Mathematics Instructional Rubrics Explanation of Score Points for
Student Anchor Paper
Date: 5/21/01 Grade Level: High School
Title of Rubric:Title of Rubric:Title of Rubric:Title of Rubric: Representation - Holistic
Set of comments: Score Point isScore Point isScore Point isScore Point is 1 because:
Although the table is correct and the graph is correct (but not labeled), the conclusion is incorrect and not supported by the representation.
Mathematics Instructional Rubrics – Section Three 122
Student Anchor Paper Score Point 1
Mathematics Instructional Rubrics – Section Three 123
Rubric: Combination - High School Level Problem: Linear Programming Problem Pilot Teacher Tips to Introduce the Rubric: To use the combination rubric, carefully select problems that require all traits of the communication, reasoning, and representation rubrics. Provide an overview and/or a review on the framework of solving the problem before students begin the problem. A suggested framework follows:
1. Assign variables, x=, y= 2. Write inequalities 3. Graph inequalities 4. Find maximum/minimum values 5. Evaluate maximum/minimum values into cost function.
After students are given the framework, instruct them to work alone or with a partner to develop a solution. Initially, all students are able to start the problem, but not solve it. Of the students who solve the problem, some include all of the necessary work and explanations, while some only include partial work and explanations. Use these student papers with the class to illustrate what additional steps they need to take to improve their work.
Additional Teacher Comments:
• Follow the procedure of presenting the rubric first and then having the students do a problem using the rubric as a guide.
• Distribute the rubric to the students and again review the 3-column idea. Remind the students when they solve a problem they are to think in terms of three columns: “How” (what we are doing); “Why” (why we use these steps) and “Representation” (the mathematics). Practice solving problems selected from SAT materials.
• As the students use the rubrics, they get better at explaining the “why.” Typical comments from students include the fact that they are able to get an expression, but are unable to explain why it works. They can understand what they are looking for, but cannot explain or show it.
• The most difficult part of this experience is trying to find additional problems that fit the rubrics. It is helpful to work out a problem first, decide what kind of answer you are looking for, and then design the rubric.
Mathematics Instructional Rubrics – Section Three 124
Whe
n th
e co
mm
unic
atio
n an
d re
ason
ing
rubr
ics a
re u
sed
sepa
rate
ly, t
hey
both
ass
ess a
stud
ents
’ abi
lity
to u
se th
e pr
oper
labe
ls, d
etai
ls, sy
mbo
ls, a
nd m
ath
term
s in
the
expl
anat
ion.
Whe
n th
e in
divi
dual
rubr
ics a
re u
nite
d to
cr
eate
the
com
bina
tion
rubr
ic th
e st
atem
ent c
once
rnin
g th
e us
e of
labe
ls, e
tc, b
ecom
es re
dund
ant.
Ther
efor
e, th
e st
atem
ent i
s lef
t in
the
reas
onin
g ru
bric
, whe
re it
is m
ost a
ppro
pria
te.
Mat
hem
atic
s Com
bina
tion
Rub
ric
– H
igh
Scho
ol L
evel
Com
mun
icat
ion
This
is h
ow I
solv
e th
e pr
oble
m.
Rea
soni
ng
My
impl
emen
tati
on in
clu
des
orga
niz
ed m
ath
emat
ical
st
eps/
proc
edu
res
and
rele
van
t de
tail
s.
Rep
rese
ntat
ion
Th
is is
how
I r
epre
sen
t th
e pr
oble
m a
nd
solu
tion
.
4
• M
y re
spon
se is
thor
ough
, wel
l org
aniz
ed, a
nd
logi
cal w
hen
desc
ribi
ng a
ll m
y pr
oced
ures
. •
I inc
lude
a co
mpl
ete
and
corr
ect e
xpla
natio
n us
ing
all o
f the
rele
vant
and
spec
ific d
etai
ls
from
the
prob
lem
whe
n de
scri
bing
the
proc
edur
es u
sed
to a
rriv
e at
the
corr
ect
solu
tion.
• M
y im
plem
enta
tion
is f
ull
y de
velo
ped
and
com
plet
ely
supp
orte
d by
rel
evan
t an
d sp
ecif
ic
deta
ils
from
th
e pr
oble
m.
• I
incl
ude
a c
orre
ct e
xpla
nat
ion
bas
ed o
n
mat
hem
atic
al t
ruth
s of
wh
y I
sele
ct t
he
con
cept
s an
d re
pres
enta
tion
s pr
esen
ted
in m
y co
rrec
t so
luti
on.
• I
use
all
of
the
labe
ls, s
ymbo
ls, c
once
pts,
te
rmin
olog
y, a
nd
repr
esen
tati
ons
accu
rate
ly,
clea
rly,
an
d su
ccin
ctly
in m
y re
spon
se.
• M
y re
pres
enta
tion
(s)
is w
ell p
rese
nte
d w
ith
al
l de
tail
s w
ell e
xecu
ted
in t
hat
it is
co
mpl
ete,
acc
ura
te, c
lear
, cor
rect
, an
d ea
sy
to in
terp
ret.
•
I re
pres
ent
the
elem
ents
in a
n in
sigh
tfu
l se
lect
ion
/for
mat
th
at il
lust
rate
s al
l of
th
e n
eces
sary
rel
atio
nsh
ips
in t
he
prob
lem
. •
I st
ate
my
corr
ect
con
clu
sion
an
d/or
ge
ner
aliz
atio
n c
omp
lete
ly s
upp
orte
d fr
om
my
repr
esen
tati
on.
3
• M
y re
spon
se is
mos
tly
orga
nize
d an
d lo
gica
l w
hen
desc
ribi
ng m
ost o
f my
proc
edur
es.
• I i
nclu
de a
corr
ect e
xpla
natio
n us
ing
mos
t of
the
rele
vant
and
spec
ific d
etai
ls fr
om th
e pr
oble
m w
hen
desc
ribi
ng th
e pr
oced
ures
use
d to
arr
ive
at th
e co
rrec
t sol
utio
n.
• M
y im
plem
enta
tion
is m
ostl
y de
velo
ped
and
supp
orte
d by
rel
evan
t an
d sp
ecif
ic d
etai
ls f
rom
th
e pr
oble
m.
• I
incl
ude
a c
orre
ct, e
xpla
nat
ion
bas
ed o
n
mat
hem
atic
al t
ruth
s of
wh
y I
sele
ct t
he
con
cept
s an
d re
pres
enta
tion
s pr
esen
ted
in m
y co
rrec
t so
luti
on.
• I
use
mos
t of
th
e la
bels
, sym
bols
, con
cept
s,
term
inol
ogy,
an
d re
pres
enta
tion
s ac
cura
tely
, cl
earl
y an
d su
ccin
ctly
in m
y re
spon
se.
• M
y re
pres
enta
tion
(s)
is a
ppro
pria
te in
th
at
it is
mos
tly
com
plet
e, a
ccu
rate
, cle
ar,
corr
ect,
an
d ea
sy t
o in
terp
ret.
•
I re
pres
ent
the
elem
ents
in a
n a
ppro
pria
te
sele
ctio
n/f
orm
at t
hat
illu
stra
tes
mos
t of
th
e n
eces
sary
rel
atio
nsh
ips
in t
he
prob
lem
. •
I st
ate
my
corr
ect
con
clu
sion
an
d/or
ge
ner
aliz
atio
n m
ostl
y su
ppor
ted
from
my
repr
esen
tati
on.
2
• M
y re
spon
se is
poo
rly
orga
nize
d or
is il
logi
cal
whe
n de
scri
bing
som
e of
my
proc
edur
es.
• I i
nclu
de a
n ex
plan
atio
n us
ing
som
e of
the
rele
vant
and
spec
ific d
etai
ls fr
om th
e pr
oble
m
whe
n de
scri
bing
the
proc
edur
es u
sed
to a
rriv
e at
a so
lutio
n.
• M
y im
plem
enta
tion
is s
omew
hat
dev
elop
ed a
nd
supp
orte
d by
rel
evan
t an
d sp
ecif
ic d
etai
ls f
rom
th
e pr
oble
m.
• I
incl
ude
a p
arti
ally
cor
rect
an
d so
mew
hat
or
gan
ized
exp
lan
atio
n o
f w
hy
I se
lect
th
e co
nce
pts
and
repr
esen
tati
ons
pres
ente
d in
my
solu
tion
. •
I u
se s
ome
of t
he
labe
ls, s
ymbo
ls, c
once
pts,
te
rmin
olog
y, a
nd
repr
esen
tati
ons
accu
rate
ly in
m
y re
spon
se.
• M
y re
pres
enta
tion
(s)
is s
omew
hat
co
mpl
ete,
acc
ura
te, a
nd/
or p
arti
ally
cor
rect
, an
d so
me
part
s ar
e ea
sy t
o in
terp
ret.
•
I pa
rtia
lly
repr
esen
t th
e el
emen
ts in
a
sele
ctio
n/f
orm
at t
hat
illu
stra
tes
som
e of
th
e n
eces
sary
rel
atio
nsh
ips
in t
he
prob
lem
. •
I st
ate
my
con
clu
sion
an
d/or
gen
eral
izat
ion
p
artl
y su
ppor
ted
from
my
repr
esen
tati
on.
1
• M
y re
spon
se is
not
org
aniz
ed a
nd is
illo
gica
l w
hen
desc
ribi
ng fe
w o
r no
ne o
f my
proc
edur
es.
• I i
nclu
de a
n ex
plan
atio
n us
ing
few
or
none
of
the
rele
vant
or s
peci
fic d
etai
ls fr
om th
e pr
oble
m w
hen
desc
ribi
ng th
e pr
oced
ures
.
• M
y im
plem
enta
tion
is r
arel
y su
ppor
ted
by
rele
van
t an
d sp
ecif
ic d
etai
ls f
rom
th
e pr
oble
m.
• I
incl
ude
an
inco
rrec
t, d
isor
gan
ized
, or
min
imal
ex
plan
atio
n o
f w
hy
I se
lect
th
e co
nce
pts
and
repr
esen
tati
ons
pres
ente
d in
my
solu
tion
. •
I u
se f
ew o
r n
one
of t
he
labe
ls, s
ymbo
ls,
con
cept
s, t
erm
inol
ogy,
or
repr
esen
tati
ons
in m
y re
spon
se.
• M
y re
pres
enta
tion
(s)
is in
com
plet
e,
inac
cura
te, i
nco
rrec
t, a
nd
is d
iffi
cult
to
inte
rpre
t.
• I
repr
esen
t th
e el
emen
ts in
a
sele
ctio
n/f
orm
at t
hat
illu
stra
te f
ew o
r n
one
of t
he
nec
essa
ry r
elat
ion
ship
s in
th
e pr
oble
m.
• I
stat
e m
y co
ncl
usi
on a
nd/
or g
ener
aliz
atio
n
wit
hou
t su
ppor
t fr
om m
y re
pres
enta
tion
.
Mathematics Instructional Rubrics – Section Three 125
Linear Programming Problem
The Industrial Arts teacher works with students after school to make tables and chairs to sell so they can raise funds to go on a trip. They can make at most five items per week. Materials cost $40 for tables and $20 for chairs. They have to stay within a budget of $120 per week for materials. Each table sells for $50 and each chair for $30. How many of each can the students make each week to maximize their funds? Show all of your work, fully explain how you solved the problem and explain why you chose the steps you used.
PromptCombination Rubric
Mathematics Instructional Rubrics – Section Three 126
Mathematics Instructional Rubrics Explanation of Score Points for
Student Anchor Paper
Date: 5/21/01 Grade Level: High School
Title of Rubric:Title of Rubric:Title of Rubric:Title of Rubric: Combination
Set of comments: Score Point isScore Point isScore Point isScore Point is 4 / 4 / 4 because:
The explanation is organized to show how the problem is solved. The implementation is complete and includes organized mathematical steps and procedures and relevant details. The problem is accurately represented graphically and algebraically. The solution is accurate and supported by the work.
Mathematics Instructional Rubrics – Section Three 127
Student Anchor Paper Score Point 4/4/4
Mathematics Instructional Rubrics – Section Three 128
Mathematics Instructional Rubrics Explanation of Score Points for
Student Anchor Paper
Date: 5/21/01 Grade Level: High School
Title of Rubric:Title of Rubric:Title of Rubric:Title of Rubric: Combination
Set of comments: Score Point isScore Point isScore Point isScore Point is 3 / 3 / 4 because:
A correct explanation using most of the procedures is included. The specific details as to how (1,4) in the solution is missing. The work is mostly developed. Explanations as to why (1,4) is the solution was not justified. All elements of the problem are represented.
Mathematics Instructional Rubrics – Section Three 129
Student Anchor Paper Score Point 3/3/4
Mathematics Instructional Rubrics – Section Three 130
Mathematics Instructional Rubrics Explanation of Score Points for
Student Anchor Paper
Date: 5/21/01 Grade Level: High School
Title of Rubric:Title of Rubric:Title of Rubric:Title of Rubric: Combination
Set of comments: Score Point isScore Point isScore Point isScore Point is 3 / 3 / 4 because:
A correct explanation using most of the relevant details is included. Most concepts are developed. Why the concepts were selected is included. The inequalities x > 0, y > 0 are not stated. The problem and solution are accurately represented graphically and algebraically.
Mathematics Instructional Rubrics – Section Three 131
Student Anchor Paper Score Point 3/3/4
Mathematics Instructional Rubrics – Section Three 132
Mathematics Instructional Rubrics Explanation of Score Points for
Student Anchor Paper
Date: 5/21/01 Grade Level: High School
Title of Rubric:Title of Rubric:Title of Rubric:Title of Rubric: Combination
Set of comments: Score Point isScore Point isScore Point isScore Point is 2 / 2 / 2 because:
The explanation includes some details from the problem in the beginning. The explanation describing the procedures used to arrive at the solution is unclear. The implementation is somewhat developed and supported. Some labels and concepts are used accurately. Elements are partially represented in a graph. Some inequalities are represented. The conclusion is partly supported by the graph.
Mathematics Instructional Rubrics – Section Three 133
Student Anchor Paper Score Point 2/2/2
Answer continued on next page
Mathematics Instructional Rubrics – Section Three 134
Mathematics Instructional Rubrics – Section Three 135
Mathematics Instructional Rubrics Explanation of Score Points for
Student Anchor Paper
Date: 5/21/01 Grade Level: High School
Title of Rubric:Title of Rubric:Title of Rubric:Title of Rubric: Combination
Set of comments: Score Point isScore Point isScore Point isScore Point is 1 / 1 / 1 because:
An explanation using a few of the relevant details is included. A few concepts and representations are included. The conclusion is stated without support form the representation.
Mathematics Instructional Rubrics – Section Three 136
Student Anchor Paper Score Point 1/1/1
Mathematics Instructional Rubrics – Section Four 137
HIGH SCHOOL PROMPTS
The New Soccer Field (from the Summer 2000 Math Assessment Pilot)
Your company is constructing a soccer field for a high school. The field is 110 yards long and 80 yards wide. You receive the following note from your boss: The Board of Zoning requires that a 6” layer of gravel must be laid on all athletic fields before top soil is added, to give adequate drainage. Our gravel supplier says that each truck load contains 12 cubic yards of gravel, at a cost of $9.25/cubic yard delivered. Calculate how many truckloads of gravel we should order for the soccer field and estimate how much it will cost. I need to have your background calculations, clearly explained, as well as the total cost in order to get budget approval.
High School PromptCommunication
Analytic and Holistic Rubrics
Mathematics Instructional Rubrics – Section Four 138
Sale (from Summer 2000 Math Assessment Pilot)
For a sale, a shopkeeper lowered the original price of an item by 30 percent. After the sale, the shopkeeper told this clerk, Mike, to raise the price of that item by 30 percent of its sale price. So Mike marked the item with its original price. Was Mike right or wrong in doing that? Present a convincing argument to support your answer: your may wish to include a specific example as part of your argument.
High School PromptCommunication
Analytic and Holistic Rubrics
Mathematics Instructional Rubrics – Section Four 139
High School PromptCommunication
Analytic and Holistic Rubrics
Mathematics Instructional Rubrics – Section Four 140
Mathematics Instructional Rubrics – Section Four 141
High School PromptCommunication
Analytic and Holistic Rubrics
Mathematics Instructional Rubrics – Section Four 142
The Budget Mystery (from Summer 2000 Math Assessment Pilot)
In 1990, the maintenance budget for a school was $30,000 out of a total budget of $500,000. In 1991, the figure was $31,200 out of a total budget of $520,000. Inflation between 1990 and 1991 was 8%. Parents complain that the money spent on maintenance increased. The maintenance manager complained that the money spent on maintenance was decreased. The principal claimed that, in fact, there has been no change in spending for maintenance.
1. Write what the parents could say to justify their claim of an increase 2. Write what the maintenance manager could say to justify his claim of a decrease. 3. Write what the principal could say to justify her claim of no change.
High School PromptReasoning
Analytic and Holistic Rubrics
Mathematics Instructional Rubrics – Section Four 143
Ebony’s Medicine (from Summer 2000 Math Assessment Pilot)
Ebony is taking medicine for an illness. Her doctor told her to begin to take 500 milligrams (mg) of this medicine at 8 a.m. The amount of the medicine still active in Ebony’s system at the end of any given hour is reduced by 25% of that amount that was present at the beginning of that hour. At 12:00 noon, how many milligrams (mg) of the medicine is still active (to the nearest milligram)? Show all of your work and explain the steps you used to justify your answer. Numeric answers must always be labeled.
High School PromptReasoning
Analytic and Holistic Rubrics
Mathematics Instructional Rubrics – Section Four 144
High School PromptReasoning
Analytic and Holistic Rubrics
(from PA Department of Education)
Mathematics Instructional Rubrics – Section Four 145
High School PromptReasoning and RepresentationAnalytic and Holistic Rubrics
Mathematics Instructional Rubrics – Section Four 146
Almanac (from Mathematics Connections Kit, Page 39)
Students will use an almanac to collect data on any of the following: Women’s Olympic 400-meter Freestyle Swim Men’s Mile Record Winning Times for the Indianapolis 500
1. Construct a scatter plot on graph paper and on a graphing calculator with “year” as the independent variable and “winning time” as the dependent variable.
2. Complete a linear regression and write the equation. 3. What is the slope of your model and what does it mean? Test your model for accuracy
with at least three of the actual data points. Comment on the accuracy. Predict the winning time for next running of your event. Is the prediction reasonable? Why or why not?
High School PromptRepresentation
Analytic and Holistic Rubrics
Mathematics Instructional Rubrics – Section Four 147
French Fries (from Summer 2000 Math Assessment Pilot)
You and your friends think that sometimes you get ripped off when you buy fries. Some portions seem to have a lot fewer french fries than others, so you decide to do a study. For a week, after school, you and your friends count the number of french fries in 20 different orders. Here is what you found:
Portion # # of Fries Portion # # off Fries 1 30 11 40 2 35 12 32 3 35 13 32 4 38 14 30 5 31 15 35 6 43 16 33 7 32 17 33 8 32 18 31 9 29 19 38 10 40 20 31
1. Construct a graph representing the information from your study. 2. About how many french fries would you expect to get next time? Explain your
reasoning. 3. Based on your date, how few fries would you need to get before feeling ripped off?
Explain your reasoning.
High School PromptRepresentation
Analytic and Holistic Rubrics
Mathematics Instructional Rubrics – Section Four 148
Cow Grazing (from Mathematics Assessment Handbook, Page 42)
A cow is tied to the long side of a barn 10 feet from the corner. The barn measures 11 feet wide and 28 feet long. If the rope is 21 feet long, what is the total area of the space in which the cow can graze? Draw a picture which includes the grazing area to show how you arrived at you answer.
High School PromptRepresentation
Analytic and Holistic Rubrics
Mathematics Instructional Rubrics – Section Four 149
Emilio’s Bookcases (from Summer 2000 Math Assessment Pilot)
Emilio has a small carpentry shop in his basement to make bookcases. He makes two sizes of bookcases, large and small. His profit on a large bookcase is $80, and his profit on a small bookcase in $50. It takes Emilio six hours to make a large bookcase and two hours to make a small one. He can spend only 24 hours each week on his carpentry work. He must make at least two of each size each week. What is the maximum weekly profit?
High School PromptRepresentation
Analytic and Holistic Rubrics
Mathematics Instructional Rubrics – Section Four 150
High School PromptRepresentation
Analytic and Holistic Rubrics
Mathematics Instructional Rubrics – Section Four 151
Mathematics Instructional Rubrics – Section Four 152
Population Survey (from Performance Events, 1993-94. Kentucky Department of Education.)
High School PromptCombination Rubric
Mathematics Instructional Rubrics – Section Four 153
RRRRESOURCESESOURCESESOURCESESOURCES
Websites National Council of Teachers of Mathematics http://www.nctm.org Pennsylvania Council of Teachers of Math http://www.pctm.org Pennsylvania Department of Education http://www.pde.psu.edu Pennsylvania Academic Math Standards http://www.pde.psu.edu/standard/mathstan.doc PDE Review & Eval. For Select.Textbooks http://www.pde.psu.edu/mathreveval.pdf PDE Math Assessment Handbook http://www.pde.psu.edu/pssa/mathbook.pdf TIMSS Released Test Items 1995 http://timss.bc.edu/timss1995i/Items.html Glenn Commission Report http://www.ed.gov/americacounts/glenn/report.doc Exemplary and Promising Programs http://www.ed.gov/offices/OERI/ORAD/KAD/expert_panel/math-science.html K-12 Math Curriculum Center http://www.edc.org/mcc/ Elementary ARC Center http://www.comap.com/elementary/projects/arc/ Middle School Show Me Center http://www.showmecenter.missouri.edu/ Secondary Compass Center http://www.ithaca.edu/compass/frames.htm Project 2061 Textbook Rating http://www.project2061.org/tools/textbook/algebra/index.htm
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Websites (continued)
Council of Chief State School Officers http://www.ccsso.org (Select tab at top of home page labeled state agencies) National Center for Educational Statistics http://www.nces.ed.gov Mathematics Forum http://www.mathforum.com TIMSS http://nces.ed.gov/TIMSS TIMSS http://timss.bc.edu/ NAEP http://nces.ed.gov/nationsreportcard/ Eisenhower National Clearinghouse http://www.enc.org National Science Foundation http://nsf.gov Research for Better Schools http://www.rbs.org Math Problems Assessment 1999/2000 http://www.apiu.k12.pa.us/downloads/map.html
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References
Badger, Elizabeth. (May 1992). “More than Testing,” Arithmetic teacher. National Council of Teachers of Mathematics. Blum, R., & Arter, J. (eds.). (1996). A handbook for student performance assessment in an era of restructuring. (Section VI). Alexandria, VA: Association for Supervision and Curriculum Development. Charles, R., and Silver, E. (eds.). (1988). The teaching and assessing of mathematical problem solving. Vol. 3. Reston, VA: National Council of Teachers of Mathematics. Goldberg, Gail, Maryland Department of Education; telephone 410-767-0100. Illinois State Board of Education. (1995). Effective scoring rubrics – A guide to their development and use. (Available from Illinois State Board of Education, 100 N. First Street, Springfield, IL 62777) Lesh, R., and Lamon, S. (eds.). (1992). Assessment of authentic performance in school mathematics. Washington, D.C.: American Association for the Advancement of Science. Marzano, R., Pickering, D., and McTighe, J. (1993). Assessing student outcomes: performance assessment using the dimensions of learning model. Alexandria, VA: Association for Supervision and Curriculum Development. National Council of Teachers of Mathematics. (2000). Principles and Standards for School Mathematics. Reston, VA. Newmann, F. Secada, W., and Wehlage, G. A guide to authentic instruction and assessment; vision, standards and scoring. Madison: Wisconsin Center for Education Research. NWREL. (1999). Seeing with new eyes (Video). IOX; telephone 310-822-3225. Resnick L.B., and Resnick, D. P. (1991). “Assessing the Thinking Curriculum: New Tools for Educational Reform. In B. Gifford (ed.), Changing assessments: alternative views of aptitude, achievement and instruction. Norwell, Mass.: Kluwer. Stenmark, Jean Kerr (ed.). (1991). “Mathematics Assessment” Myths, models, good questions, and practical suggestions. Reston, VA: National Council of Teachers of Mathematics.
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References (continued) Stiggins, R. (2001). Student-involved classroom assessment (3rd ed.). New York: Merrill. Stiggins, R. (1994). Student-centered classroom assessment. New York: Merrill. Wiggins, G. (1998). Educative assessment. San Francisco: Jossey-Bass. Wiggins, G. (1989a). “A True Test: Toward More Authentic and Equitable Assessment.” Phi Delta Kappan, 70(9), 703-713. Wiggins, G. (1989b). “Teaching to the (Authentic) Test.” Educational Leadership, 46(7), 41-47. Wiggins, G., (1992). “Creating Tests Worth Taking.” Educational Leadership, 49(8), 26-33. Wiggins, G., (1991). “Standards, Not Standardization: Evoking Quality Student Work.” Educational Leadership, 48(5), 18-25. Wolf, D., Bixby, J., Glen, Jr. III, Gardner, H. (1991). “To Use Their Minds Well: Investigating New Forms of Student Assessment.” In G. Grant (ed.), Review of Research in Education. Washington, D.C.: American Educational Research Association.
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Glossary Analytic Rubric A scoring guide that contains separate and discrete descriptive
traits with a score point for each trait; separate and discrete criteria allow a response to be viewed in a part-to-whole relationship
Anchor Papers Examples of student work that illustrate each level or point on a
rubric Combination Rubric A scoring guide that contains all the skill development traits from
the holistic rubrics Criteria The specific concepts that are important in a performance Descriptors Statements of the various levels of a performance Formative Evaluation An assessment of student work intended to provide information
and guidance for future growth and development
Holistic Rubric A scoring guide that clusters multiple traits across a range of score points; clustering the traits enables the user to assess a response as a whole
Prompt A question or problem posed to the student Rubric A scoring guide for evaluation of student work that includes
specific performance criteria in a continuum of leveled descriptions from high to low
Score Point A designated value assigned to a descriptor Summative Evaluation An assessment of student work used for the purpose of providing a
concluding or cumulative rating